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Only states within an energy shell of width roughly $k_{B}T$ around $E_F$ appreciably change occupation at low temperature. There are roughly $g(E_F)k_{B}T$ such states and each excitation costs energy of order $k_{B}T$. Thus $\Delta U\propto g(E_F)(k_{B}T{)}^2$, giving $C_{el}\propto g(E_F)k_B^{2}T$. This is the thermal-shell estimate of electronic heat capacity; filled states far below $E_F$ are blocked by the Pauli exclusion principle. A smooth DOS and fixed particle number give the more precise Sommerfeld result $C_{el}=({\pi }^2/3)g(E_F)k_B^{2}T$ to leading order.

Only a shell of energy width of order $k_{B}T$ near the Fermi energy can change occupation: much lower states are blocked by the Pauli exclusion principle. The thermal-shell estimate of electronic heat capacity countsTo recover the rough coefficient requested in the paper, assign each of these active particles an excitation energy of order $3k_{B}T/2$. ThenThe differentiation matters: the active shell itself widens as temperature rises. This argument explains the small linear-in-$T$ electronic heat capacity; its shell width and average excitation energy are heuristic, so it does not establish a precise numerical prefactor.

A controlled calculation from the Fermi-Dirac distribution gives a nearby but different result. At fixed particle number, the Sommerfeld expansion yieldsThus $9/2$ is a rough estimate of the exact free-electron coefficient ${\pi }^2/2\simeq 4.935$, not an exact consequence of the stated density of states. Both require $T\ll T_F$.