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The band structure E_n(k) is the energy-versus-wavevector relationship for electrons in a crystal, with n the band index and k the crystal momentum in the Brillouin zone. The density of states g(E) counts the number of electronic states per unit energy: g(E) = sum_n integral [delta(E - E_n(k))] d3k / (2pi)3. Peaks in the density of states (van Hove singularities) occur where the gradient nabla_k E_n(k) vanishes — at band edges, saddle points, and flat regions. The density of states at the Fermi level, g(E_F), governs essentially all low-energy properties: electronic specific heat, Pauli paramagnetism, superconducting transition temperature, and transport.
The band structure E_n(k) is the complete map of allowed electron energies as a function of crystal momentum k within the Brillouin zone. Each band n is a continuous function of k, and the set of all bands determines virtually every electronic property of the material. Band structures are typically plotted along high-symmetry paths in the Brillouin zone (for example, Gamma-X-M-Gamma in a square lattice), which captures the essential features: band widths, gap sizes, band crossings, and flat regions.
While the band structure contains full information, many properties depend only on how many states exist at each energy — the density of states g(E). This is computed by integrating over the Brillouin zone: g(E) = sum_n integral delta(E - E_n(k)) d3k / (2pi)3. For free electrons in 3D, g(E) is proportional to sqrt(E), reflecting the growing surface area of the constant-energy sphere. In a real crystal, g(E) is dramatically modified: it vanishes in band gaps, shows peaks and kinks at van Hove singularities (where the gradient of E_n(k) vanishes), and can have sharp spikes from flat bands.
Van Hove singularities are topologically guaranteed. In any 3D band, the gradient must vanish at the band minimum, the band maximum, and at least two saddle points. At these energies, the density of states has non-analytic behavior — a step discontinuity at band edges in 3D, logarithmic divergences at saddle points, and true divergences in lower dimensions. These features have direct physical consequences: if a van Hove singularity lies near the Fermi level, the high density of states enhances the electronic specific heat, magnetic susceptibility, and the tendency toward instabilities (magnetic ordering, charge density waves, superconductivity).
The Fermi surface — the constant-energy surface E_n(k) = E_F in k-space — is the single most important geometric object in the physics of metals. At low temperatures, only electrons within ~k_BT of the Fermi surface can participate in transport, thermal, or magnetic processes. The shape of the Fermi surface determines anisotropic conductivity, the de Haas-van Alphen effect (oscillations of magnetization with magnetic field that directly map out the Fermi surface cross-sections), nesting conditions that drive charge and spin density waves, and the phase space for electron-phonon scattering. Experimental techniques like ARPES (angle-resolved photoemission spectroscopy) can now measure both the band structure and the Fermi surface with remarkable precision.