Direct Numerical Solution of the Continuum Three-Body Coulomb Problem
M. Baertschy and D. Byrum, University of California, Davis
W. Isaacs and C. W. McCurdy, Lawrence Berkeley National Laboratory
T. Rescigno, Lawrence Livermore National Laboratory
Research Objectives
Our group's research centers on low-energy electron scattering from small molecules, an important problem in environments such as semiconductor processing plasmas, materials development, and toxic waste remediation.
The physically important process of electron impact ionization, where one or more electrons are knocked out of the target by the incoming electron, has not been formally solved for even the simplest case of electron-hydrogen atom scattering. We are developing an approach to this continuum three-body Coulomb problem, and seek the first direct numerical solution without nonphysical approximations.
Computational Approach
We treat the full 6-dimensional Schrödinger equation for the electron-hydrogen system. Expanding the wave function in angular momentum results in sets of coupled 2D differential equations. The use of exterior complex scaling, whereby the radial coordinates are mapped onto a complex contour, formally turns the infinite range problem into a finite range problem.
We discretize the radial coordinates using variable spacing finite differences, resulting in a large sparse matrix problem. We solve the full-coupled problem using an iterative method, the conjugate gradient squared algorithm, with a block-diagonal preconditioner. The angular momentum expansion provides a natural framework for distributing the solution vector across processors. By using a parallel version of SuperLU, a direct sparse solver, to apply the preconditioner, we can also distribute individual angular momentum blocks across several processors. This code was developed on the T3E using MPI.
Accomplishments
We have developed a code able to generate the numerical wave functions describing electron impact ionization of hydrogen for a range of scattering energies. From these wave functions we can extract scattering cross sections that describe how energy is shared between the two outgoing electrons after ionization.
A sample partial-wave component of the full e-H wave function. The peaks along the right edge correspond to excitation of the hydrogen atom. The outgoing waves represent ionization flux.
Significance
These are the first accurate calculations of electron-hydrogen ionization singly differential (energy sharing) cross sections and, indeed, represent the first "exact" solution to a quantum mechanical three-body Coulomb problem. These calculations lay the groundwork for one approach to incorporate ionization into our group's electron-molecule calculations.
Publications
C. W. McCurdy and T. N. Rescigno, "Calculating differential cross sections for electron-impact ionization without explicit use of the asymptotic form," Physical Review A 56, R4369-R4372 (1997).
C. W. McCurdy, T. N. Rescigno, and D. A. Byrum, "An approach to electron-impact ionization that avoids the three-body Coulomb asymptotic form," Physical Review A 56, 1958-1969 (1997).
T. N. Rescigno, M. D. Baertschy, D. A. Byrum, and C. W. McCurdy, "Making complex scaling work for long-range potentials," Physical Review A 55, 4253-4262 (1997).