Introduction

In the last chapter, we investigated the dynamics of liquids whose particles are free to wander about due to the reasonably weak level of inter-particle bonding. In a solid (crystal or glass), bonding between particles is stronger and the translational motion of the particles is arrested. Nevertheless, these “solid” particles continue to move and execute small, localized vibrations about a fixed point in space. In this chapter and the next, we investigate the nature of this vibrational motion and its impact on the thermal properties of a solid. Here we begin by considering a simple model of masses connected by ideal springs to demonstrate how vibrations of individual atoms are, in reality, a consequence of propagating waves traveling through the crystal lattice. In order to connect these waves with the quantum mechanical perspective of each atom behaving as a quantized harmonic oscillator, we find ourselves introducing the concept of a quantum of elastic wave, known as a phonon.

An important outcome of our development of quantized elastic waves is a growing appreciation for a special region of reciprocal space known as the Brillouin zone, which is populated by all the wave vectors, K, corresponding to allowed phonon waves in the crystal. For phonons whose K matches the edge of this zone, significant Bragg scattering results, to produce two equivalent standing wave patterns separated by an energy gap. We will revisit the Brillouin zone often in the chapters to come, and we will begin to appreciate the significance of this boundary for the motion of all waves that attempt to travel within a crystal.