Crystal families

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, pp. 40-41

Section 1.3.4.4.4. Crystal families

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

1.3.4.4.4. Crystal families

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The classification into crystal systems has many important applications, but it has the disadvantage that it is not compatible with the classification into lattice systems. Space groups that belong to the hexagonal lattice system are distributed over the trigonal and the hexagonal crystal system. Conversely, space groups in the trigonal crystal system belong to either the rhombohedral or the hexagonal lattice system. It is therefore desirable to define a further classification level in which the classes consist of full crystal systems and of full lattice systems, or, equivalently, of full geometric crystal classes and full Bravais classes. Since crystal systems already contain only geometric crystal classes with spaces of metric tensors of the same dimension, this can be achieved by the following definition.

Definition

For a space group $[{\cal G}]$ with point group $[{\cal P}]$ the crystal family of $[{\cal G}]$ is the union of all geometric crystal classes that contain a space group $[{\cal G}']$ that has the same Bravais type of lattices as $[{\cal G}]$ .

The crystal family of $[{\cal G}]$ thus consists of those geometric crystal classes that contain a point group $[{\cal P}']$ such that $[{\cal P}]$ and $[{\cal P}']$ are contained in a common supergroup $[\cal B]$ (which is a Bravais group) and such that $[{\cal P}]$ , $[{\cal P}']$ and $[\cal B]$ all act on lattices with the same number of free parameters.

In two-dimensional space, the crystal families coincide with the crystal systems and in three-dimensional space only the trigonal and hexagonal crystal system are merged into a single crystal family, whereas all other crystal systems again form a crystal family on their own.

Example

The trigonal and hexagonal crystal systems belong to a single crystal family, called the hexagonal crystal family, because for both crystal systems the number of free parameters of the corresponding lattices is 2 and a point group of type $[\bar{3}m]$ in the trigonal crystal system is a subgroup of a point group of type 6/mmm in the hexagonal crystal system.

A space group in the hexagonal crystal family belongs to either the trigonal or the hexagonal crystal system and to either the rhombohedral or the hexagonal lattice system. A group in the hexagonal crystal system cannot belong to the rhombohedral lattice system, but all other combinations of crystal system and lattice system are possible. The distribution of the space groups in the hexagonal crystal family over these different combinations is displayed in Table 1.3.4.3.

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Distribution of space-group types in the hexagonal crystal family

Crystal system	Geometric crystal class	Lattice system
Crystal system	Geometric crystal class	Hexagonal	Rhombohedral
Hexagonal	6/mmm	P6/mmm, P6/mcc, ,
	$[\bar{6}2m]$	$[P\bar{6}m2]$ , $[P\bar{6}c2]$ , $[P\bar{6}2m]$ , $[P\bar{6}2c]$
	6mm	P6mm, P6cc, ,
	622	P622, , , , ,
	6/m	P6/m,
	$[\bar{6}]$	$[P\bar{6}]$
	6	P6, , , , ,
Trigonal	$[\bar{3}m]$	$[P\bar{3}1m]$ , $[P\bar{3}1c]$ , $[P\bar{3}m1]$ , $[P\bar{3}c1]$	$[R\bar{3}m]$ , $[R\bar{3}c]$
	3m	P3m1, P31m, P3c1, P31c	R3m, R3c
	32	P312, P321, , , ,	R32
	$[\bar{3}]$	$[P\bar{3}]$	$[R\bar{3}]$
	3	P3, ,	R3

Remark : Up to dimension 3 it seems exceptional that a crystal family contains more than one crystal system, since the only instance of this phenomenon is the hexagonal crystal family consisting of the trigonal and the hexagonal crystal systems. However, in higher dimensions it actually becomes rare that a crystal family consists only of a single crystal system.

For the space groups within one crystal family the same coordinate system is usually used, which is called the conventional coordinate system (for this crystal family). However, depending on the application it may be useful to work with a different coordinate system. To avoid confusion, it is recommended to state explicitly when a coordinate system differing from the conventional coordinate system is used.

References

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 40-41