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## Über dieses Buch

The advances in and applications of x-ray and neutron crystallography form the essence of this new edition of this classic textbook, while maintaining the overall plan of the book that has been well received in the academic community since the first edition in 1977. X-ray crystallography is a universal tool for studying molecular structure, and the complementary nature of neutron diffraction crystallography permits the location of atomic species in crystals which are not easily revealed by X-ray techniques alone, such as hydrogen atoms or other light atoms in the presence of heavier atoms. Thus, a chapter discussing the practice of neutron diffraction techniques, with examples, broadens the scope of the text in a highly desirable way. As with previous editions, the book contains problems to illustrate the work of each chapter, and detailed solutions are provided. Mathematical procedures related to the material of the main body of the book are not discussed in detail, but are quoted where needed with references to standard mathematical texts. To address the computational aspect of crystallography, the suite of computer programs from the fourth edition has been revised and expanded. The programs enable the reader to participate fully in many of the aspects of x-ray crystallography discussed in the book. In particular, the program system XRAY* is interactive, and enables the reader to follow through, at the monitor screen, the computational techniques involved in single-crystal structure determination, albeit in two dimensions, with the data sets provided.

Exercises for students can be found int the book, and solutions are available to instructors.

## Inhaltsverzeichnis

### 1. Crystal Morphology and Crystal Symmetry

Abstract
Crystals, with their plane faces, sharp angles, and color, have excited interest since the earliest times. Their color and decorative qualities are recorded in the Bible [1]: we need not start as far back as that, but will consider instead some of the highlights in the build-up of the science of Crystallography.

### 2. Lattices and Space-Group Theory

Abstract
We continue our study of crystals by investigating the internal arrangements of crystalline materials. Crystals are characterized by periodicities in three dimensions. An atomic grouping, or pattern motif which, itself, may or may not be symmetrical, is repeated again and again by a symmetry mechanism, namely the space group of the crystal. There are 230 space groups, and each crystal substance belongs to one or other of them. In its simplest form, a space group may be derived from the repetition of a pattern motif by the translations of a lattice, as discussed below. It can be developed further by incorporating additional symmetry elements, as demonstrated through the following text and Problem 2.1. We now enlarge on these ideas, starting with an examination of lattices.

### 3. X-Rays and X-Ray Diffraction

Abstract
X-rays are an electromagnetic radiation of short wavelength, and can be produced by the sudden deceleration of rapidly moving electrons at a target material. If an electron falls through a potential difference of V volt, it acquires an energy eV electron-volt (eV), where e is the charge on an electron. This energy may be expressed as quanta of X-rays of wavelength λ, where each quantum is given by
$$\lambda = hc/(eV)$$
h being the Planck constant and c the speed of light in vacuum. Substitution of numerical values into (3.1) leads to
$$\lambda = 12.4/V$$
where V is measured in kilovolt and λ is given in Angstrom units (Å). The wavelength range of X-rays is approximately 0.1–100Å, but for the purposes of practical X-ray crystallography, the range used is restricted to 0.7–2.5Å.

### 4. Intensities and Intensity Statistics

Abstract
The measurement of the intensity of a diffracted X-ray beam can be carried out photographically, by camera techniques, but almost always today by quantum counting, with diffractometer instruments. We can measure either a peak intensity or an integrated intensity, the latter parameter being preferred for the expression of the intensity of X-ray reflection.

### 5. Examination of Single Crystals: Optical and X-Ray Diffraction Practice

Abstract
The preliminary optical examination of crystalline specimens is interesting and useful in its own right and is a major tool still employed by mineralogists and geologists. However, in structure determinations with modern equipment, it is not uncommon nowadays to by-pass this step and proceed immediately with X-ray studies. This is because in most cases, the X-ray technique is straightforward and test data can be quickly scanned with a single-crystal X-ray diffractometer, Sects. 5.5 and 5.6, or with area detector (see Sect. 5.7), and the suitability and quality of the crystal assessed. There are other situations, however, where complications may arise, for example, because of an unusual crystal habit, Sect. 5.3.5, pseudosymmetry, Sects. 7.2.2, 7.5.4, and Sect. 8.5.3, or twinning, Sect. 5.10. In such cases, it might be possible to extract useful information from an optical examination of a crystal before the more detailed, costly and time-consuming X-ray methods are tried.

