Schottky anomaly with unequal degeneracies ($C_V/(N{k}_{\mathrm{B}})=x^2{g}_0{g}_1{e}^{-x}/(g_0+g_1{e}^{-x}{)}^2$)

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 $q=g_0+g_1{e}^{-x}$, with $x=\Delta /(k_{\mathrm{B}}T)$. Differentiation givesThe 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 $N{k}_{\mathrm{B}}\ln g_0$ in the ideal model.