# 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.

• Classification based on crystal system (7 systems). Note that trigonal but not rhombohedral is used because this is the crystal system but not the lattice system.
• Classification based on crystal point group (32 groups). Both Hermann-Mauguin and Schönflies notations are given.
• Classification based on space group (230 groups). Both the short and full forms of Hermann-Mauguin notation are given.
• How to set the origin. There are multiple choices for some space groups.
• List of symmetric operations. The places of inversion center, rotation axes, screw axes, mirror planes, and glide planes are shown in figures. The transformations of atomic positions are also indicated.
• The minimum region where independent atomic sites exist.
• Wyckoff positions (see below).
• Reflection condition for each site (see below).
• Subgroups and supergroups (see below).

### 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

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.

• Space group in the short form of Herman-Mauguin notation.
• Lattice constants (a,b,c,α,β,γ). Obvious ones can be omitted.
• Coordinates for independent sites of each constituent element.
• Occupancy of the site, if it is less than one.
Usually, the following information is also given.
• Number of molecules in a conventional unit cell. It is often referred to as Z.
• Multiplicity and Wyckoff position for each site.
• The thermal vibration parameter of each site.

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Taka-hisa Arima