= Solution
Different measurements reveal different aspects of <electronic band structure>; a convincing determination combines their constraints.
\b[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:
$$
E_B=h\nu-\Phi-E_{\rm kin},\qquad
k_\parallel=\frac{\sqrt{2m_eE_{\rm kin}}}{\hbar}\sin\theta.
$$
Here $\Phi$ 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>.
\b[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 $E_n(\mathbf k)$.
\b[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:
$$
F=\frac{\hbar}{2\pi e}A_{\rm ext}.
$$
Rotating the <magnetic field> maps the geometry of extremal orbits. The thermal <damping> determines the <cyclotron effective mass>,
$$
m_c=\frac{\hbar^2}{2\pi}\frac{\partial A}{\partial E},
$$
with the appropriate magnitude for a hole orbit. <Cyclotron resonance> supplies the corresponding dynamical mass through $\omega_c=eB/m_c$. 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.
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