Dressing effects in the laser-assisted ($e,2e$) process in fast-electron–hydrogen-atom collisions in an asymmetric coplanar scattering geometry

We present the theoretical treatment of laser-assisted $(e,2e)$ ionizing collisions in hydrogen for fast electrons, in the framework of the first-order Born approximation at moderate laser intensities and photon energies beyond the soft-photon approximation. The interaction of the laser field with the incident, scattered, and ejected electrons is treated nonperturbatively by using Gordon-Volkov wave functions, while the atomic dressing is treated by using first-order perturbation theory. Within this semiperturbative formalism, we obtain a closed-form formula for the nonlinear triple differential cross section (TDCS), which is valid for linear as well circular polarizations. Analytical simple expressions of TDCS are derived in the weak field domain and low-photon energy limit. It was found that for nonresonant $(e,2e)$ reactions, the analytical formulas obtained for the atomic matrix element in the low-photon energy limit give a good agreement, qualitative and quantitative, with the numerical semiperturbative model calculations. We study the influence of the photon energy as well of the kinetic energy of the ejected electron on the TDCS, in the asymmetric coplanar geometry, and show that the dressing of the atomic target strongly influences the $(e,2e)$ ionization process. We discuss the exchange effects between the ejected and scattered electrons in the TDCS.

Figure 1

Schematic representation of the scattering geometry for the $(e,2e)$ reaction. The incident electron has energy $E_i$ and momentum ${\mathbf{k}}_{\mathbf{i}}$, while the scattered electron has energy $E_f$ and momentum ${\mathbf{k}}_{\mathbf{f}}$, and is detected at a fixed scattering angle ${\theta }_f$. The ejected electron has energy $E_e$ and momentum ${\mathbf{k}}_{\mathbf{e}}$, and its detection angle ${\theta }_e$ is varied. $\mathrm{\Delta }={\mathbf{k}}_{\mathbf{i}}-{\mathbf{k}}_{\mathbf{f}}$ denotes the momentum transfer vector from the incident to the scattered electron, and $\mathbf{q}=\mathrm{\Delta }-{\mathbf{k}}_{\mathbf{e}}$ is ion recoil momentum vector. For a coplanar scattering geometry lying in the $(x,z)$ plane the azimuthal angles ${\phi }_f$ and ${\phi }_e$ are equal to $0^{\circ }$ or ${180}^{\circ }$, with the incident electron propagating in the $z$-axis direction, ${\phi }_i=0^{\circ }$ and ${\theta }_i=0^{\circ }$.

Figure 2

The TDCSs, in atomic units, for the ionization of hydrogen from the ground state by electron impact in the presence of a LP laser field with $\mathit{\varepsilon }\parallel {\mathit{k}}_i$, as a function of the ejected electron angle ${\theta }_e$, for one-photon absorption. The kinetic energies of the incident and ejected electron are $E_i=2$ keV and $E_e=200$ eV, and the scattering angle is ${\theta }_f=5^{\circ }$. The laser intensity is $I=1 \mathrm{T}\mathrm{W}/{\mathrm{cm}}^2$, while the photon energy is $\omega =1.55$ eV in Fig. 2, $3.1$ eV in Fig. 2, $4.65$ eV in Fig. 2, and 9.3 eV in Fig. 2. The solid lines correspond to the laser-assisted TDCSs calculated from Eq. (48), the dashed lines correspond to TDCSs in which the atomic dressing is neglected, and dot-dashed lines correspond to TDCSs in which the atomic dressing is considered in the LFA. Figures 2–2 show the same results of the laser-assisted TDCSs as in Figs. 2–2, but in a logarithmic scale, in which the dotted line represent the TDCS in the absence of the laser field.

Figure 3

Similar result as in Figs. 2–2, but for a scattering angle ${\theta }_f={15}^{\circ }$.

