Relativistic Quantum Theory of Scattering on Arbitrary Electrostatic Potential and Stimulated Bremsstrahlung

Abstract

It is well known that in relativistic quantum theory there is not exact analytical formula for cross section of a charged particle elastic scattering on arbitrary electrostatic potential \(\varphi (\mathbf {r})\). Even in case of Coulomb field for which the exact solution of Dirac equation is known, relativistic “Coulomb wave functions”, nevertheless, because of its complex character it is impossible to obtain exact analytical expression for scattering cross section via these wave functions. The elastic scattering in relativistic domain is mainly described in approximations when the scattering potential can be considered as a perturbation (in opposite limits). These are the well-known Born and eikonal approximations corresponding to quantum perturbation theory by particle wave function (when the condition \(\left| U\right| \ll \hbar \mathrm {v}/a\) is satisfied; U is the potential energy, a is the space size of the range of effective scattering, \(\mathrm {v}\) is the particle initial velocity), and high-momentum approximation (if potential energy in the scattering field is much less than the particle initial energy: \(\left| U\right| \ll p\mathrm {v}\)), respectively. In case of Coulomb field there is also an approximation of large impact parameters (large momenta)—Farry-Sommerfeld-Maue (FSM) approximation , which describes well enough the scattering at the small angles. It is also known that the wave function in the eikonal approximation describes the particle state only in a limited space range of the scattering process (\(z\ll pa^{2}/\hbar \); z is the coordinate along the direction of particle initial momentum), i.e., the eikonal solution is not valid at large distances. The wave function of the Born approximation , in contrast to the eikonal one, describes the particle state at arbitrary point in the scattering range, particularly at asymptotically large distances. Nevertheless, the common region where both approximations under consideration are valid is very restricted. On the other hand, within the (small) potential range where both approximations are applicable, these wave functions describe the scattering by different accuracies. Practically, these wave functions describe the scattering at the opposite conditions: the eikonal wave function describes the particle state when the opposite condition of the Born approximation holds (e.g., for Coulomb field with a charge \(Z_{a}e\)—at the condition: \(Z_{a}e^{2}/\hbar \mathrm {v}\gtrsim 1\)). In these circumstances a natural question arises—is it possible to find out such an approximate solution of Dirac equation beyond the scope both perturbation theory and eikonal approximation—corresponding to more general wave function being applicable in both quantum and quasiclassical limits for relativistic potential scattering (including in particular the Born and eikonal approximations in corresponding limits—under its conditions of applicability)? Here we will try to derive such an approximate solution of the Dirac equation which satisfies the formulated requests. We will call it generalized eikonal approximation (GEA) for a spinor particle scattering on the arbitrary short-range or long-range electrostatic potential. Then we will clear up the relationship of this approximation to the known ones for electron elastic scattering on the short-range (Born and eikonal approximations) and long-range Coulomb potentials (Born, eikonal, and Farry-Sommerfeld-Maue approximations ). This GEA approximation is developed for electron inelastic scattering process too—to describe the relativistic stimulated bremsstrahlung in the field of strong and superstrong laser radiation with electrostatic potential fields of arbitrary form and finite or infinite effective radiuses (atoms, ions, etc.). The significance of GEA type wave function for description of such laser-assisted electron–atom–ion scattering processes, apart from the known restrictions of mentioned above approximations, is also conditioned with the fact that the wave functions in known approximations describe the particle state factorized by elastic scattering and induced radiation-absorption processes. In the result, we lose the phase relations at the description of electron quantum dynamics interacting with the both fields simultaneously, which have important role for induced process, in particular, for coherent part of interaction. Therefore, for description of strong and superstrong laser–matter (plasma) interaction processes, we need the electron wave function that takes into account the simultaneous influence of both scattering and radiation fields including dynamic phase relations. This induced GEA wave function may be applied specifically for the description of the above-threshold ionization (ATI) process of atoms/ions by superstrong laser fields in relativistic theory taking into account the photoelectron rescattering effect on the atomic remainder because of the action of long-range Coulomb field of atomic ion on the photoelectron final state (this process is considered in the next chapter of this book).