Statistical Physics

5.1 Heat Capacity of a Solid I – Einstein Model 69Fig. 5.1. Crystal lattice. Atoms, shown by solid circles, are arranged periodically ina lattice. The equilibrium position of each atom is called a lattice point. The atomscan be considered as connected by springsandC V = dU ( ) 2 ωdT =3Nk 1B2k B T sinh 2 (ω/2k B T ) . (5.4)The temperature dependence of the heat capacity is plotted in Fig. 5.2. WhenT →∞, ω/2k B T → 0, and C V approaches 3Nk B . The molar heat in thislimit is 3N A k B =3R =24.93 J/mol K. This is the Dulong–Petit law. Wecan understand this result by using the equipartition law; namely, a thermalenergy of (1/2)k B T is given to the kinetic energy and the potential energyin each of the x, y, andz directions in this limit. On the other hand, at lowtemperature, i.e. when T → 0, C V ∝ (1/T 2 )e −ω/kBT ; that is, it decreasesexponentially.Fig. 5.2. Heat capacity due to lattice vibrations, calculated from the Einstein model.C V/N k B is plotted as a function of k BT/ω