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Neutron powder diffraction finds extensive application in the solution, and more particularly, the refinement, of crystal structures. This chapter reviews the tools (space group symmetry, specification of atomic positions, crystallography, reflection conditions) for describing and solving crystal structures. The solution of the structure of the low temperature form of NaOD provides a simple example. There follows an account of structure refinement, particularly the Rietveld method. This involves consideration of the observed pattern, the calculated pattern, and the many factors that need to be taken into account (peak shapes, peak widths, background, structure factors, displacement parameters, preferred orientation, etc.), followed by the matching of the two to determine the crystal structure parameters. Le Bail extraction (from whole pattern fitting) is derived from the Rietveld method. The chapter concludes with a discussion of the practical aspects of the Rietveld method and a selection of illustrative examples.

The study of crystal symmetry started as a study of the symmetry of macroscopic crystalline objects, mostly minerals. When it was realised that external symmetry reflected the internal symmetry of the disposition of atoms and molecules within the crystal, mathematical methods were applied to the classification of all possible combinations of symmetry operations under the condition of periodic translational repetition, within the framework of group theory. The study of molecular crystals from a structural point of view requires neither abstract lattices nor group theory. This chapter discusses bottom-up crystallography, space group symmetry and its mathematical representation, scattering by one or two charge points, the atomic scattering factor, the molecular structure factor, the structure factor for infinite periodic systems, Miller indices and Bragg's law, and the electron density in a crystal. Some examples in which X-ray diffraction is used to study the internal structure of a condensed phase are given.

First a general description of the concept of crystalline structures is presented with some historical background information. The classical approach of periodic structures is presented along with the important topic of symmetry and its role characterizing physical properties. The limitations of the classical model are then introduced in view of the new experimental observations discovered since the 1970s. New forms of crystalline structures including incommensurately modulated and composite structures are presented along with quasicrystalline structures (quasicrystals). The necessity to extend the theory of space group symmetry is then discussed and the concept of superspace symmetry is introduced in order to describe these new forms of matters.

The generation and properties of X-rays and their diffraction by crystals are discussed. The collection of X-ray data by photographic and diffractometric methods is explained, leading to a discussion of the effects of space group symmetry on the recording of X-ray diffraction spectra. Geometrical structure factors are explained with examples, and the analytical deduction of the conditions limiting X-ray reflection according to space group symmetry described. Examples are given of the deduction of space groups from X-ray diffraction records.