Spglib dataset

Definitions and conventions

The dataset is accessible through the struct given by

dump(Dataset)

Dataset <: Spglib.AbstractDataset
  spacegroup_number::Int32
  hall_number::Int32
  international_symbol::String
  hall_symbol::String
  choice::String
  transformation_matrix::StaticArraysCore.SMatrix{3, 3, Float64, 9}
  origin_shift::StaticArraysCore.SVector{3, Float64}
  n_operations::Int32
  rotations::Vector{StaticArraysCore.SMatrix{3, 3, Int32, 9}}
  translations::Vector{StaticArraysCore.SVector{3, Float64}}
  n_atoms::Int32
  wyckoffs::Vector{Char}
  site_symmetry_symbols::Vector{String}
  equivalent_atoms::Vector{Int32}
  crystallographic_orbits::Vector{Int32}
  primitive_lattice::Lattice{Float64}
  mapping_to_primitive::Vector{Int32}
  n_std_atoms::Int32
  std_lattice::Lattice{Float64}
  std_types::Vector{Int32}
  std_positions::Vector{StaticArraysCore.SVector{3, Float64}}
  std_rotation_matrix::StaticArraysCore.SMatrix{3, 3, Float64, 9}
  std_mapping_to_primitive::Vector{Int32}
  pointgroup_symbol::String

Space group type

`spacegroup_number`

The space group type number defined in International Tables for Crystallography (ITA).

`hall_number`

The serial number from $1$ to $530$ which are found at list of space groups (Seto's web site). Be sure that this is not a standard crystallographic definition as far as the author of Spglib knows.

`international_symbol`

The (full) Hermann–Mauguin notation of space group type is given by .

`choice`

The information on unique axis, setting or cell choices.

The symmetry operations of the input unit cell are stored in rotations and translations. A crystallographic symmetry operation $(\mathbf{W}, \mathbf{w})$ is made from a pair of rotation $\mathbf{W}$ and translation $\mathbf{w}$ parts with the same index. Number of symmetry operations is given as n_operations. The detailed explanation of the values is found at get_symmetry.

Wyckoff positions and symmetrically equivalent atoms

`n_atoms`

Number of atoms in the input unit cell. This gives the numbers of elements in wyckoffs and equivalent_atoms.

`wyckoffs`

This gives the information of Wyckoff letters by integer numbers, where $0$, $1$, $2$, $\ldots$, represent the Wyckoff letters of $a$, $b$, $c$, $\ldots$ These are assigned to all atomic positions of the input unit cell in this order. Therefore the number of elements in wyckoffs is same as the number of atoms in the input unit cell, which is given by n_atoms.

This is determined from the symmetry of the primitive cell.

`site_symmetry_symbols`

This gives site-symmetry symbols. These are valid for the standard settings. For different settings and choices belonging to the same space group type, the same set of the symbols is returned.

This is determined from the symmetry of the primitive cell.

`equivalent_atoms`

This gives the mapping table from the atomic indices of the input unit cell to the atomic indices of symmetrically independent atom, such as [1, 1, 1, 1, 5, 5, 5, 5], where the symmetrically independent atomic indices are $1$ and $5$. We can see that the atoms from $1$ to $4$ are mapped to $1$ and those from $5$ to $8$ are mapped to $5$. The number of elements in equivalent_atoms is same as the number of atoms in the input unit cell, which is given by n_atoms.

Warning

You may notice that the indices here differ from those in Spglib's official documentation, where the indices start from $0$. This discrepancy arises because indices in Julia start from $1$ by default. Consequently, all indices here are incremented by $1$.

Symmetry operations found for the input cell are used to determine the equivalent atoms. equivalent_atoms and crystallographic_orbits are almost equivalent, but they can be different in a special case as written in get_symmetry.

`crystallographic_orbits`

This is almost equivalent to equivalent_atoms. But symmetry of the primitive cell is used to determine the symmetrically equivalent atoms.

