(a) Show that the free energy of a phonon mode of frequency w is kBT in [2sinh (hw/2kBT)]. It is necessary to retain the zero-point energy ½hw to obtain this result.
(b) If ? is the fractional volume change, then the free energy of the crystal may be written as F(?, T) = ½ B?2 + kBT ? In [2sinh (hwK/2kBT)] where B is the hulk modulus. Assume that the volume dependence of wK is ?w/w = ???, where ? is known as the Gruneisen constant. If ? is taken as independent of the mode K, show that F is a minimum with respect to ? when B ? = ??1/2hw coth (hw/2kBT), and show that this may be written in terms of the thermal energy density as ? = ?U(T)/B.
(c) Show that on the Debye model ? = ? ? In ?/? In V. Note: Many approximations are involved in this theory: the result (a) is valid only if w is independent of temperature; ? may he quite different for different modes.
This question was answered on: Jul 11, 2017
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