Simple crystal lattices

Atoms in molecules and solids are bound together by the valence electrons which have been supplied by them and are shared among them. In this chapter we will consider the motion of atoms under given interatomic forces without asking about their nature – the electronic origin of the bonds. We will confine ourselves to solids and study some simple characteristics of the atomic motions within them – lattice vibrations – brought about by the periodic structure of the crystal lattice. The lattice vibrations will turn out to be a prototype of elementary excitations in a solid, other examples of which will be given in later sections.

Consider an assemblage of a great number (N) of identical atoms with interatomic potential ν(r) which typically is attractive except for the hard-core repulsion, and let them be arrayed in one-dimensional space with positions x1, x2, … in increasing order. In the approximation that only the nearest neighbor potentials are considered, as is allowed when ν(r) is short ranged, the total potential energy is given by U ≃ ν(x1 – x2) + ν(x2 – x3) + …, and the most stable arrangement of the atoms is a periodic array with equal distance a at which ν(r) is a minimum.