TABLE OF CONTENTS In previous Chapters, we have considered the electrons in a crystal that consisted of a rigid lattice of atoms. This represented a good approximation because the mass of an atom is more than 2000 the mass of an electron. However, such assumptions founder when considering specific heat, thermal expansion, the temperature dependence of electron relaxation time, and thermal conductivity. In order to interpret these phenomena involving electrons and atoms, a more refined model needs to be considered, in which the atoms are allowed to move and vibrate around their equilibrium positions in the lattice. In this Chapter, we will present a simple, yet relatively accurate mathematical model to describe the mechanical vibrations of atoms in a crystal. We will first cover one-dimensional monatomic and diatomic crystals followed by three-dimensional crystals. We will then consider the collective movement or excitations of the atoms in a crystal, the so called phonons, and conclude with a section on the velocity of sound in a medium.
We saw in section 2.5, when discussing the formation of bonds in solids, that these equilibrium positions were achieved by balancing attractive and repulsive forces between individual atoms. We assumed that the attractive and repulsive forces always canceled each other and that the masses were infinite. The resulting potential U(R) curve for an atom as a function of its distance R from a neighboring atom is shown in Fig. 5.1. This figure shows a minimum...
