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This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.

This chapter describes the basis of X-ray diffraction by a three-dimensional crystal and introduces the Laue equations which describe the geometric conditions for constructive interference of X-rays diffracted by a crystal lattice. The discussion reinforces the importance of the reciprocal lattice and introduces the concepts of the reciprocal unit cell and the reciprocal lattice vector. Equations relating the real and reciprocal unit cell parameters, as well as the volumes of the real and reciprocal unit cells, are derived and the geometric relationship between the scattering vector and the corresponding reflecting plane in the crystal is established. These geometric considerations lead naturally to one of crystallography's most important rules, known as Bragg's law. Finally, the chapter introduces the concept of the Ewald sphere — an elegant geometric construction that encapsulates Bragg's law in a highly useful graphical form, which is used in many places in the subsequent text.

This chapter presents the properties of thermal neutrons. Their wavelength (from the de Broglie equation) is well suited to the investigation of condensed matter, i.e., to the study of liquids, glasses (amorphous materials), and crystalline materials with varying degrees of order. That the neutrons carry magnetic moment also makes them well suited to the study of magnetic ordering. The theory of nuclear and magnetic scattering from individual atoms and from assemblies of atoms is presented, this leading to the definition of neutron scattering length and to the concepts of coherent and incoherent scattering. The focus then shifts to the direction and intensity of diffraction from crystalline materials (Bragg's law, structure factors), and to the description of this scattering when samples are presented in polycrystalline or powder form (Debye-Scherrer cones).

The chapter starts with an historical introduction to the topics covered here. It describes Laue's analysis of X-ray diffraction, i.e. how to derive the three Laue equations. It moves on to provide W. L. Bragg's analysis and a general derivation of Bragg's law and its expression in reciprocal lattice vector rotation. The chapter then gives equivalence of Laue's equations and Bragg's law. Finally, it gives P. P. Ewald's synthesis in terms of the reflecting sphere construction and its applications.

A 1912 letter reported a German experiment that appeared to show that X-rays were waves. William discussed it with his son, who then explained all the observed spots by supposing that X-rays falling on a crystal were indeed diffracted, but contrary to Laue's analysis, the crystal planes of the face-centred ZnS crystal selected specific wavelengths from the incident radiation via reflection. Lawrence presented his analysis to the Cambridge Philosophical Society, including ‘Bragg's Law’. William modified an optical spectrometer and used it to study X-rays, which led Lawrence to deduce the atomic structure of simple crystals. Father and son then collaborated on structure determinations and began preparation of X-rays and Crystal Structure. William gave Lawrence full credit but received most of the kudos; British science couldn't applaud the achievement by a young colonial student. He was elected a Fellow of Trinity College, while William surrendered the study of X-ray to Moseley.

This chapter tells the story of the discovery of X-ray diffraction and of its immediate aftermath. The scene is set in Munich, with the institutes of Groth, Röntgen, and Sommerfeld. The story starts with Ewald’s thesis. Ewald’s answer to a question by Laue sparked Laue’s intuition: crystals are three-dimensional gratings that should diffract X-rays whose short wavelengths are of the same order as interatomic distances. The experiment was made by Friedrich and Knipping against the will of Sommerfeld. Although based on false premises, it was successful, and Laue’s partially incorrect interpretation led to the geometrical theory of X-ray diffraction. The way the news reached W. H. Bragg is recounted, as well as his first reactions and his exchange of letters with Laue. The discovery spurred the introduction of the reciprocal lattice by Ewald and the interpretation of Laue’s experiment by W. H. Bragg’s son, Lawrence (Bragg’s law). The chapter ends with a discussion of the controversy between Forman and Ewald as to the way the history of the discovery was told by its discoverers.

This chapter describes the theory behind X-ray diffraction data collection, and reduction. A diffraction pattern is the Fourier transform of the crystal structure, and can be described in terms of reciprocal space; Fourier transformation of the diffraction pattern should recover an image of the electron density of the crystal structure, but can not be directly achieved because the relative phases of the diffracted beams are lost and only the directions and amplitudes (as intensities) are available. The geometry of diffraction is described by Bragg's law and the Ewald sphere construction. Methods are explained for assigning indices to X-ray reflections and finding the orientation of the crystal. Aspects of data collection strategies include the selection of data to be measured and the impact of the crystal mosaic spread. Integration is the process of extracting reflection intensities. Corrections are necessary for various effects such as the diffraction geometry, absorption, and incident X-ray variations.

Electron diffraction comes from the combined scattering from many atoms arranged in a crystal. Bragg’s law and the Ewald sphere explain where the diffraction spots will fall on a detector and when the different reflections will be excited. Phase identification and zone-axis orientation is done from one or a few ED patterns. Determination of the unit-cell dimensions come from three main zone axes, high-order Laue zones (HOLZ) or a tilt series. Precession geometry is explained with ray paths and snap-shots from the collection of precession patterns in the TEM. Digital (by software) and analogue (by hardware) precession are compared. The geometrical factors that influence intensities, i.e. the Lorentz factor are given. The very latest techniques for collecting complete 3D ED data is shown, with the two existing geometries, automated diffraction tomography and the rotation method. Convergent-beam electron diffraction (CBED) is briefly introduced.

