# Transformations (property) This page documents an [OPTIMADE](https://www.optimade.org/) [Property Definition](https://schemas.optimade.org/#definitions). See [https://schemas.optimade.org/](https://schemas.optimade.org/) for more information. **ID: [`https://schemas.anyterial.se/defs/v0.1/datasets/transformations`](https://schemas.anyterial.se/defs/v0.1/datasets/transformations.md)** **Definition name:** `transformations` **Property name:** Transformations **Description:** Ordered table of crystallographic transformation records. Each item describes transformations and normalizer information for one concrete International Tables H-M entry. In `transformations_hm_entry.json.gz`, items are keyed for lookup by the companion top-level `indicies.index_hm_entry_to_transformations_per_hm_entry` object; that index is not an OPTIMADE property. **Type:** list **Requirements/Conventions**: - Items MUST be ordinary dictionaries rather than being keyed by H-M entry. - The `hm_entry` property identifies the International Tables H-M entry represented by the item. - The `hall_entry` property identifies the Hall setting used to generate setting-specific normalizer and transformation data. **Examples:** - `[{"hm_entry": "P 1", "hall_entry": "p_1", "centering_translations": [["0", "0", "0"]], "baernighausen": []}]` **Formats:** [[JSON](transformations.json)] [[MD](transformations.md)] **JSON definition:** ``` json { "$id": "https://schemas.anyterial.se/defs/v0.1/datasets/transformations", "$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json", "title": "Transformations", "$comment": "Dataset of transformation entries.", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "transformations", "label": "transformations_dataset" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Ordered table of crystallographic transformation records.\nEach item describes transformations and normalizer information for one concrete International Tables H-M entry.\nIn `transformations_hm_entry.json.gz`, items are keyed for lookup by the companion top-level `indicies.index_hm_entry_to_transformations_per_hm_entry` object; that index is not an OPTIMADE property.\n\n**Requirements/Conventions**:\n\n- Items MUST be ordinary dictionaries rather than being keyed by H-M entry.\n- The `hm_entry` property identifies the International Tables H-M entry represented by the item.\n- The `hall_entry` property identifies the Hall setting used to generate setting-specific normalizer and transformation data.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/entrytypes/transformations", "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "title": "Anyterial transformation fields", "x-optimade-definition": { "kind": "entrytype", "format": "1.3", "version": "0.1.0", "name": "transformations", "label": "transformations_entrytype_anyterial" }, "properties": { "id": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/id", "x-optimade-requirements": { "support": "must", "sortable": true, "query-support": "all mandatory", "response-level": "always" }, "title": "ID", "x-optimade-type": "string", "x-optimade-definition": { "label": "id_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "id" }, "type": [ "string" ], "description": "A unique string referencing a specific entry in the database.\n\n**Requirements/Conventions:**\n\n- Taken together, the ID and entry type MUST uniquely identify the entry.\n- Reasonably short IDs are encouraged and SHOULD NOT be longer than 255 characters.\n- IDs MAY change over time.", "examples": [ "db/1234567", "cod/2000000", "cod/2000000@1234567", "nomad/L1234567890", "42" ], "x-optimade-unit": "inapplicable" }, "type": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/type", "x-optimade-requirements": { "support": "must", "sortable": true, "query-support": "all mandatory", "response-level": "always" }, "title": "type", "x-optimade-type": "string", "x-optimade-definition": { "label": "type_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "type" }, "type": [ "string" ], "description": "The name of the type of an entry.\n\n**Requirements/Conventions:**\n\n- MUST be an existing entry type.\n- The entry of type and ID MUST be returned in response to a request for // under the versioned or unversioned base URL serving the API.", "examples": [ "structures" ], "x-optimade-unit": "inapplicable" }, "immutable_id": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/immutable_id", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "all mandatory", "response-level": "may" }, "title": "immutable ID", "x-optimade-type": "string", "x-optimade-definition": { "label": "immutable_id_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "immutable_id" }, "type": [ "string", "null" ], "description": "The entry's immutable ID (e.g., a UUID).\n\n**Requirements/Conventions:**\n\n- This is important for databases having preferred IDs that point to \"the latest version\" of a record, but still offer access to older variants.\n- This ID maps to the version-specific record, in case it changes in the future.", "examples": [ "8bd3e750-b477-41a0-9b11-3a799f21b44f", "fjeiwoj,54;@=%<>#32" ], "x-optimade-unit": "inapplicable" }, "last_modified": { "$id": "https://schemas.optimade.org/defs/v1.2/properties/core/last_modified", "x-optimade-requirements": { "support": "should", "sortable": false, "query-support": "all mandatory", "response-level": "must" }, "title": "last modified", "x-optimade-type": "timestamp", "x-optimade-definition": { "label": "last_modified_core", "kind": "property", "version": "1.2.0", "format": "1.2", "name": "last_modified" }, "type": [ "string", "null" ], "format": "date-time", "description": "Date and time representing when the entry was last modified.", "examples": [ "2007-04-05T14:30:20Z" ], "x-optimade-unit": "inapplicable" }, "hm_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hm_entry", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Hermann-Mauguin entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hm_entry", "label": "hm_entry_spacegroups" }, "type": [ "string", "null" ], "description": "The Hermann-Mauguin entry label for a conventional space-group setting from table A1.4.2.7 of the [International Tables for Crystallography (2006). Volume B, Reciprocal space. ISBN: 978-0-7923-6592-1, doi:10.1107/97809553602060000102](https://doi.org/10.1107/97809553602060000102).\n\nThe symbol is a full Hermann-Mauguin-style setting label, except that:\n\n* The older glide-plane letters are used, rather than the newer `e` notation introduced in the Fourth Edition of the International Tables for Crystallography (1995) for the space groups Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68).\n* When necessary to disambiguate the 530 conventional settings, an origin-choice suffix (`:1`, `:2`) is used.\n\n**Requirements/Conventions**:\n\n- The value MUST be written as a plain string with spaces between Hermann-Mauguin symbol parts.\n- The disambiguation suffix MUST be a colon `:` and an integer appended to the string without whitespace, for example `:1` or `:2`.", "x-optimade-unit": "inapplicable", "examples": [ "P 1", "C c c a:1", "C c c b:2" ] }, "hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "crystal_system": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/crystal_system", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Crystal system", "x-optimade-type": "string", "x-compatibility": [ "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group.crystal_system.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "crystal_system", "label": "crystal_system_pointgroups" }, "type": [ "string", "null" ], "description": "The crystal system of the space group or point group.\n\nValues use the conventional crystallographic system names.", "x-optimade-unit": "inapplicable", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "examples": [ "triclinic", "monoclinic" ] }, "centering_translations": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/centering_translations", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Centering translations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "centering_translations", "label": "centering_translations_spacegroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Centering translations of the conventional cell.\n\nEach list member is one exact fractional-coordinate centering translation as defined by `/properties/symmetry/centering_translation`.\nThe zero translation `(0,0,0)` is listed first.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/centering_translation", "title": "Centering translation", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "centering_translation", "label": "centering_translation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "One centering translation of a conventional crystallographic cell.\n\nThe translation is represented in fractional coordinates using exact fraction strings.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of three exact fractional-coordinate components.\n- The zero translation is included in centering-translation lists and is normally listed first.", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" }, "examples": [ [ "0", "0", "0" ], [ "1/2", "1/2", "0" ] ] }, "examples": [ [ [ "0", "0", "0" ] ], [ [ "0", "0", "0" ], [ "1/2", "1/2", "0" ] ] ] }, "euclidean_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/euclidean_normalizer", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Euclidean normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "euclidean_normalizer", "label": "euclidean_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Finite Euclidean normalizer operations for one crystallographic space-group setting.\n\nThe Euclidean normalizer consists of metric-preserving affine operations that normalize the space group in the chosen setting.\nThese operations are useful for algorithms that need to compare or enumerate equivalent descriptions of the same setting under rigid crystallographic changes of coordinates.\n\nThis object is generated from the finite Euclidean normalizer operations exposed by cctbx.\nIt is not a bounded candidate search table.\nTherefore fields such as `candidate_set`, `candidate_sets`, and bounded-search `bounds` do not belong to this property.\n\nThis property is one of three related normalizer tables for a setting:\nthis property holds the finite Euclidean normalizer operations obtained from cctbx,\n`orthogonal_affine_normalizer` holds the older bounded coset table restricted to signed-permutation linear parts,\nand `affine_normalizer` holds the bounded coset table generated from unimodular integer linear parts.\nContinuous origin-shift freedoms are described separately by `continuous_normalizer`.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n For this property the value is `euclidean`.\n\n - **n\\_centering\\_translations**: REQUIRED; Integer.\n Number of centering translations represented in the underlying Euclidean normalizer operation construction.\n\n - **n\\_pointgroup\\_symops**: REQUIRED; Integer.\n Number of point-group symmetry operations represented before centering translations are combined with them.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of Euclidean normalizer operations listed in `symops`.\n This value MUST equal the length of `symops`.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **symops**: REQUIRED; List of dictionaries.\n Finite Euclidean normalizer operations for the setting.\n Each item follows `/defs/v0.1/properties/symmetry/op`.\n\n - **symops\\_mod\\_centering**: REQUIRED; List of dictionaries.\n Euclidean normalizer operations factorized modulo the centering translations of the setting.\n This keeps one operation for each linear part and translation class modulo centering, while `symops` keeps the complete operation list.\n Each item follows `/defs/v0.1/properties/symmetry/op`.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "n_centering_translations": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_centering_translations", "title": "Number of centering translations", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_centering_translations", "label": "n_centering_translations_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of centering translations in the conventional cell of the space-group setting.\n\nWhen the entry contains a `centering_translations` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_pointgroup_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/n_pointgroup_symops", "title": "Number of pointgroup symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_pointgroup_symops", "label": "n_pointgroup_symops_pointgroups" }, "type": [ "integer", "null" ], "description": "Number of point-group symmetry operations.\n\nFor a space-group entry this is the number of operations of the point group of the space group, and it MUST equal the length of the `symops_representative` list when present.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Finite Euclidean normalizer operations for the setting.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "title": "Operation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op", "label": "op_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Information related to a crystallographic operation acting within one coordinate setting.\n\nRepresents an affine_transformation that is a crystallographic operation within one setting.\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe remaining fields classify the operation crystallographically, for example by rotation type, axis, sense, and screw or glide component.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the operation.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n\n - **rot\\_type**: OPTIONAL; String.\n Crystallographic operation-type label for the linear part.\n\n - **axis**: OPTIONAL; List of 3 Integers.\n Operation axis or invariant direction using the integer-vector convention returned by the generator.\n\n - **sense**: OPTIONAL; Integer.\n Rotation sense/sign convention returned by the generator; `0` is used when no handed rotation sense is applicable.\n\n - **screw\\_glide**: OPTIONAL; List of 3 Fractions (String).\n Screw-axis or glide-plane component associated with a space-group affine operation.