# Variables

Variables frequently used in C-API are explained. In interfaces of
other languages, the definition may be different, e.g., `lattice`
for Python interface is transposed.

(variables_lattice)=
## `lattice`

**Attention**: In Python interface, `lattice` is transposed
({ref}`py_variables_crystal_structure`).

Basis vectors (in Cartesian)

```
[ [ a_x, b_x, c_x ],
  [ a_y, b_y, c_y ],
  [ a_z, b_z, c_z ] ]
```

Cartesian position {math}`\mathbf{x}_\mathrm{Cart}` is obtained by

```{math}
  \mathbf{x}_\mathrm{Cart} = \mathrm{L}\cdot\mathbf{x}_\mathrm{frac}
```
where {math}`\mathrm{L}` is the basis vectors defined above and is
{math}`\mathrm{L}=(\mathbf{a},\mathbf{b},\mathbf{c})`, and
{math}`\mathbf{x}_\mathrm{frac}` is the atomic position in fractional
coordinates given below.


## `position`

Atomic points (in fractional coordinates with respect to basis vectors)

```
[ [ x1_a, x1_b, x1_c ],
  [ x2_a, x2_b, x2_c ],
  [ x3_a, x3_b, x3_c ],
  ...                   ]
```

## `types`

Atomic species are differentiated by integers. The numbers are just
used to distinguishes atoms, and is not related to atomic numbers.

```
[ type_1, type_2, type_3, ... ]
```
The number of elements is same as the number of atoms.

## `rotation`

Rotation matrices (integer) of symmetry operations.

```
[ [ r_aa, r_ab, r_ac ],
  [ r_ba, r_bb, r_bc ],
  [ r_ca, r_cb, r_cc ] ]
```
## `translation`

Translation vectors corresponding to symmetry operations in fractional
coordinates.

```
[ t_a, t_b, t_c ]
```

(variables_symprec)=
## `symprec`


Distance tolerance in Cartesian coordinates to find crystal
symmetry.

For atomic positions, roughly speaking, two position vectors `x` and `x'` in
Cartesian coordinates are considered to be the same if `|x' - x| < symprec`.
For more details, see the [spglib paper](https://arxiv.org/abs/1808.01590>),
Sec. II-A.

The angle distortion between basis vectors is converted to a length
and compared with this distance tolerance. For more details, see the
[spglib paper](https://arxiv.org/abs/1808.01590), Sec. IV-A. It is
possible to specify angle tolerance explicitly, see
{ref}`variables_angle_tolerance`.

(variables_angle_tolerance)=
## `angle_tolerance`

Tolerance of angle between basis vectors in degrees to be tolerated
in the symmetry finding. To use angle tolerance, another set of
functions are prepared for C-API as follows

```
spgat_get_dataset
spgat_get_symmetry
spgat_get_symmetry_with_collinear_spin
spgat_get_multiplicity
spgat_find_primitive
spgat_get_international
spgat_get_schoenflies
spgat_refine_cell
```

(variables_mag_symprec)=
## `mag_symprec`

Tolerance for magnetic symmetry search in the unit of spins or magmoms. By
specifying a negative value, the same value as `symprec` is used to compare
magnetic moments.

The following functions are called by the same way with an additional argument
of `const double angle_tolerance` in degrees. By specifying a negative value,
the behavior becomes the same as usual functions (e.g.
`spg_get_magnetic_dataset`). For python-API, the magnetic tolerance is given
with a keyword argument of `mag_symprec`, where the default value of
`mag_symprec` is a negative value.

```
spgms_get_magnetic_dataset
spgms_get_symmetry_with_collinear_spin
spgms_get_symmetry_with_site_tensors
```
