Valence-band Auger line shapes for Si surfaces: Simplified theory and corrected numerical results

A reduced version of the independent-electron theory of Auger line shapes is given that takes advantage of the general smallness of those components of a solid's local density of states (LDOS) matrix which are off-diagonal in angular momentum. Thus we present approximate formulae which relate the valence-band Auger line shapes of a soslid simply to its angular-momentum projected LDOS near a surface. Using these simplified formulae, we have located and corrected a number of numerical errors in our previous calculations of the Auger line shapes of Si. We present corrected results here, showing that the independent-electron theory, coupled with the use of atomic Auger matrix elements for Si, actually predicts too large a probability of $L_{2,3}{M}_1{M}_{2,3}$ decay relative to that for $L_{2,3}{M}_{2,3}{M}_{2,3}$ decay to give good agreement with measured Si $L_{2,3}\mathrm{VV}$ Auger line shapes. Possible reasons for the experimental smallness of the $L_{2,3}{M}_1{M}_{2,3}$ Auger decay rate are discussed. The Si $L_1{L}_{2,3}V$ Coster-Kronig line is also recalculated and shown to be in correspondence with the simplified theory of Auger transitions.