Theory of two-electron rearrangementKx-ray transitions

A two-configuration interaction model is developed to account for the branching ratios of the $K{L}_{2,3}^n$ radiative electron rearrangement ($K{L}^n$ RER) x-ray transitions. The mixing between the quasidegenerate $2p^{-m}$ and $2s^{-2\mathrm{}}2p^{-m+2\mathrm{}}$ $L$-shell configurations is considered both for the initial ($m=n$) and final ($m=n+1\mathrm{}$) states. The branching ratio is shown to be proportional to the square of the $2p^{-n-1\mathrm{}}$ mixing coefficient in the final $2s^{-2\mathrm{}}2p^{-n+1\mathrm{}}$ states for any number of initial $2p$ vacancies. In cases where both initial- and final-state mixing occurs, the corresponding amplitudes are shown to add destructively. Branching ratios are calculated in $\mathrm{LS}$ coupling for $Z$ varying from 10 to 30 by using the multiconfiguration Hartree-Fock method. Calculated ratios are in close accord with available experimental data for Mg, Al, and Si. The validity of the two-configuration interaction model is examined, and it is found that the $K{L}^n$ RER transitions are almost entirely due to the $2s-2p$ quasidegeneracy. The $2s\to 1s$ shake down which has recently been found to be important for two-electron, one-photon $K^2-L^2$ x-ray transitions is shown to have a negligible influence on the $K{L}^n$ RER transitions. The large widths of the $K{L}^n$ RER lines are attributed to the short lifetime of the $2s^{-2\mathrm{}}2p^{-n+1\mathrm{}}$ final states which decay by $L_1^2-L_1{L}_{2,3}M$ Coster-Kronig transitions.