Ejection of innershell electrons induced by recollision in a laser-driven carbon atom

Ejection of core electrons as a result of recollision in a laser-driven carbon atom is theoretically investigated with a quasiclassical model. The model, called “fermionic molecular dynamics,” gives rise to a ground-state carbon atom where the six electrons are paired in shells, with different binding energies. This feature renders possible, on a classical level, the discussion of the ejection of electrons from different shells. By analyzing a large number of trajectories of a carbon atom exposed to an intense, few-cycle laser pulse, we reveal a class of recollision trajectories where the recolliding electron is recaptured into the atomic core after ejecting a core electron. We also discuss the difference between quadruple ionization trajectories leading to a final ${\mathrm{C}}^{4+}$ ion where the two bound electrons have opposite spin, and the trajectories where the bound electrons have equal spin.

Figure 1

Coordinates ${\mathbf{r}}_i^{\left(g\right)}$ of the electrons in the FMD ground state of (a) neutral C, (b) ${\mathrm{C}}^{+}$, (c) ${\mathrm{C}}^{2+}$, (d) ${\mathrm{C}}^{3+}$, (e) ${\mathrm{C}}^{4+}$, and (f) ${\mathrm{C}}^{5+}$. In all panels, all electrons are located in the $xy$ plane. $\alpha$ spin electrons are shown with red, solid circles, $\beta$ electrons are shown with open circles, and the C nucleus is shown as a slightly smaller, black solid circle. The arrows indicate the direction and magnitude of the momenta ${\mathbf{p}}_i^{\left(g\right)}$ (all lying in the $xy$ plane) in the ground state; the scale for the momentum vectors is set by the value $|{\mathbf{p}}_{\mathrm{core}}^{\left(g\right)}|\approx 5.95$ a.u. for neutral C [the longest momentum vectors in (a)]. Not shown is the momentum ${\mathbf{P}}^{\left(g\right)}$ of the C nucleus, which is ${\mathbf{P}}^{\left(g\right)}\approx (0,9.63,0)$ a.u. for neutral C, and similar for the other charge states.

Figure 2

Total probability $P_n$ to ionize to different final charge states, as indicated in the legend (the line labeled by $n+$ indicates that $n$ electrons were ejected), as a function of the laser intensity. Statistical error bars $\pm \sqrt{k_n}/k_{\mathrm{tot}}$ $(\pm$ one standard deviation) are shown when their length exceeds the size of the curve symbols.

Figure 3

(a) Distribution of final bound-state energies ${\varepsilon }_{\mathrm{ion}}$ and final momentum ${\Phi }_x$ in the laser polarization direction, for quadruple ionization trajectories. The overlaid thin, solid lines represent the distribution summed over either ${\varepsilon }_{\mathrm{ion}}$ or ${\Phi }_x$, on arbitrary scales. (b) Distribution of final bound-state energies ${\varepsilon }_{\mathrm{ion}}$, divided into two groups: with and without core-electron ejection. The error bars on the total distribution show the estimated statistical error (one standard deviation). The vertical lines indicate the minimum energy of a ${\mathrm{C}}^{4+}$ ion with one $\alpha$ and one $\beta$ electron (solid line), and a ${\mathrm{C}}^{4+}$ ion with two $\alpha$ electrons (dashed line). In both (a) and (b), the laser frequency is ${\omega }_0=0.057$ a.u., ${\phi }_0=0$, and the intensity is $I=4\times {10}^{15}$ W/${\mathrm{cm}}^2$.

Figure 4

Recollision trajectories with core-electron ejection in category 1 (see Table 2). The single-electron energies ${\epsilon }_i,i=1,2,...,6$, are shown as a function of time $t$. Spin $\alpha$ electrons are shown with solid lines and spin $\beta$ electrons with broken lines. Red (gray) curves: valence electrons; green (light gray) curves: inner valence electrons; blue (black) curves: core electrons. For reference, solid, horizontal lines are drawn at ${\epsilon }_i=0$ and ${\epsilon }_i=3U_p\approx 26$ a.u. The laser field $\widehat{\mathbf{x}}·\mathbf{E}(t,\mathbf{0})=E_0\Psi \left(t\right)$ is shown (on an arbitrary scale) in the background with a thick line. The laser parameters are ${\omega }_0=0.057$ a.u. and $I=4\times {10}^{15}$ W/${\mathrm{cm}}^2$ (same as in Fig. 3). The trajectory displayed in (a) has final bound energy ${\varepsilon }_{\mathrm{ion}}=-33.0$ a.u., the trajectory in (b) has ${\varepsilon }_{\mathrm{ion}}=-28.1$ a.u., and (c) has ${\varepsilon }_{\mathrm{ion}}=-19.5$ a.u.

Figure 5

Contributions to the single-electron energy ${\epsilon }_i={\epsilon }_i^{\left(\mathrm{kin}\right)}+{\epsilon }_i^{\left(\mathrm{pot}\right)}+{\epsilon }_i^{\left(\mathrm{aux}\right)}+\frac{1}{2}{\epsilon }_i^{\left(2e\right)}$ [see Eq. (6) for definition] for the two valence electrons [valence electron 1: red (gray) curves; valence electron 2: black curves]. For both electrons, ${\epsilon }_i^{\left(\mathrm{aux}\right)}<0.04$ a.u. for all $t$, and ${\epsilon }_i^{\left(\mathrm{aux}\right)}$ is therefore not shown. The laser parameters and the initial conditions are the same as in Fig. 4. The laser field $\Psi \left(t\right)$ is shown (in arbitrary units) as a thick line in the background.

Figure 6

Recollision trajectories with core-electron ejection in category 2 (see Table 2), recapture of the recolliding electron. The line styles and color coding of the curves are the same as in Fig. 4. The laser parameters are ${\omega }_0=0.057$ a.u. and $I=4\times {10}^{15}$ W/${\mathrm{cm}}^2$. The trajectory shown in (a) has final bound energy ${\varepsilon }_{\mathrm{ion}}=-32.5$ a.u., the trajectory in (b) has ${\varepsilon }_{\mathrm{ion}}=-30.3$ a.u., and (c) has ${\varepsilon }_{\mathrm{ion}}=-22.6$ a.u.

Figure 7

Recollision trajectories with core-electron ejection in category 3 (see Table 2), where the bound electrons in the final state have equal spin. The line styles and color coding of the curves are the same as in Fig. 4, and the laser parameters are ${\omega }_0=0.057$ a.u. and $I=4\times {10}^{15}$ W/${\mathrm{cm}}^2$. The trajectory in (a) has final bound energy ${\varepsilon }_{\mathrm{ion}}=-24.5$ a.u. and the trajectory in (b) has ${\varepsilon }_{\mathrm{ion}}=-19.7$ a.u.

Figure 8

Recollision trajectories without core-electron ejection, where both core electrons stay bound. The line styles and color coding of the curves are the same as in Fig. 4, and the laser parameters are ${\omega }_0=0.057$ a.u. and $I=4\times {10}^{15}$ W/${\mathrm{cm}}^2$. The trajectory in (a) has final bound energy ${\varepsilon }_{\mathrm{ion}}=-32.6$ a.u. and the trajectory in (b) has ${\varepsilon }_{\mathrm{ion}}=-19.8$ a.u.