= Solution
The <Fermi-Dirac distribution> gives the mean occupancy of a single <fermion> state:
$$
\boxed{f(\epsilon)=\frac1{e^{(\epsilon-\mu)/(k_BT)}+1}}.
$$
Its maximum occupancy of one expresses the <Pauli exclusion principle>. A spatial orbital with two <spin> states can hold two <electrons> because they are different single-particle states. At zero <temperature>, the occupation is a step at the <Fermi energy> $E_F$; at low nonzero <temperature>, occupation changes appreciably only in an <energy> shell of width of order $k_BT$ about the <chemical potential>.
Consequently, most conduction <electrons> cannot absorb small thermal <energies> into already occupied states. Only a fraction of order $T/T_F$ participates, where $T_F=E_F/k_B$. For a normal <metal> with a smooth <density of states> $g(E_F)$, including both <spin> states and using the total-system normalization, the <Sommerfeld expansion> at fixed <particle number> gives
$$
U(T)-U(0)=\frac{\pi^2}{6}g(E_F)(k_BT)^2+O(T^4),
$$
and therefore
$$
\boxed{C_{\rm el}=\gamma_{\rm el}T,\qquad
\gamma_{\rm el}=\frac{\pi^2}{3}k_B^2g(E_F)}.
$$
For a three-dimensional <free electron gas>, $g(E_F)=3N/(2E_F)$, so $C_{\rm el}=(\pi^2/2)Nk_B(T/T_F)$. This is much smaller than a classical equipartition estimate at $T\ll T_F$. The <chemical potential> must adjust with <temperature> to maintain $N$; treating it as fixed while differentiating an <energy> <integral> gives the wrong fixed-$N$ calculation in general.
The lattice adds a Debye contribution $C_{\rm ph}=\beta_D T^3$ at <temperatures> well below the <Debye temperature>. Hence
$$
\boxed{\frac CT\simeq\gamma_{\rm el}+\beta_D T^2}.
$$
A plot of measured $C/T$ against $T^2$ can separate the electronic intercept from the <phonon> slope. <calorimetry> must account for the <heat capacity> of the sample holder, heat leaks, thermal equilibration and <thermometer> calibration. A magnetic impurity contribution or a superconducting transition can invalidate this two-term fit; the prediction is for the normal metallic state in an appropriate low-temperature interval. Differences from a free-electron coefficient can reveal <quasiparticle> effective-mass enhancement or nontrivial electronic structure rather than a failure of Fermi statistics.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2015/paper-1-quantum-gas-statistics.png]
{title=Original ideal-box condensate fraction and Fermi-Dirac occupation curves}
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