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## Homework Statement

Show that, in the Debye theory, the number of excited vibrational modes in the frequency range ##\nu## to ##\nu+d\nu##, at temperature T, is proportional to x

^{2}e

^{-x}, where ##x=h\nu/kT##. The maximum in this function occurs at a frequency ##\nu'=2kT/h##; hence ##\nu'→0## as ##T→0##.

**2. The attempt at a solution**

The Debye theory gives that ##g(\nu)\sim α\nu^2##. Since the problem comtains the term of ##e^{-x}##. I think it might be related to the partition equation of single excited classic oscillator(##n>0##):

$$

q(\theta_i)=e^{-\frac{3\theta_i}{2T}}\Sigma_{n=0}^{\infty}(e^{-\theta_i/T})^n=\frac{e^\frac{3\theta_i}{2T}}{1-e^{-\theta_i/t}}

$$

where ##\theta_i=h\nu_i/k##. So the partition equation for the monoatomic crystal becoms:

$$

Q=e^{-N\phi(0)/2kT}\Pi_{i=1}^{3N}q(\theta_i)

$$

and

$$

-lnQ = \frac{N\phi(0)}{2kT}+\int^\infty_0[ln(1-e^{-h\nu/kT})+\frac{3h\nu}{2kT}]g(\nu)d\nu

$$

where ##\phi(0)## is the internal energy of all the atoms in the crystal are at equilibrium locations and ##N## the number of atoms. With these equations and:

$$

\int^\infty_0g(\nu)d\nu=3N

$$

How should I proceed to get ##g(\nu)d\nu## ?