\n\n - **origin\\_shift**: OPTIONAL; List of 3 Fractions (String).\n Origin shift associated with the screw/glide decomposition of a space-group affine operation.\n\n- Whether the operation is proper follows from the sign of the `det` field of `affine_transformation`; no separate properness flag is stored.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "rot_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Symbolic crystallographic operation-type label for the linear part.", "enum": [ "1", "-1", "2", "m", "3", "-3", "4", "-4", "6", "-6" ] }, "axis": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant-direction descriptor for the operation.", "items": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "One integer component of the axis vector." } }, "sense": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Rotation sense/sign convention returned by the generator." }, "screw_glide": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Screw-axis or glide-plane component represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "origin_shift": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Origin-shift descriptor represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true }, "rot_type": "2", "sense": 0, "axis": [ 0, 0, 1 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] } }, "symops_mod_centering": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/symops_mod_centering", "title": "Symmetry operations modulo centering translations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "symops_mod_centering", "label": "symops_mod_centering_spacegroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Representative symmetry-operation descriptors modulo centering translations.\n\nEach list member is an operation on the format defined by the property definition: https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "title": "Operation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op", "label": "op_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Information related to a crystallographic operation acting within one coordinate setting.\n\nRepresents an affine_transformation that is a crystallographic operation within one setting.\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe remaining fields classify the operation crystallographically, for example by rotation type, axis, sense, and screw or glide component.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the operation.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n\n - **rot\\_type**: OPTIONAL; String.\n Crystallographic operation-type label for the linear part.\n\n - **axis**: OPTIONAL; List of 3 Integers.\n Operation axis or invariant direction using the integer-vector convention returned by the generator.\n\n - **sense**: OPTIONAL; Integer.\n Rotation sense/sign convention returned by the generator; `0` is used when no handed rotation sense is applicable.\n\n - **screw\\_glide**: OPTIONAL; List of 3 Fractions (String).\n Screw-axis or glide-plane component associated with a space-group affine operation.\n\n - **origin\\_shift**: OPTIONAL; List of 3 Fractions (String).\n Origin shift associated with the screw/glide decomposition of a space-group affine operation.\n\n- Whether the operation is proper follows from the sign of the `det` field of `affine_transformation`; no separate properness flag is stored.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "rot_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Symbolic crystallographic operation-type label for the linear part.", "enum": [ "1", "-1", "2", "m", "3", "-3", "4", "-4", "6", "-6" ] }, "axis": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant-direction descriptor for the operation.", "items": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "One integer component of the axis vector." } }, "sense": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Rotation sense/sign convention returned by the generator." }, "screw_glide": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Screw-axis or glide-plane component represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "origin_shift": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Origin-shift descriptor represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true }, "rot_type": "2", "sense": 0, "axis": [ 0, 0, 1 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,z" }, "rot_type": "1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] ] } }, "examples": [ { "normalizer_kind": "euclidean", "n_centering_translations": 1, "n_pointgroup_symops": 1, "n_symops": 2, "n_linear_parts": 2, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "rot_type": "-1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ], "operation_kind": "euclidean" } ], "symops_mod_centering": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "rot_type": "-1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ], "operation_kind": "euclidean" } ] } ] }, "hall_to_it_std_transform": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/hall_to_it_std_transform", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Hall to IT standard transform", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_to_it_std_transform", "label": "hall_to_it_std_transform_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Exact basis and origin transform from one stored Hall setting to the International Tables standard Hall setting of the same space-group type.\n\nThis transform is useful when data generated or detected in an arbitrary Hall setting needs to be compared with a conventional IT-standard reference setting.\nThe transform is represented by `matrix` and `vector`, following the same affine-transformation convention as the other generated transformation tables.\n\nIf `x_to_ref_hall` is a fractional coordinate column vector in the target IT-standard Hall setting, and `x_from_hall` is the corresponding fractional coordinate column vector in the source Hall setting keyed by the containing map, then the stored transform satisfies:\n`x_from_hall = matrix * x_to_ref_hall + vector`.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary describing one exact transform from the containing Hall setting to the IT-standard Hall setting of the same `it_number`.\n- The `index` value is always `1`, because this is a same-space-group setting transform rather than a proper subgroup transform.\n- Matrix and vector entries MUST be exact strings, using integer strings or fraction strings as appropriate.\n- It MUST be a dictionary with the following keys:\n\n - **hall\\_entry**: REQUIRED; String.\n Source Hall-entry key for the setting transformed by this object.\n\n - **it\\_number**: REQUIRED; Integer.\n International Tables space-group number shared by the source and target Hall settings.\n\n - **to\\_hall\\_entry**: REQUIRED; String.\n Target Hall-entry key for the IT-standard Hall setting of the same space-group type.\n\n - **index**: REQUIRED; Integer.\n Transform index.\n For this table the value is `1` because the transform maps between settings of the same space group.\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for this setting transform.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n The transform convention is `x_from_hall = matrix * x_to_ref_hall + vector`, where `matrix` and `vector` are the fields inside `affine_transformation`.", "properties": { "hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "to_hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/to_hall_entry", "title": "To Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "to_hall_entry", "label": "to_hall_entry_transformations" }, "type": [ "string", "null" ], "description": "Target Hall-entry key to which a setting transform maps the current Hall setting.\n\nThe value is a normalized Hall-entry key following the same convention as `/defs/v0.1/properties/spacegroups/hall_entry`, i.e., derived from the Hall symbol by using lowercase letters and underscores in place of spaces.\nThe corresponding display Hall symbol can be recovered by looking up the key in the spacegroups dataset.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "index": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/index", "title": "Subgroup or transform index", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "index", "label": "index_transformations" }, "type": [ "integer", "null" ], "description": "Subgroup or transform index.\n\nFor subgroup transforms it is the crystallographic subgroup index `[G:H]`, equal to the determinant factor of the basis transformation when applicable.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] } }, "examples": [ { "hall_entry": "p_1", "it_number": 1, "to_hall_entry": "p_1", "index": 1, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ] } } ] }, "orthogonal_affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/orthogonal_affine_normalizer", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Orthogonal affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "orthogonal_affine_normalizer", "label": "orthogonal_affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Orthogonal affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains the signed-permutation subset of affine normalizer representatives.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `signed_permutation_matrices`, meaning 3 by 3 integer matrices with exactly one nonzero entry in each row and column and each nonzero entry equal to `-1` or `1`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\nthis property holds the older bounded table restricted to signed-permutation linear parts,\nand `affine_normalizer` holds the bounded table generated from unimodular integer linear parts, which is a superset of this one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds satisfied by the signed-permutation candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed orthogonal affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds satisfied by the signed-permutation candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Absolute determinant of every signed-permutation candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value in the signed-permutation candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed orthogonal affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "normalizer_kind": "orthogonal_affine", "representation": "orthogonal_coset_representatives", "candidate_set": "signed_permutation_matrices", "n_symops": 47, "n_linear_parts": 47, "n_raw_candidates": 48, "n_unique_candidates": 48, "n_coset_representatives": 47, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 }, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "orthogonal_affine" } ] } ] }, "affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer", "label": "affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains representatives generated from bounded unimodular integer linear parts. It is a finite bounded representative table, not a complete infinite affine normalizer.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `bounded_unimodular_integer_matrices`, meaning unimodular 3 by 3 integer matrices satisfying the recorded `bounds`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\n`orthogonal_affine_normalizer` holds the older bounded table restricted to signed-permutation linear parts,\nand this property holds the bounded table generated from unimodular integer linear parts, which is a superset of the orthogonal one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds defining the bounded unimodular integer candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds defining the bounded unimodular integer candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Required absolute determinant of every candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value allowed in candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "normalizer_kind": "affine", "representation": "bounded_coset_representatives", "candidate_set": "bounded_unimodular_integer_matrices", "n_symops": 63, "n_linear_parts": 63, "n_raw_candidates": 6960, "n_unique_candidates": 6960, "n_coset_representatives": 6959, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 }, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } ] }, "continuous_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/continuous_normalizer", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Continuous normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "continuous_normalizer", "label": "continuous_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Parameterized continuous normalizer subspace for a setting.\n\nIt describes continuous origin-shift freedoms by dimension and fractional-coordinate basis vectors rather than by enumerating infinitely many operations.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **dimension**: OPTIONAL; Integer.\n Dimension of the continuous parameter subspace.\n When present, it MUST equal the length of `basis_vectors`.\n\n - **basis\\_vectors**: REQUIRED; List of vectors.\n Basis vectors spanning the continuous normalizer parameter space.\n Each basis vector is represented as exact fractional-coordinate components.\n\n - **coordinate\\_system**: OPTIONAL; String.\n Coordinate system used for the parameter vectors.\n\n - **representation**: OPTIONAL; String.\n Textual description of the parameterized representation.", "properties": { "dimension": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Dimension of the continuous parameter subspace. When present, it MUST equal the length of `basis_vectors`." }, "basis_vectors": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Basis vectors spanning the continuous normalizer parameter space.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One basis vector in fractional coordinates.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "coordinate_system": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate system used for the parameter vectors." