4. Generalize what we did in class for the 3d Debye solid for a two-dimensional
$L \times L$ square solid of $N$ atoms. For this, you can assume that there are
only two-independent polarization states for the phonons (and that the total
number of modes is $2N$ instead of $3N$). Both polarizations have the same
speed of sound $c_s$.
(a) Derive an expression for the Debye frequency cutoff $\omega_D$ expressed in
terms of $N$, $L$, and $c_s$.
(b) Derive an expression for the average energy $E$, expressed as an integral
over the dimensionless variable $x = \hbar \omega / \tau$.
(c) Evaluate the leading low-temperature ($\tau << \hbar \omega_D$) and high-temperature
($\tau >> \hbar \omega_D$) behavior of $E$ as a function of $\tau$.
