# 1998 International Union of Crystallography Acta Crystallographica Section B Printed in Great Britain ± all rights reserved ISSN 0108-7681 # 1998 782 Acta Cryst. (1998). B54, 782±789 Group-Theoretical Analysis of Octahedral Tilting in Perovskites Christopher J. Howard a * and Harold T. Stokes b a Australian Nuclear Science and Technology Organisation, Private Mail Bag 1, Menai NSW 2234, Australia, and b Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602-4675, USA. E-mail: cjh@ansto.gov.au (Received 9 January 1998; accepted 18 March 1998 ) Abstract A group-theoretical analysis is made of the structures derived from the aristotype cubic perovskite (Pm3 Å m) by the simple tilting of rigid octahedral units. The tilting is mediated by the irreducible representations R  4 and M  3 or the two in combination. These result in 15 possible structures, compared with the 23 possibilities suggested previously by Glazer [Acta Cryst. (1972), B28, 3384± 3392]. The analysis makes the group±subgroup relation- ships apparent. 1. Introduction Structures in the perovskite family ABX 3 have held the interest of crystallographers over many years (Kay & Bailey, 1957; Glazer, 1972, 1975; Megaw, 1973; Thomas, 1989, 1996; Burns & Glazer, 1990; Woodward, 1997a,b) and continue to attract a wider interest on account of their fascinating electrical and magnetic properties. Perovskite compounds exhibit ferroelectricity, piezo- electricity and non-linear optical behaviour (Newnham & Ruschau, 1991), and certain closely related compounds are famous as high-temperature cuprate superconductors (Cava et al., 1987; Capponi et al., 1987). The majority of materials displaying giant magneto- resistive effects are compounds with the perovskite structure (Gong et al., 1995). The ideal perovskite, being cubic, in space group Pm3 Å m, is a particularly simple structure, but it is also a demanding one, because, aside from the lattice para- meter, there are no variable parameters in the structure. Consequently, the majority of perovskites are in fact distorted perovskites (hettotypes). Three different types of distortions have been identi®ed (Megaw, 1973): distortions of the BX 6 octahedral units, B-cation displacements within the octahedra, and the tilting of the BX 6 octahedra relative to one another as practically rigid corner-linked units. The third type of distortion, octahedral tilting, is the most common type of distortion and forms the subject of this paper. The most promising approach to the classi®cation of perovskites with octahedral tilting is to consider ®rst the possible tilting patterns and then to ®nd the corre- sponding subgroups. This approach has been used by Glazer (1972), who developed a description of the different tilting patterns and then obtained space groups by inspection. We review Glazer's work, brie¯y, in the next section, because we shall make use of his notation in this paper. The group-theoretical analysis adopted in this paper and described in the third section is devel- oped along similar lines. The different possible tilting patterns are ®rst described by different vectors in a representation space and then for each tilting pattern (vector) the required space group is the isotropy subgroup, comprising the operations which leave that vector invariant. The analysis yields a list of 15 possible space groups for perovskites derived through octahedral tilting. A connection is made to the (23) tilt systems given previously by Glazer (1972, 1975). The group±subgroup relationships are derived and displayed. It is interesting to note that all known perovskites based on octahedral tilting conform with the 15 space groups on our list, with the exception of one high-temperature structure which seems poorly determined. 2. Octahedral tilting The ideal cubic perovskite is commonly visualized as a three-dimensional network of regular corner-linked BX 6 octahedra, the B cations being at the centres of these octahedra and the A cations being centrally located in the spaces between them. The tetrad axes of the octa- hedra coincide with the crystallographic cubic axes. The majority of distorted perovskites are derived from the cubic aristotype by the practically rigid tilting of the octahedral units. By this we mean the tilting of octahedra around one or more of their symmetry axes, maintaining both the regularity of the octahedra (any distortions at most second order in the tilt angle) and their corner connectivity (strictly). Such tilting allows greater ¯exibility in the coordination of the A cation, while leaving the environment of the B cation essentially unchanged. Glazer (1972) found it convenient to describe octa- hedral tilting in terms of component tilts around `pseudo-cubic' axes, that is, the cubic axes of the aris- totype. It becomes apparent that a tilt around one of