httk property definitionsProperty definitions using the OPTIMADE property definition format, hosted by httk
This page documents an OPTIMADE Property Definition. See https://schemas.optimade.org/ for more information.
ID: https://schemas.httk.org/defs/v0.1/properties/transformations/euclidean_normalizer
Definition name: euclidean_normalizer
Property name: Euclidean normalizer
Description: Finite Euclidean normalizer operations for one crystallographic space-group setting.
Type: dictionary
The Euclidean normalizer consists of metric-preserving affine operations that normalize the space group in the chosen setting. These operations are useful for algorithms that need to compare or enumerate equivalent descriptions of the same setting under rigid crystallographic changes of coordinates.
This object is generated from the finite Euclidean normalizer operations exposed by cctbx.
It is not a bounded candidate search table.
Therefore fields such as candidate_set, candidate_sets, and bounded-search bounds do not belong to this property.
This property is one of three related normalizer tables for a setting:
this property holds the finite Euclidean normalizer operations obtained from cctbx,
orthogonal_affine_normalizer holds the older bounded coset table restricted to signed-permutation linear parts,
and affine_normalizer holds the bounded coset table generated from unimodular integer linear parts.
Continuous origin-shift freedoms are described separately by continuous_normalizer.
Requirements/Conventions:
It MUST be a dictionary with the following keys:
normalizer_kind: REQUIRED; String.
Kind label for this normalizer contribution.
For this property the value is euclidean.
n_centering_translations: REQUIRED; Integer. Number of centering translations represented in the underlying Euclidean normalizer operation construction.
n_pointgroup_symops: REQUIRED; Integer. Number of point-group symmetry operations represented before centering translations are combined with them.
n_symops: REQUIRED; Integer.
Number of Euclidean normalizer operations listed in symops.
This value MUST equal the length of symops.
n_linear_parts: REQUIRED; Integer.
Number of distinct linear matrix parts represented in symops.
symops: REQUIRED; List of dictionaries.
Finite Euclidean normalizer operations for the setting.
Each item follows /defs/v0.1/properties/symmetry/op.
symops_mod_centering: REQUIRED; List of dictionaries.
Euclidean normalizer operations factorized modulo the centering translations of the setting.
This keeps one operation for each linear part and translation class modulo centering, while symops keeps the complete operation list.
Each item follows /defs/v0.1/properties/symmetry/op.
This is the finite operation list of the group obtained by expanding cctbx's additional Euclidean-normalizer generators, represented modulo integer translations of the recorded cell.
It includes the original space-group operations and the identity, unlike the nontrivial coset lists in the two affine-normalizer properties.
The normalizer's n_centering_translations counts its own pure-translation representatives, which can outnumber the centering translations of the original space group.
Its n_pointgroup_symops equals n_linear_parts, and n_symops = n_centering_translations * n_pointgroup_symops.
However, symops_mod_centering is factored by the original space group's centering_translations supplied in the containing H-M record.
Its length is therefore n_symops divided by the length of that outer list and need not equal the normalizer's n_pointgroup_symops.
