The Einstein Model

3.8 The Einstein Model

For \(\omega = \omega_0\) we find for the inner energy

\begin{equation*} U = 3 \int d\omega D(\omega) \frac{\hbar \omega}{\exp\left(\frac{\hbar \omega}{k T}\right)-1} = \frac{\hbar \omega_0}{\exp\left(\frac{\hbar \omega_0}{k T}\right)-1} 3 N \qquad . \end{equation*}

(3.43)

and:

\begin{equation*} C_V = \frac{dU}{dT}= \frac{3 N \hbar \omega}{\left(e^{\frac{\hbar \omega}{k T}} - 1 \right)^2} \frac{e^{\frac{\hbar \omega}{k T}} \hbar \omega}{k T} \qquad . \end{equation*}

(3.44)

For high temperatures we again get:

\begin{equation*} C_V = 3 N k \qquad . \end{equation*}

(3.45)