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Textual description of the parameterized representation." } }, "examples": [ { "dimension": 3, "coordinate_system": "fractional", "basis_vectors": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ] } ] }, "isomorphic_subgroups": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/isomorphic_subgroups", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Isomorphic subgroup transforms", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "isomorphic_subgroups", "label": "isomorphic_subgroups_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Isomorphic subgroup transforms of bounded index for one parent setting or space-group type.\n\nAn isomorphic subgroup has the same space-group type as the parent but is embedded with a finite index, usually corresponding to an enlarged unit cell or a sublattice choice.\nThese transforms are useful for algorithms that need to enumerate same-type subgroup embeddings, compare structures under supercell changes, or construct bounded same-space-group refinement paths.\n\nThe transform records use the basis-transform convention represented by `matrix` and `vector`.\nThe `index` field is the subgroup index and equals the determinant factor of the basis transformation.\nThe generator currently searches indices up to its configured maximum index and deduplicates equivalent transforms under normalizer equivalence.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary containing an `items` list.\n- Each item in `items` MUST describe one exact isomorphic subgroup transform.\n- Matrix and vector entries MUST be exact strings, using integer strings or fraction strings as appropriate.", "properties": { "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Isomorphic subgroup transform records for one parent key.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "items": [ { "index": 2, "wyckoff_splitting": { "a": [ [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "1/2" ] ] ], [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "0" ] ] ] ] }, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] }, "baernighausen": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/baernighausen", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "B\u00e4rnighausen subgroup transforms", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "baernighausen", "label": "baernighausen_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "B\u00e4rnighausen subgroup transform table for one parent setting or space-group type.\n\nEntries describe generated embeddings of subgroup settings into the containing parent setting.\nEach list item groups transform records for one target subgroup, with the target stored as ordinary data rather than as a JSON dictionary key.\nTransform records follow `/properties/symmetry/basis_transform`.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain either `target_hall_entry` or `target_it_number`, depending on whether the containing dataset is Hall-setting keyed or IT-number keyed.\n- Each dictionary MUST contain `transforms`, a list of basis-transform records.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "B\u00e4rnighausen transform group for one target subgroup.", "required": [ "transforms" ], "properties": { "target_hall_entry": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Target Hall entry when the containing dataset is Hall-setting keyed." }, "target_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Target International Tables number when the containing dataset is IT-number keyed." }, "transforms": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Basis transforms for this target subgroup.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } } }, "examples": [ [ { "target_hall_entry": "p_1", "transforms": [ { "index": 2, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] ] }, "backward_lift_criteria": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/backward_lift_criteria", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Backward lift criteria", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "backward_lift_criteria", "label": "backward_lift_criteria_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Criteria table for one supergroup IT number used to lift occupied Wyckoff data from a subgroup back to that supergroup along a chosen B\u00e4rnighausen transform.\n\nEach list item groups transform records for one target subgroup IT number.\nThe target subgroup IT number is stored in `target_it_number` rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `target_it_number`, the subgroup IT number.\n- Each dictionary MUST contain `transforms`, a list of basis-transform records carrying backward-lift criteria.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift criteria group for one target subgroup IT number.", "required": [ "target_it_number", "transforms" ], "properties": { "target_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Target subgroup International Tables number." }, "transforms": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Basis transforms and criteria for this target subgroup.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } } }, "examples": [ [ { "target_it_number": 2, "transforms": [ { "index": 2, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] }, "criteria": [ { "parent": "a", "constraints": [ { "roles": [ [ "a", 0 ] ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0", "0", "0" ] } ] } ] } ] } ] ] }, "same_space_group_affine_images_std": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/same_space_group_affine_images_std", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Same space group affine images std", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "same_space_group_affine_images_std", "label": "same_space_group_affine_images_std_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Same-space-group affine-image record for one International Tables standard setting.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **it\\_number**: REQUIRED; Integer.\n International Tables space-group number for the standard setting.\n\n - **hall**: REQUIRED; String.\n Hall-entry key of the reference setting used for the standard setting.\n\n - **affine\\_images**: REQUIRED; List of dictionaries.\n Same-space-group affine images represented by exact `matrix` and `vector` transforms.", "properties": { "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "hall": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "affine_images": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Same-space-group affine image transforms.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "it_number": 1, "hall": "p_1", "affine_images": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ] } } ] } ] }, "affine_normalizer_coset_data": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer_coset_data", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Affine normalizer coset data", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer_coset_data", "label": "affine_normalizer_coset_data_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Ordered table of bounded affine normalizer coset-representative data for crystallographic space groups, with one item for each Hall setting.\n\nThe affine normalizer of a space group describes affine mappings that send the space group to itself.\nIn practical algorithms this information is useful after a candidate space-group setting has been identified, because additional normalizer representatives can be applied to explore equivalent descriptions, equivalent origin choices, or equivalent embeddings without changing the underlying space group.\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of normalizer operations rather than every operation in that class.\n\nThe `orthogonal_affine_normalizer_cosets` list is the signed-permutation subset of representatives.\nThe `affine_normalizer_cosets` list is generated from a bounded set of unimodular integer linear parts and may include non-orthogonal representatives.\nBoth lists are finite bounded tables and MUST NOT be interpreted as complete infinite affine normalizers.\nEach listed representative carries `compatible_systems`, which states the crystal metric systems for which the representative preserves the metric constraints.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- The list order MUST match the generator's Hall-setting order.\n- The companion top-level `indicies.hall_symbol_to_affine_normalizer_coset_data` lookup maps normalized Hall entries to zero-based positions in this list; it is not an OPTIMADE property.\n- Each dictionary MUST contain a `hall_entry` value equal to the Hall-entry key used by the companion index.\n- Count fields MUST equal the length of their corresponding representative lists.\n- Matrix and vector entries inside representatives MUST be exact strings, using integer strings or fraction strings as appropriate.\n- The `candidate_sets` dictionary records generation bookkeeping for the finite candidate searches and is not itself a crystallographic normalizer object.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One affine-normalizer coset-data row for a single Hall setting.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **hall\\_entry**: REQUIRED; String.\n Normalized Hall entry identifying the Hall setting represented by this row.\n\n - **it\\_number**: REQUIRED; Integer.\n International Tables space-group number of the represented Hall setting.\n\n - **crystal\\_system**: REQUIRED; String.\n Crystal system of the represented Hall setting.\n\n - **orthogonal\\_affine\\_normalizer\\_cosets**: REQUIRED; List of dictionaries.\n Signed-permutation affine normalizer coset representatives modulo the space group.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n\n - **affine\\_normalizer\\_cosets**: REQUIRED; List of dictionaries.\n Bounded affine normalizer coset representatives modulo the space group.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n\n - **n\\_orthogonal\\_cosets**: REQUIRED; Integer.\n Number of representatives in `orthogonal_affine_normalizer_cosets`.\n\n - **n\\_cosets**: REQUIRED; Integer.\n Number of representatives in `affine_normalizer_cosets`.\n\n - **candidate\\_sets**: REQUIRED; Dictionary.\n Generator bookkeeping for the finite candidate sets used to produce the two representative lists.", "properties": { "hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "crystal_system": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/crystal_system", "title": "Crystal system", "x-optimade-type": "string", "x-compatibility": [ "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group.crystal_system.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "crystal_system", "label": "crystal_system_pointgroups" }, "type": [ "string", "null" ], "description": "The crystal system of the space group or point group.\n\nValues use the conventional crystallographic system names.", "x-optimade-unit": "inapplicable", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "examples": [ "triclinic", "monoclinic" ] }, "orthogonal_affine_normalizer_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/orthogonal_affine_normalizer_cosets", "title": "Orthogonal affine normalizer cosets", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "orthogonal_affine_normalizer_cosets", "label": "orthogonal_affine_normalizer_cosets_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Runtime list of orthogonal signed-permutation affine normalizer coset representatives modulo the space group.\n\nEach item is one finite listed representative and follows `/properties/symmetry/basis_transform`.\nThe list is a bounded representative table, not a complete infinite affine normalizer.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } } ] ] }, "affine_normalizer_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer_cosets", "title": "Affine normalizer cosets", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer_cosets", "label": "affine_normalizer_cosets_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Runtime list of bounded affine normalizer coset representatives modulo the space group.\n\nEach item is one finite listed representative and follows `/properties/symmetry/basis_transform`.\nThe list is a bounded representative table, not a complete infinite affine normalizer.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } } ] ] }, "n_orthogonal_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_orthogonal_cosets", "title": "Number of orthogonal cosets", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_orthogonal_cosets", "label": "n_orthogonal_cosets_transformations" }, "type": [ "integer", "null" ], "description": "Number of orthogonal affine normalizer coset representatives stored for the setting.\n\nThis value MUST equal the length of the `orthogonal_affine_normalizer_cosets` list of the containing record.", "x-optimade-unit": "inapplicable", "examples": [ 47, 23 ] }, "n_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_cosets", "title": "Number of cosets", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_cosets", "label": "n_cosets_transformations" }, "type": [ "integer", "null" ], "description": "Number of affine normalizer coset representatives stored for the setting.\n\nThis value MUST equal the length of the `affine_normalizer_cosets` list of the containing record.", "x-optimade-unit": "inapplicable", "examples": [ 63, 31 ] }, "candidate_sets": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Generator bookkeeping for the finite candidate sets used to produce the stored affine-normalizer coset representatives.\nThis dictionary records how many candidate affine operations were considered before and after quotienting modulo the space group and before applying metric-compatibility filters.", "properties": { "orthogonal_affine_normalizer_cosets": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Candidate-set metadata for the orthogonal signed-permutation representative list.", "properties": { "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the linear candidate set used by the generator." }, "n_linear_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of linear matrices in the raw candidate set." }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates generated before deduplication modulo the space group." }, "n_coset_representatives_before_metric_filter": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of coset representatives before filtering by crystal metric compatibility." } } }, "affine_normalizer_cosets": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Candidate-set metadata for the bounded unimodular-integer representative list.", "properties": { "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the linear candidate set used by the generator." }, "n_linear_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of linear matrices in the raw candidate set." }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates generated before deduplication modulo the space group." }, "n_coset_representatives_before_metric_filter": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of coset representatives before filtering by crystal metric compatibility." