the free-electron model

The free-electron model makes a brazen simplification that turns out to work astonishingly well: treat the conduction electrons of a metal as a gas of free particles rattling around in a box, ignoring both the crystal's ions and the electrons' mutual repulsion. Strip away all the complexity of the lattice and, remarkably, you still capture many of a metal's essential properties. It is the physicist's first and most fruitful lie about metals.

In its quantum (Sommerfeld) form, each electron is a particle in a three-dimensional box with energy E = hbar^2 k^2 / (2m), a simple parabola in the wavevector k. Because electrons are fermions, the Pauli exclusion principle forbids two from sharing a state, so they stack up filling every level from the bottom to the Fermi energy E_F, with occupation set by the Fermi-Dirac distribution. From this alone follow real results: an electronic heat capacity that is linear in temperature (not the classical constant), the temperature-independent Pauli paramagnetism, and the Wiedemann-Franz law relating thermal and electrical conductivity. The density of states rises as the square root of energy, g(E) proportional to sqrt(E), in three dimensions.

The model is the baseline against which real metals are measured; adding a weak periodic potential upgrades it to the nearly-free-electron model, where gaps open at the Brillouin zone boundary and genuine bands appear. Its honesty lies in what it admits it ignores: the periodic lattice potential (the job of band theory) and electron-electron interactions. That it works at all, despite the strong Coulomb repulsion between electrons, is explained by screening and Landau's Fermi-liquid theory, in which the low-energy excitations behave like weakly interacting quasiparticles.