Consider a crystal that can exist in either of two structures, denoted by and . We
Question:
(a) Show from chapter 4 that the free energy density contributed by the phonons in a solid at a temperature much less than the Debye temperature is given by –π2τ4/30v3/h3, in the Debye approximation with v taken as the velocity of all phonons.
(b) Show that at the transformation temperature
τc4 = (30h3/π2)[Uβ(0) – Uα(0)]/(vβ-3 – vα-3).
There will be a finite real solution of vβ < vα. This example is a simplified model of a class of actual phase transformations in solids.
(c) The latent heat of transformation is defined as the thermal energy that must be supplied to carry the system through the transformation. Show that the latent heat for this model is
L = 4[Uβ(0) – Uα(0)].
In (84) and (85), U refers to unit volume.
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