Symmetry Group

Description

Contains a description of a symmetry group.

Project files

This format is used for all symmetry group files in the CASM project, including the lattice_point_group.json, factor_group.json, and crystal_point_group.json files.

JSON Attributes List

Symmetry Group attributes:

Name	Description	Format
group_classification	Group classification information, such as the point group	Symmetry Group Classification
group_operations	Descriptions of group operations	dict of Symmetry Operation
group_structure	Multiplication table and conjugacy classes	Symmetry Group Structure

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Symmetry Group Classification attributes:

Name	Description	Format
latex_name	Point group name, formatted for Latex	string
name	Point group name	string
periodicity	Whether there is translation symmetry	string

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Symmetry Operation attributes:

Name	Description	Format
CART	Cartesian coordinate transformation representation	Coordinate Transformation Representation
FRAC	Fractional coordinate transformation representation	Coordinate Transformation Representation
info	Symmetry operation information	Symmetry Operation Information
master_group_index	Index of this operation in the master group	int

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Coordinate Transformation Representation attributes:

Name	Description	Format
matrix	Coordinate transformation matrix	2d array of number
tau	Coordinate translation vector	array of number
time_reversal	Whether symmetry includes time reversal invariance	bool

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Symmetry Operation Information attributes:

Name	Description	Format
brief	Brief string descriptions of the symmetry operation	dict of string
conjugacy_class	Index of the conjugacy class containing the operation	int
invariant_point	Coordinate of an invariant point of the operation	dict
inverse_operation	Index in the group of the inverse operation	int
mirror_normal	Unit normal vector of mirror and glide planes	dict
rotation_angle	Rotation angle of the operation	number
rotation_axis	Rotation axis of the operation	dict
shift	Screw or glide shift vector	dict
type	Type of symmetry operation	string

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Symmetry Group Structure attributes:

Name	Description	Format
conjugacy_classes	Conjugacy class elements and type	dict
multiplication_table	Group multiplication table	2d array of int

JSON Attributes Description

Symmetry Group JSON Object

• group_classification: [Symmetry Group Classification]Group Classification

  Group classification information, such as the point group.

• group_operations: dict of Symmetry Operation

  Descriptions of group operations. The dictionary keys are of the form op_01, op_02, …, or op_1, op_2, …, if there are less than 10 elements in the group.

• group_structure: Symmetry Group Structure

  Contains the group multiplication table and conjugacy classes.

Symmetry Group Classification JSON Object

• latex_name: string

  Point group name, formatted for Latex

• name: string

  Point group name.

• periodicity: string

  Has the value “PERIODIC” if there is translation symmetry, or the value “APERIODIC” if there is not translation symmetry.

Symmetry Operation JSON Object

• CART: Coordinate Transformation Representation

  Cartesian coordinate transformation representation.

• FRAC: Coordinate Transformation Representation

  Fractional coordinate transformation representation

• info: Symmetry Operation Information

  Symmetry operation information

• master_group_index: int

  Index of this operation in the master group. The master group is the group from which all subgroups are formed. Most commonly, the master group is the factor group of the prim.

Coordinate Transformation Representation JSON Object

The symmetery operations transform a spatial coordinate $x \rightarrow x’$ according to $x’ = R*x+\tau$, where $R$ is the 3x3 coordinate transformation matrix and $\tau$ is the coordinate translation vector. The values $R$ and $\tau$ for the same symmetry operation may be expressed in either Cartesian or fractional coordinates using the conversion, $x^{cart} = L * x^{frac}$, where $L$ is the lattice vector column matrix.

• matrix: 2d array of number (shape=(3,3))

  Coordinate transformation matrix.

• tau: array of number (shape=(3,))

  Coordinate translation vector.

• time_reversal: bool

  Whether symmetry includes time reversal invariance.

Symmetry Operation Information JSON Object

• brief: dict

  Brief string descriptions of the symmetry operation, following the conventions of (International Tables for Crystallography (2015). Vol. A. ch. 1.4, pp. 50-59). Includes Cartesian and fractional coordinate representations.

  Example:

  "brief" : {
    "CART" : "6⁺ (0.0000000 0.0000000 2.5843392) 0, 1.867143, z",
    "FRAC" : "6⁺ (0.0000000 0.0000000 0.5000000) 0.3333333, 0.6666667, z"
  }
  

  Examples:

  The following examples are given in the fractional coordinate representation.

  • 1: Identity operation.

