Complete Numerical Solution of Electron-Hydrogen Model Collision Problem above the Ionization Threshold

Abstract

Benchmark results are presented for electrons colliding with hydrogen atoms in the S-wave (TemkinPoet) model collision problem, which neglects angular momentum. Complete results (elastic, inelastic, and ionization), accurate to 1%, are obtained by numerically integrating Schrödinger's equation subject to correct asymptotic boundary conditions. This marks the first time direct matching to asymptotic boundary conditions has been shown to yield convergent ionization amplitudes for a Coulomb three-body problem. Results are presented for impact energies of 54.4 and 40.8 eV, where comparison with other theories is available.PACS numbers: 34.80. Dp, 31.15.Fx, 34.10. + x, 34.80.Bm The Temkin-Poet [1,2] model of electron-hydrogen scattering is now widely regarded as an ideal testing ground for the development of general methods intended for the full three-body Coulomb problem. Although only s states are included for both the projectile and the atomic electrons, this model problem still contains most of the features that make the real scattering problem hard to solve. Indeed, even in this simplified model, converged energy distributions for ionization cannot generally be obtained via the close-coupling formalism [3]. Any general method that cannot obtain complete, converged results for this model problem will face similar difficulties when applied to the full electron-hydrogen system. For this reason, we believe it is essential to develop a general numerical method capable of solving the Temkin-Poet model completely before angular momentum is included. Here we report such a method. Complete, precision results for e 2

A complete numerical approach to electron–hydrogen collisions

“…In an attempt to further reduce the computational overhead of the e-H problem we used the propagation method, based on Poet [30], that was recently used by Jones and Stelbovics [31] for model e-H ionization problems. However, to allow for the inhomogeneous term χ in the scattering wave (16), this procedure required modification.…”

“…The results of the present work were obtained from a numerical grid in coordinate space using both exterior complex scaling and an enhancement of a propagation algorithm [30,31] previously used for model e-H calculations. So as to differentiate this method from the ECS method [24], we will refer to the present work as the propagating exterior complex scaling method, and use the acronym PECS.…”

Theory of Electron Impact Ionization of Atoms

“…This is because of the ambiguity of the Peterkop wave function used in [51]. As a result in [51] one has to further add some hyperradiusdependent phase [namely, Q͑k Ͼ ,1͒ of [51]] to reproduce the phase obtained in the exact numerical integration [7] and other calculations [50] of the Temkin-Poet model. Apart from this, the agreement between our ionization amplitudes in the Temkin-Poet and collinear S-wave models and results of [51] indicates that the simple and transparent approach to calculating the partial waves and amplitudes presented in this work leads to the correct answer.…”

“…exact Temkin-Poet model for ionization has been numerically solved in [7]. The agreement between the corresponding benchmark ionization amplitude and T 0,0,0,0…”

“…Subsequently, numerical methods were extended to include more partial waves and applied to the full problem at energies between the first and the second excitation thresholds [3][4][5][6]. More recently, Jones and Stelbovics [7] used a direct integration method to calculate electron-impact ionization within the framework of the TP model. A major advance in the direct solution of the Schrödinger equation was made using the exterior complex scaling (ECS) technique [8][9][10] when calculations reached the stage of yielding quantitative agreement with measurements of e-H fully differential ionization cross sections.…”