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This chapter discusses the basic properties of phonons. Topics covered include solid matter as a gas of excitations, the crystal lattice, dynamics of a linear lattice, dynamics of space lattices, the reciprocal lattice, normal coordinates in three dimensions, general properties of lattice waves, dispersion of lattice waves, lattice specific heat and the frequency spectrum, vibrations of an elastic continuum, and the Debye theory.

This chapter begins with a definition of space lattice. For most pure materials, the lowest energy state at low temperatures is a crystalline solid, where the atoms or molecules making up the substance bind together in a manner which periodically repeats itself in space. The atoms are not points but have a finite spatial extent, and they execute thermal and quantum mechanical zero-point motions. If the crystal structure involves only one kind of atom and if there is only one atom per unit cell, then we may locate each atom at the origin of a unit cell. However, if there are multiple atoms per unit cell we must specify their location within the unit cell. The collection of atom coordinates is referred to as a basis and together with the lattice defines a crystal structure: lattice + basis = crystal structure. The remainder of the chapter covers point groups, Bravais lattices, and space groups in two dimensions; point groups, Bravais lattices, and space groups in three dimensions; common crystal structures; Miller indices; and the Wigner–Seitz polyhedra and coordination polyhedra.

The various approaches to the concept of space lattices are described in this chapter. The first one is the space-filling approach, which dates back to Aristotle and Kepler. It was applied to the structure of crystals by the French school of Haüy and Delafosse and by Seeber in Germany, and was further developed by Bravais, Sohncke, Schoenflies, and Fedorov. The contribution of the German school of Weiss and Mohs was to introduce the concept of crystal systems and crystal classes. The second approach is the close-packing approach, first considered by Kepler and Hooke in the seventeenth century, and developed by Wollaston and Barlow in the nineteenth century. In the last part of the chapter the molecular theories used by the early-nineteenth-century physicists to explain the optical and elastic properties of matter are briefly reviewed. The problem of the validity of the Cauchy relations is discussed.

This chapter stresses the significance of the discovery of X-ray diffraction by Laue, Friedrich, and Knipping, at the time of the discovery and afterwards. The discovery confirmed the wave nature of X-rays, and the reaction of the supporters of the corpuscular theory, such as W. H. Bragg, is evoked. Laue’s discovery also confirmed the concept of space lattice, and the knowledge crystallographers had of that concept at that time is discussed. The discovery played a major role in the studies of the atomic structure of materials and of the structure of atoms. Its impact on X-ray spectroscopy and on the chemical, biochemical, physical, material, and mineralogical sciences is sketched in the last part of the chapter.

The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.