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

  • #1
7
1

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## ?
 

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