Einstein high-temperature heat-capacity expansion ($C_V=3N{k}_B[1-T_E^2/(12T^2)+O(T^{-4})]$)

Expanding the Einstein model of a solid internal energy gives $U=3N{k}_B[T+T_E^2/(12T)+O(T^{-3})]$, because the constant term in the thermal occupation cancels the zero-point energy. Differentiation yields the displayed heat capacity and proves absence of a $1/T$ term. The ratio of the $T^{-2}$ and constant coefficients determines $T_E$ independently of sample size.