The metal puzzle, and the free-electron sea
Drude's classical picture — electrons rattling like billiard balls through the ions — successfully explained Ohm's law and roughly gave the ratio of thermal to electrical conductivity. But it made a disastrous prediction: if electrons were a classical gas, equipartition would make them contribute an extra \tfrac32 k_B per electron to the heat capacity. Experiment sees almost none of it. The fix is the same quantum idea as for phonons, applied to fermions.
The free-electron (Sommerfeld) model keeps Drude's boldest idealization — treat the valence electrons as a gas of non-interacting particles in a box, ignoring both the ion cores and the electrons' mutual repulsion — but obeys quantum statistics. Confined to a box of side L, each electron occupies a standing-wave state labelled by a wavevector \mathbf{k}, with the free-particle energy E=\hbar^2k^2/2m.
The free-electron dispersion: energy grows as the square of the wavevector.
Filling the sea: the Fermi energy and Fermi surface
Electrons are fermions, so the Pauli exclusion principle forbids two of them (of the same spin) from sharing a state. At absolute zero the electrons cannot all crowd into the lowest level; they stack up, filling every \mathbf{k}-state from zero up to a sharp cutoff. In k-space the occupied states fill a sphere — the Fermi sphere — whose surface is the Fermi surface, and whose energy is the Fermi energy E_F. For a density n of electrons the answer follows purely from counting states.
The Fermi energy of a free-electron gas, set entirely by the electron density n.
Put in numbers for a typical metal (n \sim 10^{29}\,\mathrm{m^{-3}}) and E_F comes out at a few electronvolts. Divided by k_B, that is a Fermi temperature of 10^4–10^5 K — far above room temperature. The crucial consequence: at any ordinary temperature the electron gas is highly degenerate (T \ll T_F); it is nothing like a hot classical gas. The idealizations to keep honest — no ion potential, no electron-electron interaction — are severe, yet the model works startlingly well because Pauli exclusion dominates the energetics.
Fermi-Dirac occupancy and the density of states
At finite temperature the sharp cutoff softens. The probability that a state of energy E is occupied is the Fermi-Dirac distribution: a step at the chemical potential \mu \approx E_F, smeared over a width of only \sim k_BT. Combined with the density of states g(E) — how many states lie in each energy interval, which for free electrons in 3D grows like \sqrt{E} — it determines every thermal and transport property of the metal.
The Fermi-Dirac distribution: a step at μ, thermally smeared over ~k_BT.
The puzzle resolved: heat capacity and conduction
Because only a fraction \sim T/T_F of the electrons can absorb thermal energy, the electronic heat capacity is not the classical \tfrac32 Nk_B but a tiny amount linear in T. This is why metals' electrons stay almost invisible in calorimetry, and why the total low-temperature heat capacity of a metal splits neatly into an electronic \gamma T plus the phonon \beta T^3 of Guide 2.
The electronic heat capacity is linear in T and small by the factor T/T_F ≪ 1.
Conduction is likewise a Fermi-surface story: an applied field displaces the whole Fermi sphere slightly in k-space, and the net imbalance of electrons near the surface carries the current. What limits it? Not the ordered lattice itself — a perfect periodic crystal offers no resistance, the surprising result we prove in Guide 4 — but scattering off phonons (rising with temperature) and off defects (the residual low-T resistance). The Hall effect then reveals the sign and density of the carriers, confirming the whole picture. Next we restore the ingredient the free-electron model threw away — the periodic potential — and discover why some solids don't conduct at all.