= Solution
<Angle-resolved photoemission spectroscopy> illuminates a surface with <photons> of known <energy> and measures the <kinetic energy> and emission direction of the escaping <electrons>. Referencing the initial binding <energy> to the <Fermi level> gives
$$
\boxed{E_B=h\nu_{\rm light}-\Phi-E_{\rm kin},\qquad
k_\parallel=\frac{\sqrt{2m_eE_{\rm kin}}}{\hbar}\sin\theta.}
$$
Here $\Phi$ is the sample/analyser work-function reference appropriate to the calibrated <energy> scale, and $\theta$ is the emission angle from the surface normal. Parallel <crystal momentum> is conserved modulo a surface <reciprocal lattice> vector; measuring two angular coordinates maps the two in-plane <momentum> components.
Sweeping angle and <energy> traces the occupied spectral weight and its dispersion. In a quasi-two-dimensional sheet, no detailed free-<electron> model of the perpendicular final-state <momentum> is required to map the in-plane bands. Peaks reflect the occupied single-particle <spectral function>, multiplied by Fermi occupation and photoemission <matrix> elements and broadened by finite lifetimes and resolution. Ordinary photoemission does not directly measure empty conduction bands; a gap may need complementary information. A clean surface, vacuum and charge control are especially important for an insulating specimen.
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