这是一份warwick华威大学PX366的成功案例

When we also define a characteristic temperature, $\Theta_{\mathrm{E}}$, the Einstein-temperature, via
$$
\Theta_{\mathrm{E}} \equiv \frac{h v_{\mathrm{E}}}{k_{\mathrm{B}}}
$$
We get the following expressions for the molar energy of the crystal and it’s specific heat.
and
$$
\begin{gathered}
\widetilde{E}^{\mathrm{E}}=\frac{3}{2} R \Theta_{\mathrm{E}}+\frac{3 R \Theta_{\mathrm{E}}}{\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1} \
\widetilde{C}{V}^{\mathrm{E}}=\left(\frac{\partial \widetilde{E}^{\mathrm{E}}}{\partial T}\right){V}=3 R \frac{\left(\frac{\Theta_{\mathrm{E}}}{T}\right)^{2} \exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)}{\left[\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1\right]^{2}}
\end{gathered}
$$

PX366 COURSE NOTES ：

The density of frequencies, $N(v) \mathrm{d} v$, can be deduced from the density of states
$$
N(k) \mathrm{d} k=\frac{k^{2}}{2 \pi^{2}} V \mathrm{~d} k \quad \text { with } k=\frac{2 \pi}{\lambda} \quad \text { and } v \lambda=c
$$
Hence, with $k=\frac{2 \pi}{c} v$ and $\mathrm{d} k=\frac{2 \pi}{c} \mathrm{~d} v$
$$
N(v) \mathrm{d} v=\frac{4 \pi^{2} v^{2}}{2 \pi^{2} c^{2}} V \frac{2 \pi}{c} \mathrm{~d} v=\frac{4 \pi V v^{2}}{c^{3}} \mathrm{~d} v
$$