Statistical Physics

5.2 Heat Capacity of a Solid II – Debye Model 73Fig. 5.5. Wave vector space. The allowed values of k = (πn x/L x, πn y/L y,πn z/L zn z) form a lattice in this spaceThe final term in the right-hand side is the contribution from the zero-pointoscillation, which is independent of the temperature.To evaluate this equation, we perform the summation over k as an integralin wave vector space. In the one-dimensional case k n =(π/L)n, thereis one point corresponding to an allowed wave number for each interval∆k = π/L. In the three-dimensional case, there is one point correspondingto an allowed wave vector in each volume π 3 /L x L y L z . Therefore, there aredk x dk y dk z /(π 3 /V ) allowed wave vectors in a volume of dk x dk y dk z ,andsoU is given in integral form byU = ∑ α∫ π/a0= V8π 3 ∑α∫ π/a ∫ π/adk x dk y∫ π/a−π/a00dk zVπ 3ω α (k)exp [βω α (k)] − 1 +const.∫ π/a ∫ π/aω α (k)dk x dk y dk z−π/a −π/a exp [βω α (k)] − 1 +const.(5.16)In the second line of this equation, the integrals for each direction have beenextended to negative wave vector components, and the expression has beendivided by 2 3 = 8. This is allowed because the frequency is a function ofthe absolute value of k. The last term is the contribution from the zeropointoscillation, which is independent of temperature. On the basis of thisexpression for the internal energy, we shall now investigate the temperaturedependence of the internal energy and the heat capacity.5.2.2 Heat Capacity at High TemperatureThe high-temperature region is defined as that in which βω α (k) ≪ 1issatisfied for all of the normal modes. In this case the following approximationcan be used:exp[βω α (k)] ≃ 1+βω α (k) =1+ ω α (k)k B T . (5.17)