Introduction to Solid State Physics · Problem 5

PHONONS II. THERMAL PROPERTIES — Charles Kittel

Physics — Solid State · gruneisen-constant

Problem

*Grüneisen constant. (a) Show that the free energy of a phonon mode of frequency ω\omega is kBTln⁡[2sinh⁡(ℏω/2kBT)]k_BT \ln [2 \sinh (\hbar\omega/2k_BT)]. It is necessary to retain the zero-point energy 12ℏω\frac{1}{2}\hbar\omega to obtain this result. (b) If Δ\Delta is the fractional volume change, then the free energy of the crystal may be written as F(Δ,T)=12BΔ2+kBT∑ln⁡[2sinh⁡(ℏωK/2kBT)] ,F(\Delta, T) = \tfrac{1}{2}B\Delta^2 + k_BT\sum \ln [2\sinh(\hbar\omega_{\mathbf{K}}/2k_BT)] \ , where BB is the bulk modulus. Assume that the volume dependence of ωK\omega_{\mathbf{K}} is δω/ω=−γΔ\delta\omega/\omega = -\gamma\Delta, where γ\gamma is known as the Grüneisen constant. If γ\gamma is taken as independent of the mode K\mathbf{K}, show that FF is a minimum with respect to Δ\Delta when BΔ=γΣ12ℏωB\Delta = \gamma\Sigma\frac{1}{2}\hbar\omega coth (ℏω/2kBT)(\hbar\omega/2k_BT), and show that this may be written in terms of the thermal energy density as Δ=γU(T)/B\Delta = \gamma U(T)/B. (c) Show that on the Debye model γ=−∂ln⁡θ/∂ln⁡V\gamma = -\partial \ln \theta/\partial \ln V. Note: Many approximations are involved in this theory: the result (a) is valid only if ω\omega is independent of temperature; γ\gamma may be quite different for different modes.

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