## book reviews

**X-Ray Crystallography**. By Gregory S. Girolami. University Science Books, 2015. Pp. 500. Price (hardcover) USD 88.00. ISBN 978-1-891389-77-1 .

^{a}Université de Lorraine, CRM2, UMR 7036, Vandoeuvre-les-Nancy, F-54506, France, and ^{b}CNRS, CRM2, UMR 7036, Vandoeuvre-les-Nancy, F-54506, France^{*}Correspondence e-mail: massimo.nespolo@crm2.uhp-nancy.fr

Keywords: book review; X-ray crystallography.

*X-Ray Crystallography* is a textbook (available also as an e-book) with a wide wingspan, composed of three sections covering basic symmetry notions, X-ray diffraction, and structure solution and methods. It clearly addresses a public having essentially a background in chemistry and is written in a clear, didactic style. Because not everything can be treated even in a quite sizeable text (500 pp.), a clever choice has been made not to repeat what is already available *ad libitum* in countless books. The content concentrates instead on those subjects that too often are either absent or presented with a too specialistic approach for the intended audience. The result is an up-to-date textbook, pleasant and easy to read, which indeed responds to a clear need. Each chapter presents a topic with a good compromise between completeness and synthesis, supplemented by a series of exercises (with solutions available from the publisher). Unfortunately, a few but important slips, discussed below, cast some shadows on an otherwise excellent text. The choice of not including the is worth criticism: a few long formulas that show up as a sort of *deux ex machina* would have been self-evident had the been used.

The first section, *Symmetry and space groups* (139 pp.), is divided into 12 chapters and aims at giving a robust, although necessarily limited, background to point- and space-group theory with special emphasis on those aspects that a chemist needs to master in their daily work (the author had actually sent me a previous version of this section to get my critical opinion and many, although not all, of my remarks were apparently been considered useful enough to modify the text before it went to press). After a brief but appreciable historical introduction (Chapter 1), and point groups are presented in two chapters, the rest being devoted to plane and space groups. One can particularly appreciate the distinction between and that is ignored in so many texts but clearly explained here. It is therefore even more surprising and hard to understand the presence of some terminological liberties and imprecisions. The unnecessary neologism `travel has been introduced to indicate fixed-point-free operations (*i.e.* operations including an intrinsic translational component). The term `mirror', used both for the and the operation (instead of `reflection'), is a misuse of language that could and should have been avoided. The difference between *inversion point* and *centre of symmetry*, incorrectly treated as synonyms, should have been explained. One may also wonder why orthographic, instead of the standard stereographic, projections are used as a graphic tool to explain operations (p. 19*ff*); why the generators (which are *operations*) of a group have been called `characteristic symmetry *elements*' (p. 29); why cyclic groups are not called as such (p. 30). Despite these defects, the presentation is pedagogical yet more rigorous than in many other textbooks.

Periodicity is the object of the fourth chapter, where the notions of lattice and *sic*) symmetry operations where a significant slip occurs in footnote 5 on p. 60, the *e* glide being defined as two glide *planes* mutually perpendicular: the confusion between and is unexpected and surprising. Chapters 6 and 7 introduce two-dimensional and three-dimensional lattices, respectively. In the lower crystal families, the lack of restrictions imposed by symmetry is correctly indicated by leaving the corresponding cell parameter `arbitrary', instead of assigning it a value `different from', a typical mistake affecting many other textbooks. Chapter 8 introduces plane groups in a graphical and intuitive way, without the use of matrices: the same is true in the following chapters on space groups. Although drawings and analogy are powerful didactic tools, there is a clear limit to what can be obtained in this way; the complete absence of any matrix treatment is a clear shortcoming here.

Chapter 9 has the title `Equivalent positions' and discusses general and special Wyckoff positions, but actually starts with a detailed description of *a*), but apparently without much effect.]

The first section ends with three chapters on space groups, where the reader is guided through a step-by-step interpretation of the space-group diagrams with several general (abstract) and concrete (molecular) examples, but without any real calculation.

Section 2, *X-rays and diffraction* (152 pp.), consists of 15 chapters and overall is one of the most efficient presentations of the subject I have recently read, avoiding the typical traps of cloning old stuff. After a short introduction about the generation of X-rays (Chapter 13), where the essential matter is presented in a clear and concise way, the reader is taken on a short but effective journey through goniometers, (modern) detectors and data treatment, to land on the island of data reduction, where concrete directions for obtaining exploitable data from the measured intensities (and the physical reasons behind the necessary corrections) are presented. At this point one would have expected an explanation about from the previous sections; but the text has not covered morphological aspects and thus are still unknown, so the author is obliged to refer to Chapter 21 (we are still in Chapter 14). The poor expression `lattice defects' (p. 173) to indicate what are actually *structural* defects echoes some die-hard laboratory jargon.

The next four chapters present diffraction of X-rays from an electron, an atom and arrays of atoms. Mathematical details are kept to a minimum, yet the presentation is clear and detailed. *via* the diffraction cones (*à la Buerger*), which leads naturally to the and lattice. The four reflection indices (read *Laue indices*) for trigonal and hexagonal crystals are confined to a footnote on p. 222, which is insufficient and unsatisfactory. A convincing presentation requires the use of Bravais–Miller indices but, as already noted, morphology is not covered in this textbook.

Chapter 21 introduces Bragg's law and *d* spacing, and here finally appear, with an unavoidable reference to the notions of lattice planes, crystal faces and interplanar distances. With only six and a half pages and despite the already manifest talent of the author for concentrating quite a lot of notions in a short space, here the reader is left gasping for breath. To be noted is the incorrect statement (p. 230) according to which for crystal faces are always reduced to their lowest terms, as this is not true for lattice planes in centred cells (Nespolo, 2015*b*); the same actually applies to a crystal face, which is the last plane of the family. Also to be noted is the use of parentheses for reflection (Laue) indices: this use should actually be restricted to to indicate families of lattice planes.