Examples:
{"normalizer_kind": "euclidean", "n_centering_translations": 1, "n_pointgroup_symops": 2, "n_symops": 2, "n_linear_parts": 2, "symops": [{"rot_type": "-1", "sense": 0, "axis": [0, 0, 0], "screw_glide": ["0", "0", "0"], "origin_shift": ["0", "0", "0"], "operation_kind": "euclidean", "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": "1", "sense": 0, "axis": [0, 0, 0], "screw_glide": ["0", "0", "0"], "origin_shift": ["0", "0", "0"], "operation_kind": "euclidean", "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}}], "symops_mod_centering": [{"rot_type": "-1", "sense": 0, "axis": [0, 0, 0], "screw_glide": ["0", "0", "0"], "origin_shift": ["0", "0", "0"], "operation_kind": "euclidean", "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": "1", "sense": 0, "axis": [0, 0, 0], "screw_glide": ["0", "0", "0"], "origin_shift": ["0", "0", "0"], "operation_kind": "euclidean", "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}}]}JSON definition:
{
"$id": "https://schemas.httk.org/defs/v0.1/properties/transformations/euclidean_normalizer",
"$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json",
"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`.\n\nThis is the finite operation list of the group obtained by expanding cctbx's additional Euclidean-normalizer generators, represented modulo integer translations of the recorded cell.\nIt includes the original space-group operations and the identity, unlike the nontrivial coset lists in the two affine-normalizer properties.\nThe normalizer's `n_centering_translations` counts its own pure-translation representatives, which can outnumber the centering translations of the original space group.\nIts `n_pointgroup_symops` equals `n_linear_parts`, and `n_symops = n_centering_translations * n_pointgroup_symops`.\nHowever, `symops_mod_centering` is factored by the original space group's `centering_translations` supplied in the containing H-M record.\nIts length is therefore `n_symops` divided by the length of that outer list and need not equal the normalizer's `n_pointgroup_symops`.",
"properties": {
"normalizer_kind": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string",
"null"
],
"description": "Kind label for this normalizer contribution.",
"enum": [
"euclidean",
null
]
},
"n_centering_translations": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/spacegroups/n_centering_translations",
"title": "Number of centering translations",
"x-optimade-type": "integer",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "n_centering_translations",
"label": "n_centering_translations_spacegroups"
},
"type": [
"integer",
"null"
],
"description": "Number of centering translations in the conventional cell of the space-group setting.\n\nWhen the entry contains a `centering_translations` list, this value MUST equal its length.",
"x-optimade-unit": "inapplicable",
"examples": [
1,
2
]
},
"n_pointgroup_symops": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/pointgroups/n_pointgroup_symops",
"title": "Number of point-group operations",
"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.\n\nPoint-group entries do not carry this field; their operation count is `order`.\nFor a space-group entry it is the order of the quotient by the full translation subgroup and MUST equal `n_symops / n_centering_translations`.\nInversion and other improper point operations are included.",
"x-optimade-unit": "inapplicable",
"examples": [
1,
2
]
},
"n_symops": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/spacegroups/n_symops",
"title": "Number of symops",
"x-optimade-type": "integer",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "n_symops",
"label": "n_symops_spacegroups"
},
"type": [
"integer",
"null"
],
"description": "Number of symmetry operations in the finite operation list of the generated entry.\n\nWhen the entry contains a `symops` list, this value MUST equal its length.",
"x-optimade-unit": "inapplicable",
"examples": [
1,
2
]
},
"n_linear_parts": {
"$id": "https://schemas.httk.org/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.\n\nThis value MUST equal the number of distinct `affine_transformation.matrix` values in the containing `symops` list when that list is present; translation differences do not increase it.",
"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.httk.org/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 - **operation\\_kind**: OPTIONAL; String.\n The value `euclidean` identifies an operation emitted within a Euclidean-normalizer table; ordinary point-group and space-group operation records omit it.\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.\n\nWriting the affine part as `(W,w)`, the intrinsic translation `screw_glide = v` is defined by `(W,w)^n = (I,n*v)`, where `n` is the order of `W`.\nThe reported `origin_shift = q` satisfies `(I-W)*q = w-v`; moving the coordinate origin to `q` leaves the intrinsic translation `v`.\nIt locates the symmetry element and is not a second translation to add to `w`.\nFor a proper rotation, `axis` is the integer direction fixed by `W`; for a rotoinversion it is the rotation axis fixed by `-W`, hence for a mirror it is the plane-normal direction.\nThese are direct-lattice direction components, not Cartesian unit vectors or reciprocal-plane indices.\nThe identity and inversion use `[0,0,0]` because neither has a unique axis.\nThe signed `sense` follows cctbx's rotation/rotoinversion convention about that reported axis; converting a rotoinversion to a Schoenflies rotation-reflection symbol can reverse the rotation sense.",
