Electronic heat capacity coefficient ($\gamma ={\lim }_{T\to 0}{C}_{\mathrm{el}}/T$)

For a Fermi liquid, $C_{\mathrm{el}}=\gamma T+o(T)$ at low temperature. In a simple quasiparticle description, $\gamma =({\pi }^2/3)k_B^{2}g(E_F)$, where the density of states includes both spin states and uses the same per-volume or per-mole normalization as $\gamma$. A fit $C/T=\gamma +\beta T^2$ separates this contribution from a Debye lattice term over an appropriate low-temperature range. Magnetic or other low-energy excitations can require extra fit terms.

For an isotropic three-dimensional liquid at fixed particles per mole, the molar coefficient scales as ${\gamma }_{\mathrm{mol}}\propto m^{\ast }{V}_{\mathrm{mol}}^{2/3}$. Density and mass renormalization both affect the slope; the slope ratio is not simply the effective-mass ratio.

Only single-particle states within order $k_{B}T$ of the Fermi energy can change occupation at low temperature. Their number is order $g(E_F)k_{B}T$, and their excitation energy is order $k_{B}T$, giving $U-U_0\propto T^2$ and electronic heat capacity proportional to $T$.