Compton scattering of two x-ray photons by an atom

The process of inelastic nonresonant two x-ray free-electron laser (XFEL) photon Compton scattering by a free atom is theoretically investigated. The object of the study is the He atom. We obtain the absolute values and the shape of the double differential scattering cross section. The quantum effect of creation of “hot” scattered photons with maximum energy $2\hslash \omega -I_{1s}$ is predicted ($\hslash \omega$ is the energy of the incident XFEL photon on the atom, and $I_{1s}$ is the energy of the ionization threshold of an atomic $1s^2$ shell).

Figure 1

Partial probability amplitudes for the inelastic nonresonant Compton scattering process of two XFEL photons by a He atom in the Feynman diagram representation. Arrow right, electron $\left(x\equiv x{l}^{\prime },\varepsilon \equiv \varepsilon l\right)$; arrow left, vacancy $\left(1s\right)$. Filled circle, interaction vertex via the radiative transition operator ${\widehat{H}}_1$; open circle, interaction vertex via the contact transition operator ${\widehat{H}}_2$. $\omega$ $\left({\omega }_C\right)$ is the incident (scattered) photon. Time direction of the process is from left to right $\left(t_1<t_2\right)$. $\left(a,b,c,d\right)$; see Sec. 2 of the paper.

Figure 2

Double differential cross section for the inelastic nonresonant Compton scattering process of two linearly polarized (perpendicular to the scattering plane, $\perp )$ XFEL photons by the He atom in the approximation ${\mathrm{\Gamma }}_{1s}=0$. (a) Dependence of the scattering cross section on the incident photon energy with $\theta ={90}^{\circ }$. (b) Angular distribution of scattered photons with $\omega =500$ eV. ${\mathrm{\Gamma }}_{\text{beam}}=1.0$ eV, $I_{1s}=23.45$ eV. $\omega$ $\left({\omega }_C\right)$ is the energy of the incident (scattered) photon, and $\theta$ is the scattering angle.

Figure 3

Same as Fig. 2, but for ${\omega }_C\ge 650$ eV (the region of “hot” scattered photons). ${\omega }_C^{\text{max}}=2\omega -I_{1s},\omega =500$ eV.