Excitons in large-gap insulators: Solid argon

The theory of excitons for the range intermediate between the Frenkel-Peierls and the Wannier-Mott models is formulated in terms of the band structure $E_n\left(\overrightarrow{\mathrm{k}}\right)$ and the Coulomb and exchange integrals involving the Wannier functions of the valence and conduction bands. When the band structure is that appropriate to solid rare gases and alkali halides, the problem simplifies greatly if the electron and the hole can be assumed to be confined to the same unit cell. Taking full account of the symmetry, the problem in this case reduces to the diagonalization of a simple second-order Fredholm determinant. Numerical calculations are performed for the lowest transverse and longitudinal exciton states of solid argon. The Wannier function for the valence bands is replaced by the $3p$ atomic orbitals, and that for the conduction band is computed by approximating the Bloch functions with an orthogonalized plane wave. Very good agreement with experimental results is obtained for the lowest exciton doublet. The doublet splitting and the relative intensities of the two peaks are computed in terms of the ratio of the exchange integrals to the spin-orbit splitting of the valence bands. The conclusions are extended to the other solid rare gases.