### 6. Fourier Series and Fourier Transforms

Abstract
In Sect. 5.3, we touched upon an analogy between the scattering of X-rays and that of visible light. Here, we extend that discussion to consider aspects of Fourier series and Fourier transforms that are germane to our study of X-ray diffraction.

### 7. Fourier Techniques in X-Ray Structure Determination

Abstract
We have reached the stage where we can consider how to attack the solving of a crystal structure. After the earliest trial and error determinations in the 1920s with very simple and highly symmetrical structures, it was found that the application of Fourier series, initially in one dimension, led to the electron density function, in which peak maxima in the electron density corresponded to atomic positions. As we have seen in the previous chapters, it is necessary to have the phases of the structure factors for a Fourier synthesis to be carried out meaningfully. One way in which phase information may be obtained is through the Patterson function of vector density, a function of interatomic vectors in the crystal structure.

### 8. Direct Methods and Refinement

Abstract
In this chapter, we consider direct methods, also known as phase probability methods, of solving the phase problem, together with Patterson search techniques, least-squares refinement, and other important procedures that are involved in the overall investigation of crystal and molecular structure.

### 9. Examples of Crystal Structure Determination

Abstract
In this chapter we draw together, by means of actual examples, some of the material presented earlier in the book. It may be desirable for the reader to refer back to the previous chapters for descriptions of the techniques used, since we shall present here mainly the results obtained at each stage.

### 10. Proteins and Macromolecular X-Ray Analysis

Abstract
In this chapter, we take a more detailed look at methods of X-ray analysis that are particularly applicable to large biological molecules. It will involve some useful reiteration of concepts and ideas discussed in previous chapters. We would also remind readers that although there are definite distinctions between large and small molecules in the crystallographic arena, there is no reason to exclude one from the other, and in fact, there are many advantages in being familiar with both. The major differences should become clearer as we progress through this chapter. It follows that while we deal mainly with macromolecules here, much of the information provided in this chapter is applicable to all areas of crystal structure analysis.

### 11. Neutron Diffraction from Single Crystals

Abstract
When we discussed the variation of observable symmetry according to the nature of the examining probe, Sect. 1.4, we stated that one possible probe is a neutron beam. As an example, when the structure of elemental chromium is examined by X-ray diffraction, it shows a body-centered cubic arrangement, Fig. 11.1a. Elemental chromium has the electronic configuration (Ar 3d5 4s1); it is antiferromagnetic at room temperature, and its electron spins, arising from the unpaired electrons, give rise to the magnetic structure shown in Fig. 11.1b. The magnetic moment of the neutron interacts with the permanent dipole of chromium to form this structure, and consequently the diffraction record shows a primitive cubic structure.

### 12. Powder Diffraction

Abstract
The powder method was devised by Hull soon after the discovery of X-ray diffraction, and developed in detail by Hull [1] and by Debye and Scherrer [2]. Since that time, X-ray diffraction from powdered specimens has been used in divers investigations of materials. The main interest in this book is structure determination for which powder methods have, until recent years, been inappropriate, mainly because of the problem of overlap of the X-ray reflections which causes three-dimensional data information to collapse on to a one-dimensional powder record. The vast improvement in instrument technology in recent years has led to powder photographs and diffractograms of sufficient precision to be interpretable in terms of the underlying crystal structures, and powder techniques have now been developed as a very significant tool in X-ray structure determination. Before launching into this topic, however, we summarize here some of the many applications of X-ray powder diffraction other than in crystal structure determination.

### 13. Computer-Aided Crystallography

Abstract
This chapter has been designed to further the knowledge gained from a study of the earlier chapters of this book. The computer programs that are supplied as the Web Program Packages and described here are complementary to that work, and enable the reader to gain practical experience of concepts and methods germane to X-ray structure analysis. While these stand-alone programs are provided for quick and easy access for problem solving within the context of this book, we emphasize that the serious structure analyst must also refer to the other important program systems readily available.