Transformation matrix and origin shift

`transformation_matrix` and `origin_shift`

transformation_matrix ($\mathbf{P}$) and origin_shift ($\mathbf{p}$) are obtained as a result of space-group-type matching under a set of unique axis, setting and cell choices. These are operated to the basis vectors and atomic point coordinates of the input unit cell as

\[\begin{align} \begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix} &= \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{P}^{-1},\\ \mathbf{x}_\text{s} &= \mathbf{P} \mathbf{x} + \mathbf{p} \ (\mathrm{mod}\ \mathbf{1}), \end{align}\]

by which the basis vectors are transformed to those of a standardized unit cell. Atomic point coordinates are shifted so that symmetry operations have one of possible standard origins. The detailed definition is presented at Definitions and conventions.

Standardized crystal structure after idealization

`n_std_atoms`, `std_lattice`, `std_types`, and `std_positions`

The standardized crystal structure after idealization corresponding to a Hall symbol is stored in n_std_atoms, std_lattice, std_types, and std_positions. These output usually contains the rotation in Cartesian coordinates and rearrangement of the order atoms with respect to the input unit cell.

`std_rotation_matrix`

Rotation matrix that rotates the standardized crystal structure before idealization $\begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}$ to that after idealization $\begin{bmatrix} \bar{\mathbf{a}}_\text{s} & \bar{\mathbf{b}}_\text{s} & \bar{\mathbf{c}}_\text{s} \end{bmatrix}$ in Cartesian coordinates of the given input unit cell. The rotation matrix $\mathbf{R}$ is defined by

\[\begin{bmatrix} \bar{\mathbf{a}}_\text{s} & \bar{\mathbf{b}}_\text{s} & \bar{\mathbf{c}}_\text{s} \end{bmatrix} = \begin{bmatrix} \mathbf{R} \mathbf{a}_\text{s} & \mathbf{R} \mathbf{b}_\text{s} & \mathbf{R} \mathbf{c}_\text{s} \end{bmatrix}\]

More precisely, this rotation matrix is an orthonormal matrix. Since $\begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}$ can be distored, to make $\mathbf{R}$ orthonormal, it is calculated as follows. Make cubes of $\begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}$ and $\begin{bmatrix} \bar{\mathbf{a}}_\text{s} & \bar{\mathbf{b}}_\text{s} & \bar{\mathbf{c}}_\text{s} \end{bmatrix}$ by

\[\mathbf{L} = \begin{bmatrix} \dfrac{\mathbf{a}}{\lvert\mathbf{a}\rvert} & \dfrac{(\mathbf{a} \times \mathbf{b}) \times \mathbf{a}}{\lvert(\mathbf{a} \times \mathbf{b}) \times \mathbf{a}\rvert} & \dfrac{\mathbf{a} \times \mathbf{b}}{\lvert\mathbf{a} \times \mathbf{b}\rvert} \end{bmatrix}.\]

Watching $\mathbf{L}_\text{s}$ as $3\times 3$ matrices, $\mathbf{R}$ is obtained by solving

\[\bar{\mathbf{L}}_\text{s} = \mathbf{R} \mathbf{L}_\text{s}.\]

`std_mapping_to_primitive`

This gives a list of atomic indices in the primitive cell of the standardized crystal structure, where the same number presents the same atom in the primitive cell. By collective the atoms having the same number, a set of relative lattice points in the standardized crystal structure is obtained.

Crystallographic point group

`pointgroup_symbol`

pointgroup_symbol is the symbol of the crystallographic point group in the Hermann–Mauguin notation. There are 32 crystallographic point groups

\[\{1,\ \bar{1},\ 2,\ m,\ 2/m,\ 222,\ mm2,\ mmm,\ 4,\ \bar{4},\ 4/m,\ 422,\ 4mm,\ \bar{4}2m,\ 4/mmm,\ 3,\ \bar{3},\ 32,\ 3m,\ \bar{3}m,\ 6,\ \bar{6},\ 6/m,\ 622,\ 6mm,\ \bar{6}m2,\ 6/mmm,\ 23,\ m\bar{3},\ 432,\ \bar{4}3m,\ m\bar{3}m\}\]

Intermediate data in symmetry search

A primitive cell is searched from the translational symmetry. This primitive cell is given by primitive_lattice and mapping_to_primitive below.

`primitive_lattice`

Non-standardized basis vectors of a primitive cell in the input cell.

`mapping_to_primitive`

This gives a list of atomic indices in the primitive cell of the input crystal structure, where the same number presents the same atom in the primitive cell. By collective the atoms having the same number, a set of relative lattice points in the input crystal structure is obtained.