Theory of Deep Minima in $(e,2e)$ Measurements of Triply Differential Cross Sections

Deep minima in $\mathrm{He}(e,2e){\mathrm{He}}^{+}$ triply differential cross sections are traced to vortices in atomic wave functions. Such vortices have been predicted earlier, but the present calculations show that they have also been observed experimentally, although not recognized as vortices. Their observation in $(e,2e)$ measurements shows that vortices play an important role in electron correlations related to the transfer of angular momentum between incident and ejected electrons. The vortices significantly extend the list of known features that summarize the general picture of electron correlations in impact ionization.

Figure 1

($e\mathrm{\text{:}}2e$) coplaner symmetric geometry, showing initial ${\mathit{K}}_i$ and final ${\mathit{k}}_a,{\mathit{k}}_b$ momentum vectors along with ${\mathit{k}}_{+}$ and ${\mathit{k}}_{ab}$ and the angles ${\theta }_{+}$, and $\Theta$. The vector ${\mathit{k}}_{ab}$ is perpendicular to the $xz$ plane and ${\mathit{k}}_{+}$ lies in that plane.

Figure 2

(a) The experimental results for $\mathrm{He}(e,2e){\mathrm{He}}^{+}$ with $E_a=E_b=20\mathrm{eV}$ and ${\theta }_{+}=67.5°$ from Ref. 13 (blue dots) compared with the TDCS calculated using the DS3C wave function (red dashed line). Also shown is a theory curve for ${\theta }_{+}=59.3°$ which goes through the DS3C vortex. (b) Comparison of the deconvolved TDCS of Ref. 13 with a fit to the vortex of Eq. (2). The horizontal lines indicate the width of the convolution function of Ref. 13. Also shown is a fit obtained by varying parameters in the DS3C amplitude.

Figure 3

Contour plot of $|A({\mathit{k}}_a,{\mathit{k}}_b)|$ on a logarithmic scale. Vectors are drawn in the direction of the current $\mathit{v}$ showing that the minimum in Fig. 2 is near a vortex. The diagonal lines trace the vector ${\mathbf{k}}_{+}$ as the angle $\Theta$ is varied to produce the DS3C curves in Fig. 2.