Constant-pressure gas versus electronic metal heat capacity ($C_{p,\mathrm{gas}}/C_{e,\mathrm{metal}}=5E_F/({\pi }^2{k}_{B}T)$)

For equal atom counts, a classical ideal monatomic gas has $C_p=5N{k}_B/2$. A monovalent three-dimensional free-electron metal has electronic contribution $C_e=({\pi }^2/2)N{k}_B{k}_{B}T/E_F$. Their ratio is the displayed expression. Replacing the gas's constant-pressure heat capacity with its $C_V=3N{k}_B/2$ changes the ratio by3/5. Neither ratio includes the metal's lattice heat capacity or gas nonideality.

An electron-only comparison omits the lattice contribution to a metal's measured heat capacity. In a simple low-temperature normal metal, the same sample normalization giveswhere the linear term is electronic heat capacity and the cubic term is the leading Debye model lattice contribution. Both coefficients must use compatible units and amounts of material.

If a reference gas has heat capacity $C_{\mathrm{gas}}$, its ratio to the total metal value isThe electron-only ratio is therefore reduced when lattice phonons contribute appreciably. The crossover estimate is $T=\sqrt{\gamma /\beta }$, where the two terms are equal. Superconductivity, magnetic excitations or other low-energy structure can add further terms. A ratio that is valid for electronic degrees of freedom alone must consequently be distinguished from a ratio of whole-sample calorimetric measurements.