Absolute differential bremsstrahlung cross sections for 0.4–2-keV electrons scattered by Ar, Kr, and Xe atoms

Absolute cross sections of ultra-soft x-ray radiation in electron scattering upon atoms and the soft-photons approximation

“…In equation ( 2) averaging over the electron spins in the initial state, summation over its spins in the final state and summation over the photon polarizations are carried out. If the direction of flight of the electrons is not fixed as in the experiment [16,17], then it is necessary to perform integration over this direction in the equation (2) . To do this, we direct the axis z along the vector v and write dσ S as…”

“…where Θ is the polar angle of the vector k (the angle between the vectors v and k), the azimuthal angle of the vector k is assumed to be zero, ϕ is the azimuthal angle of the vector v . In experiments [16,17] the angle Θ 83 • . After integration by angle ϕ we obtain,…”

“…Since, as already noted, the greatest discrepancy between the experimental data presented in [5,9] and the theory is exists in the small photon energies region, the data obtained in Refs. [16,17] are of particular interest. These works present results of absolute, not relative, as in [1], measurements of BS cross sections values for ultrasoft x-ray photons (λ = 7..18 nm) generated in scattering of electrons with energies of 0.4..2 keV on Ar, Kr, and Xe atoms.…”

“…The experimental data differ by a factor 3..4 from the calculations of Pratt et al [10,11], carried out with an accurate account of the phases of the electron wave functions in the static atomic potential, but with the radial dependence of these functions that corresponds to the free motion of the scattered electron. Note, however, that the papers [16,17] present BS data for electrons of intermediate energies, where the Born approximation is invalid. Let us remind that the first Born approximation is sufficient only if the speed of incoming electron is much higher than the average speed of all atomic electrons, including the innermost ones.…”

“…The important results reported in Refs. [16,17], have no theoretical explanation yet. In this paper, some of these results, namely, the values of BS cross sections, including the limiting extrapolated values at infinitely large wavelengths of BS photons, we interpret on the basis of the soft-photon approximation (SPA) of quantum electrodynamics [3],…”