Electron-impact ionization of atomic hydrogen

Abstract

. DISCLAIMER DISCLAIMERPortions of this document may be illegible in electronic image products. Images are produced from the best avaiiabie original document.,-. LBNL-4521O Electron-Impact Ionization of Atomic Hydrogen 2000_-------.---.... . . . . . . . . . . . . . . . 72 6.4.2 SDCSfor 17.6,20,25, and30eV . . . . . . . . . . . . . . . . 73 6.5 Integral Ionization Cross Sections . . . . . . . . . . . . . . . . . . . . 74 7 The Three-Body Electron-Impact In the early 1970's Burke and Mitchell [15, 14] showed that cross sections for the elastic and excitation channels could be calculated at energies above the ionization threshold by including positive-energy pseudostates in the expansion. This work was extended in the 1980's by Oza and Callaway [23,22] (Fl, 72) Electron-Impact Ionization of Atomic Hydrogen List of Tables Nature of Ionization Electron-impact ionization is the process in which a target atom or molecule is ionized by a collision with an electron. Scattering theory calculations have progressed to the point of being able to accurately treat non-breakup processes for an electron scattering from relatively complicated target molecules. However, ionization represents a fundamentally different class of problems characterized by a final state in which three particles that interact via long-range by rearranging the Schrodinger equation (Equation 2.1).Since lli~ (?'l, 72) represents the scattered part of the wave function at large distances it must be an outgoing wave in rl and r2. Thus, we define W; (71, 72) Exterior Complex Scaling Application to long-range potentials Finite Difference Implementation ECS on a grid Under ECS, the scattered wave @& (z(rI), Z(T2)) is a continuous function but has discontinuous first derivatives along the lines rl or r2 equal to&.There is no problem representing the wave function on a two-dimensional grid in rl and r2, but in order to correctly approximate its derivatives on each grid point we will require that~be one of the grid points. The scattered wave will be calculated directly on to the ECS contour by solving Equation 3.5 on the two-dimensional, complex-scaled grid. Functions whose analytic forms are known, such as the right-hand side of Equation 3.5 and the potentials in the Hamiltonian, are mapped on to the ECS contour by simply evaluating them on the contour z(r) for both rl and r2. The non-analytic two-electron potential $ is scaled in this way by noting that it is piece-wise analytic and scaling the rl < r2 and rl > r2 regions separately. The potential is unchanged on the real part of the grid and, as will be demonstratedlater in this chapter, the potentials beyond~have very little effect on the wave function in the interior region. Finite difference approximations to derivatives Dimension of the problem Properties of the Calculated Wave FunctionsThe figure on the left shows the absolute value of #~P (rl, Tz) ..................................................j .......................... Total Cross Section Differential Cross ...

A complete numerical approach to electron–hydrogen collisions

“…Previous model calculations [55] showed that finer grid spacing is required near the nucleus (where Coulomb potentials are stronger) than for the outer region, which is consistent with ECS calculations [48].…”

“…Baertschy et al [48] found that a conjugate gradient squared (CGS) algorithm could iteratively solve their matrix equation with O(N 3/2 ) operations by using an approximate solution as a preconditioner. They also found that solutions to the uncoupled scattering wave equations (which require significantly less computational resources) could be used for these preconditioners.…”

“…and C L0 00L0 = 1, whereĵ L is the Riccati-Bessel function, and our derivation reduces to that given in [48].…”

“…In fact, these formulae were used by Baertschy et al [48] for their ECS implementation, though adapted for variable grid spacing. We used a very accurate 3-point Numerov finite-difference formula that can evaluate second-order differential equations that have no first-derivative terms, as in (16).…”

“…, (11) where to ensure that our derivation is equivalent to Baertschy et al [48] for Z = 1, the Coulomb wave is normalized so that…”