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. 1.6, p. 44
Section 1.6.1. Introduction^{a}Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany |
Symmetry relations using crystallographic group–subgroup relations have proved to be a valuable tool in crystal chemistry and crystal physics. Some important applications include:
Bärnighausen (1975, 1980) presented a standardized procedure to set forth structural relations between crystal structures with the aid of symmetry relations between their space groups. For a review on this subject see Müller (2004). Short descriptions are given by Chapuis (1992) and Müller (2006). The main concept is to start from a simple, highly symmetrical crystal structure and to derive more and more complicated structures by distortions and/or substitutions of atoms. Similar to the `diagrams of lattices of subgroups' used in mathematics, a tree of group–subgroup relations between the space groups involved, now called a Bärnighausen tree, serves as the main guideline. The highly symmetrical starting structure is called the aristotype after Megaw (1973) or basic structure after Buerger (1947, 1951); other terms used in the literature on phase transitions in physics are prototype or parent structure. The derived structures are the hettotypes or derivative structures. In Megaw's (1973) terminology, the structures mentioned in the tree form a family of structures.
The structure type to be chosen as the aristotype depends on the specific problem and, therefore, the term aristotype cannot be defined in a strict manner. For example, a body-centred packing of spheres (space group ) can be chosen as the aristotype for certain intermetallic structures. By symmetry reduction due to a loss of the centring, the CsCl type (space group ) can be derived. However, if all the structures considered are ionic, there is no point in starting from the body-centred packing of spheres and one can choose the CsCl type as the aristotype.
References
Aizu, K. (1970). Possible species of ferromagnetic, ferroelectric, and ferroelastic crystals. Phys Rev. B, 2, 754–772.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. (1975). Group–subgroup relations between space groups as an ordering principle in crystal chemistry: the `family tree' of perovskite-like structures. Acta Cryst. A31, part S3, 01.1–9.
Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175.
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 V_{3}Si. Phys. Rev. Lett. 17, 1216–1219.
Birman, J. L. (1978). Group-theoretical methods in physics, edited by P. Kramer & A. Rieckers, pp. 203–222. New York: Springer.
Buerger, M. J. (1947). Derivative crystal structures. J. Chem.Phys. 15, 1–16.
Buerger, M. J. (1951). Phase transformations in solids, ch. 6. New York: Wiley.
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/Pergamon.
Di Costanzo, L., Forneris, F., Geremia, S. & Randaccio, L. (2003). Phasing protein structures using the group–subgroup relation. Acta Cryst. D59, 1435–1439.
Howard, C. J. & Stokes, H. T. (2005). Structures and phase transitions in perovskites – a group-theoretical approach. Acta Cryst. A61, 93–111.
Igartua, J. M., Aroyo, M. I. & Perez-Mato, J. M. (1996). Systematic search of materials with high-temperature structural phase transitions: Application to space group P2_{1}2_{1}2_{1}. Phys. Rev. B, 54, 12744–12752.
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, 3rd ed., Part 1, 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.)
Lyubarskii, G. Ya. (1960). Group Theory and its Applications in Physics. London: Pergamon. (Russian: Teoriya grupp i ee primenenie v fizike. Moskva: Gostekhnizdat, 1957; German: Anwendungen der Gruppentheorie in der Physik. Berlin: Deutscher Verlag der Wissenschaften, 1962.)
Megaw, H. D. (1973). Crystal Structures: A Working Approach. Philadelphia: Saunders.
Müller, U. (2004). Kristallographische Gruppe-Untergruppe-Beziehungen und ihre Anwendung in der Kristallchemie. Z. Anorg. Allg. Chem. 630, 1519–1537.
Müller, U. (2006). Inorganic Structural Chemistry, 2nd ed., pp. 212–225. Chichester: Wiley. (German: Anorganische Strukturchemie, 6. Aufl., 2008, S. 308–327. Wiesbaden: Teubner.)
Salje, E. K. H. (1990). Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press.
Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientific.
Tolédano, J.-C. & Tolédano, P. (1987). The Landau Theory of Phase Transitions. Singapore: World Scientific.