Debye Approximation of Heat Capacity in 1D

[jkthejetplane]

Homework Statement

Using the Debye approximation, illustrate how the phonon heat capacity changes with
respect to temperature in 1D. Discuss your results in the low and high temperature limit
respectively.

Relevant Equations

equations of Cv from book in 3d listed below

So really i am just unsure how to answer the last part of the question. I am unsure how to apply the low and high temperature limits the way i have done it. Do i set upper/lower limits on the integral and solve? If so i am not sure what to put

Here is what he book has for 3d

 

[pai535]

After you change your variable, the upper limit of integral will be $\hbar\omega_D/k_B T$. Then put $T\to 0$ so $\hbar\omega_D/k_B T \to \infty$ and integrate.

For high-temperature behaviour, $T \to \infty$, $\hbar\omega/k_B T \to 0$. In this case maybe you don't change your variable and keep the original $\hbar\omega/k_B T$, then $\exp(\hbar\omega/k_B T)\to 1$. Then integrate. (in denominator of integral maybe can perform a Taylor expansion)