Figure 4

The TDCSs for the ionization of hydrogen by electron impact in the presence of a LP laser field as a function of the ejected electron angle, ${\theta }_e$, at the photon energy $\omega =3.1$ eV for no photon exchange (full line) and one- and two-photon absorption (dot-dashed and dashed lines). The dotted lines represent the TDCSs in the absence of the laser field (FF). The scattering angle is ${\theta }_f=5^{\circ }$ in Fig. 4 and ${\theta }_f={15}^{\circ }$ in Fig. 4. The other parameters concerning the scattering geometry, incident and ejected electron energies, and laser intensity are the same as in Fig. 2.

Figure 5

The TDCSs for the ionization of hydrogen by electron impact in the presence of a LP laser field as a function of the ejected electron angle ${\theta }_e$, for one-photon absorption. The photon energy is $\omega =4.65$ eV, while the kinetic energy of the ejected electron is $E_e=100$ eV in Fig. 5, 200 eV in Fig. 5, 400 eV in Fig. 5, and 800 eV in Fig. 5. The other parameters concerning the scattering geometry, kinetic energy of the incident projectile, angle of the scattered electron, and laser intensity are the same as in Fig. 2. Figures 5–5 show the same results of the laser-assisted TDCSs as in Fig. 5–5, but in a logarithmic scale. The dotted lines in Figs. 5–5 represent the TDCSs in the absence of the laser field.

Figure 6

Similar result as in Figs. 5–5, but for a scattering angle ${\theta }_f={15}^{\circ }$.

Figure 7

The TDCSs for the ionization of hydrogen by electron impact in the presence of a LP laser field as a function of the photon energy $\omega$, for one-photon absorption. The solid lines correspond to the laser-assisted TDCSs calculated from Eq. (48), the dashed lines correspond to TDCSs in which the atomic dressing is neglected, and dot-dashed lines correspond to TDCSs in which the atomic dressing is considered in the LFA. The angles of the scattered and ejected electrons are ${\theta }_f={15}^{\circ }$ and ${\theta }_e=-{55}^{\circ }$, while the incident and ejected electron kinetic energies, and laser intensity are the same as in Fig. 2. Figure 7 shows the detailed structure of the resonance peaks which occur at photon energies $\omega =|E_1|(1-1/n^2)$, with $n\ge 2$. Figure 7 shows, in a linear scale, the detailed structure of the oscillatory behavior of TDCS at small photon energies, $\omega \le 1.6$ eV.

Figure 8

The TDCSs for the ionization of hydrogen by electron impact in the presence of a LP laser field as a function of the kinetic energy of the ejected electron $E_e$, for one-photon absorption. The solid lines represent the total laser-assisted TDCSs calculated from Eq. (48), the dot-dashed lines correspond to TDCSs in which the exchange contribution is neglected (direct TDCSs), and the dashed lines correspond to TDCSs in which the direct contribution is neglected (exchange TDCSs). The angles of the scattered and ejected electrons are ${\theta }_f=0^{\circ }$ in Figs. 8 and 8, $5^{\circ }$ in Figs. 8 and 8, and ${15}^{\circ }$ in Figs. 8 and 8, while ${\theta }_e=5^{\circ }$ and $-{55}^{\circ }$ in the left- and right-hand panels, respectively. The rest of the parameters, incident electron energy, photon energy, and laser intensity, are the same as in Fig. 5.

Figure 9

The TDCSs for the ionization of hydrogen by electron impact in the presence of a LP laser field as a function of the photon energy $\omega$, for one-photon absorption. The solid lines correspond to the total laser-assisted TDCSs calculated from Eq. (48), the dot-dashed lines correspond to TDCSs in which the exchange contribution is neglected (direct TDCSs), and the dashed lines correspond to TDCSs in which the direct contribution is neglected (exchange TDCSs). The dotted lines represent the TDCSs in the absence of the laser field. The kinetic energy of the ejected electron is $E_e=200$ eV in Fig. 9, 600 eV in Fig. 9, and 1200 eV in Fig. 9. The angles of the ejected and scattered electrons are ${\theta }_e=5^{\circ }$ and ${\theta }_f=0^{\circ }$, while the kinetic energy of the incident electron and laser intensity are the same as in Fig. 8.