Approach to electron capture into arbitrary principal shells of energetic projectiles

Abstract

An approach for treating electron capture into arbitrary principal shells or into continuum states of energetic projectiles is developed. The approach is based on the momentum-density matrix of the captured electron summed over all substates $l$, $m$ or integrated over all emission angles, respectively. In conjunction with the kinematics of energetic capture reactions it leads to drastic simplifications in capture theories. Within the eikonal approximation, the present approach yields an exact result for the cross section describing capture of hydrogenic $1s$ electrons into hydrogenic bound or continuum states of the projectile. The final result is a simple analytical expression factorizing into the Oppenheimer-Brinkman-Kramers cross section times a scaling factor between 0.1 and 0.4. The theoretical scaling factor for the total cross section turns out to be a function of $\frac{v}{v_K}$ (the ratio of projectile velocity to target $K$-shell velocity) which is approximately independent of the target and projectile charges. For hydrogen and helium targets surprisingly good agreement is obtained with a large body of experimental data.