Different measurements reveal different aspects of electronic band structure; a convincing determination combines their constraints.

Angle-resolved photoemission. In angle-resolved photoemission spectroscopy, photons of known energy eject occupied Electrons. Measuring their kinetic energy and emission angle reconstructs binding energy and the component of crystal momentum parallel to the surface:

E_{B} = h ν - Φ - E_{kin}, k_{∥} = \frac{2 m _{e} E _{kin}}{ℏ} sin θ .

Here

Φ

is the sample work function when kinetic energy is referenced to the nearby vacuum; an analyzer energy scale requires its own work-function calibration. Momentum is conserved modulo a surface reciprocal lattice vector, neglecting photon momentum. Tracking the peaks maps occupied dispersion relations and the Fermi surface. The measured intensity also contains the matrix element, the Fermi-Dirac distribution and the spectral function, so an absent bright feature does not establish an absent band. For a three-dimensional solid, the perpendicular momentum requires a final-state model. Inverse photoemission spectroscopy probes unoccupied states by injecting Electrons and detecting emitted photons.

Optical spectroscopy. Photon absorption and ellipsometry measure interband transitions, optical selection rules and the complex dielectric response. Because optical photon momentum is small on the scale of the Brillouin zone, direct transitions approximately preserve crystal momentum. Their onset constrains a direct band gap; a threshold involving a phonon can reveal an indirect band gap. Peak positions constrain energy differences between bands, but the response also depends on the joint density of states and optical matrix elements, so it does not uniquely determine every

E_{n} (k)

Quantum oscillations and cyclotron resonance. In clean samples at low temperature, the de Haas-van Alphen effect measures oscillations in magnetization and the Shubnikov-de Haas effect measures oscillations in electrical resistivity as the inverse magnetic field is varied. Their period in inverse magnetic field is

1/ F

, where Onsager quantization relates

F

to an extremal Fermi-surface area perpendicular to the field:

F = \frac{ℏ}{2 π e} A_{ext} .

(2)

Rotating the magnetic field maps the geometry of extremal orbits. The thermal damping determines the cyclotron effective mass,

m_{c} = \frac{ℏ ^{2}}{2 π} \frac{\partial A}{\partial E},

(3)

with the appropriate magnitude for a hole orbit. Cyclotron resonance supplies the corresponding dynamical mass through

ω_{c} = e B / m_{c}

. These are detailed probes near the Fermi level, rather than measurements of all occupied bands. Diffraction supplies the reciprocal lattice needed to label the measured momenta, but diffraction by itself is not a measurement of electronic dispersion.