"properties": {
"affine_transformation": {
"$id": "https://schemas.httk.org/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 invertible affine transformation preserves collinearity and parallelism, but need not preserve Euclidean distances or angles.\nA singular affine map can collapse a line or plane to a lower-dimensional image.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nWith column vectors, the map is `u_out = matrix * u_in + vector`; matrix rows specify the three output components.\nThe containing property identifies whether `u_in` denotes fractional coordinates or abstract Wyckoff parameters and identifies the input and output settings.\nNo wrapping modulo lattice translations is implicit in this equation; apply any required periodic reduction only in the specified output setting.\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 It MUST express the same affine map as `matrix` and `vector`, using `x,y,z` for the input components.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether the linear part preserves the crystallographic metric family specified by the containing setting; this is not a test of the fractional matrix against the Cartesian identity metric.",
"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": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/core/fraction",
"title": "Fraction",
"x-optimade-type": "string",
"x-optimade-definition": {
"label": "fraction_core",
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "fraction"
},
"type": [
"string",
"null"
],
"description": "A numerical representation formed as the quotient of two numbers represented as a string.",
"examples": [
"2/3",
"5/42",
"10",
"0"
],
"x-optimade-unit": "inapplicable"
}
}
},
"vector": {
"x-optimade-type": "list",
"x-optimade-unit": "inapplicable",
"x-optimade-dimensions": {
"names": [
"dim_lattice"
],
"sizes": [
3
]
},
"type": [
"array",
"null"
],
"description": "Exact fractional-coordinate vector part of the affine transformation.",
"items": {
"$id": "https://schemas.httk.org/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": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/symmetry/op_xyz",
"title": "Operation xyz",
"x-optimade-type": "string",
"x-compatibility": [
"https://schemas.optimade.org/defs/v1.2/properties/optimade/common/symmetry_operation_xyz",
"https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html"
],
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "op_xyz",
"label": "op_xyz_symmetry"
},
"x-optimade-unit": "inapplicable",
"type": [
"string",
"null"
],
"description": "Coordinate operation expressed in the algebraic xyz form, also known as Jones' faithful representation (Bradley & Cracknell, 1972: pp. 35-37; adapted for computer strings).\n\nThe following definition is adapted from (and meant to be compatible with) the IUCr symCIF version 1.0.1 dictionary definition of `_space_group_symop.operation_xyz` referenced to: International Tables for Crystallography (2002). Volume A, Space-group symmetry, edited by Th. Hahn, 5th. ed. (Kluwer Academic Publishers).\nIt is available at: https://www.iucr.org/__data/iucr/cifdic_html/2/cif_sym.dic/Ispace_group_symop.operation_xyz.html\n\nIf W is a matrix representation of the rotational part of the symmetry operation defined by the positions and signs of x, y and z, and w is a column of translations defined by the fractions, an equivalent position X' is generated from a given position X by the equation: X' = WX + w.",
"x-undef-pattern": "^([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?),([-+]?[xyz]([-+][xyz])?([-+](1/2|[12]/3|[1-3]/4|[1-5]/6))?|[-+]?(1/2|[12]/3|[1-3]/4|[1-5]/6)([-+][xyz]([-+][xyz])?)?)$",
"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.\nThis optional integer annotation MUST equal the exact determinant of `matrix`; its absence does not imply determinant one.\nRational matrices can have noninteger determinants, in which case this integer annotation is omitted."
},
"is_orthogonal": {
"x-optimade-type": "boolean",
"x-optimade-unit": "inapplicable",
"type": [
"boolean",
"null"
],
"description": "Whether the matrix part is an isometry of the setting's metric, i.e. it preserves every metric tensor of the setting's crystal family expressed in this basis.\nFor a same-setting matrix `M` and metric tensor `g`, the criterion is `M^T g M = g` for every positive-definite metric in that family.\nThis is orthogonality with respect to the actual (generally non-Cartesian) lattice metric, not orthogonality of the matrix as a plain array: hexagonal sixfold rotations are isometries, whereas a cell-enlarging transform is not."
