Calculation of electron-loss-to-continuum cusps: An algebraic approach

An algebraic approach for evaluation of the bound-free transition form factor $〈{\overrightarrow{\mathrm{v}}}^{-}\left|\exp \right(i\overrightarrow{\mathrm{Q}}·\overrightarrow{\mathrm{r}}\left)\right|n,l,m〉$ between arbitrary hydrogenic initial states $|n,l,m〉$ and continuum final states $|{\overrightarrow{\mathrm{v}}}^{-}〉$ in the low-velocity limit $v\ll \frac{Z}{n}$ is developed. This form factor determines the initial-state dependence of the electron-loss-to-continuum (ELC) cusp in the Born approximation. The method extends the well-known algebraic O(4,2) approach for bound-bound transitions to lowlying continuum states by exploiting the continuity across the ionization limit. A correspondence between scattering states and Rydberg bound states is established using the fact that the Runge-Lenz vector $\overrightarrow{\mathrm{A}}$ and the velocity vector $\overrightarrow{\mathrm{v}}$ become collinear near the ionization threshold. The present method takes explicitly into account the dynamical symmetry of the Coulomb field. We use the result for a systematic investigation of the doubly differential cross section and the shape of ELC cusps as a function of the initial state, its binding energy, the target, and the projectile velocity ($v_P$) within the Born approximation. Comparison is made with recent experimental data from our laboratory for highly charged projectiles. We find qualitative—and sometimes quantitative—agreement with the data.