Tables for
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 14.3, p. 874

Section 14.3.4. Phase transitions

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

14.3.4. Phase transitions

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If a crystal undergoes a phase transition from a high- to a low-symmetry modification, the transition may be connected with a group–subgroup degradation. In such a case, the comparison of the lattice complexes corresponding to the Wyckoff positions of the original space group on the one hand and of its various subgroups on the other hand very often shows which of these subgroups are suitable for the low-symmetry modification.

This kind of procedure will be demonstrated with the aid of a space group [R\bar{3}m] and its three translation-equivalent subgroups with index 2, namely R32, [R\bar{3}] and R3m. In the course of the subgroup degradation, the Wyckoff positions of [R\bar{3}m] behave differently:

The descriptive symbols R and [00{1 \over 2}\ R] refer to Wyckoff positions [R\bar{3}m\ 3a] and 3b as well as to Wyckoff positions R32 3a and 3b and [R\bar{3}\ 3a] and 3b. Therefore, all corresponding point configurations and atomic arrangements remain unchanged in these subgroups. In subgroup R3m, however, the respective Wyckoff position is 3a with descriptive symbol R[z], i.e. a shift parallel to [001] of the entire point configuration is allowed.

The descriptive symbol R2z for [R\bar{3}m\ 6c] occurs also for R32 6c and [R\bar{3}\ 6c]. Again both subgroups do not allow any deformations of the corresponding point configurations or atomic arrangements. Symmetry reduction to R3m, however, yields a splitting of each R2z configuration into two R[z] configurations. The two z parameters may be chosen independently.

As M and [00{1 \over 2}\ M] are the descriptive symbols not only of [R\bar{3}m\ 9e] and 9d but also of [R\bar{3}\ 9e] and 9d, [R\bar{3}] does not enable any deformation of the corresponding atomic arrangements. In R32 and in R3m, however, the respective point configurations may be deformed differently, as the descriptive symbols show: R3x and [00{1 \over 2}\; R3x] (R32 9d and 9e), [R3x\bar{x}\hbox{[}z\hbox{]}] (R3m 9b).

Wyckoff positions [R\bar{3}m\ 18f] and 18g (R6x and [00{1 \over 2}\ R6x]) correspond to R32 9d and 9e (R3x and [00{1 \over 2}\ R3x]), to [R\bar{3}\ 18f\ (R6xyz)], and to R3m 18c [(R3x\bar{x}2y\hbox{[}z\hbox{]})]. In R32, the hexagons 6x around the points of the R lattice are split into two oppositely oriented triangles 3x, which may have different size. In [R\bar{3}] and in R3m, the hexagons may be deformed differently.

Wyckoff position [R\bar{3}m\ 18h\ (R6x\bar{x}z)] corresponds to sets of trigonal antiprisms around the points of an R lattice. These antiprisms may be distorted in R32 18f (R3x2yz) or rotated in [R\bar{3}] 18f (R6xyz). In R3m 9b [(R3x\bar{x}\hbox{[}z\hbox{]})], each antiprism is split into two parallel triangles that may differ in size.

In each of the three subgroups, any point configuration belonging to the general position [R\bar{3}m] 36i splits into two parts. Each of these parts may be deformed differently.

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