Effective-momenta approach for the four-body Coulomb problem

We propose approximate separable solutions for the Schrödinger equation corresponding to the four-body Coulomb problem. By exploring several asymptotic limits, we find configurations where the nondiagonal terms of the Hamiltonian can be introduced in the two-body Coulomb-like distortion factors leading to modified momenta. The present effective momenta model is used to study the double ionization of He by ion impact at the fully differential level. Possible differences arising from proton and antiproton impact are explored at impact energies in the range 700 keV/amu–6 MeV/amu. The results are represented in terms of contour plots as a function of the electrons’ emission angles, and the four-body dynamics for the double-ionization process is analyzed. Finally, we compare diverse representations for the fully differential cross section. A detailed interpretation of the features of the process is done by overlapping the recoil level lines over the contour plots. Ternary plots are tested as an alternative tool to explore the different physical mechanisms involved in the double emission process at the fully differential level.

Figure 1

Coordinate scheme for the four-body problem: (a) Jacobi generalized coordinates and (b) relative coordinates.

Figure 2

Angular distributions of the two ejected electrons in the scattering plane. The polar angles are measured from the direction of the incident beam. The calculated ionization FDCS are given in units of ${10}^{-5}$ a.u. for proton and antiproton impinging on He at $E_i=700$ keV. The momentum transfer $Q$ is 0.9 a.u. The contour plots (a)–(c) correspond to proton impact and (d)–(f) to antiproton impact. The (a) and (d) plots correspond to ejected electrons with 3 eV in equal energy regime, (b) and (e) to 10 eV electron ejection energies, and (c) and (f) to the case of 15 eV electron energies.

Figure 3

Contour plots of the angular distributions of the two electrons for double ionization of He by proton impact (a), (b) and antiproton (c), (d) impact. The impact energies considered are $E_i=2000$ keV (a) and (c), and $E_i=6000$ keV (b) and (d). The emission geometry considered is that of Fig. 2. The FDCS are in units of ${10}^{-5}$ a.u.

Figure 4

Contour plots of the angular distributions of the two electrons for double ionization of He by proton impact at 700 keV/amu. The electrons are ejected with equal energies of 10 eV in the collision plane and the momentum transfer $Q$ is (a) 1.2 a.u. and (b) 1.6 a.u. The FDCS are in units of ${10}^{-5}$ a.u.

Figure 5

Angular distributions of the FDCS for two electrons drawn over the corresponding absolute values of the target recoil momentum, for the same dynamical conditions and projectiles considered in Figs. 2b and 2e. The scale on each plot indicates the absolute values for the target recoil momentum.

Figure 6

Ternary plots for the distributions corresponding to Figs. 2b and 2e. The left panel corresponds to proton impact and the right one to antiproton, respectively. The FDCS are expressed in atomic units.

Figure 7

Ternary plots for the distributions corresponding to Figs. 4a and 4b, respectively. FDCS are expressed in atomic units.