Two-electron cusp in the double ionization of helium

We analyze the double ionization of He under the impact of 100 keV ${\mathrm{He}}^{2+}$ projectiles. The process is described within the framework of the impact parameter and frozen-correlation approximations where the one-electron events are treated by the continuum distorted wave method. Correlation between the emitted electrons, which plays an important role in forming the shape of the differential distribution of the electron emission, is described by the Coulomb density of states approximation (CDS). Special attention is paid to the region of the two-electron cusp that has been observed in a recent experiment for 100 keV ${\mathrm{He}}^0+\mathrm{He}$ collisions [L. Sarkadi and A. Orbán, Phys. Rev. Lett. 100, 133201 (2008)]. In the cusp region the correlated motion of the two electrons is influenced dominantly by the outgoing projectile, that is, the correlation function of the CDS treatment is expected to depend on the electron momenta measured relative to the projectile rather than to the target nucleus. A qualitative agreement with the experiment is achieved with a CDS model based on the use of such a projectile-centered correlation function that applies effective charges as given in the dynamically screened three-Coulomb wave function.

Figure 1

Panels (a–d): Contour plots of calculated relative six-fold differential cross sections for DI of He under 100 keV ${\mathrm{He}}^{2+}$ impact as functions of the electron ejection energies $E_1$ and $E_2$ at fixed ejection angles: ${\theta }_1={\theta }_2=0^{°}$ and ${\phi }_1={\phi }_2=0^{°}$. Theories: (a) Independent particle model (IPM); (b) Coulomb density of states model (CDS); (c) Dynamically screened CDS with a correlation function centered on the projectile (DSCDS); (d) Convoluted DSCDS (see text). Panel (e): Experimental data of Sarkadi and Orbán [19] for the process given by Eq. (1). The scale of the plots is logarithmic, the cross section between the neighboring contour levels changes by a factor of 2. The intensity increases with increasing tone of darkness.

Figure 2

Two-electron emission distributions following DI of He under 100 keV ${\mathrm{He}}^{2+}$ impact as a function of the electron ejection energies $E_1$ and $E_2$ at ${\theta }_1={\theta }_2=0^{°}$. Left panel: Convoluted DSCDS (see text). Right panel: Experimental data of Sarkadi and Orbán [19] for the process given by Eq. (1).

Figure 3

Contour plots of calculated six-fold differential cross sections for DI of He under 100 keV ${\mathrm{He}}^{2+}$ impact in the projectile-centered reference frame as functions of the momentum coordinates $k_{2x}^{\prime }$ and $k_{2z}^{\prime }$ of the electron $e_2$ at a fixed momentum vector of the electron $e_1$, ($k_{1x}^{\prime },$$k_{1z}^{\prime })=(0,-0.086$), and at relative azimuthal angle ${\phi }_k{{}_1}^{\prime }-{\phi }_k{{}_2}^{\prime }=0^{°}$. Left, middle, and right panel: IPM, CDS, and DSCDS theory, respectively. The arrow shows the momentum vector of $e_1$.

Figure 4

Energy spectra of the electron $e_1$ in the laboratory frame at ${\theta }_1={\theta }_2=0^{°}$ for 100 keV ${\mathrm{He}}^{2+}+\mathrm{He}$ collision. The curves are the results of the present calculations obtained by convoluting the theoretical cross sections with the energy and angular resolution of the experiment by Sarkadi and Orbán [19]. Solid line: DSCDS; dashed line: IPM. Closed circles: Experimental data for the process shown in Eq. (1). For the spectra in panels (a), (b), and (c) the energy of $e_2$ lies in the intervals 11.0–12.5 eV, 13.0–14.2 eV, and 14.5–16.0 eV.

Figure 5

The ECC cusp for single ionization of He by impact of 100 keV ${\mathrm{He}}^{2+}$ ions. The curve and the points are results of CDW-EIS calculations and experimental data [31], respectively. The experimental data are normalized at the maximum of the theoretical curve.

Figure 6

Four-fold differential cross sections in the laboratory frame as a function of ejection energy $E_1$ for 100 keV ${\mathrm{He}}^{2+}+\mathrm{He}$ collisions at $E_2=12$ eV and at ${\theta }_2=0^{°}$. The curves are results obtained in the framework of DSCDS and IPM. DSCDS: ${\theta }_1=0^{°}$, solid line; ${\theta }_1={0.125}^{°}$, dashed line; ${\theta }_1={0.25}^{°}$, dotted line; ${\theta }_1={0.5}^{°}$, dot-dashed line. IPM: ${\theta }_1=0^{°}$, dot-dot-dashed line.