Book Review: International Tables for Crystallography Vol D

Crystals are anisotropic media and each one possesses a certain structural symmetry. The physical properties of crystals are therefore described by tensors, the elements of which are determined by the particular symmetry. Crystals may also transform from one structural phase to another, grow as twins or have domain structures. The vibrations of their constituent atoms (phonons) and the behaviour of electrons in crystals are also subject to the effects of crystal symmetry. All these important topics, and their useful applications, are treated in this new and weighty volume, in the world-renowned series published for the International Union of Crystallography.

As the series title implies, the volumes contain many useful tables, but they also comprise comprehensive explanatory chapters. The present tome brings together the widest ranging and latest crystallographic information, distilled by 28 international experts (from 9 different countries). Terry Willis originally proposed the idea for this volume and André Authier has skilfully orchestrated the whole project. The high standard of the IUCr has been maintained: the book is beautifully produced, together with fine illustrations, some in colour, and numerous references to the extensive literature; and it is supported by a CD-ROM with a further 140 pages of explanation (including, amongst other things, algebraic details of groups and subgroups) as well as useful software. The latter contains calculations involving tensors, irreducible group representations and structural phase transformations: (for example, the changes in tensor properties of physical quantities during ferroic phase transitions).

The volume is divided into three parts, of which Part 1 is the largest, comprising half the book. This describes the tensorial aspects of physical properties and sets up the necessary mathematics. Tensors of rank 2, 3 and 4 are treated, together with all the intricacies that the effects of crystal symmetry have upon their constituent elements. Strain and stress tensors are described, together with the geometrical interpretation of their coefficients; and some nice pictures of the representation surfaces of the inverse of Young's modulus for several simple crystals are shown. Experiments for the determination of elastic constants and of thermal expansion are described. Various types of magnetism, disordered and ordered, are discussed; and many diagrams of magnetic lattices are displayed. The detailed theories of linear and non-linear crystal optics are given, as well as much helpful practical advice on the use of the polarizing microscope.

As is well known, light and matter interact through their electric fields, inducing a dipole. Electric polarization is a vector, which is related to the applied electric field via the dielectric susceptibility tensor. The latter is tabulated here for all crystallographic point groups. Transport properties - the flow of electricity or of heat through crystals - are also described. The site symmetry places restrictions on the tensor coefficients of atomic displacement parameters, either from thermal motion or positional disorder. There are informative tables on these, as well as colourful graphical representations of density modulations. Superspace groups and tensors in higher-dimensional space are introduced to describe quasiperiodic structures - incommensurate or modulated structures, or quasicrystals.

Symmetry aspects of excitations are discussed in Part 2. The collective motion of atomic displacements, small compared with interatomic distances, are described by phonons. Here again, symmetry plays a key rôle and matrices are used for the calculations. Several examples are given of phonon dispersion. The electron energy band structure in crystals is described with the aid of group theory, Brillouin zones and Bloch functions. The electric field gradient tensor is introduced and, as an example, the various oxygen sites in the high-temperature superconductor YBa₂Cu₃O₇ are discussed. Phonons give rise to Raman scattering and tables provide the details of the Raman tensor in each of the 32 crystal classes. The longitudinal and transverse modes of Brillouin scattering are also tabulated in the various Laue classes.

The many different types of ferroelectric, ferroelastic and ferroic structural phase transitions are examined in Part 3, together with their symmetries and thermodynamics. There are tables giving detailed information on the lowering of point group symmetry associated with irreducible representations and there are many examples and diagrams of particular crystals. The subject of twinning usually occupies only a short section of textbooks on crystallography, and usually only one aspect is given, whether it be growth twinning, transformation twinning or mechanical twinning. Here the fascinating early history of twinning is outlined, together with all the necessary lattice aspects and group-theoretical mathematics needed for understanding twinning and domain structures. Many examples of twins, twin boundaries and fault vectors are given, beautifully illustrated with diagrams of the various crystal structures and atomic-resolution transmission electron micrographs. Domain structures are examined in detail and a large table of group-subgroup symmetry descents is provided. Domain pairs, domain twin laws and domain walls are all comprehensively discussed.

Crystallographers, materials scientists, mineralogists and solid-state physicists will all welcome this up-to-date, handsome and most distinguished reference book. Our grateful thanks go to the editor, André Authier, to all the authors and to the IUCr technical editors, Nicola Ashcroft and Amanda Berry, for all their painstaking work.

Moreton Moore