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Bounds that define the finite bounded unimodular-integer linear candidate set.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Required absolute determinant of every bounded linear candidate." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute value allowed for entries of the bounded integer linear candidates." } } } } } } } } }, "examples": [ [ { "hall_entry": "p_1", "it_number": 1, "crystal_system": "triclinic", "orthogonal_affine_normalizer_cosets": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } ], "affine_normalizer_cosets": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } ], "n_orthogonal_cosets": 47, "n_cosets": 63, "candidate_sets": { "orthogonal_affine_normalizer_cosets": { "candidate_set": "signed_permutation_matrices", "n_linear_candidates": 48, "n_raw_candidates": 96, "n_coset_representatives_before_metric_filter": 47 }, "affine_normalizer_cosets": { "candidate_set": "bounded_unimodular_integer_matrices", "n_linear_candidates": 6960, "n_raw_candidates": 13920, "n_coset_representatives_before_metric_filter": 63, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 } } } } ] ] }, "orthogonal_affine_normalizer_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/orthogonal_affine_normalizer_cosets", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Orthogonal affine normalizer cosets", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "orthogonal_affine_normalizer_cosets", "label": "orthogonal_affine_normalizer_cosets_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Runtime list of orthogonal signed-permutation affine normalizer coset representatives modulo the space group.\n\nEach item is one finite listed representative and follows `/properties/symmetry/basis_transform`.\nThe list is a bounded representative table, not a complete infinite affine normalizer.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } } ] ] }, "affine_normalizer_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer_cosets", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Affine normalizer cosets", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer_cosets", "label": "affine_normalizer_cosets_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Runtime list of bounded affine normalizer coset representatives modulo the space group.\n\nEach item is one finite listed representative and follows `/properties/symmetry/basis_transform`.\nThe list is a bounded representative table, not a complete infinite affine normalizer.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } } ] ] }, "maximal_subgroup_relations": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/maximal_subgroup_relations", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Maximal subgroup relations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "maximal_subgroup_relations", "label": "maximal_subgroup_relations_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Maximal non-isomorphic subgroup relations for International Tables space-group types.\n\nEach list item describes all known maximal non-isomorphic subgroup relations for one supergroup IT number.\nThe supergroup IT number is stored in `supergroup_it_number` rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `supergroup_it_number`, an International Tables space-group number.\n- Each dictionary MUST contain `relations`, a list of subgroup-relation records for that supergroup.\n- Each relation record MUST include the subgroup IT number, subgroup index, subgroup type, and optional klassengleiche subtype.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Maximal subgroup relations for one supergroup IT number.", "required": [ "supergroup_it_number", "relations" ], "properties": { "supergroup_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "International Tables number of the supergroup." }, "relations": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Maximal subgroup relation records for this supergroup.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One maximal subgroup relation.", "required": [ "subgroup_it_number", "index", "subgroup_type" ], "properties": { "subgroup_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "International Tables number of the subgroup." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Index of the subgroup in the supergroup." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "ITA maximal subgroup class, such as `t` or `k`." }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Further classification for klassengleiche subgroup relations, when applicable." } } } } } }, "examples": [ [ { "supergroup_it_number": 5, "relations": [ { "subgroup_it_number": 1, "index": 2, "subgroup_type": "t" }, { "subgroup_it_number": 3, "index": 2, "subgroup_type": "k", "k_subtype": "loss_of_centering_translation" } ] } ] ] }, "affine_images": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_images", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Affine images", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_images", "label": "affine_images_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Same-space-group affine images for a standard setting.\n\nThe list combines Euclidean normalizer operations and isomorphic subgroup transforms to enumerate alternative affine images of the same space group used by runtime search code.\nMatrix/vector transform fields follow `/properties/symmetry/basis_transform`.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] }, "examples": [ [ { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ] } ] ] }, "index": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/index", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Subgroup or transform index", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "index", "label": "index_transformations" }, "type": [ "integer", "null" ], "description": "Subgroup or transform index.\n\nFor subgroup transforms it is the crystallographic subgroup index `[G:H]`, equal to the determinant factor of the basis transformation when applicable.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "subgroup_type": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/subgroup_type", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Maximal subgroup type", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "subgroup_type", "label": "subgroup_type_transformations" }, "type": [ "string" ], "description": "International Tables maximal subgroup class.\n\nThe value is `t` for a translationengleiche subgroup and `k` for a klassengleiche subgroup.\nThe field is omitted when the enclosing record is not a maximal subgroup relation.", "enum": [ "t", "k" ], "x-optimade-unit": "inapplicable", "examples": [ "k", "t" ] }, "k_subtype": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/k_subtype", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Klassengleiche subgroup subtype", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "k_subtype", "label": "k_subtype_transformations" }, "type": [ "string", "null" ], "description": "Subtype of a klassengleiche (`k`) subgroup relation.\n\nThe value distinguishes loss of centering translations from enlarged-unit-cell subgroups.\nThe value is null for non-`k` relations and for records that are not subgroup relations.", "x-optimade-unit": "inapplicable", "enum": [ "loss_of_centering_translation", "enlarged_unit_cell" ], "examples": [ "enlarged_unit_cell", "loss_of_centering_translation" ] }, "target": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/target", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Criterion target", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "target", "label": "target_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Exact target vector or scalar of a generated linear criterion.\n\nThis field appears in `criteria` items of basis-transform records, for example in backward-lift criteria, where it gives the value that the linear form evaluated on the constrained coordinates must equal.\nThe components are exact rational values serialized as fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" }, "examples": [ [ "0", "0", "0" ], [ "1/2", "0", "0" ] ] }, "to_hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/to_hall_entry", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "To Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "to_hall_entry", "label": "to_hall_entry_transformations" }, "type": [ "string", "null" ], "description": "Target Hall-entry key to which a setting transform maps the current Hall setting.\n\nThe value is a normalized Hall-entry key following the same convention as `/defs/v0.1/properties/spacegroups/hall_entry`, i.e., derived from the Hall symbol by using lowercase letters and underscores in place of spaces.\nThe corresponding display Hall symbol can be recovered by looking up the key in the spacegroups dataset.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "transformations_per_hm_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/transformations_per_hm_entry", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Transformations per H-M entry", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "transformations_per_hm_entry", "label": "transformations_per_hm_entry_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Transformation data grouped by H-M entry.\n\nEach list item contains the H-M entry as ordinary data together with all transformation sections generated for that setting.\nThis representation avoids using H-M entries as JSON dictionary keys in the OPTIMADE-described payload.\nThe accompanying `hall_entry` field identifies the Hall symbol used to compute the symmetry operations and normalizer data.\nThe `centering_translations` field gives the setting's centering translations directly in each record, so modulo-centering operation lists can be interpreted without joining against another dataset.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `hm_entry`.\n- Each dictionary SHOULD contain `hall_entry` and the generated transformation sections for that setting.\n- If `euclidean_normalizer.symops_mod_centering` is present, the same dictionary SHOULD contain `centering_translations`.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Transformation record for one H-M entry.", "required": [ "hm_entry" ], "properties": { "hm_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hm_entry", "title": "Hermann-Mauguin entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hm_entry", "label": "hm_entry_spacegroups" }, "type": [ "string", "null" ], "description": "The Hermann-Mauguin entry label for a conventional space-group setting from table A1.4.2.7 of the [International Tables for Crystallography (2006). Volume B, Reciprocal space. ISBN: 978-0-7923-6592-1, doi:10.1107/97809553602060000102](https://doi.org/10.1107/97809553602060000102).\n\nThe symbol is a full Hermann-Mauguin-style setting label, except that:\n\n* The older glide-plane letters are used, rather than the newer `e` notation introduced in the Fourth Edition of the International Tables for Crystallography (1995) for the space groups Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68).\n* When necessary to disambiguate the 530 conventional settings, an origin-choice suffix (`:1`, `:2`) is used.\n\n**Requirements/Conventions**:\n\n- The value MUST be written as a plain string with spaces between Hermann-Mauguin symbol parts.\n- The disambiguation suffix MUST be a colon `:` and an integer appended to the string without whitespace, for example `:1` or `:2`.", "x-optimade-unit": "inapplicable", "examples": [ "P 1", "C c c a:1", "C c c b:2" ] }, "hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "centering_translations": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/centering_translations", "title": "Centering translations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "centering_translations", "label": "centering_translations_spacegroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Centering translations of the conventional cell.\n\nEach list member is one exact fractional-coordinate centering translation as defined by `/properties/symmetry/centering_translation`.\nThe zero translation `(0,0,0)` is listed first.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/centering_translation", "title": "Centering translation", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "centering_translation", "label": "centering_translation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "One centering translation of a conventional crystallographic cell.\n\nThe translation is represented in fractional coordinates using exact fraction strings.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of three exact fractional-coordinate components.\n- The zero translation is included in centering-translation lists and is normally listed first.", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" }, "examples": [ [ "0", "0", "0" ], [ "1/2", "1/2", "0" ] ] }, "examples": [ [ [ "0", "0", "0" ] ], [ [ "0", "0", "0" ], [ "1/2", "1/2", "0" ] ] ] }, "euclidean_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/euclidean_normalizer", "title": "Euclidean normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "euclidean_normalizer", "label": "euclidean_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Finite Euclidean normalizer operations for one crystallographic space-group setting.\n\nThe Euclidean normalizer consists of metric-preserving affine operations that normalize the space group in the chosen setting.