Free-free electron scattering from a model atom: Comparison of one- versus Two-photon absorption and Test of the Soft-photon Approximation

We have calculated the cross sections for one- and two-photon absorption by an electron scattering from the Kr\"uger-Schulz model atom in the presence of a low-frequency (linearly or circularly polarized) laser. Let $\ensuremath{\xi}\ensuremath{\equiv}\frac{e{A}_{0}(\stackrel{^}{\ensuremath{\epsilon}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}})}{(mc\ensuremath{\omega})}$, where ${A}_{0}$, $\stackrel{^}{\ensuremath{\epsilon}}$ and $\ensuremath{\omega}$ are the amplitude, polarization, and frequency of the laser field, and where $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}$ is the momentum transfer to the electron of mass $m$ and charge $e$. If $|\ensuremath{\xi}|\ensuremath{\lesssim}1$, one-photon dominates over two-photon absorption except perhaps over some range of angles, and except over a range of energies near a resonance. We examined in detail the applicability of the soft-photon approximation.