}
},
"examples": [
{
"matrix": [
[
"-1",
"0",
"0"
],
[
"0",
"-1",
"0"
],
[
"0",
"0",
"1"
]
],
"vector": [
"0",
"0",
"0"
],
"xyz": "-x,-y,z",
"det": 1,
"is_orthogonal": true
}
]
},
"operation_kind": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string"
],
"enum": [
"euclidean"
],
"description": "Euclidean-normalizer context when present; this does not change the operation's affine action."
},
"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",
null
]
},
"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; zero for identity, inversion, twofold rotation, and mirror operations.",
"enum": [
-1,
0,
1,
null
]
},
"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": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/core/fraction",
"title": "Fraction",
"x-optimade-type": "string",
"x-optimade-definition": {
"label": "fraction_core",
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "fraction"
},
"type": [
"string",
"null"
],
"description": "A numerical representation formed as the quotient of two numbers represented as a string.",
"examples": [
"2/3",
"5/42",
"10",
"0"
],
"x-optimade-unit": "inapplicable"
}
},
"origin_shift": {
"x-optimade-type": "list",
"x-optimade-unit": "inapplicable",
"x-optimade-dimensions": {
"names": [
"dim_lattice"
],
"sizes": [
3
]
},
"type": [
"array",
"null"
],
"description": "Origin-shift descriptor represented exactly as a list of fraction strings.",
"items": {
"$id": "https://schemas.httk.org/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": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/spacegroups/symops_mod_centering",
"title": "Symmetry operations modulo centering translations",
"x-optimade-type": "list",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "symops_mod_centering",
"label": "symops_mod_centering_spacegroups"
},
"x-optimade-unit": "inapplicable",
"type": [
"array",
"null"
],
"description": "Representative symmetry-operation descriptors modulo centering translations.\n\nEach list member is an operation on the format defined by the property definition: https://schemas.httk.org/defs/v0.1/properties/symmetry/op\n\nTwo listed full operations are equivalent here when their matrices agree and their translation vectors differ by a centering translation modulo integer cell translations.\nThere is one representative for each point-group operation, including inversion-related operations.\nFor a space-group record, the list length MUST equal `n_pointgroup_symops`.\nCombining each representative with every `centering_translations` vector and reducing translations modulo integers recovers `symops` as a set.",
"items": {
"$id": "https://schemas.httk.org/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 - **operation\\_kind**: OPTIONAL; String.\n The value `euclidean` identifies an operation emitted within a Euclidean-normalizer table; ordinary point-group and space-group operation records omit it.\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.\n\nWriting the affine part as `(W,w)`, the intrinsic translation `screw_glide = v` is defined by `(W,w)^n = (I,n*v)`, where `n` is the order of `W`.\nThe reported `origin_shift = q` satisfies `(I-W)*q = w-v`; moving the coordinate origin to `q` leaves the intrinsic translation `v`.\nIt locates the symmetry element and is not a second translation to add to `w`.\nFor a proper rotation, `axis` is the integer direction fixed by `W`; for a rotoinversion it is the rotation axis fixed by `-W`, hence for a mirror it is the plane-normal direction.\nThese are direct-lattice direction components, not Cartesian unit vectors or reciprocal-plane indices.\nThe identity and inversion use `[0,0,0]` because neither has a unique axis.\nThe signed `sense` follows cctbx's rotation/rotoinversion convention about that reported axis; converting a rotoinversion to a Schoenflies rotation-reflection symbol can reverse the rotation sense.",
"properties": {
"affine_transformation": {
