Spectral momentum density of graphite from (e,2e) spectroscopy: Comparison with first-principles calculation

We have measured the spectral momentum density ρ(E,q) of graphite by (e,2e) spectroscopy for momentum parallel and perpendicular to the crystal c axis. In the independent-electron approximation, ρ(E,q)=${\mathrm{\Sigma }}_{\mathrm{G}}$‖${\mathit{U}}_{\mathrm{k}}$(G)${\mathrm{\Vert }}^2$ δ(q-k-G)δ(E-E(k)) where the one-electron wave function is ${\mathrm{\Psi }}_{\mathrm{k}}$(r)=${\mathit{e}}^{\mathit{i}\mathrm{k}\mathrm{\cdot }\mathrm{r}}$${\mathrm{\Sigma }}_{\mathrm{G}}$${\mathit{U}}_{\mathrm{k}}$(G)${\mathit{e}}^{\mathit{i}\mathrm{G}\mathrm{\cdot }\mathrm{r}}$) and G is a reciprocal-lattice vector. The measurements covered a range of momentum parallel to the c axis equal to 0≤‖q‖≤1.84 A${̊}^{\mathrm{-}1}$ and a range of momentum perpendicular to the c axis equal to 0≤‖q‖≤2.35 A${̊}^{\mathrm{-}1}$. The energy range spanned the valence band of graphite from 4.4 eV above the Fermi energy to 27.6 eV below the Fermi energy. The momentum resolution was 0.47 and 0.73 A${̊}^{\mathrm{-}1}$ (full width at half maximum) for momentum parallel and perpendicular to the c axis, respectively. The energy resolution was 8.6 eV. The maximum coincidence rate was ∼0.02 counts/sec. The band structure E(k) and spectral density ‖$U_{\mathrm{k}}$(G)${\Vert }^2$ have been calculated from first principles using a self-consistent density-functional theory in the local-density approximation with a mixed-basis pseudopotential technique. The agreement within experimental uncertainties between measurement and theory is excellent.