Einstein Model

The Einstein model assumes that all vibrational modes in a solid lattice have the same frequency $\omega_E$ and, therefore,

$$g(\omega) = DN \delta (\omega - \omega_E).$$ (183)

Substituting Eq. (183) into Eq. (182) we obtain

$$Z = -ND \Bigg (e^{+\beta \hbar \omega_E /2} -e^{-\beta \hbar \omega_E /2} \Bigg).$$ (184)

The average internal energy of the lattice can be computed from Eq. (184) as follows,

$$E = \frac{\partial \text{ln} Z}{\partial (-\beta)} = +DN \frac{\hbar \omega_E}{2} \text{coth} (\beta \hbar \omega_E/2).$$ (185)

Note that in the high temperature limit,

$$\lim_{\beta \rightarrow 0} E = \frac{DN}{\beta} = DN k T.$$ (186)

The heat capacity at constant volume $C_v$ can also be obtained from Eq. (184) as follows,

$$C_v = \frac{\partial E}{\partial T} \Bigg )_v = \Bigg (-\frac{ND}......igg)}{\Bigg(e^{\beta \hbar \omega_E /2}-e^{-\beta \hbar \omega_E /2} \Bigg)^2}.$$ (187)

The expression introduced by Eq. (187) can be simplified to obtain

$$C_v = \Bigg( \frac{\theta}{T} \Bigg)^2 \frac{ND k}{\Bigg(e^{-\fra......T} \Bigg)^2 \frac{e^{\frac{\theta}{T}}}{\Bigg(e^{\frac{\theta}{T}}-1 \Bigg)^2},$$ (188)

with $\theta \equiv \frac{\hbar \omega_E}{k}$.

Limiting cases (i) At high temperature, $\theta << T$ and $e^{\frac{\theta}{T}} \approx 1 + \frac{\theta}{T}$. Therefore,

$$C_v= ND k \Bigg( \frac{\theta}{T} \Bigg)^2 \frac{1+\frac{\theta}{T} + ...}{(1+\frac{\theta}{T}-1 + ...)^2} = ND k.$$ (189)

Eq. (189) is the Dulong-Petit's law --i.e., the experimental limit for $C_v$ at high temperature. (ii). At low temperature, $\theta >> T$. Therefore,

$$C_v= ND k \Bigg( \frac{\theta}{T} \Bigg)^2 e^{-\frac{\theta}{T}}.$$ (190)

It is important to note that Eq. (190) does not predict the correct limit for $C_v$ at low temperature, since $C_v$ should be proportional to $T^3$ when $T \rightarrow 0$.