The number of vibrational modes g(v)dv in Debye theory

[Sizhe]

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 x²e^-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 $g(\nu)d\nu$ ?

 

[Sizhe]

Lonely Prof. on PhysicExchange gave me the right answer. I attach his answer here for general interests:
https://physics.stackexchange.com/q...vibrational-modes-g-nud-nu-proportional-to-x2