Heat-capacity activation estimate ($\Delta /k_B=\ln [C(T_2)/C(T_1)]/(T_1^{-1}-T_2^{-1})$)

For gapped excitations with $C(T)\simeq A\exp [-\Delta /(k_{B}T)]$ and approximately constant prefactor, two heat capacity measurements at $T_2>T_1$ give the displayed activation-energy estimate. A more general prefactor $A{T}^p$ changes its numerator to $\ln [C(T_2)/C(T_1)]-p\ln (T_2/T_1)$. The two-point estimate measures an effective exponential scale; it does not independently determine the prefactor or establish a detailed excitation density of states.