\nThese operations are useful for algorithms that need to compare or enumerate equivalent descriptions of the same setting under rigid crystallographic changes of coordinates.\n\nThis object is generated from the finite Euclidean normalizer operations exposed by cctbx.\nIt is not a bounded candidate search table.\nTherefore fields such as `candidate_set`, `candidate_sets`, and bounded-search `bounds` do not belong to this property.\n\nThis property is one of three related normalizer tables for a setting:\nthis property holds the finite Euclidean normalizer operations obtained from cctbx,\n`orthogonal_affine_normalizer` holds the older bounded coset table restricted to signed-permutation linear parts,\nand `affine_normalizer` holds the bounded coset table generated from unimodular integer linear parts.\nContinuous origin-shift freedoms are described separately by `continuous_normalizer`.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n For this property the value is `euclidean`.\n\n - **n\\_centering\\_translations**: REQUIRED; Integer.\n Number of centering translations represented in the underlying Euclidean normalizer operation construction.\n\n - **n\\_pointgroup\\_symops**: REQUIRED; Integer.\n Number of point-group symmetry operations represented before centering translations are combined with them.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of Euclidean normalizer operations listed in `symops`.\n This value MUST equal the length of `symops`.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **symops**: REQUIRED; List of dictionaries.\n Finite Euclidean normalizer operations for the setting.\n Each item follows `/defs/v0.1/properties/symmetry/op`.\n\n - **symops\\_mod\\_centering**: REQUIRED; List of dictionaries.\n Euclidean normalizer operations factorized modulo the centering translations of the setting.\n This keeps one operation for each linear part and translation class modulo centering, while `symops` keeps the complete operation list.\n Each item follows `/defs/v0.1/properties/symmetry/op`.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "n_centering_translations": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_centering_translations", "title": "Number of centering translations", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_centering_translations", "label": "n_centering_translations_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of centering translations in the conventional cell of the space-group setting.\n\nWhen the entry contains a `centering_translations` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_pointgroup_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/pointgroups/n_pointgroup_symops", "title": "Number of pointgroup symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_pointgroup_symops", "label": "n_pointgroup_symops_pointgroups" }, "type": [ "integer", "null" ], "description": "Number of point-group symmetry operations.\n\nFor a space-group entry this is the number of operations of the point group of the space group, and it MUST equal the length of the `symops_representative` list when present.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Finite Euclidean normalizer operations for the setting.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "title": "Operation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op", "label": "op_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Information related to a crystallographic operation acting within one coordinate setting.\n\nRepresents an affine_transformation that is a crystallographic operation within one setting.\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe remaining fields classify the operation crystallographically, for example by rotation type, axis, sense, and screw or glide component.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the operation.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n\n - **rot\\_type**: OPTIONAL; String.\n Crystallographic operation-type label for the linear part.\n\n - **axis**: OPTIONAL; List of 3 Integers.\n Operation axis or invariant direction using the integer-vector convention returned by the generator.\n\n - **sense**: OPTIONAL; Integer.\n Rotation sense/sign convention returned by the generator; `0` is used when no handed rotation sense is applicable.\n\n - **screw\\_glide**: OPTIONAL; List of 3 Fractions (String).\n Screw-axis or glide-plane component associated with a space-group affine operation.\n\n - **origin\\_shift**: OPTIONAL; List of 3 Fractions (String).\n Origin shift associated with the screw/glide decomposition of a space-group affine operation.\n\n- Whether the operation is proper follows from the sign of the `det` field of `affine_transformation`; no separate properness flag is stored.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "rot_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Symbolic crystallographic operation-type label for the linear part.", "enum": [ "1", "-1", "2", "m", "3", "-3", "4", "-4", "6", "-6" ] }, "axis": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant-direction descriptor for the operation.", "items": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "One integer component of the axis vector." } }, "sense": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Rotation sense/sign convention returned by the generator." }, "screw_glide": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Screw-axis or glide-plane component represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "origin_shift": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Origin-shift descriptor represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true }, "rot_type": "2", "sense": 0, "axis": [ 0, 0, 1 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] } }, "symops_mod_centering": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/symops_mod_centering", "title": "Symmetry operations modulo centering translations", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "symops_mod_centering", "label": "symops_mod_centering_spacegroups" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Representative symmetry-operation descriptors modulo centering translations.\n\nEach list member is an operation on the format defined by the property definition: https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op", "title": "Operation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op", "label": "op_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Information related to a crystallographic operation acting within one coordinate setting.\n\nRepresents an affine_transformation that is a crystallographic operation within one setting.\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe remaining fields classify the operation crystallographically, for example by rotation type, axis, sense, and screw or glide component.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the operation.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n\n - **rot\\_type**: OPTIONAL; String.\n Crystallographic operation-type label for the linear part.\n\n - **axis**: OPTIONAL; List of 3 Integers.\n Operation axis or invariant direction using the integer-vector convention returned by the generator.\n\n - **sense**: OPTIONAL; Integer.\n Rotation sense/sign convention returned by the generator; `0` is used when no handed rotation sense is applicable.\n\n - **screw\\_glide**: OPTIONAL; List of 3 Fractions (String).\n Screw-axis or glide-plane component associated with a space-group affine operation.\n\n - **origin\\_shift**: OPTIONAL; List of 3 Fractions (String).\n Origin shift associated with the screw/glide decomposition of a space-group affine operation.\n\n- Whether the operation is proper follows from the sign of the `det` field of `affine_transformation`; no separate properness flag is stored.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "rot_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Symbolic crystallographic operation-type label for the linear part.", "enum": [ "1", "-1", "2", "m", "3", "-3", "4", "-4", "6", "-6" ] }, "axis": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Integer-vector axis or invariant-direction descriptor for the operation.", "items": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "One integer component of the axis vector." } }, "sense": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Rotation sense/sign convention returned by the generator." }, "screw_glide": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Screw-axis or glide-plane component represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "origin_shift": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Origin-shift descriptor represented exactly as a list of fraction strings.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true }, "rot_type": "2", "sense": 0, "axis": [ 0, 0, 1 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] }, "examples": [ [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,z" }, "rot_type": "1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ] } ] ] } }, "examples": [ { "normalizer_kind": "euclidean", "n_centering_translations": 1, "n_pointgroup_symops": 1, "n_symops": 2, "n_linear_parts": 2, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "rot_type": "-1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ], "operation_kind": "euclidean" } ], "symops_mod_centering": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "rot_type": "-1", "sense": 0, "axis": [ 0, 0, 0 ], "screw_glide": [ "0", "0", "0" ], "origin_shift": [ "0", "0", "0" ], "operation_kind": "euclidean" } ] } ] }, "hall_to_it_std_transform": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/hall_to_it_std_transform", "title": "Hall to IT standard transform", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_to_it_std_transform", "label": "hall_to_it_std_transform_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Exact basis and origin transform from one stored Hall setting to the International Tables standard Hall setting of the same space-group type.\n\nThis transform is useful when data generated or detected in an arbitrary Hall setting needs to be compared with a conventional IT-standard reference setting.\nThe transform is represented by `matrix` and `vector`, following the same affine-transformation convention as the other generated transformation tables.\n\nIf `x_to_ref_hall` is a fractional coordinate column vector in the target IT-standard Hall setting, and `x_from_hall` is the corresponding fractional coordinate column vector in the source Hall setting keyed by the containing map, then the stored transform satisfies:\n`x_from_hall = matrix * x_to_ref_hall + vector`.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary describing one exact transform from the containing Hall setting to the IT-standard Hall setting of the same `it_number`.\n- The `index` value is always `1`, because this is a same-space-group setting transform rather than a proper subgroup transform.\n- Matrix and vector entries MUST be exact strings, using integer strings or fraction strings as appropriate.\n- It MUST be a dictionary with the following keys:\n\n - **hall\\_entry**: REQUIRED; String.\n Source Hall-entry key for the setting transformed by this object.\n\n - **it\\_number**: REQUIRED; Integer.\n International Tables space-group number shared by the source and target Hall settings.\n\n - **to\\_hall\\_entry**: REQUIRED; String.\n Target Hall-entry key for the IT-standard Hall setting of the same space-group type.\n\n - **index**: REQUIRED; Integer.\n Transform index.\n For this table the value is `1` because the transform maps between settings of the same space group.\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for this setting transform.\n It MUST follow `/defs/v0.1/properties/symmetry/affine_transformation`.\n The transform convention is `x_from_hall = matrix * x_to_ref_hall + vector`, where `matrix` and `vector` are the fields inside `affine_transformation`.", "properties": { "hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "to_hall_entry": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/to_hall_entry", "title": "To Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "to_hall_entry", "label": "to_hall_entry_transformations" }, "type": [ "string", "null" ], "description": "Target Hall-entry key to which a setting transform maps the current Hall setting.\n\nThe value is a normalized Hall-entry key following the same convention as `/defs/v0.1/properties/spacegroups/hall_entry`, i.e., derived from the Hall symbol by using lowercase letters and underscores in place of spaces.\nThe corresponding display Hall symbol can be recovered by looking up the key in the spacegroups dataset.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "index": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/index", "title": "Subgroup or transform index", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "index", "label": "index_transformations" }, "type": [ "integer", "null" ], "description": "Subgroup or transform index.\n\nFor subgroup transforms it is the crystallographic subgroup index `[G:H]`, equal to the determinant factor of the basis transformation when applicable.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] } }, "examples": [ { "hall_entry": "p_1", "it_number": 1, "to_hall_entry": "p_1", "index": 1, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ] } } ] }, "orthogonal_affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/orthogonal_affine_normalizer", "title": "Orthogonal affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "orthogonal_affine_normalizer", "label": "orthogonal_affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Orthogonal affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains the signed-permutation subset of affine normalizer representatives.