Equal-degeneracy two-level energy and heat capacity ($C=N{k}_B{x}^2{e}^x/(1+e^x{)}^2$)

For N independent units with nondegenerate levels 0 and $\Delta >0$, the canonical partition function is $(1+e^{-\beta \Delta }{)}^N$. Its energy is $N\Delta /(1+e^{\beta \Delta })$ and heat capacity is $N{k}_B{x}^2{e}^x/(1+e^x{)}^2$, with $x=\Delta /(k_{B}T)$. Both are activated at low temperature. At high temperature the energy saturates at $N\Delta /2$ and heat capacity decays as $T^{-2}$. The equal-degeneracy Schottky peak is at $x\simeq 2.39936$ and height $0.43923N{k}_B$. A common energy shift leaves heat capacity unchanged; unequal level degeneracies require a different partition function.