5. Grüneisen constant. (a) Show that the free energy of a phonon mode of frequency 𝜔 is 𝑘𝐵&#1198

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5. Grüneisen constant. (a) Show that the free energy of a phonon
mode of frequency 𝜔 is
𝑘𝐵𝑇 𝐼𝑛 [2 𝑠𝑖𝑛h (ℏ𝜔/2𝑘𝐵𝑇)]. It is necessary to retain the zero-point
energy 1/2(ℏ𝜔) to obtain
this result. (b) If A is the fractional volume change, then the
free energy of the crystal may be
written as 𝐹(Δ, T) = (1/2) 𝐵Δ^2 + 𝑘𝐵𝑇 ∑𝑙𝑛 [2 𝑠𝑖𝑛h (ℏ𝜔𝐾/2𝑘𝐵𝑇)] where
B is the bulk modulus. Assume that the volume dependence of 𝜔𝑘 is
𝛿𝜔/𝜔 = −𝛾Δ, where 𝛾 is known as the Grüneisen constant. If 𝛾 is
taken as independent of the mode K, show that F is a minimum
with respect to A when 𝐵Δ = 𝛾∑ ( 1/2 ℏ𝜔) coth ( ℏ𝜔/2𝑘𝐵𝑇 ), 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 = − 𝜕ln𝜃/𝜕ln𝑉 .
Note: Many approximations are involved in this theory: the result
(a) is valid only if 𝜔 is independent of temperature; 𝛾 may be
quite for different Inodes.