# Homework Help: By equating the high-temperature limit of the above expression to the classical

1. Feb 15, 2012

### blueyellow

1. The problem statement, all variables and given/known data

According to the Debye theory of the specific heat of a three-dimensional solid, the internal vibrational energy of a volume V of a solid containing N atoms is:

U=A(T)$\int^{x_{D}}_{0}$$\frac{x^{3}dx}{e^{x}-1}$

where x=$\frac{\hbar\omega}{k_{B}T}$ is the dimensionless form of the vibration frequency $\omega$

a) What assumptions are made in the Debye theory about the distribution of frequency modes as a function of their wavevecdotr K?
b) Derive an expression for the (dimensionless) Debye cutoff frequency x$_{D}$ in terms of these assumptions.
c) By equating the high-temperature limit of the above expression to the classical three-dimensional result U=3Nk$_{B}$T, deduce the unknown function A(T)
d) Hence derive an expression for the low-temperature specific heat.

3. The attempt at a solution

I have done questions a) and b). I am stuck on c).

As T tends to infinity, surely x$_{D}$ tends to 0 because x$_{D}$=$\frac{\hbar\omega_{D}}{k_{B}T}$ according to my notes.

But surely they cannot expect me to integrate the above expression for U from 0 to 0?