Phonons and Thermal Properties

Abstract

In the 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 of 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.