# Spglib dataset#

At version 1.9.4, SpglibDataset was modified. The member name setting is changed to choice and pointgroup_number is removed.

The dataset is accessible through the C-structure given by

typedef struct {
int spacegroup_number;
int hall_number;
char international_symbol[11];
char hall_symbol[17];
char choice[6];
double transformation_matrix[3][3];
double origin_shift[3];
int n_operations;
int (*rotations)[3][3];
double (*translations)[3];
int n_atoms;
int *wyckoffs;
char (*site_symmetry_symbols)[7];
int *equivalent_atoms;
int *crystallographic_orbits;
double primitive_lattice[3][3];
int *mapping_to_primitive;
int n_std_atoms;
double std_lattice[3][3];
int *std_types;
double (*std_positions)[3];
double std_rotation_matrix[3][3];
int *std_mapping_to_primitive;
char pointgroup_symbol[6];
} SpglibDataset;

## 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 .

### hall_symbol#

The Hall symbol is stored here.

### choice#

The information on unique axis, setting or cell choices.

## Symmetry operations#

### rotations, translations, and n_operations#

The symmetry operations of the input unit cell are stored in rotations and translations. A crystallographic symmetry operation $$(\boldsymbol{W}, \boldsymbol{w})$$ is made from a pair of rotation $$\boldsymbol{W}$$ and translation $$\boldsymbol{w}$$ parts with the same index. Number of symmetry operations is given as n_operations. The detailed explanation of the values is found at spg_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 [0, 0, 0, 0, 4, 4, 4, 4], where the symmetrically independent atomic indices are 0 and 4. We can see that the atoms from 0 to 3 are mapped to 0 and those from 4 to 7 are mapped to 4. 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.

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 spg_get_symmetry.

### crystallographic_orbits#

New at version 1.15

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 ($$\boldsymbol{P}$$) and origin_shift ($$\boldsymbol{p}$$) 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

$( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} ) = ( \mathbf{a} \; \mathbf{b} \; \mathbf{c} ) \boldsymbol{P}^{-1} \;\; \text{and} \;\; \boldsymbol{x}_\mathrm{s} = \boldsymbol{P}\boldsymbol{x} + \boldsymbol{p} \;\;(\mathrm{mod}\; \mathbf{1}),$

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.

At versions 1.7.x and 1.8 or before, the set of transformation_matrix ($$\boldsymbol{P}_\text{old}$$) and origin_shift ($$\boldsymbol{p}_\text{old}$$) was differently defined from the current definition as follows:

$( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} ) = ( \mathbf{a} \; \mathbf{b} \; \mathbf{c} ) \boldsymbol{P}_\text{old} \;\; \text{and} \;\; \boldsymbol{x}_\mathrm{s} = \boldsymbol{P}^{-1}_\text{old}\boldsymbol{x} - \boldsymbol{p}_\text{old} \;\;(\mathrm{mod}\; \mathbf{1}).$

## 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.

At versions 1.7.x and 1.8 or before, the variable names of the members corresponding to those above are n_brv_atoms, brv_lattice, brv_types, and brv_positions, respectively.

### std_rotation_matrix#

New in version 1.11

Rotation matrix that rotates the standardized crystal structure before idealization $$( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )$$ to that after idealization $$( \bar{\mathbf{a}}_\mathrm{s} \; \bar{\mathbf{b}}_\mathrm{s} \; \bar{\mathbf{c}}_\mathrm{s} )$$ in Cartesian coordinates of the given input unit cell. The rotation matrix $$\boldsymbol{R}$$ is defined by

(1)#$( \bar{\mathbf{a}}_\mathrm{s} \; \bar{\mathbf{b}}_\mathrm{s} \; \bar{\mathbf{c}}_\mathrm{s} ) = ( \boldsymbol{R} \mathbf{a}_\mathrm{s} \; \boldsymbol{R} \mathbf{b}_\mathrm{s} \; \boldsymbol{R} \mathbf{c}_\mathrm{s} ).$

More precisely, this rotation matrix is an orthonormal matrix. Since $$( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )$$ can be distored, to make $$\boldsymbol{R}$$ orthonormal, it is calculated as follows. Make cubes of $$( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )$$ and $$( \bar{\mathbf{a}}_\mathrm{s} \; \bar{\mathbf{b}}_\mathrm{s} \; \bar{\mathbf{c}}_\mathrm{s} )$$ by $$\mathrm{L} = \begin{pmatrix}\frac{\mathbf{a}}{|\mathbf{a}|} & \frac{(\mathbf{a} \times \mathbf{b}) \times \mathbf{a}}{|(\mathbf{a} \times \mathbf{b}) \times \mathbf{a}|} & \frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|} \end{pmatrix}$$. Watching $$\mathrm{L}$$s as $$3\times 3$$ matrices, $$\boldsymbol{R}$$ is obtained by solving $$\bar{\mathrm{L}}_\mathrm{s} = \boldsymbol{R} \mathrm{L}_\mathrm{s}$$.

### std_mapping_to_primitive#

This is available at versions 1.10 or later. 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#

New in version 1.8.1

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

1, -1, 2, m, 2/m, 222, mm2, mmm, 4, -4, 4/m, 422, 4mm, -42m, 4/mmm,
3, -3, 32, 3m, -3m, 6, -6, 6/m, 622, 6mm, -6m2, 6/mmm, 23, m-3,
432, -43m, m-3m