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Acta Cryst. (1997). A53, 251-252

Introduction to crystallographic statistics (IUCr Monographs on Crystallography, No. 6)

By Uri Shmueli and George H. Weiss

Pp. ix + 173. Oxford University Press/International Union of Crystallography, 1995
Price £45.00. ISBN 0-19-855926-7

This book is concerned with the derivation of probability density functions (p.d.f.s) for structure amplitudes and in part for phases. It is an overview of the research work of the two authors, augmented with all necessary introductory material and overviews of related work. The authors' significant contribution in this field has been the application of Fourier representations to the derivation and calculation of exact p.d.f.s.

The Introductory Material leads the reader through an explanation of structure-factor representations (F, U, E), the notion of rational independence, random variables and fundamental ideas in probability theory such as the p.d.f., moments, characteristic functions, cumulants and conditional p.d.f.s. A complete derivation of the p.d.f.s of |E| in space groups P1 and P1¯ follows. This leads naturally to the need for some approximate solutions, so the authors present a derivation of the `ideal' distributions, based upon the use of the central limit theorem, for the same two cases. As an extension of these two approximate solutions, the authors analyse the effects of non-crystallographic symmetry (bicentric and subcentric distributions) and end Chapter III by deriving the ideal conditional p.d.f. of a three-phase invariant. Chapter IV deals with the conventional higher-order approximations to a p.d.f. obtained from the ideal p.d.f.s by the use of various orthogonal polynomial expansions, illustrating the need for improved representations of the p.d.f. of |E| with several examples. I have the impression that the authors' hearts are not in these elaborate although approximate expansions, and Chapter V begins their pièce de résistance, the derivation and use of the Fourier representation of exact p.d.f.s. They treat these representations in detail and compare them with the results for ideal p.d.f.s based on the central limit theorem.

The text is written in a uniform style where words and diagrams have been carefully chosen and are used sparingly to great effect. The ideas involved are clearly exposed without an excess of detail obscuring the underlying principles. The text has been composed by the authors themselves, using LaTeX, resulting in few typographic errors but with its characteristic poor hyphenation of words. To my taste, titles and subtitles are used too sparingly with wording that often does not enable the content to be easily identified, and it is disappointing that the references do not contain the terminal page numbers of the works cited. The authors provide an internet document correcting known typographic errors of the book (http://crystal.tau.ac.il/xtal/corstat/index.html).

A reader needs a good grounding in mathematics to find a way through the Fourier transforms and other algebra necessary for an understanding of intensity and phase statistics. A knowledge of probability theory and statistics is not necessary. I found this text most valuable to read but requiring a high level of concentration. I thoroughly recommend its study.

H. D. Flack

Laboratoire de Cristallographie
24, quai E. Ansermet
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