4. Diffraction

In order to understand diffraction from a three-dimensional crystal, a new type of lattice, constructed from the real lattice used so far, is defined. It is the reciprocal lattice proposed by Paul Ewald around 1911. ‘Diffraction’ considers how this lattice is constructed and investigates its uses in understanding diffraction. Reflection intensities and amplitudes are also explained along with Friedel’s Law; the convolution theorem; Fourier Transformation; powder diffraction, which has many uses in industry and in academic research; incommensurate or modulated crystals; and quasicrystals, often seen in metal alloys, which cannot be explained by conventional crystal symmetry ideas. Finally, disorder in crystals is discussed.

2. Symmetry

In order to explain what crystals are and how their structures are described, we need to understand the role of symmetry, for this lies at the heart of crystallography. ‘Symmetry’ explains the different types of symmetry: rotational, mirror or reflection, point, chiral, and translation. There are thirty-two point groups and seven crystal systems, according to which symmetries are present. These are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Miller indices, lattices, crystal structure, and space groups are described in more detail. Any normal crystal belongs to one of the 230 space group types. Crystallographers generally use the International Notation system to denote these space groups.

5. Seeing atoms

It is clear that knowledge of the relative phases is essential if we wish to find the atoms in a crystal. So what do we do if we do not have phase information? ‘Seeing atoms’ describes the phase problem and the different methods of phase determination used by crystallographers: a difference Fourier map; the Patterson method; electron density maps; multiple isomorphous replacement; multiple-wavelength anomalous dispersion using synchrotron radiation, which is often used in macromolecular crystallography; molecular replacement, commonly used in protein crystallography; the Sayre equation, a mathematical relationship that enables probable values for the phases of some diffracted beams to be found; and a new technique called charge flipping.

6. Sources of radiation

To observe diffraction from crystals it is necessary to have a source of radiation whose wavelength is of the same order as the atomic spacings. ‘Sources of radiation’ shows that the electromagnetic spectrum’s X-ray region does this nicely and describes the use of X-ray tubes. Another source of radiation is synchrotron radiation, which exhibits a number of special properties: the radiation emitted ranges from the hard X-ray region, through the ultraviolet and infra-red wavelengths up to visible light; the X-ray beam is plane-polarized within the horizontal plane; and the radiation is highly collimated in the vertical plane. Radiation from free-electron lasers, neutron sources, and electron diffraction is also discussed.

1. A long history

‘A long history’ explains that it was during the 17th-century Enlightenment that saw the systematic study of crystals or ‘crystallography’ by key scientists including Johannes Kepler, Robert Hooke, Christian Huygens, Nicolas Steno, and Abbé René-Just Haüy—the true father of crystallography, who postulated that crystals must be made up of regular arrangements of polyhedral units. The 19th century saw new theories of crystals with the identification of thirty-two crystal classes, fourteen Bravais lattices, and 230 possible space groups. A new era of crystallography emerged with the discovery of X-ray diffraction by crystals by Max Theodor Felix Laue. William Henry Bragg and his son William Lawrence Bragg went on to identify many crystal structures.

3. Crystal structures

‘Crystal structures’ describes the different types of close packing—hexagonal, cubic, face-centred cubic, and body-centred cubic—used to describe many simple inorganic structures, especially those of the elements. The reason for atoms to pack so closely together is to form the densest array possible to provide a stable structure. The ability of a chemical substance to adopt different crystal structures is called polymorphism, as displayed by carbon. Examples of simple inorganic structures, such as common salt, are explained along with organic crystal structures, and the different methods of crystal growth. Crystallography has also played a major part in determining the structures and activities of large biological molecules like DNA, RNA, proteins, and viruses.