A non-mathematical account of the discovery of X-ray diffraction by von Laue and its use as a new kind of high-resolution microscopy by W. L. Bragg is given. There follows a simple explanation of how the electron densities in various regions of any molecule that can be crystallized can be retrieved from its X-ray diffraction pattern. Also, it is explained how the molecular weight of the molecule can be determined from straightforward measurements of the diffraction and the density of the crystal. The identity of the elements in a crystal, as well as the nature of the chemical bonding between them, may also be derived from measurement of the electron density distribution within it. The importance of Bragg’s Law, relating X-ray pattern to interatomic distance, is demonstrated, and initial applications of it by Bragg and Pauling are given.

This chapter analyzes the diffraction of X-rays from crystals. The wavelengths of X-rays can be of the same order as the interatomic spacings in crystals. In optics we are familiar with the fact that a collection of equally spaced lines (a line grating) scatters (or diffracts) light at discrete angles relative to an incoming beam, when the wavelength becomes smaller than the line spacing. This suggests that a similar phenomenon will occur with crystal lattices; i.e., crystal lattices can function as three-dimensional diffraction gratings. The utility of X-ray diffraction is based on the direction and intensity of the scattered waves. We can obtain information about the atomic positions within the unit cell of a crystal and the distribution of the electron charge density. The chapter also discusses Bragg’s law; Laue equations; the reciprocal lattice; the Ewald construction; the Brillouin zone; the geometrical structure factor; the atomic scattering factor; the Debye–Waller factor; sources of X-rays; and experimental methods to study X-ray diffraction. The chapter finishes with some problems.

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a crystalline material, are derived, and the structure factor and differential cross-section are obtained in the static approximation. It is explained how the presence of defects, short-range order, and reduced dimensionality causes diffuse scattering. For non-crystalline materials, such as liquids and glasses, the pair distribution function and density-density correlation function are introduced, and their relation to the static structure factor established. For molecular fluids, the form factor is defined and calculated for a diatomic molecule, and the separation of intra- and inter-molecular scattering is discussed. The principles of small-angle neutron scattering are described.

This chapter covers the preliminary characterization of the crystals in order to determine if they are suitable for a full structure determination. Probably more frustrating than failure to produce crystals at all, is the growth of beautiful crystals which do not diffract, which have very large unit cell dimensions, or which decay very rapidly in the X-ray beam, though this last problem has been largely overcome by freezing the sample. It is impossible in one brief chapter to give more than a flavour of what the X-ray crystallographic technique entails and it is assumed that the protein chemist growing the crystals will have contact with a protein crystallographer, who will carry out the actual structure determination and in whose laboratory state-of-the-art facilities exist. However, preliminary characterization can often be carried out with little more than the equipment which is widely available in Chemistry and Physics Departments and so the crystal grower remote from a protein crystallography laboratory can monitor the success of their experiments. The reader should refer to the first edition for protocols useful for photographic characterization but such techniques are seldom used nowadays. It must be remembered, in any case, that X-rays are dangerous and the inexperienced should not try to X-ray protein crystals without help. It is necessary to provide an overview of X-ray crystallography, to put the preliminary characterization in context. For a general description of the technique the reader should refer to Glusker et al. (1) or Stout and Jensen (2). For protein crystallography in particular, the books by McRee (3) and Drenth (4) describe many of the advances since the seminal work of Blundell and Johnson (5). Amongst many excellent introductory articles, those by Bragg (6), published years ago, and Glusker (7) are particularly recommended. The scattering or diffraction of X-rays is an interference phenomenon and the interference between the X-rays scattered from the atoms in the structure produces significant changes in the observed diffraction in different directions. This variation in intensity with direction arises because the path differences taken by the scattered X-ray beams are of the same magnitude as the separation of the atoms in the molecule.

A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of “reflection” of X rays from crystal lattice planes (Bragg, 1913). While the Bragg equation, which implies this “reflection,” has proved extremely useful, it does not really help in understanding the process of X-ray diffraction. Therefore we will proceed instead by way of an elementary consideration of diffraction phenomena generally, and then diffraction from periodic structures (such as crystals), making use of optical analogies (Jenkins and White, 1957; Taylor and Lipson, 1964; Harburn et al., 1975). The eyes of most animals, including humans, comprise efficient optical systems for forming images of objects by the recombination of visible radiation scattered by these objects. Many things are, of course, too small to be detected by the unaided human eye, but an enlarged image of some of them can be formed with a microscope—using visible light for objects with dimensions comparable to or larger than the wavelength of this light (about 6 × 10−7 m), or using electrons of high energy (and thus short wavelength) in an electron microscope. In order to “see” the fine details of molecular structure (with dimensions 10−8 to 10−10 m), it is necessary to use radiation of a wavelength comparable to, or smaller than, the dimensions of the distances between atoms. Such radiation is readily available (1) in the X rays produced by bombarding a target composed of an element of intermediate atomic number (for example, between Cr and Mo in the Periodic Table) with fast electrons, or from a synchrotron source, (2) in neutrons from a nuclear reactor or spallation source, or (3) in electrons with energies of 10–50 keV. Each of these kinds of radiation is scattered by the atoms of the sample, just as is ordinary light, and if we could recombine this scattered radiation, as a microscope can, we could form an image of the scattering matter.