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

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

Section 1.3.1. Introduction

Ulrich Müllera*

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

1.3.1. Introduction

| top | pdf |

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

First citation 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.Google Scholar
 First citation Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Comm. Math. Chem. 9, 139–175.Google Scholar
 First citation Baur, W. H. (1994). Rutile type derivatives. Z. Kri­stallogr. 209, 143–150.Google Scholar
 First citation Baur, W. H. & McLarnan, T. J. (1982). Observed wurtzite derivatives and related tetrahedral structures. J. Solid State Chem. 42, 300–321.Google Scholar
 First citation Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782.Google Scholar
 First citation Birman, J. L. (1966b). Simplified theory of symmetry change in second–order phase transitions: application to V3Si. Phys. Rev. Lett. 17, 1216–1219.Google Scholar
 First citation Bock, O. & Müller, U. (2002a). Symmetrieverwandtschaften bei Varianten des Perowskit-Typs. Acta Cryst. B58, 594–606.Google Scholar
 First citation Bock, O. & Müller, U. (2002b). Symmetrieverwandtschaften bei Varianten des ReO3-Typs. Z. Anorg. Allg. Chem. 628, 987–992.Google Scholar
 First citation 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.Google Scholar
 First citation Cracknell, A. P. (1975). Group theory in solid state physics. New York: Taylor and Francis Ltd. and Pergamon.Google Scholar
 First citation Izyumov, Y. A. & Syromyatnikov, V. N. (1990). Phase transitions and crystal symmetry. Dordrecht: Kluwer Academic Publishers.Google Scholar
 First citation 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).Google Scholar
 First citation McLarnan, T. J. (1981a). Mathematical tools for counting polytypes. Z. Kristallogr. 155, 227–245.Google Scholar
 First citation McLarnan, T. J. (1981b). The numbers of polytypes in close packings and related structures. Z. Kristallogr. 155, 269–291.Google Scholar
 First citation McLarnan, T. J. (1981c). The combinatorics of cation-deficient close-packed structures. J. Solid State Chem. 26, 235–244.Google Scholar
 First citation Meyer, A. (1981). Symmmetriebeziehungen zwi­schen Kristallstrukturen des Formeltyps AX2, ABX4 und AB2X6 sowie deren Ordnungs- und Leerstellenvarianten. Dissertation, Universität Karlsruhe.Google Scholar
 First citation Müller, U. (1978). Strukturmöglichkeiten für Pentahalogenide mit Doppeloktaeder-Molekülen (MX5)2 bei dichtester Packung der Halogenatome. Acta Cryst. A34, 256–267. Google Scholar
 First citation Müller, U. (1980). Strukturverwandtschaften unter den [EPh_4^+]- Salzen. Acta Cryst. B36, 1075–1081.Google Scholar
 First citation Müller, U. (1981). MX4-Ketten aus kantenverknüpften Okta­edern: mögliche Kettenkonfigurationen und mögliche Kristallstrukturen. Acta Cryst. B37, 532–545.Google Scholar
 First citation Müller, U. (1986). MX5-Ketten aus eckenverknüpften Oktaedern. Mögliche Kettenkonfigurationen und mögliche Kristallstrukturen bei dichtester Packung der X-atome. Acta Cryst. B42, 557–564.Google Scholar
 First citation Müller, U. (1992). Berechnung der Anzahl möglicher Strukturtypen für Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178.Google Scholar
 First citation Müller, U. (1993). Inorganic structural chemistry, pp. 233–246. Chichester, New York: Wiley. (German: Anorganische Strukturchemie, 3. Aufl., 1996, S. 296–309. Stuttgart: Teubner.)Google Scholar
 First citation Müller, U. (1998). Strukturverwandtschaften zwischen trigonalen Verbindungen mit hexagonal-dichtester Anionenteilstruktur und besetzten Oktaederlücken. Z. Anorg. Allg. Chem. 624, 529–532.Google Scholar
 First citation Müller, U. (2002). Kristallpackungen mit linear koordinierten Atomen. Z. Anorg. Allg. Chem. 628, 1269–1278.Google Scholar
 First citation 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.Google Scholar
 First citation Pöttgen, R. & Hoffmann, R.-D. (2001). AlB2-related intermetallic compounds – a comprehensive view based on a group–subgroup scheme. Z. Kristallogr. 216, 127–145.Google Scholar
 First citation Salje, E. K. H. (1990). Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press.Google Scholar
 First citation Stokes, H. T. & Hatch, D. M. (1988). Isotropy subgroups of the 230 crystallographic space groups. Singapore: World Scientific.Google Scholar
 First citation Tendeloo, G. van & Amelinckx, S. (1974). Group-theoretical considerations concerning domain formation in ordered alloys. Acta Cryst. A30, 431–440.Google Scholar
 First citation Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.Google Scholar
 First citation Wondratschek, H. & Jeitschko, W. (1976). Twin domains and antiphase domains. Acta Cryst. A32, 664–666.Google Scholar








































to end of page
to top of page