Information in this page is valid for spglib 1.8.1 or later. The definitions of transformation matrix and origin shift were different in the previous versions.
Some references about crystallographic definitions and conventions are shown below. Though spglib may not follow them fully, it doesn’t mean spglib doesn’t respect them, rather it is due to the lack of understanding by the author of spglib.
In spglib, basis vectors are represented by three column vectors:
in Cartesian coordinates. Depending on the situation, is used instead of .
Coordinates of an atomic point are represented as three fractional values relative to basis vectors as follows,
where . A position vector in Cartesian coordinates is obtained by
A symmetry operation consists of a pair of the rotation part and translation part , and is represented as in the spglib document. The symmetry operation transfers to as follows:
The transformation matrix changes choice of basis vectors as follows
where and are the basis vectors of an arbitrary system and of a starndardized system, respectively. Transformation matrix doesn’t rotate a crystal in Cartesian coordinates, but just changes the choices of basis vectors.
The origin shift gives the vector from the origin of the standardized system to the origin of the arbitrary system ,
Origin shift doesn’t move a crystal in Cartesian coordinates, but just changes the origin to measure the coordinates of atomic points.
A change of basis is described by the combination of the transformation matrix and the origin shift denoted by where first the transformation matrix is applied and then origin shift. The points in the standardized system and arbitrary system are related by
A graphical example is shown below.
(click the figure to enlarge)
In this example,
Using the APIs
starndardized unit cell is obtained. The “starndardized unit cell” in
this document means that the (conventional) unit cell structure is
standardized by the crystal symmetry and lengths of basis vectors.
Crystals are categorized by Hall symbols in 530 different types in
terms of 230 space group types, unique axes, settings, and cell
choices. Moreover in spglib, lengths of basis vectors are used to
choose the order of if
the order can not be determined only by the symmetrical conventions.
In the standardized unit cells, there are five different centring types available, base centrings of A and C, rhombohedral (R), body centred (I), and face centred (F). The transformation is applied to the standardized unit cell by
where , , and are the basis vectors of the primitive cell and is the transformation matrix from the standardized unit cell to the primitive cell. for centring types are given as follows:
For rhombohedral lattice systems with the choice of hexagonal axes, is applied.
Spglib allows tolerance parameters to match a slightly distorted unit cell structure to a space group type with some higher symmetry. Using obtained symmetry operations, the distortion is removed to idealize the unit cell structure. The coordinates of atomic points are idealized using respective site-symmetries (Grosse-Kunstleve et al. (2002)). The basis vectors are idealized by forceing them into respective lattice shapes as follows. In this treatment, except for triclinic crystals, crystals can be rotated in Cartesian coordinates, which is the different type of transformation from that of the change-of-basis transformation explained above.