Chapter 22 deals with the limiting sphere (the *d*. Apart from these questionable choices, the chapter is clear, concise and easy to follow. One may remark that resolution is defined in ångstroms only, whereas the complementary definition Å^{−1}, directly related to the region of the explored and to the decay of the is commonly used by the experimentalist: in a practical textbook one may expect that the reader's attention is drawn to these details. The next chapter on structure factors follows naturally and is complemented by an introduction to Argand diagrams, which will be used later. It prepares the way for a chapter on and symmetry-related data, with the emphasis on the meaning of data merging.

Chapter 25, `Centrosymmetry and *P*1, and `achiral' groups, like *Pm*, does not make sense. Firstly, polar groups (which contain at least one polar direction) are 68 in number, not 65 (the latter are the here incorrectly called `chiral'). Secondly, neither *P*1 nor *Pm* are chiral groups (they do not belong to the 11 pairs of enantiomorphic groups). If the reader survives this terminological storm, they will then land on a happy island where a smiling team of absolute configurations and absolute structures greet them. It's a pity that when the wrong is picked up by the structure-solution software and the user needs to invert it, the solution in some cases is just described as `more complicated'. Here (but later too) the use of normalizers would have brought the necessary light, but unfortunately this fundamental topic even today is considered too specialist.

Chapter 26 introduces *basis* vector. The intuitive description of will certainly be rather obscure for a newcomer in the field; more convincing is the following description as a result of destructive interference. The analytical derivation *via* the is classical stuff, but when it comes to zonal and serial conditions the description is affected by incorrect use of the (Fig. 26.4), where planes of the same family with both coprime and non-coprime indices – like (001) and (002) – coexist as if they belonged to different families of lattice planes. The section ends with a chapter about space-group determination from the and intensity distribution.

The last section, *Solving and Refining Crystal Structures* (157 pp.), includes 15 chapters with a very practical approach, spanning from Fourier theory, to solution methods (trial-and-error, Patterson, heavy atom, protein structures, modelling electron density), to strategies and and ending with chapters on powder X-ray diffraction, electron and neutron diffraction. Almost 100 pp. are devoted to methods, and two entire chapters to apart from specialistic texts, this fundamental topic often receives insufficient space, whereas here we get a clear picture, although without all the details that may appear somewhat intimidating for the target audience. A few words about multipolar and the maximum-entropy method would have had their place here. The reader will especially enjoy Chapter 40 (`mistakes and pitfalls'), where several cases of wrong structure reports from the literature are discussed. I cannot refrain from pointing out a mistake on p. 428, where the lattice built on a is called a *super*lattice instead of *sub*lattice: being based on a larger cell, the lattice has lost translations and the group of translations is therefore a of that of the original lattice; it cannot evidently be a *super*lattice! Another curious mistake occurs in the first of the two chapters (No. 34) devoted to where a one-dimensional example is chosen (p. 361) but treated with *three* (although two are zeroed): if the space has one dimension, then obviously also its reciprocal is one-dimensional. The description of here and there is imprecise, with confusion between and and the use of non-crystallographic rotations, arising from the misorientation of the reciprocal lattices (the misorientation is kept into account in the or misfit, but the is crystallographic) (Nespolo, 2015*c*). A clear misunderstanding occurs in the example of the twin of ornithine-5′-monophosphate dehydrogenase, where a `non-standard *C*2_{1}' is mentioned. In a *b*-unique monoclinic setting, as in this case, *C*2_{1} is the same as *C*2: a *C*-centring vector added to *P*2_{1} generates twofold axes. No mention of this is made in the original article (Wittmann & Rudolph, 2007), where of a *P*2_{1} crystal simulating *C*222_{1} symmetry is presented. Apart from these errors, the section is very well written and the special emphasis put on methods typically used in protein crystallography (SIR, MIR, SIRAS, MAD *etc*.) is a valuable aspect not often present in textbooks targeting a readership of chemists. One can, however, once again regret the absence of normalizers, which would have been of much help in the example on p. 338 as well as in the discussion about trial-and-error methods.

A set of six short appendices (vector and complex-number tutorials, the

the analytical derivation of atomic scattering factors and of and an in-class demonstration of diffraction), followed by a (much too short) bibliography and a detailed index end the book.Some typographical errors occur here and there in the text. An *erratum* is available from the publisher (https://www.uscibooks.com ) which corrects most of these. Besides those, one should remark on the dots in the graphical symbols of glide planes (Fig. 5.5), which are far too big; some confusion between periodicity and dimension occurring in Chapter 6, where three-dimensional diperiodic structures are described as two-dimensional (p. 67); and the presence of only two equivalent directions, instead of three, for the second and third positions in the trigonal and hexagonal crystal system.

Overall, and despite some defects pointed out above, this is perhaps one of the best textbooks addressing a readership of chemists published in recent years. If an updated version is planned, one can only encourage the author to correct the errors described above, and add some matrix algebra and a chapter on normalizers: the result would be a reference text for several years to come.

### References

Nespolo, M. (2015*a*). *Cryst. Res. Technol.* **50**, 413. Web of Science CrossRef Google Scholar

Nespolo, M. (2015*b*). *J. Appl. Cryst.* **48**, 1290–1298. Web of Science CrossRef CAS IUCr Journals Google Scholar

Nespolo, M. (2015*c*). *Cryst. Res. Technol.* **50**, 362–371. Web of Science CrossRef CAS Google Scholar

Wittmann, J. G. & Rudolph, M. G. (2007). *Acta Cryst.* D**63**, 744–749. Web of Science CrossRef IUCr Journals Google Scholar

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