Geometric and arithmetic crystal classes

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

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

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, p. 939

Section 9.8.4.2.2. Geometric and arithmetic crystal classes

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.4.2.2. Geometric and arithmetic crystal classes

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According to the previous section, in the case of modulated structures a standard basis can be chosen (for M* and correspondingly for $[\Sigma^*]$ ). According to equation (9.8.4.15), for each three-dimensional point-group operation R that leaves the diffraction pattern invariant, there is a point-group transformation [R_E] in the external space (the physical one, so that [R_E=R] ) and a point-group transformation [R_I] in the internal space, such that the pair [(R,R_I)] is a (3 + d)-dimensional orthogonal transformation [R_s] leaving a (3 + d)-dimensional lattice $[\Sigma]$ invariant. For incommensurate crystals, this internal transformation is unique and follows from the transformation by R of the modulation wavevectors [see equations (9.8.4.15) and (9.8.4.18) for the $[{\bf a}^*_{3+j}]$ basis vectors]: there is exactly one [R_I] for each R. This is so because in the incommensurate case the correspondence between M* and $[\Sigma^*]$ is uniquely fixed by the embedding rule (9.8.4.10) (see Subsection 9.8.4.1). Because the matrices Γ(R) and the corresponding transformations in the (3 + d)-dimensional space form a group, this implies that there is a mapping from the group [K_E] of elements [R_E] to the group [K_I] of elements [R_I] that transforms products into products, i.e. is a group homomorphism. A point group [K_s] of the (3 + d)-dimensional lattice constructed for an incommensurate crystal, therefore, consists of a three-dimensional crystallographic point group [K_E] , a d-dimensional crystallographic point group [K_I] , and a homomorphism from [K_E] to [K_I] .

Definition 2. Two (3 + d)-dimensional point groups [K_s] and [K'_s] are geometrically equivalent if they are connected by a pair of orthogonal transformations [(T_E,T_I)] in [V_E] and [V_I] , respectively, such that for every [R_s] from the first group there is an element [R'_s] of the second group such that [R_ET_E=T_ER'_E] and [R_IT_I=T_IR'_I] .

A point group determines a set of groups of matrices, one for each standard basis of each lattice left invariant.

Definition 3. Two groups of matrices are arithmetically equivalent if they are obtained from each other by a transformation from one standard basis to another standard basis.

The arithmetic equivalence class of a (3 + d)-dimensional point group is fully determined by a three-dimensional point group and a standard basis for the vector module M* because of relation (9.8.4.15).

In three dimensions, there are 32 geometrically non-equivalent point groups and 73 arithmetically non-equivalent point groups. In one dimension, these numbers are both equal to two. Therefore, one finds all (3 + 1)-dimensional point groups of incommensurately modulated structures by considering all triples of one of the 32 (or 73) point groups, for each one of the two one-dimensional point groups and all homomorphisms from the first to the second.

Analogously, in (3 + d) dimensions, one takes one of the 32 (73) groups, one of the d-dimensional groups, and all homomorphisms from the first to the second. If one takes all triples of a three-dimensional group, a d-dimensional group, and a homomorphism from the first to the second, one finds, in general, groups that are equivalent. The equivalent ones still have to be eliminated in order to arrive at a list of non-equivalent groups.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, p. 939