Two-Electron Correlations ine+H→e+e+pNear Threshold

Abstract

We present an ab initio calculation of the ionization cross section of atomic hydrogen near threshold with precision that compares excellently with the Shah-Elliot-Gilbody experiment [J. Phys. B 20, 3501 (1987)]. This fills the gap between theory and experiment down to 0.1 a.u. above threshold, complementing the recent spectacular work of Bray and Stelbovics [Phys. Rev. Lett. 70, 746 (1993)].The angular momentum distributions of the secondary electron display an evolution in correlation patterns toward the threshold. PACS numbers: 34.50.Fa, 34.80.Dp Electron-impact ionization of hydrogen is one of the most fundamental processes involving one heavy and two light particles in disintegration without charge transfer. Its particular simplicity allows one to estimate the order of magnitude of the total cross section readily [1].Nonetheless, theory has been considered largely incomplete because theoretical methods [2,3] could evaluate cross sections with a far less precision than can the experiment [4]. A terse historical account of this can be found in the recent Letter by Bray and Stelbovics that has succeeded in bridging this long-standing gap from intermediate to high energies by means of the convergent close-coupling (CCC) method [5]. However, they left a gap on the low-energy side unresolved. The current techniques are well suited to the study of threshold ionization whereas the CCC method is not, though an enlargement of the basis set might as well improve the convergence of the CCC method even at low energies. The present work investigates the low-energy region rather close to threshold within the framework of the time-independent scattering theory. Throughout this Letter, we use atomic units, i.e. , 1 a.u. of length = 5.29 X 10 " m (Bohr radius ao) and 1 a.u. of energy = 27.2 eV.For clarifying the spirit, let us employ an index K = 1/$2~E~, an analog of the single-electron wavelength, for both below and above threshold. Here and throughout, E is measured from the threshold so that K represents the distance in energy from the threshold and serves as a rough measure of the spatial range where the two-electron correlation [6] is important; the characteristic distance associated with the correlation is of the order of K according to Wannier [7]. The energy range may be divided into three regions. The high-energy region (~ (( 1) is where the perturbative treatment yields moderately reliable results. As a matter of fact, the controversial ratio of the single-to-double ionization of He by photon impact at K « 1 appears to be reasonably well represented by perturbation theory [8]. Here, the electron-electron correlations occur at rather short mutual distances during a short time interval, thus the two electrons may be consid-ered largely independent. The low-energy region (~~1) is, on the contrary, where the correlations become particularly important because the incident electron has ample time to see the detailed level structure of the target atom;the target in turn has ample time to respond to the electric field ...

A complete numerical approach to electron–hydrogen collisions

“…However, to allow for the inhomogeneous term χ in the scattering wave (16), this procedure required modification.…”

“…where the propagation vector − → E (i) is introduced into our derivation to allow for the inhomogeneous term in (16).…”

“…This one-dimensional formula is readily extended to two dimensions using similar techniques to Poet [30]. There is, however, an additional complexity when reforming the scattering wave equation (16) into the form of (42); for l = 0 there is a 1/r singularity and for l > 0 there is also a 1/r 2 singularity. We resolved this problem in the same way as Wang and Callaway [53], using the known limiting behaviour of the e-H scattering wavefunction…”

“…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). In one dimension, these equations have the form…”

“…We should note that this matrix element does not connect states of different total angular momentum or parity, so the summation over l 1 l 2 is limited to those that satisfy the parity relation (−1) = (−1) L+l 1 +l 2 and |l 1 −l 2 | L |l 1 +l 2 |. The PhD thesis relating to this tutorial [32] gives a detailed description of the algebraic and quantum-mechanical steps used to obtain (16) along with all the necessary angular-momentum relations.…”