Pages = ["definitions.md"]
Depth = 2
using Spglib
The dataset is accessible through the struct
given by
dump(Dataset)
The space group type number defined in International Tables for Crystallography (ITA).
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.
The (full) Hermann–Mauguin notation of space group type is given by .
The Hall symbol is stored here.
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
.
Number of atoms in the input unit cell. This gives the numbers of
elements in wyckoffs
and equivalent_atoms
.
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.
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.
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
.
This is almost equivalent to equivalent_atoms
. But symmetry of the
primitive cell is used to determine the symmetrically equivalent atoms.
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
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.
The standardized crystal structure after [idealization](@ref 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.
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
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
Watching \mathbf{L}_\text{s}
as 3\times 3
matrices, \mathbf{R}
is
obtained by solving
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.
pointgroup_symbol
is the symbol of the crystallographic point
group in the Hermann–Mauguin notation. There are 32 crystallographic
point groups
A primitive cell is searched from the translational symmetry. This
primitive cell is given by primitive_lattice
and
mapping_to_primitive
below.
Non-standardized basis vectors of a primitive cell in the input cell.
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.