= Schottky anomaly with unequal degeneracies
{c}
{title2=$C_V/(Nk_{\rm B})=x^2g_0g_1e^{-x}/(g_0+g_1e^{-x})^2$}
For independent units with lower and upper <energy levels> separated by $\Delta>0$, let their degeneracies be $g_0$ and $g_1$. Their shifted <partition function> is $q=g_0+g_1e^{-x}$, with $x=\Delta/(k_{\rm B}T)$. Differentiation gives
$$
\frac{C_V}{Nk_{\rm B}}=x^2\frac{g_0g_1e^{-x}}{(g_0+g_1e^{-x})^2}.
$$
The <heat capacity> vanishes at both low and high <temperature>. Degeneracy changes both the maximum and its <temperature>; the equal-degeneracy result is not universal. Independent lower-level degeneracy also leaves <residual entropy> $Nk_{\rm B}\ln g_0$ in the ideal model.
Back to article page
