 Ideal Bose-gas heat capacity below condensation

For quadratic three-dimensional dispersion below condensation, the excited-state thermal energy is proportional to $T^{5/2}$. Differentiating at fixed particle number and volume gives the displayed heat capacity, with limiting value approximately $1.926N{k}_B$ at $T_c^{-}$. The macroscopic ground occupation carries no excitation energy in the shifted reference. The exponent differs from the linear-phonon result of an interacting gas.