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `signed_permutation_matrices`, meaning 3 by 3 integer matrices with exactly one nonzero entry in each row and column and each nonzero entry equal to `-1` or `1`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\nthis property holds the older bounded table restricted to signed-permutation linear parts,\nand `affine_normalizer` holds the bounded table generated from unimodular integer linear parts, which is a superset of this one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds satisfied by the signed-permutation candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed orthogonal affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds satisfied by the signed-permutation candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Absolute determinant of every signed-permutation candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value in the signed-permutation candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed orthogonal affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "normalizer_kind": "orthogonal_affine", "representation": "orthogonal_coset_representatives", "candidate_set": "signed_permutation_matrices", "n_symops": 47, "n_linear_parts": 47, "n_raw_candidates": 48, "n_unique_candidates": 48, "n_coset_representatives": 47, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 }, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "orthogonal_affine" } ] } ] }, "affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer", "title": "Affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer", "label": "affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains representatives generated from bounded unimodular integer linear parts. It is a finite bounded representative table, not a complete infinite affine normalizer.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `bounded_unimodular_integer_matrices`, meaning unimodular 3 by 3 integer matrices satisfying the recorded `bounds`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\n`orthogonal_affine_normalizer` holds the older bounded table restricted to signed-permutation linear parts,\nand this property holds the bounded table generated from unimodular integer linear parts, which is a superset of the orthogonal one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds defining the bounded unimodular integer candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds defining the bounded unimodular integer candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Required absolute determinant of every candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value allowed in candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "normalizer_kind": "affine", "representation": "bounded_coset_representatives", "candidate_set": "bounded_unimodular_integer_matrices", "n_symops": 63, "n_linear_parts": 63, "n_raw_candidates": 6960, "n_unique_candidates": 6960, "n_coset_representatives": 6959, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 }, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } ] }, "continuous_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/continuous_normalizer", "title": "Continuous normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "continuous_normalizer", "label": "continuous_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Parameterized continuous normalizer subspace for a setting.\n\nIt describes continuous origin-shift freedoms by dimension and fractional-coordinate basis vectors rather than by enumerating infinitely many operations.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **dimension**: OPTIONAL; Integer.\n Dimension of the continuous parameter subspace.\n When present, it MUST equal the length of `basis_vectors`.\n\n - **basis\\_vectors**: REQUIRED; List of vectors.\n Basis vectors spanning the continuous normalizer parameter space.\n Each basis vector is represented as exact fractional-coordinate components.\n\n - **coordinate\\_system**: OPTIONAL; String.\n Coordinate system used for the parameter vectors.\n\n - **representation**: OPTIONAL; String.\n Textual description of the parameterized representation.", "properties": { "dimension": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Dimension of the continuous parameter subspace. When present, it MUST equal the length of `basis_vectors`." }, "basis_vectors": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Basis vectors spanning the continuous normalizer parameter space.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One basis vector in fractional coordinates.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "coordinate_system": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate system used for the parameter vectors." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Textual description of the parameterized representation." } }, "examples": [ { "dimension": 3, "coordinate_system": "fractional", "basis_vectors": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ] } ] }, "isomorphic_subgroups": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/isomorphic_subgroups", "title": "Isomorphic subgroup transforms", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "isomorphic_subgroups", "label": "isomorphic_subgroups_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Isomorphic subgroup transforms of bounded index for one parent setting or space-group type.\n\nAn isomorphic subgroup has the same space-group type as the parent but is embedded with a finite index, usually corresponding to an enlarged unit cell or a sublattice choice.\nThese transforms are useful for algorithms that need to enumerate same-type subgroup embeddings, compare structures under supercell changes, or construct bounded same-space-group refinement paths.\n\nThe transform records use the basis-transform convention represented by `matrix` and `vector`.\nThe `index` field is the subgroup index and equals the determinant factor of the basis transformation.\nThe generator currently searches indices up to its configured maximum index and deduplicates equivalent transforms under normalizer equivalence.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary containing an `items` list.\n- Each item in `items` MUST describe one exact isomorphic subgroup transform.\n- Matrix and vector entries MUST be exact strings, using integer strings or fraction strings as appropriate.", "properties": { "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Isomorphic subgroup transform records for one parent key.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "items": [ { "index": 2, "wyckoff_splitting": { "a": [ [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "1/2" ] ] ], [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "0" ] ] ] ] }, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] }, "baernighausen": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/baernighausen", "title": "B\u00e4rnighausen subgroup transforms", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "baernighausen", "label": "baernighausen_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "B\u00e4rnighausen subgroup transform table for one parent setting or space-group type.\n\nEntries describe generated embeddings of subgroup settings into the containing parent setting.\nEach list item groups transform records for one target subgroup, with the target stored as ordinary data rather than as a JSON dictionary key.\nTransform records follow `/properties/symmetry/basis_transform`.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain either `target_hall_entry` or `target_it_number`, depending on whether the containing dataset is Hall-setting keyed or IT-number keyed.\n- Each dictionary MUST contain `transforms`, a list of basis-transform records.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "B\u00e4rnighausen transform group for one target subgroup.", "required": [ "transforms" ], "properties": { "target_hall_entry": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Target Hall entry when the containing dataset is Hall-setting keyed." }, "target_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Target International Tables number when the containing dataset is IT-number keyed." }, "transforms": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Basis transforms for this target subgroup.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } } }, "examples": [ [ { "target_hall_entry": "p_1", "transforms": [ { "index": 2, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] ] } } }, "examples": [ [ { "hm_entry": "P 1", "hall_entry": "p_1", "centering_translations": [ [ "0", "0", "0" ] ], "baernighausen": [] } ] ] }, "transformations_per_it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/transformations_per_it_number", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Transformations per IT number", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "transformations_per_it_number", "label": "transformations_per_it_number_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Standard-setting transformation data grouped by International Tables space-group number.\n\nEach list item contains the IT number as ordinary data together with all standard-setting transformation sections generated for that space-group type.\nThis representation avoids using IT numbers as JSON dictionary keys in the OPTIMADE-described payload.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `it_number`.\n- Each dictionary SHOULD contain the generated standard-setting transformation sections for that IT number.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Transformation record for one International Tables number.", "required": [ "it_number" ], "properties": { "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "baernighausen": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/baernighausen", "title": "B\u00e4rnighausen subgroup transforms", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "baernighausen", "label": "baernighausen_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "B\u00e4rnighausen subgroup transform table for one parent setting or space-group type.\n\nEntries describe generated embeddings of subgroup settings into the containing parent setting.\nEach list item groups transform records for one target subgroup, with the target stored as ordinary data rather than as a JSON dictionary key.\nTransform records follow `/properties/symmetry/basis_transform`.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain either `target_hall_entry` or `target_it_number`, depending on whether the containing dataset is Hall-setting keyed or IT-number keyed.\n- Each dictionary MUST contain `transforms`, a list of basis-transform records.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "B\u00e4rnighausen transform group for one target subgroup.", "required": [ "transforms" ], "properties": { "target_hall_entry": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Target Hall entry when the containing dataset is Hall-setting keyed." }, "target_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Target International Tables number when the containing dataset is IT-number keyed." }, "transforms": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Basis transforms for this target subgroup.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } } }, "examples": [ [ { "target_hall_entry": "p_1", "transforms": [ { "index": 2, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] ] }, "backward_lift_criteria": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/backward_lift_criteria", "title": "Backward lift criteria", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "backward_lift_criteria", "label": "backward_lift_criteria_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Criteria table for one supergroup IT number used to lift occupied Wyckoff data from a subgroup back to that supergroup along a chosen B\u00e4rnighausen transform.\n\nEach list item groups transform records for one target subgroup IT number.\nThe target subgroup IT number is stored in `target_it_number` rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `target_it_number`, the subgroup IT number.\n- Each dictionary MUST contain `transforms`, a list of basis-transform records carrying backward-lift criteria.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift criteria group for one target subgroup IT number.", "required": [ "target_it_number", "transforms" ], "properties": { "target_it_number": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Target subgroup International Tables number." }, "transforms": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Basis transforms and criteria for this target subgroup.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } } }, "examples": [ [ { "target_it_number": 2, "transforms": [ { "index": 2, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] }, "criteria": [ { "parent": "a", "constraints": [ { "roles": [ [ "a", 0 ] ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0", "0", "0" ] } ] } ] } ] } ] ] }, "isomorphic_subgroups": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/isomorphic_subgroups", "title": "Isomorphic subgroup transforms", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "isomorphic_subgroups", "label": "isomorphic_subgroups_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Isomorphic subgroup transforms of bounded index for one parent setting or space-group type.\n\nAn isomorphic subgroup has the same space-group type as the parent but is embedded with a finite index, usually corresponding to an enlarged unit cell or a sublattice choice.\nThese transforms are useful for algorithms that need to enumerate same-type subgroup embeddings, compare structures under supercell changes, or construct bounded same-space-group refinement paths.\n\nThe transform records use the basis-transform convention represented by `matrix` and `vector`.\nThe `index` field is the subgroup index and equals the determinant factor of the basis transformation.\nThe generator currently searches indices up to its configured maximum index and deduplicates equivalent transforms under normalizer equivalence.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary containing an `items` list.\n- Each item in `items` MUST describe one exact isomorphic subgroup transform.\n- Matrix and vector entries MUST be exact strings, using integer strings or fraction strings as appropriate.", "properties": { "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Isomorphic subgroup transform records for one parent key.