# Schottky anomaly in the heat capacity $C_{V}(T)$ of a 2-level system

Why does the Schottky anomaly in the heat capacity ($$C_{V}(T)$$ against $$T$$) associated with a 2-level system with energy spacing $$E$$ appear at $$k_{B}T\approx\frac{E}{2}$$ (a precise calculation gives $$k_{B}T=0.4165 E$$) and not at $$k_{B}T\approx E$$ where the thermal energy is precisely equal to the level spacing, which would allow for the most efficient thermal excitation?

I am looking for an intuitive explanation, as I understand the mathematical framework of calculating the temperature dependence of the heat capacity and no further information is provided by standard statistical mechanics/thermodynamics textbooks...