## book reviews

**Mathematical techniques in crystallography and materials science.** Third edition. By Edward Prince. Pp. vii + 224. Berlin and Heidelberg: Springer-Verlag, 2004. Price EUR 39.95 (soft cover). ISBN 3-540-21111-X.

^{a}Depto de Fisica de la Materia Condensata, Facultad de Ciencia y Technologia, 480480 Bilbao, Spain^{*}Correspondence e-mail: wmparam@lg.ehu.es

Keywords: book review.

The first edition of this book was published about 20 years ago. It is based on the rich experience from the extensive research performed by the author over many years. The carefully selected topics concern common and less familiar mathematical procedures used in structural crystallography. The book is intended to provide a tutorial and a practical guide to novices, and a review to professionals on the theoretical background and on the advantages and disadvantages of the applied numerical procedures.

The book is divided into nine chapters. The first five chapters develop the appropriate mathematical tools related to basic geometrical and algebraic components of structural crystallography. Chapter 1 is devoted to matrices, their definitions and fundamental operations. Chapter 2 treats point-group symmetry while Chapter 3 discusses space-group symmetry. Vectors and tensors including their definitions, algebra and important examples are dealt with in Chapters 4 and 5.

The next several chapters study the mathematics underlying data-fitting problems and modelling. In Chapter 6, the form and the general features of the function of adjustable parameters, whose minimum value corresponds to the best fit of the data, are discussed in greater detail than usually appears in treatments of model fitting. Various fitting algorithms, their robustness and resistance are presented and compared. In the subsequent two chapters, the author considers different methods of evaluating the precision and accuracy in structural experiments. Finally, Chapter 9 focuses on different techniques that may be used to apply constraints to

refinements.The book closes with a short bibliography, a subject index and several appendices, the last of which contains Fortran source codes of subroutines including important statistical distribution functions.

The first edition of the book is reviewed in detail by R. Diamond [*Acta Cryst.* (1984), A**40**, 86–87 ]. The overall estimation of the book is positive apart from the several inconsistencies discussed by the reviewer. However, with one of them I completely disagree. It refers to the character tables of the point groups (Chapter 2) which for the reviewer remain `the biggest mystery'. Obviously, he has not recognized that the listed data refer to the irreducible representations of the point groups and not to their vector representations which in general are reducible ones.

The slightly enlarged second edition of the book appeared in 1994. The author made corrections and clarified a number of points. Two additional topics were treated: a section on the projection matrix and its use in studying the influence of individual data points, and a new chapter with a very clear presentation of the method of fast Fourier transform. The review of the second edition by Ewa Gałdecka [*Acta Cryst.* (1995), A**51**, 590 ] concentrates mainly on this new chapter.

The third edition of the book appeared recently. According to the author, it does not represent considerable changes from the contents of the second edition. Apart from the corrections, the author has included further explanations on a few topics, namely, the eigenvalue problem of a 3 × 3 symmetric matrix and the method of conjugate gradients, which is a variation of the method of steepest descents. It is applied to the search for the minimum of the function of adjustable model parameters. One of the main advantages of such numerical procedures is that it is not necessary to construct and invert the entire Hessian matrix in each iteration. These methods usually use the gradient of the function to be minimized as a search direction in the multidimensional parameter space. However, the method of steepest descents is efficient only when the eigenvalues of the Hessian matrix are of the same order. Otherwise, the method is rather slow and it is not guaranteed to converge to a minimum in a finite number of steps. The inefficiency of the method is due to the fact that the consecutive search lines are always perpendicular to the previous ones but only the current steepest descent direction is used. By contrast, the method of conjugate gradients takes information from previous steps into account. The method is guaranteed to find a minimum of a quadratic function in no more than *p* steps, where *p* is the dimension of the parameter space.

In my opinion, one of the main strengths of the book is its clear and pedagogical presentation. The reader benefits from the author's clear grasp of what he wishes to say and care has been taken that the essence of the topic is presented in an educational and concise fashion. Part of the heavy mathematics is truncated or greatly abbreviated since the algebraic details (available elsewhere) can often obscure the salient facts. The author offers the reader a number of illustrative examples and graphical presentations that add a qualitative perspective on the treated problem. Limits in the applications of the numerical procedures are carefully discussed and there are frequent calls for caution in the interpretation of the results.

However, the book is not free from some small imperfections. The text has been composed by the author himself, using L^{A}T_{E}X, resulting in a few typographic errors, although none of them serious.

Further, the user may find some notation confusing as it differs from the usage adopted in *International Tables for Crystallography*, Volume A (hereafter referred to as *IT*A). The author uses a 1 × 3 column-matrix presentation of the basis vectors in (*e.g.* last section of Chapter 2). A user accustomed to the *IT*A notation (direct-space basis vectors as 1 × 3 row matrices) should apply transformation matrices transposed to those listed in Appendix *C* in order to generate the superlattices of order two, three and four. Additional misunderstandings could occur as there is no indication that the listed matrices refer to primitive bases only.

Another example of similar nature can be found in Appendix *F* where the elements of specific tensors of rank two, three and four (after imposing the corresponding symmetry restrictions) are given. However, some of the listed data are either incomplete or could be at least misleading. Two examples: (i) it would be helpful if it were specified that the given form of the third-rank piezoelectric tensor for the 2 (and *m*) refers to a special orientation of the twofold axis (and the plane *m*), and how the form of the tensor can change for different orientations; (ii) there is only one listed entry for the elasticity fourth-rank tensor for the tetragonal system while one has to distinguish between two tensor forms: one for the point groups 4*m**m*, , 422, 4/*m**m**m*, and the other for the point groups 4, , 4/*m*.

I would add that it is a pity that in the present third edition the author has not used the opportunity to update the presentation of *IT*A (Chapter 3) in accordance with the new fifth edition of the tables or to improve the quality of some of the figures (*e.g.* Fig. 2.1).

Leaving these defects aside, I believe that the book will be useful both to graduate students in physics, chemistry and engineering who have an interest in structural crystallography and to experts seeking a review of numerical methods and implementation strategies. A great advantage of the book is that it is written in a style that makes it accessible also to material scientists whose background does not include structural crystallography. The main messages are clearly stated and not buried beneath technical details.