Past exam of the physics A course of the University of Cambridge / 2012 / Paper 2 / Section C / C11

With periodic boundary conditions, each $k$ state occupies volume $(2\pi {)}^3/V$ and has two spin states. Filling a sphere gives $n=k_F^3/(3{\pi }^2)$ andHere $D$ is density of states per volume and per energy; the extensive $g(E)$ is $VD(E)$. Both spin states are already included.

For divalent magnesium, $n=8.6\times 10^{28}{\mathrm{m}}^{-3}$, givingAt $T\ll T_F$, only electrons within about $k_{B}T$ of the Fermi surface change occupancy. The Sommerfeld heat capacity isEquivalently $C_{e,m}=0.995T\mathrm{mJ}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, with $T$ in kelvin, so at room temperature it is about $0.299\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. A temperature was not specified for the estimate; the linear coefficient is the general result.

At sufficiently low temperature, an ordinary nonsuperconducting metal has $C\simeq \gamma T+\beta T^3$: electrons dominate at the lowest temperatures, acoustic phonons produce the cubic contribution, and the lattice approaches $3R$ per mole as temperature exceeds its characteristic Debye scale. Anharmonic expansion can distinguish measured $C_p$ from $C_V$. In the room-temperature regime, $T/T_F\simeq 0.0036$ since $T_F\simeq 8.24\times 10^4\mathrm{K}$, so electron excitation is strongly Pauli-limited. The free-electron estimate is only about $1.2\%$ of the classical $3R$ lattice scale. Calling it negligible means small relative to the lattice contribution, not identically zero or negligible as $T\to 0$. Real band structure and interactions can alter the coefficient.