"$id": "https://schemas.httk.org/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 invertible affine transformation preserves collinearity and parallelism, but need not preserve Euclidean distances or angles.\nA singular affine map can collapse a line or plane to a lower-dimensional image.\nThe transformation is represented by a 3 by 3 matrix and a 3-vector, both serialized with exact string entries.\nWith column vectors, the map is `u_out = matrix * u_in + vector`; matrix rows specify the three output components.\nThe containing property identifies whether `u_in` denotes fractional coordinates or abstract Wyckoff parameters and identifies the input and output settings.\nNo wrapping modulo lattice translations is implicit in this equation; apply any required periodic reduction only in the specified output setting.\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 It MUST express the same affine map as `matrix` and `vector`, using `x,y,z` for the input components.\n\n - **det**: OPTIONAL; Integer.\n Determinant of `matrix` when the generator emits it.\n\n - **is\\_orthogonal**: OPTIONAL; Boolean.\n Whether the linear part preserves the crystallographic metric family specified by the containing setting; this is not a test of the fractional matrix against the Cartesian identity metric.",
"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.httk.org/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": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/core/fraction",
"title": "Fraction",
"x-optimade-type": "string",
"x-optimade-definition": {
"label": "fraction_core",
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "fraction"
},
"type": [
"string",
"null"
],
"description": "A numerical representation formed as the quotient of two numbers represented as a string.",
"examples": [
"2/3",
"5/42",
"10",
"0"
],
"x-optimade-unit": "inapplicable"
}
},
"xyz": {
"$id": "https://schemas.httk.org/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.\nThis optional integer annotation MUST equal the exact determinant of `matrix`; its absence does not imply determinant one.\nRational matrices can have noninteger determinants, in which case this integer annotation is omitted."
},
"is_orthogonal": {
"x-optimade-type": "boolean",
"x-optimade-unit": "inapplicable",
"type": [
"boolean",
"null"
],
"description": "Whether the matrix part is an isometry of the setting's metric, i.e. it preserves every metric tensor of the setting's crystal family expressed in this basis.\nFor a same-setting matrix `M` and metric tensor `g`, the criterion is `M^T g M = g` for every positive-definite metric in that family.\nThis is orthogonality with respect to the actual (generally non-Cartesian) lattice metric, not orthogonality of the matrix as a plain array: hexagonal sixfold rotations are isometries, whereas a cell-enlarging transform is not."
}
},
"examples": [
{
"matrix": [
[
"-1",
"0",
"0"
],
[
"0",
"-1",
"0"
],
[
"0",
"0",
"1"
]
],
"vector": [
"0",
"0",
"0"
],
"xyz": "-x,-y,z",
"det": 1,
"is_orthogonal": true
}
]
},
"operation_kind": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string"
],
"enum": [
"euclidean"
],
"description": "Euclidean-normalizer context when present; this does not change the operation's affine action."
},
"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",
null
]
},
"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; zero for identity, inversion, twofold rotation, and mirror operations.",
"enum": [
-1,
0,
1,
null
]
},
"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.httk.org/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.httk.org/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": 2,
"n_symops": 2,
"n_linear_parts": 2,
"symops": [
{
"rot_type": "-1",
"sense": 0,
"axis": [
0,
0,
0
],
"screw_glide": [
"0",
"0",
"0"
],
"origin_shift": [
"0",
"0",
"0"
],
"operation_kind": "euclidean",
"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": "1",
"sense": 0,
"axis": [
0,
0,
0
],
"screw_glide": [
"0",
"0",
"0"
],
"origin_shift": [
"0",
"0",
"0"
],
"operation_kind": "euclidean",
"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
}
}
],
"symops_mod_centering": [
{
"rot_type": "-1",
"sense": 0,
"axis": [
0,
0,
0
],
"screw_glide": [
"0",
"0",
"0"
],
"origin_shift": [
"0",
"0",
"0"
],
"operation_kind": "euclidean",
"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": "1",
"sense": 0,
"axis": [
0,
0,
0
],
"screw_glide": [
"0",
"0",
"0"
],
"origin_shift": [
"0",
"0",
"0"
],
"operation_kind": "euclidean",
"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
}
}
]
}
]
}