  • 3⁺ 0, 0, z: Positive 3-fold rotation around the axis with coordinates 0, 0, z, for any z.

  • 2 ( 0.5000000 -0.0000000 0.0000000) x, 0.1666667, 0.25: 2-fold screw rotation, with shift vector (0.5000000 -0.0000000 0.0000000), around the axis with coordinates x, 0.1666667, 0.25, for any x.

  • m x, -x, z: Mirror plane, with coordinates x, -x, z, for any x, z.

  • g (0.5000000 0.5000000 0.5000000) 0.08333334+x, -0.08333334+x, z: Glide reflection, with shift vector (0.5000000 0.5000000 0.5000000), and glide plane with coordinates 0.08333334+x, -0.08333334+x, z for any x, z.

  • -3⁺ 0.3333333, -0.3333333, z; 0.3333333 -0.3333333 0.2500000: Positive 3-fold rotoinversion around the axis 0.3333333, -0.3333333, z, for any z, and the invariant point 0.3333333 -0.3333333 0.2500000.

  • -1 0.3333333 0.1666666 0.2500000: Inversion, with the invariant point 0.3333333 0.1666666 0.2500000.

  • m′ x, y, y: Mirror plane, with coordinates x, y, y, for any x, y, and time reversal invariance (indicated by the prime).

• conjugacy_class: int

  Index of the conjugacy class containing the operation.

• invariant_point: dict (conditional)

  Coordinate of an invariant point of the operation, if applicable. Includes Cartesian and fractional coordinate representations.

  Example:

  "invariant_point" : {
    "CART" : [ 0.000000000000, 1.867143135410, 0.000000000000 ],
    "FRAC" : [ 0.333333325000, 0.666666650000, 0.000000000000 ]
  }
  

• inverse_operation: int

  Index in the group of the inverse operation.

• mirror_normal: dict (conditional)

  Unit normal vector of mirror and glide planes, if applicable. Includes Cartesian and fractional coordinate representations.

  Example:

  "mirror_normal" : {
    "CART" : [ 0.500000000000, -0.866025403784, 0.000000000000 ],
    "FRAC" : [ 0.000000000000, -1.000000000000, 0.000000000000 ]
  }
  

• rotation_angle: number (conditional)

  Rotation angle of the operation, in degrees, if applicable.

• rotation_axis: dict (conditional)

  Vector lying along the rotation axis of the operation, if applicable. Includes Cartesian and fractional coordinate representations.

  Example:

  "rotation_axis" : {
    "CART" : [ 0.500000000000, 0.866025403784, -0.000000000000 ],
    "FRAC" : [ 0.707106781187, 0.707106781187, 0.000000000000 ]
  }
  

• shift: dict (conditional)

  Screw or glide shift vector, if applicable. Includes Cartesian and fractional coordinate representations.

  Example:

  "shift" : {
    "CART" : [ 0.808496714096, 1.400357386566, -0.000000000000 ],
    "FRAC" : [ 0.500000000000, 0.500000000000, 0.000000000000 ]
  }
  

• type: string

  Type of symmetry operation. One of “identity”, “rotation”, “screw”, “mirror”, “glide”, “rotoinversion”, or “inversion”.

Symmetry Group Structure JSON Object

• conjugacy_classes: dict

  Conjugacy class operations, type, and rotation angle (if applicable).

  Example:

  "conjugacy_classes" : {
    "class_01" : {
      "operation_type" : "identity",
      "operations" : [ 1 ]
    },
    "class_02" : {
      "operation_type" : "screw",
      "operations" : [ 2, 3 ],
      "rotation_angle" : 60.000000000000
    },
    "class_03" : {
      "operation_type" : "rotation",
      "operations" : [ 4, 5 ],
      "rotation_angle" : 120.000000000000
    },
    ...
  }
  

• multiplication_table: 2d array of int

  Group multiplication table. The symmetry operation products $g_k = g_i g_j$ are specified by $M_{ij}=k$, where $g_i$, $g_j$, and $g_k$ are the $i$-th, $j$-th, and $k$-th symmetry operations in the group, and $M$ is the multiplication table.

Examples

Example Symmetry Group files:

1. Factor group, for HCP Zr, with octahedral interstitial O disorder: [factor_group_ex1.json]
2. Factor group, for an FCC system with collinear magnetic spin (Cmagspin) [DoF] (which has time reversal symmetry): [factor_group_ex2.json]