# Show that when the temperature is such that T Debye temperature, the specific

## Homework Statement

Consider phonons propagating on a one-dimensional chain of N identical atoms of mass M interacting by nearest-neighoour spring constants of magnitude C.

a) Show that the Debye frequency can be written as $\omega$$_{D}$=$\pi$(C/M)$^{1/2}$.
b) Show that when the temperature is such that T<<$\Theta$$_{D}$, where $\Theta$$_{D}$=$\hbar$$\omega$$_{D}$/k$_{B}$ is the Debye temperature, the specific heat can be written as C$_{V}$$\propto$Nk$_{B}$(T/$\Theta_{D}$)

## The Attempt at a Solution

I have done part a), but for part b):

my notes say that for T<<$\Theta$$_{D}$, C$_{V}$$\approx$($\frac{T}{\Theta_{D}}$)$^{3}$

so how can C$_{V}$$\propto$Nk$_{B}$(T/$\Theta_{D}$)

when there is a power of 3? And where do the N and Boltzmann's constant come from?

I have also looked in other places, but nowhere has told me why the specific heat can be written in the form they ask you to show it can be written in.

Thanks if you help.