Intermediate coupling in ${\mathit{L}}_2$-${\mathit{L}}_3$ core excitons of MgO, ${\mathrm{Al}}_2$${\mathrm{O}}_3$, and ${\mathrm{SiO}}_2$

The ${\mathit{L}}_{23}$ reflection spectra for single-crystal MgO, α-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$, and α-${\mathrm{SiO}}_2$ are measured. These reflection spectra are used to generate absorption spectra that agree well with electron-total-yield and electron-energy-loss measurements. Exciton features are shown to exist in the band gap by comparing the valence-band maximum plus optical band gap to the observed transition energies. The energy splitting of the ${\mathit{L}}_{2\mathrm{-}}$ ${\mathit{L}}_3$ excitons, as seen in reflection, is clearly resolved for ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ and ${\mathrm{SiO}}_2$, while in electron-total-yield measurements the doublets are not resolved for ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$. This resolution allows the calculation of the exchange energy using intermediate-coupling theory.