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The nearly free electron model treats the crystal potential as a weak perturbation on free electrons. Free electron energy parabolas E = ħ^2k2/2m, when folded into the first Brillouin zone, cross at zone boundaries. The periodic potential lifts these degeneracies through Bragg scattering, opening energy gaps of magnitude 2|V_G| at each zone boundary, where V_G is the Fourier component of the potential at reciprocal lattice vector G. This model explains the origin of band gaps from first principles and shows that even a weak potential qualitatively changes the electronic structure from continuous to banded.
The nearly free electron model asks: what happens to free electrons when you turn on a weak periodic potential? For truly free electrons, the energy is a simple parabola E = hbar2 k2 / 2m, and there are no gaps — every energy is allowed. But when this parabola is "folded" into the first Brillouin zone (by shifting k by reciprocal lattice vectors G), the parabolas from different zones overlap and cross. At the crossing points, which occur at zone boundaries where k = G/2, two plane wave states are degenerate.
The periodic potential V(r) = sum_G V_G eiG·r lifts these degeneracies through Bragg scattering. Near a zone boundary, the states eikr and ei(k-G)r are nearly degenerate and are strongly mixed by V_G. Degenerate perturbation theory gives two new eigenstates — standing waves that are symmetric and antisymmetric combinations — with energies split by 2|V_G|. The symmetric standing wave (cos type) piles charge density on the ion cores where the potential is most attractive, lowering its energy. The antisymmetric one (sin type) piles charge between ions, raising its energy. The energy difference is the band gap.
The size of each gap is controlled by the Fourier component V_G of the potential at the corresponding reciprocal lattice vector. This is physically sensible: if the potential has a strong component at wavevector G, the electrons at the corresponding zone boundary scatter strongly and the gap is large. If V_G is small, the gap is small and the band structure looks nearly free-electron-like. This is why the model works well for simple metals like sodium, potassium, and aluminum, where the valence electrons are delocalized s/p electrons that see a weak effective potential (screened by other electrons).
The nearly free electron model provides the clearest picture of how band gaps arise and why some materials are metals while others are insulators. A metal has a Fermi energy that falls within a band (partially filled states available for conduction). An insulator has a Fermi energy in a gap (no states available at the Fermi level). Whether the bands are partially or completely filled depends on the electron count per unit cell, the gap sizes, and the Brillouin zone geometry. This model is complementary to the tight-binding approach: NFE starts from delocalized electrons and adds a weak lattice, while tight-binding starts from localized atomic orbitals and adds inter-atomic hopping. Real band structures interpolate between these limits.