## X-Ray Crystallography

Gregory S. Girolami University of Illinois at Urbana-Champaign

### Preface

This textbook was written to satisfy several needs.|

• First, to describe the fundamentals of crystallography so that they can be understood by a graduate student or advanced undergraduate student encountering them for the first time.

• Second, to offer mental images of and physical insights into the phenomena and equations behind crystallography, so as to bring them to life and make them more understandable.

• Third, to provide a balanced treatment of symmetry, the physics of diffraction, and the methods to solve and refine structures. Many existing textbooks omit or skimp on one or more of these topics, whereas all three are necessary for a full understanding of the beauty and power of crystallography.

This book focuses on those topics most useful to those who are interested in crystallography as a research tool, although I hope that it may also be of use to those who aim to become practicing crystallographers. It emphasizes single crystal X‐ray diffraction but also includes sections on powder, neutron, and electron diffraction.

The book is organized into 42 relatively short chapters, each containing roughly the amount of material that can be covered in one 90 minute lecture. This organization makes it easier for students to digest the material, and also gives instructors more flexibility to pick and choose among the topics and to change the order of presentation. Extensive problem sets are included that help the student master the ideas presented in the chapters.

Some notable features of this book are as follows.

Sections are devoted to questions that might occur to a thoughtful student, but which are seldom covered in existing textbooks. For example:

• Why can the magnetic component of electromagnetic radiation be ignored when calculating the scattering of X‐rays by electrons, which, after all, have magnetic moments?

• How are reflection intensities measured from CCD and image plate frames?

• What is the justification for using structure factors and complex exponentials as representations of electromagnetic (i.e., simple cosine) waves?

• How are the Fourier transform equations that relate structure factors and electron density derived?

• Why are the distributions of reflection intensities from centrosymmetric and non‐centrosymmetric crystals different?

Two‐dimensional lattices and two‐dimensional plane groups are discussed before introducing the three‐dimensional versions of these topics. Students generally find it far easier to learn how to think about and use symmetry in three dimensions if they first master the relevant concepts in two dimensions, which after all is far easier to see.

Some advanced and emerging topics are included that are not always found in textbooks, such as disorder, twinning, microfocus sources, charge flipping, and the maximum likelihood method of refinement. Some topics with a mineralogical, purely practical, or materials science focus are not included; among these are the classification of macroscopic crystal shapes, the art of growing crystals, the interpretation of Weissenberg images, the tensor properties of crystals, and aperiodic structures. Instructors and students interested in these topics can find them in other texts, and the highly modular nature of the present book makes it easy to add such material to a course.

I have coined some new terms where I felt that needed terminology was lacking. One of these is ‘travel symmetry operation,’ which is a collective term for translations, screw rotations, and glide reflections. Another is ‘auxiliary symmetry element’ for rotoinversion axes, screw axes, and glide planes, which are not true symmetry elements for their respective operations (a distinction that too few are aware of).

Although this book does not contain a laboratory component, the course I teach at the University of Illinois does. In the laboratory, students learn how to use the SHELX software package to solve crystal structures. They are led through the process with the help of *A Guide to Using the SHELX Crystallographic Software Package*, which can be downloaded free of charge at http://scs.illinois.edu/x‐ray/software2/shelx.php. The students practice on one structure, and then solve several others for their lab grade.

The mathematics behind crystallography – vector algebra, complex numbers, triple integrals, Fourier analysis, and reciprocal space – can be very intimidating, even to good students. For this reason, it is doubly important that crystallographic textbooks be written clearly. As mentioned at the beginning of this Preface, I have taken special pains to achieve this goal and hope that the effort has been at least partially successful.

I wish to thank Iain Paul, who taught the X‐ray crystallography course at Illinois before me and indirectly shaped my own approach to the topic. Holger Hellwig, who taught the course from my lecture notes for three years while I was away in Administration Land, made a few of the notes available to students along with some additions of his own, and thereby created the seed from which this book grew. And I am most grateful to the many students, colleagues, and friends – especially Dan Grayson and Ken Raymond – who provided valuable help.

Gregory S. Girolami

Urbana, Illinois