How to Use "International Tables for Crystallography"

International Tables for Crystallography Vol. A (hereafter simply referred to as International Tables) is necessary to analyze the crystal structure by diffraction data and to understand the crystal structure described in the paper.

What are described

The crystal structure must be described according to the rules of crystallography, which are provided in International Tables.
There are 230 crystallographic space groups. International Tables provide the following information for each space group.

Wyckoff Position

A site of coordinates x,y,z is transferred to another position with a symmetry operation. Considering all the symmetric operations for the space group, you can make a list of the sites equivalent to x,y,z. However, all the listed sites are not always different with each other. That is, multiple symmetric operations may transfer x,y,z to the same positions. In other words, the atomic sites can be classified by the independent symmetry operations. This classification of the coordinates x,y,z is called the Wyckoff position. The Wyckoff position is represented by multiplicity and a lower-case letter (Wyckoff Letter). Wyckoff position is important in group theory. In material science, multiplicity and site symmetry is of importance.


Multiplicity is the number of the equivalent sites present in a conventional unit cell.

Site symmetry

Site symmetry describes the combination of the inversion center, symmetry axes, and mirror planes containing the site.
The site symmetry determines the degeneracy of the wave functions of the atom at the site and the presence or absence of hybridization between wave functions. Therefore, the site symmetry is closely related to the energy level, optical response, and so on.

Reflection Conditions for the Site

The condition for a group of equivalent sites to contribute the Bragg hkl reflection is dependent on the Wyckoff position. It is described only if the reflection conditions of the space group is not enough.

Subgroups and Supergroups

Consider a second-order structural phase transition. The second-order phase transition is caused by some symmetry breaking. The change in space group across the phase transition is limited. If the phase transition between a higher-symmetry space group A and a lower-symmetry space group B is of second order, B is a subgroup of A, and A is a supergroup of B.

Description of Crystal Structure

The crystal structure is described in a relatively simple form, if one follows International Tables, The minimum information necessary for describing the crystal structure is as follows.

Usually, the following information is also given.

Taka-hisa Arima