Gruneisen constant 

(a) Show that the free energy of a phonon mode of frequency w is k_BT in [2sinh (hw/2k_BT)]. It is necessary to retain the zero-point energy ½hw to obtain this result. 

(b) If Δ is the fractional volume change, then the free energy of the crystal may be written as F(Δ, T) = ½ BΔ² + k_BT Σ In [2sinh (hw_K/2k_BT)] where B is the hulk modulus. Assume that the volume dependence of w_K is δw/w = —γΔ, where γ is known as the Gruneisen constant. If γ is taken as independent of the mode K, show that F is a minimum with respect to Δ when B Δ = γΣ1/2hw coth (hw/2k_BT), and show that this may be written in terms of the thermal energy density as Δ = γU(T)/B.

(c) Show that on the Debye model γ = — ∂ In θ/∂ In V. Many approximations are involved in this theory: the result (a) is valid only if w is independent of temperature; γ may he quite different for different modes.