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

An algebraic approach for evaluation of the bound-free transition form factor $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{-}}|\mathrm{exp}(i\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|n,l,m〉$ between arbitrary hydrogenic initial states $|n,l,m〉$ and continuum final states $|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{-}}〉$ in the low-velocity limit $v\ensuremath{\ll}\frac{Z}{n}$ 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 $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}$ and the velocity vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$ 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.