International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 |
International Tables for Crystallography (2011). Vol. A1, ch. 1.7, p. 66
Section 1.7.3.2.3. The program COMMONSUPER^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain, and ^{b}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
The program COMMONSUPER calculates the space-group types of common supergroups of two space groups and for a given maximal lattice index . The procedure used is analogous to the one implemented in the program COMMONSUBS (cf. Section 1.7.3.1.5). The two sets of supergroups of and of are determined by the program CELLSUPER. The intersection of the sets of supergroups gives the set of the space-group types of the common supergroups of and with . A relation between the indices and is obtained by imposing the structural requirement of equal numbers of formula units in the (primitive) unit cell of the common supergroup obtained from the numbers of the formula units and of and : The program COMMONSUPER selects and lists those supergroups of the set whose indices and satisfy the above condition.
The input data for COMMONSUPER include the specification of and , the numbers of formula units per conventional unit cell, and the maximum lattice index . The output data of COMMONSUPER are:
Example 1.7.3.2.3
The program COMMONSUPER is useful in the search for structural relationships between structures whose symmetry groups and are not group–subgroup related. The derivation of the two structures as different distortions from a basic structure is a clear manifestation of such relationships. The symmetry group of the basic structure is a common supergroup of and . Consider the ternary intermetallic compound CeAuGe. At 8.7 GPa a first-order phase transition is observed from a hexagonal arrangement (space group , No. 186, two formula units per unit cell, ) into an orthorhombic high-pressure modification of symmetry , No. 62, (Brouskov et al., 2005). There is no group–subgroup relation between the symmetry groups of the high- and low-pressure structures. For the program finds two common supergroups of , and , : (i) the group with and , and (ii) , with and . The common basic structure of the AlB_{2} type, proposed by Brouskov et al. (2005), corresponds to the common supergroup found by COMMONSUPER.
References
Brouskov, V., Hanfland, M., Pöttgen, R. & Schwarz, U. (2005). Structural phase transitions of CeAuGe at high pressure. Z. Kristallogr. 220, 122–127.