Lattice systems

Souvignier, B.

doi:10.1107/97809553602060000921

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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International Tables for Crystallography (2016). Vol. A. ch. 1.3, p. 39

Section 1.3.4.4.2. Lattice systems

B. Souvignier^a ^*

^a Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

1.3.4.4.2. Lattice systems

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It is sometimes convenient to group together those Bravais types of lattices for which the Bravais groups belong to the same holohedry.

Definition

Two lattices belong to the same lattice system if their Bravais groups belong to the same geometric crystal class (which is thus a holohedry).

Remark : The lattice systems were called Bravais systems in earlier editions of this volume.

Example

The primitive cubic, face-centred cubic and body-centred cubic lattices all belong to the same lattice system, because their Bravais groups all belong to the holohedry with symbol $[m\bar{3}m]$ .

On the other hand, the hexagonal and the rhombohedral lattices belong to different lattice systems, because their Bravais groups are not even of the same order and lie in different holohedries (with symbols 6/mmm and $[\bar{3}m]$ , respectively).

From the definition it is obvious that lattice systems classify lattices because they consist of full Bravais types of lattices. On the other hand, the example of the geometric crystal class $[\bar{3}m]$ shows that lattice systems do not classify point groups, because depending on the chosen basis a point group in this geometric crystal class belongs to either the hexagonal or the rhombohedral lattice system.

However, since the translation lattices of space groups in the same Bravais class belong to the same Bravais type of lattices, the lattice systems can also be regarded as a classification of space groups in which full Bravais classes are grouped together.

Definition

Two Bravais classes belong to the same lattice system if the corresponding Bravais arithmetic crystal classes belong to the same holohedry.

More precisely, two space groups $[\cal G]$ and $[{\cal G}']$ belong to the same lattice system if the point groups $[{\cal P}]$ and $[{\cal P}']$ are contained in Bravais groups $[{\cal B}]$ and $[{\cal B}']$ , respectively, such that $[{\cal B}]$ and $[{\cal B}']$ belong to the same holohedry and such that $[{\cal P}]$ , $[{\cal P}']$ , $[{\cal B}]$ and $[{\cal B}']$ all have spaces of metric tensors of the same dimension.

Every lattice system contains the lattices of precisely one holohedry and a holohedry determines a unique lattice system, containing the lattices of the Bravais arithmetic crystal classes in the holohedry. Therefore, there is a one-to-one correspondence between holohedries and lattice systems. There are four lattice systems in dimension 2 and seven lattice systems in dimension 3. The lattice systems in three-dimensional space are displayed in Table 1.3.4.1. Along with the name of each lattice system, the Bravais types of lattices contained in it and the corresponding holohedry are given.

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Lattice systems in three-dimensional space

Lattice system	Bravais types of lattices	Holohedry
Triclinic (anorthic)	aP	$[\bar{1}]$
Monoclinic	mP, mS	2/m
Orthorhombic	oP, oS, oF, oI	mmm
Tetragonal	tP, tI	4/mmm
Hexagonal	hP	6/mmm
Rhombohedral	hR	$[\bar{3}m]$
Cubic	cP, cF, cI	$[m\bar{3}m]$

References

International Tables for Crystallography (2016). Vol. A. ch. 1.3, p. 39