International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 1.7, p. 66

Section 1.7.3.2.3. The program COMMONSUPER

Mois I. Aroyo,a* J. Manuel Perez-Mato,a Cesar Capillasa and Hans Wondratschekb

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain, and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.7.3.2.3. The program COMMONSUPER

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The program COMMONSUPER calculates the space-group types of common supergroups [{\cal G}] of two space groups [{\cal H}_1] and [{\cal H}_2] for a given maximal lattice index [[(i_{L})_{\rm max}]]. The procedure used is analogous to the one implemented in the program COMMONSUBS (cf. Section 1.7.3.1.5[link]). The two sets of supergroups of [{\cal H}_1] and of [{\cal H}_2] 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 [\{{\cal G}\}] of [{\cal H}_1] and [{\cal H}_2] with [[i_{L}]\leq [(i_{L})_{\rm max}]]. A relation between the indices [[i_1]=|{\cal G}|/|{\cal H}_1|] and [[i_2]=|{\cal G}|/|{\cal H}_2|] is obtained by imposing the structural requirement of equal numbers of formula units in the (primitive) unit cell of the common supergroup [{\cal G}] obtained from the numbers of the formula units [Z_1] and [Z_2] of [{\cal H}_1] and [{\cal H}_2]: [ [i_{2}] = [i_{1}] \cdot {{Z_{2}}\over{Z_{1}}} \cdot {{|{\cal P}_{{\cal H}_{1}}|}\over{|{\cal P}_{{\cal H}_{2}}|}}. \eqno(1.7.3.2)]The program COMMONSUPER selects and lists those supergroups of the set [\{{\cal G}\}] whose indices [[i_1]=|{\cal G}|/|{\cal H}_1|] and [[i_2]=] [|{\cal G}|/|{\cal H}_2|] satisfy the above condition.

The input data for COMMONSUPER include the specification of [{\cal H}_1] and [{\cal H}_2], the numbers of formula units per conventional unit cell, and the maximum lattice index [[(i_{L})_{\rm max}]]. The output data of COMMONSUPER are:

  • (i) The space-group types of the common supergroups [{\cal G}] of [{\cal H}_1] and [{\cal H}_2] with the indices [[i_1]] and [[i_2]], [[(i_{L})_{1}]] and [[(i_{L})_{2}]], and [[(i_{P})_{1}]] and [[(i_{P})_{2}]]. Optional links to the programs POINT and GENPOS give access to data for the point group [{\cal P}_{{\cal G}}] and the general positions of the supergroup [{\cal G}].

  • (ii) A link to the SUPERGROUPS module (see Section 1.7.3.2.1[link]) enables the calculation of all different supergroups [{\cal G}_{r}] of [{\cal H}_1] and [{\cal H}_2] of a space-group type [{\cal G}] and indices [[i_1]] and [[i_2]]. Each supergroup [{\cal G}_{r}] is specified by the corresponding transformation matrix relating the conventional bases of the supergroup and the group, and the representatives of the coset decomposition of [{\cal G}_{r}] relative to [{\cal H}_{1}] or [{\cal H}_{2}].

Example 1.7.3.2.3

The program COMMONSUPER is useful in the search for structural relationships between structures whose symmetry groups [{\cal H}_1] and [{\cal H}_2] 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 [{\cal H}_1] and [{\cal H}_2]. Consider the ternary intermetallic compound CeAuGe. At 8.7 GPa a first-order phase transition is observed from a hexagonal arrangement (space group [{P}{6_3mc}], No. 186, two formula units per unit cell, [Z_1=2]) into an orthorhombic high-pressure modification of symmetry [{P}{nma}], No. 62, [Z_2=4] (Brouskov et al., 2005[link]). There is no group–subgroup relation between the symmetry groups of the high- and low-pressure structures. For [[(i_{L})_{\rm max}]=4] the program finds two common supergroups of [{\cal H}_1 = {P}{6_3mc}], [Z_1=2] and [{\cal H}_2={P}{nma}], [Z_2=4]: (i) the group [{P}{6_3/mmc}] with [[i_1]=2] and [[i_2]=6], and (ii) [{P}{6/mmm}], with [[i_1]=4] and [[i_2]=12]. The common basic structure of the AlB2 type, proposed by Brouskov et al. (2005[link]), corresponds to the common supergroup [{P}{6/mmm}] 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.








































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