Geometry of Crystals, Polycrystals, and Phase Transformations . By Harshad K. D. H. Bhadeshia. CRC Press, 2018. Hardcover, Pp. xv + 251. Price GBP 37.59. ISBN 9781138070783.

Geometry of Crystals, Polycrystals, and Phase Transformations is a relatively small book (available also as an eBook) whose pretentious title does not faithfully represent the actual content. Indeed, the book does not address crystals in general, but metallic crystals (metals and alloys), i.e. only the higher-symmetry cases, whose geometry is particularly simple to deal with. The reader interested in a more general approach, valid for lowsymmetry crystals as well, will not find this book of much help. It clearly suffers from the bias coming from the author’s background; in fact, it is hard to understand the rationale behind giving detailed (and often incorrect) definitions of basic concepts like ‘equivalent’, ‘symmetry’, ‘lattice’, while terms like ‘Peierls barrier’, ‘Burgers vector’, ‘deviatoric’ and ‘shear modulus’ are used without first being defined. The large number of solved exercises would certainly be a strong point of the book; unfortunately, the background necessary for solving them is not given in the text but as part of the solution, which makes the whole set of exercises unsuitable for self-study. The text is divided into two parts, of comparable length: ‘Basic crystallography’ (eight chapters) and ‘A few advanced methods’ (five chapters), followed by two short appendices on matrix algebra. The choice of the title of the second part suggests a lack of inspiration. In fact, the titles of the chapters composing it (‘Orientation relations’, ‘Homogeneous deformation’, ‘Invariant-plane strains’, ‘Martensite’, ‘Interfaces’) do no fit the idea of ‘methods’, even ‘advanced’ ones. But this is far from being the only flaw of the book. Fundamental and inexcusable crystallographic misunderstandings tessellate the path of the reader through the text. A constant, most disturbing confusion between lattice and structure pervades the whole text; for the expert reader, this is most annoying; for the beginner, it is a trap which will lead to lifelong confusions and mistakes. It is therefore not surprising to find another serious mistake in the list of acronyms with which the book begins, namely assigning the term ‘trigonal’ to the letter ‘R’ (which, by the way, should have been in italics). The fundamental difference between trigonal and rhombohedral seems to be beyond the scope of this book. Chapter 1 has the title ‘Introduction and point groups’ and is a short introduction to lattices, structures, symmetry operations and point groups. This chapter is a collection of mistakes and imprecise terms which should never have been written: reference to well established texts would have been a much wiser choice. Besides the confusion between ‘trigonal’ and ‘rhombohedral’ already mentioned above, we find the term ‘Miller indices’ also used to indicate the direction indices; crystal systems are imprecisely defined by the restrictions on cell parameters (Table 1.1), where unnecessary restrictions, unrelated to the symmetry, appear as well, although, to be fair, in a different colour (black) with respect to those imposed by symmetry (in red). Figure 1.6 shows the projection of 14 types of unit cell, which however correspond to only 12 types of Bravais lattices. In fact, the projection of a b-unique mB unit cell (equivalent to mP) is incorrectly labelled mC (the in-plane angle should have been right to get an mC unit cell) and the ‘trigonal’ (which does not exist: read ‘rhombohedral’ instead) unit cell is shown without any centring nodes, which means that the hP unit cell is shown twice. Miller indices (the correct ones: indices of lattice planes) are given as h1; h2; h3 instead of h; k; l and are incorrectly defined as never containing common factors, which is true only for primitive ISSN 2053-2733