# Heat capacity using Debye dispersion relation

1. Mar 10, 2014

### torch000

1. The problem statement, all variables and given/known data
Using the Debye dispersion approximation, calculate the heat capacity of a harmonic, monatomic, 1D lattice. Next, find the temperature dependence in the low temperature limit. (Assume that the longitudinal mode has spring constant CL = C, and the two transverse modes both have spring constant CT = 0.2C. )

2. Relevant equations
$C_{lattice}=k_{B}\sum$p ∫d$\omega$Dp($\omega$) $\frac{x^{2}e^{x}}{(e^{x}-1)^{2}}$ , p is over all of the modes
$d\omega$ $D_{p}( \omega)$ = $\frac{Lk_{B}T}{hv\pi}dx$
where $D_{p}( \omega)$ = density of states
$c_p$ = the spring constants for longitudinal and transverse
3. The attempt at a solution
So I should end up with 3 heat capacities to add together, two transverse and one longitudinal.

$C_{lattice}=k_{B}\sum_p \int^{\inf}_{0} \frac{Lk_{B}T}{hv\pi}\frac{x^{2}e^{x}}{(e^{x}-1)^{2}} \omega D_{p}(\omega) = \sum_{p}\frac{Lk_{B}T}{hv\pi}\frac{\pi^{2}}{3}K\sqrt{\frac{M}{c_{p}}}$

$C_{lattice}$ = $\frac{\pi L k_{B}}{3h}\sqrt{\frac{M}{c_{p}}}KT(2\sqrt{5}+1)$
So this right here is my final solution, and I'm trying to see how far off base I am.

Last edited: Mar 11, 2014
2. Mar 11, 2014

### maajdl

3. Mar 11, 2014

### torch000

That was a mistake and should've been $(e^{x}-1)$, and the expression for $C_{lattice}$ is from Kittel if youre wondering where I got it.