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "items": [ { "index": 2, "wyckoff_splitting": { "a": [ [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "1/2" ] ] ], [ "a", "x,y,z", [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1/2", "0" ] ] ] ] }, "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ] } } ] } ] }, "same_space_group_affine_images_std": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/same_space_group_affine_images_std", "title": "Same space group affine images std", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "same_space_group_affine_images_std", "label": "same_space_group_affine_images_std_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Same-space-group affine-image record for one International Tables standard setting.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **it\\_number**: REQUIRED; Integer.\n International Tables space-group number for the standard setting.\n\n - **hall**: REQUIRED; String.\n Hall-entry key of the reference setting used for the standard setting.\n\n - **affine\\_images**: REQUIRED; List of dictionaries.\n Same-space-group affine images represented by exact `matrix` and `vector` transforms.", "properties": { "it_number": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/it_number", "title": "International Tables space-group number", "x-optimade-type": "integer", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/structures/space_group_it_number" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "it_number", "label": "it_number_spacegroups" }, "type": [ "integer", "null" ], "description": "The International Tables space-group number.\n\nThis integer identifies the space-group type numbered 1 through 230 in International Tables for Crystallography. Multiple Hall settings can share the same `it_number`.", "x-optimade-unit": "inapplicable", "examples": [ 1, 5 ] }, "hall": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/hall_entry", "title": "Hall entry", "x-optimade-type": "string", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "hall_entry", "label": "hall_entry_spacegroups" }, "type": [ "string", "null" ], "description": "Normalized Hall-table entry key used internally by the generated datasets.\n\nThe value is derived from the Hall symbol by using lowercase letters and underscores in place of spaces. It is stable for lookup within these data files, while the display Hall symbol is provided separately by `hall` and its formatted variants.\n\n**Requirements/Conventions**:\n\n- This field identifies a concrete Hall setting, not only an IT space-group type.\n- The same value is normally used as the key of the containing `spacegroups` map.", "x-optimade-unit": "inapplicable", "examples": [ "p_1", "-p_1" ] }, "affine_images": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Same-space-group affine image transforms.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "it_number": 1, "hall": "p_1", "affine_images": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ] } } ] } ] }, "continuous_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/continuous_normalizer", "title": "Continuous normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "continuous_normalizer", "label": "continuous_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Parameterized continuous normalizer subspace for a setting.\n\nIt describes continuous origin-shift freedoms by dimension and fractional-coordinate basis vectors rather than by enumerating infinitely many operations.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **dimension**: OPTIONAL; Integer.\n Dimension of the continuous parameter subspace.\n When present, it MUST equal the length of `basis_vectors`.\n\n - **basis\\_vectors**: REQUIRED; List of vectors.\n Basis vectors spanning the continuous normalizer parameter space.\n Each basis vector is represented as exact fractional-coordinate components.\n\n - **coordinate\\_system**: OPTIONAL; String.\n Coordinate system used for the parameter vectors.\n\n - **representation**: OPTIONAL; String.\n Textual description of the parameterized representation.", "properties": { "dimension": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Dimension of the continuous parameter subspace. When present, it MUST equal the length of `basis_vectors`." }, "basis_vectors": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Basis vectors spanning the continuous normalizer parameter space.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One basis vector in fractional coordinates.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "coordinate_system": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate system used for the parameter vectors." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Textual description of the parameterized representation." } }, "examples": [ { "dimension": 3, "coordinate_system": "fractional", "basis_vectors": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "1" ] ] } ] }, "orthogonal_affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/orthogonal_affine_normalizer", "title": "Orthogonal affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "orthogonal_affine_normalizer", "label": "orthogonal_affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Orthogonal affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains the signed-permutation subset of affine normalizer representatives.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `signed_permutation_matrices`, meaning 3 by 3 integer matrices with exactly one nonzero entry in each row and column and each nonzero entry equal to `-1` or `1`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\nthis property holds the older bounded table restricted to signed-permutation linear parts,\nand `affine_normalizer` holds the bounded table generated from unimodular integer linear parts, which is a superset of this one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds satisfied by the signed-permutation candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed orthogonal affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds satisfied by the signed-permutation candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Absolute determinant of every signed-permutation candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value in the signed-permutation candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed orthogonal affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } } }, "examples": [ { "affine_transformation": { "matrix": [ [ "1", "0", "0" ], [ "0", "1", "0" ], [ "0", "0", "2" ] ], "vector": [ "0", "0", "0" ], "xyz": "x,y,2z", "det": 2, "is_orthogonal": false }, "index": 2, "subgroup_type": "k", "k_subtype": "enlarged_unit_cell", "wyckoff_splitting": [ { "parent": "a", "splits": [ { "letter": "a", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ], "criteria": [ { "parent": "a", "constraints": [ { "roles": [ { "letter": "a", "index": 0 } ], "coeffs": [ [ [ "1", "0", "0" ] ] ], "target": [ "0" ] } ] } ] }, { "affine_transformation": { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,-z", "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "affine" } ] } } }, "examples": [ { "normalizer_kind": "orthogonal_affine", "representation": "orthogonal_coset_representatives", "candidate_set": "signed_permutation_matrices", "n_symops": 47, "n_linear_parts": 47, "n_raw_candidates": 48, "n_unique_candidates": 48, "n_coset_representatives": 47, "bounds": { "det_abs": 1, "max_abs_linear_entry": 1 }, "symops": [ { "affine_transformation": { "xyz": "-x,-y,-z", "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "-1" ] ], "vector": [ "0", "0", "0" ], "det": -1, "is_orthogonal": true }, "compatible_systems": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ], "operation_kind": "orthogonal_affine" } ] } ] }, "affine_normalizer": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/affine_normalizer", "title": "Affine normalizer", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_normalizer", "label": "affine_normalizer_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Affine normalizer coset representatives for one crystallographic space-group setting.\n\nThe representatives are listed modulo the space group itself, so each listed operation represents an equivalence class of affine normalizer operations rather than every operation in that class.\nThis property contains representatives generated from bounded unimodular integer linear parts. It is a finite bounded representative table, not a complete infinite affine normalizer.\n\nThe `candidate_set` field belongs in this property because the listed representatives are produced from a deliberately restricted finite candidate set.\nFor this property `candidate_set` is `bounded_unimodular_integer_matrices`, meaning unimodular 3 by 3 integer matrices satisfying the recorded `bounds`.\nThe plural field `candidate_sets` is not part of the emitted data and MUST NOT be used here.\n\nThis property is one of three related normalizer tables for a setting:\n`euclidean_normalizer` holds the finite Euclidean normalizer operations obtained from cctbx,\n`orthogonal_affine_normalizer` holds the older bounded table restricted to signed-permutation linear parts,\nand this property holds the bounded table generated from unimodular integer linear parts, which is a superset of the orthogonal one.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **normalizer\\_kind**: REQUIRED; String.\n Kind label for this normalizer contribution.\n\n - **representation**: REQUIRED; String.\n Representation label for the listed data.\n\n - **candidate\\_set**: REQUIRED; String.\n Name of the finite linear candidate set used for generation.\n\n - **n\\_symops**: REQUIRED; Integer.\n Number of listed coset representatives after metric-compatibility filtering.\n\n - **n\\_linear\\_parts**: REQUIRED; Integer.\n Number of distinct linear matrix parts represented in `symops`.\n\n - **n\\_raw\\_candidates**: REQUIRED; Integer.\n Number of affine candidates found before deduplication modulo the space group.\n\n - **n\\_unique\\_candidates**: REQUIRED; Integer.\n Number of unique affine candidates before quotienting by the space group.\n\n - **n\\_coset\\_representatives**: REQUIRED; Integer.\n Number of non-trivial coset representatives before metric-compatibility filtering.\n\n - **bounds**: REQUIRED; Dictionary.\n Simple numerical bounds defining the bounded unimodular integer candidate matrices.\n\n - **symops**: REQUIRED; List of dictionaries.\n Listed affine normalizer coset representatives.\n Each item follows `/defs/v0.1/properties/symmetry/basis_transform`.\n Each item MUST carry `compatible_systems`, the list of crystal systems for which the representative's linear part is compatible with a crystallographic metric.", "properties": { "normalizer_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Kind label for this normalizer contribution." }, "representation": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Representation label for the listed normalizer data." }, "candidate_set": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Name of the finite linear candidate set used for generation." }, "n_symops": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/spacegroups/n_symops", "title": "Number of symops", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_symops", "label": "n_symops_spacegroups" }, "type": [ "integer", "null" ], "description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.", "x-optimade-unit": "inapplicable", "examples": [ 1, 2 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_raw_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of affine candidates found before deduplication modulo the space group." }, "n_unique_candidates": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of unique affine candidates before quotienting by the space group." }, "n_coset_representatives": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Number of non-trivial coset representatives before metric-compatibility filtering." }, "bounds": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "Simple numerical bounds defining the bounded unimodular integer candidate matrices.", "properties": { "det_abs": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Required absolute determinant of every candidate matrix." }, "max_abs_linear_entry": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Maximum absolute entry value allowed in candidate matrices." } } }, "symops": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Listed affine normalizer coset representatives.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/basis_transform", "title": "Basis transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "basis_transform", "label": "basis_transform_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "One crystallographic transform between coordinate descriptions, settings, cells, or related group embeddings.\n\nThe affine map itself is stored in the embedded `affine_transformation` field.\nThe parent property defines the source and target coordinate systems and the precise role of the transform.\nUseful, for example, for representing setting changes, subgroup embeddings, isomorphic subgroup transforms, normalizer representatives, and same-space-group affine images.\nThis property is not limited to symmetry operations within one fixed setting; the matrix may be non-orthogonal or have determinant different from one when the transform changes cell or basis.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **affine\\_transformation**: REQUIRED; Dictionary.\n Exact affine map for the transform, using exact rational matrix and vector entries.\n The coordinate convention and source/target interpretation are supplied by the parent property.\n\n - **index**: OPTIONAL; Integer or null.\n Index metadata whose interpretation is defined by the parent property.\n Common uses include the subgroup index for subgroup embeddings, the cell index for isomorphic subgroup transforms, or an ordinal representative index in a finite transform table.\n\n - **subgroup\\_type**: OPTIONAL; String.\n International Tables subgroup-type label when the transform describes a maximal subgroup embedding.