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

International Tables for Crystallography (2006). Vol. A1, ch. 1.3, p. 24   | 1 | 2 |

Section 1.3.1. Introduction

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: mueller@chemie.uni-marburg.de

1.3.1. Introduction

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Symmetry relations using crystallographic group–subgroup relations have proved to be a valuable tool in crystal chemistry and crystal physics. Some important applications include:

  • (1) Structural relations between crystal-structure types can be worked out in a clear and concise manner by setting up family trees of group–subgroup relations (Bärnighausen, 1980[link]; Baur, 1994[link]; Baur & McLarnan, 1982[link]; Bock & Müller, 2002a[link],b[link]; Chapuis, 1992[link]; Meyer, 1981[link]; Müller, 1993[link], 2002[link]; Pöttgen & Hoffmann, 2001[link]).

  • (2) Elucidation of problems concerning twinned crystals and antiphase domains (cf. Section 1.2.7[link] , p. 18; Bärnighausen, 1980[link]; van Tendeloo & Amelinckx, 1974[link]; Wondratschek & Jeitschko, 1976[link]).

  • (3) Changes of structures and physical properties taking place during phase transitions: applications of Landau theory (Aroyo & Perez-Mato, 1998[link]; Birman, 1966a[link],b[link]; Cracknell, 1975[link]; Izyumov & Syromyatnikov, 1990[link]; Landau & Lifshitz, 1980[link]; Salje, 1990[link]; Stokes & Hatch, 1988[link]; Tolédano & Tolédano, 1987[link]).

  • (4) Prediction of crystal-structure types and calculation of the numbers of possible structure types (McLarnan, 1981a[link],b[link],c[link]; Müller, 1978[link], 1980[link], 1981[link], 1986[link], 1992[link], 1998[link], 2003[link]).

All of these applications require consideration of the relations between the atomic sites in a space group and in the corresponding subgroups.

References

Aroyo, M. I. & Perez-Mato, J. M. (1998). Symmetry-mode analysis of displacive phase transitions using International Tables for Crystallography. Acta Cryst. A54, 19–30.
Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Comm. Math. Chem. 9, 139–175.
Baur, W. H. (1994). Rutile type derivatives. Z. Kri­stallogr. 209, 143–150.
Baur, W. H. & McLarnan, T. J. (1982). Observed wurtzite derivatives and related tetrahedral structures. J. Solid State Chem. 42, 300–321.
Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782.
Birman, J. L. (1966b). Simplified theory of symmetry change in second–order phase transitions: application to V3Si. Phys. Rev. Lett. 17, 1216–1219.
Bock, O. & Müller, U. (2002a). Symmetrieverwandtschaften bei Varianten des Perowskit-Typs. Acta Cryst. B58, 594–606.
Bock, O. & Müller, U. (2002b). Symmetrieverwandtschaften bei Varianten des ReO3-Typs. Z. Anorg. Allg. Chem. 628, 987–992.
Chapuis, G. C. (1992). Symmetry relationships between crystal structures and their practical applications. Modern perspectives in inorganic chemistry, edited by E. Parthé, pp. 1–16. Dordrecht: Kluwer Academic Publishers.
Cracknell, A. P. (1975). Group theory in solid state physics. New York: Taylor and Francis Ltd. and Pergamon.
Izyumov, Y. A. & Syromyatnikov, V. N. (1990). Phase transitions and crystal symmetry. Dordrecht: Kluwer Academic Publishers.
Landau, L. D. & Lifshitz, E. M. (1980). Statistical physics, Part 1, 3rd ed., pp. 459–471. London: Pergamon. (Russian: Stati­sticheskaya Fizika, chast 1. Moskva: Nauka, 1976; German: Lehrbuch der theoretischen Physik, 6. Aufl., Bd. 5, Teil 1, S. 436–447. Berlin: Akademie-Verlag, 1984).
McLarnan, T. J. (1981a). Mathematical tools for counting polytypes. Z. Kristallogr. 155, 227–245.
McLarnan, T. J. (1981b). The numbers of polytypes in close packings and related structures. Z. Kristallogr. 155, 269–291.
McLarnan, T. J. (1981c). The combinatorics of cation-deficient close-packed structures. J. Solid State Chem. 26, 235–244.
Meyer, A. (1981). Symmmetriebeziehungen zwi­schen Kristallstrukturen des Formeltyps AX2, ABX4 und AB2X6 sowie deren Ordnungs- und Leerstellenvarianten. Dissertation, Universität Karlsruhe.
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Müller, U. (1992). Berechnung der Anzahl möglicher Strukturtypen für Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178.
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Müller, U. (2002). Kristallpackungen mit linear koordinierten Atomen. Z. Anorg. Allg. Chem. 628, 1269–1278.
Müller, U. (2003). Die Zahl der Substitutions- und Leerstellenvarianten des NaCl-Typs bei verdoppelter Elementarzelle (a, b, 2c). Z. Anorg. Allg. Chem. 629, 487–492.
Pöttgen, R. & Hoffmann, R.-D. (2001). AlB2-related intermetallic compounds – a comprehensive view based on a group–subgroup scheme. Z. Kristallogr. 216, 127–145.
Salje, E. K. H. (1990). Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press.
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