Parametrization of Electron-Impact Ionization Cross Sections from Laser-Excited and Aligned Atoms

A set of parameters describing electron-impact ionization from laser-aligned atoms are reported, which define the “length”, “width”, and “direction” of the quadruple differential cross section (QDCS) as a function of target alignment ${\mathbf{k}}_{\mathit{B}}$ for fixed ingoing electron momentum ${\mathbf{k}}_0$ and outgoing momenta ${\mathbf{k}}_1$, ${\mathbf{k}}_2$. ${}^{24}\mathrm{Mg}$ was used, with ${\mathbf{k}}_0$, ${\mathbf{k}}_1$, ${\mathbf{k}}_2$, and ${\mathbf{k}}_{\mathit{B}}$ in the same plane. The parameters are derived for a range of ${\mathbf{k}}_2$ angles, with ${\mathbf{k}}_1$ set at 30° to ${\mathbf{k}}_0$. The QDCS is then determined for all ${\mathbf{k}}_{\mathit{B}}$. The parameters are very angle sensitive, the QDCS direction varying more than 90° as the length to width ratio varied more than an order of magnitude.

Figure 1

The experimental apparatus, showing different components. See text for details.

Figure 2

Comparison of new data from (a) the ground and (b) the excited state for ${ϵ}_B=90°$, with previous measurements [4].

Figure 3

Polar plot of the QDCS obtained for $({\theta }_1,{\theta }_2)=(30°,50°)$ as a function of ${ϵ}_B$. A parametric fit to the data is shown. The anisotropy $Q$-parameter $P_{{\mathbf{k}}_0{\mathbf{k}}_1{\mathbf{k}}_2}^Q$ is defined by the length $l^Q$ and width $w^Q$ of the fit. The direction of the major axis of the fit with respect to ${\mathbf{k}}_0$ is given by ${\kappa }_{{\mathbf{k}}_0{\mathbf{k}}_1{\mathbf{k}}_2}^Q$. The calculated excited $P$-state charge cloud is shown as ${ϵ}_B$ varies [14]. The recoil momentum direction $\mathbf{q}={\mathbf{k}}_0-{\mathbf{k}}_1$ is also shown.

Figure 4

The fitted $Q$-parameters as a function of ${\theta }_2$, defined by Eq. (1). Normalized polar plots of the fitted functions with respect to ${\mathbf{k}}_0$ are shown, demonstrating their rapid change in direction and shape as ${\theta }_2$ varies. The ratios $l^Q/w^Q$ equivalent to the $P_{{\mathbf{k}}_0{\mathbf{k}}_1{\mathbf{k}}_2}^Q$ values given on the left axis are also shown in (a) to two significant figures.

Figure 5

QDCS derived from the $Q$-parameters and data in Fig. 2. Examples at different ${ϵ}_B$ are shown, normalized to the TDCS in Fig. 2. The data at ${ϵ}_B=90°$ are the weighted average of the data in Fig. 2.