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2 electron/3 electron continuum

We have a strong collaboration with Mitch Pindzola, James Colgan and co-workers studying processes that lead to double ionization of atoms and molecules. The problem of two or more electrons in the continuum is both difficult and interesting. The difficulty arises because the the 2 or more electrons can interact over extremely large distances. This problem is interesting for the same reason because we would like to know how the electrons develop correlated motion in the continuum.

The main tool we use to explore this problem is a direct, numerical solution of the time dependent Schrodinger equation (the last reference below is to a review article about this method). For electron impact ionization, we start the wave function with one electron in the ground state and the other in a continuum wave packet with momentum sending it to the atom/ion. Here is a movie of a model problem (tpa.avi) where all angular momenta are set to 0. The wave packet is such that electron 1 is the continuum electron localized at about 50 a.u. and electron 2 is in the ground state so is confined to small r. As time goes forward, the packet moves to small r1. At about 25 a.u. of time, electron 1 is at its smallest distance. At later times, the packet has three interesting pieces: (1) a part that goes back out with electron 1 going to large distance and electron 2 confined to small distances (this is elastic scattering and inelastic scattering to 2s, 3s states), (2) a part where electron 2 goes to large distances and electron 1 is confined to a small region (this is exchange scattering), and (3) a wide swath where both electrons 1 and 2 are going to large distances (this is ionization). To get accurate scattering probabilities, the wave function at late times is properly symmetrized and projected onto continuum final states.

This basic numerical method has been applied to photo-double ionization of atoms/ions/molecules, electron impact ionization of atoms/ions/molecules, photo-triple ionization of atoms/ions, and electron impact double ionization of atoms/ions. Below is a brief description of results in two recent publications.

J. Colgan, M.S. Pindzola, F. Robicheaux, C. Kaiser, A.J. Murray, and D.H. Madison, “Differential cross sections for the ionization of oriented H2 molecules by electron impact,” Phys. Rev. Lett. 101, 233201 (2008). PDF (369 kB)

This paper presents calculations of the differential cross section for electron impact ionization of oriented H2 molecules at an energy 35.4 eV. The approach can be used for any orientation of the molecule. Because we investigate particular orientations there are some new features compared to measurements/calculations where the molecular orientation is random.
This image shows the triple differential cross section averaged over all orientations as a function of the incident electron direction compared to the plane formed by the two outgoing electrons. This symbols show the results from an experiment. The good agreement between calculation and experiment suggests the quality of the calculation.
This image shows the triple differential cross section for different cases of molecular orientation. The red line is for averaging over all molecular orientations and the symbols are from an experiment. The green dot-dashed, blue solid, and magenta dashed lines are the calculated cross sections for various orentiations of the molecule with respect to the beam direction.

J. Colgan, M.S. Pindzola, and F. Robicheaux, "Lattice calculations of the photoionization of Li," Phys. Rev. Lett. 93, 053201 (2004). PDF (96 kB)

This paper presents the results of calculations for the double photoionization (with excitation) and triple photoionization of the Li atom. This was the first fully quantum calculation for this system. The motion of all three electrons is treated equally by solving the time-dependent Schrodinger equation in 9 dimensions. A radial lattice is used for the 3 electrons with coupled spherical harmonics to account for the radial and angular correlations.
This image shows the photo double ionization cross sections as a function of incident photon energy. The upper panel is the total double ionization cross section compared to experiment. The read and green symbols are for two different calculations that include all angular momenta up to 2 or up to 3. The lower panel shows the partial cross sections leaving the Li2+ in different states: 1s, 2s or 2p.
This image shows the triple ionization cross section versus the photon energy. The different colored symbols are for different levels of convergence. Since the cross section is so small, the convergence with angular momenta is not as fast as for double ionization. The black symbols are from a measurement and indicate that the calculations (although not definitively converged) are approximately correct.

Five Recent Publications

M.S. Pindzola, F. Robicheaux, and J. Colgan, “Energy and angle differential cross sections for the electron-impact double ionization of helium,” J. Phys. B 41, 235202 (2008). PDF (118 kB)

M.S. Pindzola, F. Robicheaux, C.P. Ballance, and J. Colgan, “Electron-impact ionization of Li2 and Li2+,” Phys. Rev. A 78, 042703 (2008). PDF (86 kB)

J. Colgan, M.S. Pindzola, and F. Robicheaux, “Two-photon double ionization of the hydrogen molecule,” J. Phys. B 41, 121002 (2008). PDF (193 kB)

J. Colgan, M. Foster, M.S. Pindzola, and F. Robicheaux, “The evolution of the triple differential cross section for the double photoionization of He and H2,” J. Phys. B 40, 4391 (2007). PDF (274 kB)

M. S. Pindzola, F. Robicheaux, S. D. Loch, J. C. Berengut, T. Topcu, J. Colgan, M. Foster, D. C. Griffin, C. P. Balance, D. R. Schultz, T. Minami, N. R. Badnell, M. C. Witthoeft, D. R. Plante, D. M. Mitnik, J. A. Ludlow, and U. Kleiman, “The time-dependent close-coupling method for atomic and molecular collision processes,” J. Phys. B 40, R39 (2007). PDF (386 kB)

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