(a) Referring to Equation XXXXXXXXXXand Problem 16-2, show that the internal energy of the assembly of phonons can be written where D( 0 /T) is one of several Debye functions. Its definition is (b)...

(a) Referring to Equation (16.14) and Problem 16-2, show that the internal energy of the assembly of phonons can be written

where D(
₀/T) is one of several Debye functions. Its definition is

(b) For very low temperatures. the upper limit of the integral can be taken to be infinity. Show that

by expanding the integrand in powers of e^-r
and integrating term by term. The result can be expressed as an infinite series whose sum is well known. Note that D () = 0.

(c) Show that at low temperatures