International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. C. ch. 1.4, p. 16

Section 1.4.1.2. Arithmetic crystal classes in one, two and higher dimensions

A. J. C. Wilsona‡

a St John's College, Cambridge CB2 1TP, England

1.4.1.2. Arithmetic crystal classes in one, two and higher dimensions

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In one dimension, there are two geometric crystal classes, 1 and m, and a single Bravais lattice, [{\scr p}]. Two arithmetic crystal classes result, [{\scr p}] and [m{\scr p}]. In two dimensions, there are ten geometric crystal classes, and two Bravais lattices, p and c; 13 arithmetic crystal classes result. The two-dimensional geometric and arithmetic crystal classes are listed in Table 1.4.1.1[link].

Table 1.4.1.1| top | pdf |
The two-dimensional arithmetic crystal classes

Crystal systemCrystal classSpace group
GeometricArithmetic
NumberSymbolNumberSymbol
Oblique111p1p1
222p2p2
Rectangularm3mp3pm
4pg
4mc5cm
2mm52mmp6p2mm
7p2mg
8p2gg
62mmc9c2mm
Square474p10p4
4mm84mmp11p4mm
12p4gm
Hexagonal393p13p3
3m103m1p14p3m1
1131mp15p31m
6126p16p6
6mm136mmp17p6mm

The number of arithmetic crystal classes increases rapidly with increasing dimensionality; there are 710 (plus 70 enantiomorphs) in four dimensions (Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, 1978[link]), but those in dimensions higher than three are not needed in this volume.

References

First citation Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: Wiley.Google Scholar








































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