\n The value MUST be `t` for a translationengleiche subgroup or `k` for a klassengleiche subgroup.\n It MUST be omitted when the transform is not a maximal subgroup embedding.\n\n - **k\\_subtype**: OPTIONAL; String or null.\n Klassengleiche subtype when the transform describes a klassengleiche subgroup relation.\n The value MUST be `loss_of_centering_translation` or `enlarged_unit_cell` for klassengleiche relations and null or omitted otherwise.\n\n - **compatible\\_systems**: OPTIONAL; List of strings.\n Crystal metric systems for which the transform is compatible.\n This is used for bounded affine normalizer representatives.\n\n - **operation\\_kind**: OPTIONAL; String.\n Categorical label for normalizer-type representatives.\n The value MUST be `euclidean` for Euclidean normalizer operations, `orthogonal_affine` for the signed-permutation affine normalizer subset, or `affine` for the bounded unimodular affine normalizer table.\n It MUST be omitted when the transform is a setting transform, subgroup embedding, or other transform for which no normalizer operation class applies.\n\n - **wyckoff\\_splitting**: OPTIONAL; List.\n Wyckoff-position splitting metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.\n\n - **criteria**: OPTIONAL; List.\n Backward-lift constraint metadata induced by the transform when available.\n The list is grouped by explicit parent Wyckoff letter.", "properties": { "affine_transformation": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/affine_transformation", "title": "Affine transformation", "x-optimade-type": "dictionary", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "affine_transformation", "label": "affine_transformation_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "object", "null" ], "description": "An affine transformation acting on fractional crystallographic coordinates.\n\nAn affine transformation is a geometric transformation preserving points, straight lines, and parallelism (collinearity), but may not preserve Euclidean distances and angles.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nThe transformation may, for example, represent an operation within one setting, a setting transform, a subgroup embedding, a normalizer representative, or a parametric coordinate map for a Wyckoff-position orbit representative.\nWhen used as a parametric coordinate map, the matrix may be singular because special Wyckoff positions can constrain or identify parameters.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **matrix**: REQUIRED; Exact 3x3 matrix.\n Matrix part of the affine transformation.\n It MUST be represented as a list of three row lists, each containing three exact rational entries represented as strings.\n\n - **vector**: REQUIRED; List of 3 Fractions (String).\n Translation or origin-shift vector of the affine transformation in fractional coordinates.\n\n - **xyz**: OPTIONAL; String.\n Coordinate expression for the affine transformation in `x,y,z` notation when available.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether `matrix` is orthogonal in the exact representation used by the generator.", "properties": { "matrix": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_lattice" ], "sizes": [ 3, 3 ] }, "type": [ "array", "null" ], "description": "Exact 3 by 3 matrix part of the affine transformation.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One row of the exact 3 by 3 matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } }, "vector": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array", "null" ], "description": "Exact fractional-coordinate vector part of the affine transformation.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } }, "xyz": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/symmetry/op_xyz", "title": "Operation xyz", "x-optimade-type": "string", "x-compatibility": [ "https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz", "https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html" ], "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "op_xyz", "label": "op_xyz_symmetry" }, "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.", "x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$", "examples": [ "-x,-y,z", "x,1/2-y,1/2+z" ] }, "det": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Determinant of the matrix part when emitted by the generator." }, "is_orthogonal": { "x-optimade-type": "boolean", "x-optimade-unit": "inapplicable", "type": [ "boolean", "null" ], "description": "Whether the matrix part is orthogonal." } }, "examples": [ { "matrix": [ [ "-1", "0", "0" ], [ "0", "-1", "0" ], [ "0", "0", "1" ] ], "vector": [ "0", "0", "0" ], "xyz": "-x,-y,z", "det": 1, "is_orthogonal": true } ] }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer", "null" ], "description": "Index metadata whose interpretation is supplied by the parent property." }, "subgroup_type": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "International Tables subgroup-type label when applicable.", "enum": [ "t", "k" ] }, "k_subtype": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string", "null" ], "description": "Klassengleiche subtype when applicable." }, "compatible_systems": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Crystal metric systems compatible with the transform.", "items": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "One compatible crystal-system label.", "enum": [ "triclinic", "monoclinic", "orthorhombic", "tetragonal", "trigonal", "hexagonal", "cubic" ] } }, "operation_kind": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Categorical label for normalizer-type representatives.", "enum": [ "euclidean", "orthogonal_affine", "affine" ] }, "wyckoff_splitting": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/wyckoff_splitting", "title": "Wyckoff splitting", "x-optimade-type": "list", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "wyckoff_splitting", "label": "wyckoff_splitting_transformations" }, "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Wyckoff-position splitting data associated with a subgroup or same-space-group transform.\n\nEach list item gives the split of one parent Wyckoff position.\nThe parent Wyckoff letter is stored in the `parent` field rather than as a JSON dictionary key.\n\n**Requirements/Conventions**:\n\n- It MUST be a list of dictionaries.\n- Each dictionary MUST contain `parent`, the Wyckoff letter in the parent setting.\n- Each dictionary MUST contain `splits`, an ordered list of subgroup Wyckoff-position assignments or coordinate expressions emitted by the generator.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Splitting data for one parent Wyckoff position.", "required": [ "parent", "splits" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "splits": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Ordered split records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One Wyckoff split record.", "required": [ "letter", "xyz", "affine" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Subgroup Wyckoff letter assigned by this split branch." }, "xyz": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Coordinate expression for the split branch." }, "affine": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice", "dim_affine" ], "sizes": [ 3, 4 ] }, "type": [ "array" ], "description": "Exact affine representation for the split branch as a 3 by 4 augmented matrix.\nEach row holds the three linear coefficients followed by the translation component, all as exact fraction strings.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_affine" ], "sizes": [ 4 ] }, "type": [ "array" ], "description": "One row of the augmented affine matrix.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } } } } } }, "examples": [ [ { "parent": "c", "splits": [ { "letter": "e", "xyz": "x,y,z", "affine": [ [ "1", "0", "0", "0" ], [ "0", "1", "0", "0" ], [ "0", "0", "1", "0" ] ] } ] } ] ] }, "criteria": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array", "null" ], "description": "Backward-lift constraint metadata induced by the transform, grouped by explicit parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "Backward-lift constraints for one parent Wyckoff position.", "required": [ "parent", "constraints" ], "properties": { "parent": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Parent Wyckoff letter." }, "constraints": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Constraint records for this parent Wyckoff letter.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One linear backward-lift constraint record.", "properties": { "roles": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Wyckoff-position role references entering the constraint.", "items": { "x-optimade-type": "dictionary", "x-optimade-unit": "inapplicable", "type": [ "object" ], "description": "One role reference.", "required": [ "letter", "index" ], "properties": { "letter": { "x-optimade-type": "string", "x-optimade-unit": "inapplicable", "type": [ "string" ], "description": "Wyckoff letter for the referenced role." }, "index": { "x-optimade-type": "integer", "x-optimade-unit": "inapplicable", "type": [ "integer" ], "description": "Zero-based occurrence index for the role." } } } }, "coeffs": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact coefficient vectors for the constraint.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Coefficients associated with one role.", "items": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "x-optimade-dimensions": { "names": [ "dim_lattice" ], "sizes": [ 3 ] }, "type": [ "array" ], "description": "One exact coefficient vector.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": "fraction_core", "kind": "property", "version": "0.1.0", "format": "1.3", "name": "fraction" }, "type": [ "string", "null" ], "description": "A numerical representation formed as the quotient of two numbers represented as a string.", "examples": [ "2/3", "5/42", "10", "0" ], "x-optimade-unit": "inapplicable" } } } }, "target": { "x-optimade-type": "list", "x-optimade-unit": "inapplicable", "type": [ "array" ], "description": "Exact target vector or scalar for the constraint.", "items": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/core/fraction", "title": "Fraction", "x-optimade-type": "string", "x-optimade-definition": { "label": 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"baernighausen": [] } ] ] }, "n_coset_representatives": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_coset_representatives", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of coset representatives", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_coset_representatives", "label": "n_coset_representatives_transformations" }, "type": [ "integer", "null" ], "description": "Number of nontrivial coset representatives retained after deduplication modulo the space group.\n\nThis count is taken before metric-compatibility filtering, so it MAY exceed the length of the stored `symops` list of the containing record.", "x-optimade-unit": "inapplicable", "examples": [ 47, 23 ] }, "n_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_cosets", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of cosets", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_cosets", "label": "n_cosets_transformations" }, "type": [ "integer", "null" ], "description": "Number of affine normalizer coset representatives stored for the setting.\n\nThis value MUST equal the length of the `affine_normalizer_cosets` list of the containing record.", "x-optimade-unit": "inapplicable", "examples": [ 63, 31 ] }, "n_linear_parts": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_linear_parts", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of linear parts", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_linear_parts", "label": "n_linear_parts_transformations" }, "type": [ "integer", "null" ], "description": "Number of distinct linear matrix parts represented in a normalizer or transform table.\n\nDistinctness is determined by exact element-wise comparison of the 3 by 3 matrix parts of the listed operations.", "x-optimade-unit": "inapplicable", "examples": [ 2, 4 ] }, "n_orthogonal_cosets": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_orthogonal_cosets", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of orthogonal cosets", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_orthogonal_cosets", "label": "n_orthogonal_cosets_transformations" }, "type": [ "integer", "null" ], "description": "Number of orthogonal affine normalizer coset representatives stored for the setting.\n\nThis value MUST equal the length of the `orthogonal_affine_normalizer_cosets` list of the containing record.", "x-optimade-unit": "inapplicable", "examples": [ 47, 23 ] }, "n_raw_candidates": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_raw_candidates", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of raw candidates", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_raw_candidates", "label": "n_raw_candidates_transformations" }, "type": [ "integer", "null" ], "description": "Number of candidate affine operations considered before filtering and deduplication.\n\nThe counted candidate set itself is not stored; this value documents the size of the bounded search.", "x-optimade-unit": "inapplicable", "examples": [ 48, 6960 ] }, "n_unique_candidates": { "$id": "https://schemas.anyterial.se/defs/v0.1/properties/transformations/n_unique_candidates", "x-optimade-requirements": { "support": "may", "sortable": false, "query-support": "none", "response-level": "may" }, "title": "Number of unique candidates", "x-optimade-type": "integer", "x-optimade-definition": { "kind": "property", "version": "0.1.0", "format": "1.3", "name": "n_unique_candidates", "label": "n_unique_candidates_transformations" }, "type": [ "integer", "null" ], "description": "Number of candidate affine operations remaining after exact duplicate removal.\n\nThe counted candidate set itself is not stored; this value documents the size of the bounded search.", "x-optimade-unit": "inapplicable", "examples": [ 48, 16 ] } } }, "examples": [ [ { "hm_entry": "P 1", "hall_entry": "p_1", "centering_translations": [ [ "0", "0", "0" ] ], "baernighausen": [] } ] ] } ```