X-ray-atom scattering in the presence of a laser field

We consider x-ray-hydrogen atom scattering in the presence of a monochromatic linearly polarized laser field. The S matrix of this process is presented and an expression for the differential cross section (DCS) is derived. We show that the time-dependent Wentzel-Kramers-Brillouin approximation can be applied to the present problem. The presented numerical results for the DCS as a function of the number n of photons exchanged with the laser field show a characteristic behavior. The number n can only be even. For n=0,±2 we have pronounced maxima in the DCS, followed by sharp minima at n=±4. After that, we have a plateau that is different for negative and positive values of n. The plateau for negative values of n is much more extended than for the positive ones. The structure of the plateau and the positions of the minima and maxima of the DCS do not depend on the laser field intensity, while the height of the plateaus strongly depends on it. We also analyze the dependence of the matrix elements of the x-ray spectra on the incident x-ray photon energies by using both the saddle-point method and the numerical evaluation, and we show that the shape and position of the plateau are determined by the simple relation nω=I₀-ωK. This condition connects the number of absorbed or emitted laser field photons n, the laser field photon energy ω, the atomic ionization potential I₀, and the energy of the incident x-ray photon ωK.