Effect of a vortex in the triply differential cross section for electron-impact $K$-shell ionization of carbon

Vortices are an inherent property of the velocity fields of complex, time-dependent, Schrödinger wave functions $\psi$ occurring where both the real and the imaginary parts of $\psi$ vanish. They have been known since the early work of Dirac on magnetic monopoles and have frequently been studied theoretically. The possibility of observing them by exploiting an “imaging theorem” that relates atomic wave functions to measured electron momentum distributions has recently been proposed. Using the Coulomb-Born approximation, we examine ionization of a $K$-shell electron of a model carbon atom by fast electron impact. For an incident electron energy of 1801.2 eV and a scattering angle of $4^{\circ }$, we find a vortex in the velocity field associated with a zero in the ionization $T$-matrix element and hence in the triply differential cross section, and we obtain a segment of the vortex line. Angular momentum transfer is essential to produce the vortex in the velocity field and the corresponding zero in the ionization $T$-matrix element and in the triply differential cross section.

Figure 1

Plot of angular distributions of the TDCS computed in the CB1 [solid (blue) curve] and the B1 [dashed (red) curve] for electron-impact ionization of the $K$ shell of carbon for the kinematic conditions $E_i=1801.2$ eV, ${\theta }_f=4^{\circ }$, and $E_{\mathit{k}}=5.5$ eV.

Figure 2

Real [solid (blue) curve] and imaginary [dashed (red) curve] parts of the Coulomb-Born (CB1) ionization $T$-matrix element for electron-impact ionization of the $K$ shell of carbon under the kinematic conditions $E_i=1801.2$ eV, ${\theta }_f=4^{\circ }$, and $E_{\mathit{k}}=5.5$ eV.

Figure 3

Density plot of $ln|T_{\mathit{k},1s}^{\mathrm{CB}1}|$ as a function of the $x$ and $z$ components $(k_x,k_z)$ of momentum $\mathit{k}$ of the ejected electron for electron-impact ionization of the $K$ shell of carbon. The nodal lines where $\mathrm{Re}\left[T_{\mathit{k},1s}^{\mathrm{CB}1}\right]=0$ and $\mathrm{Im}\left[T_{\mathit{k},1s}^{\mathrm{CB}1}\right]=0$ are shown by solid blue and green curves, respectively. Arrows show the direction of the velocity field $\mathit{v}={\nabla }_{\mathit{k}}\mathrm{Im}[ln{T}_{\mathit{k},1s}^{\mathrm{CB}1}]$.

Figure 4

TDCS for $K$-shell ionization of carbon by electron impact computed using different multipole components in the expansion of the CB1 ionization $T$-matrix element. Kinematic conditions are $E_i=1801.2$ eV,  ${\theta }_f=4^{\circ }$, and $E_{\mathit{k}}=5.5$ eV. The TDCS we compute with ${\ell }_{\max }=1$ is shown by the dotted (red curve); that with ${\ell }_{\max }=2$, by the dashed (blue) curve; and that with ${\ell }_{\max }=4$, by the solid (green) curve.

Figure 5

Angular distributions of the TDCS for $K$-shell ionization of carbon by electron impact. The $z$ axis is parallel to the incident beam direction. The kinematic conditions ($E_i=1801.2$ eV, ${\theta }_f=4^{\circ }$, and $E_{\mathit{k}}=5.5$ eV) are for a vortex. The TDCS computed with the CB1 $\ell =0\to 4,-\ell \le m\le \ell$ components is shown by the solid (red) curve; the TDCS computed with the CB1 ${\ell }_{\max }=4,m=0$ plus $\ell =1,m=\pm 1$ components, by the long-dashed (green) curve; and the TDCS computed with the CB1 $\ell =0\to 4,m=0$ components, by the dotted (magenta) curve. The TDCS computed with the B1 multipole components $\ell =0\to 4,m=0$ [dot-dashed (blue) curve] is also shown.

Figure 6

Angular distributions of the TDCS for $K$-shell ionization of carbon by electron impact. The $z^{\prime }$ axis is taken parallel to the momentum transfer vector. Kinematic conditions are $E_i=1801.2$ eV, ${\theta }_f=4^{\circ }$, and $E_{\mathit{k}}=5.5$ eV. The TDCS we compute with the CB1 $\ell =0\to 4,-\ell \le m\le \ell$ components is shown by the solid (red) curve; the TDCS we compute with the CB1 $\ell =0\to 2,m=0$ plus $\ell =1,m=\pm 1$ components, by the long-dashed (green) curve; the TDCS we compute with the CB1 $\ell =0\to 2,m=0$ components only, by the dotted (brown) curve; and the TDCS we compute with the CB1 $\ell =1,m=\pm 1$ components, by the dashed (magenta) circles.

Figure 7

A segment of the vortex line for the $K$-shell ionization of carbon by electron impact for an incident energy $E_i$ of $1801.2$ eV and a scattering angle ${\theta }_f$ of $4^{\circ }$. The positions of the vortex for different values of the $\mathrm{y}$ component of the momentum of the ejected electron, $k_y$, are denoted by filled (red) circles. The loci of the positions of the vortex form a segment of the vortex line.