temperature. Although Einstein’s model had great success in explaining the main features of this behaviour, the model is a great oversimplification and has limita- tions. This is because it assumes that the atoms vibrate totally independently of each other about fixed lattice sites. In reality, they do not because the atoms are coupled together. A macroscopic mechanical analogue of a crystal lattice would consist of billiard balls connected together with identical springs. Figure 4.13 shows a two-dimensional picture of this. If one ball is set vibrating, say the one labelled A in Figure 4.13, a disturbance will propagate throughout the whole system until all the balls are vibrating. Similarly, the atoms in a crystal are coupled rather than independent oscillators. Einstein’s theory can be improved by describing the N atoms in a crystal in terms of the 3N normal modes of vibration of the whole crys- tal, each with its own characteristic angular frequency ω1, ω2, . . . , ω3N . In terms of these normal modes, the lattice vibrations are equivalent to 3N independent harmonic oscillators with these angular frequencies (see also Mandl,2 Section 6.3).