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Physical Review A

Variational Calculations on Electron-Helium Scattering

A. L. Sinfailam, ROBERT K. NESBET

Published Dec 1, 1972• 113 citations
Publisher

Abstract

Calculations have been made on the elastic scattering of low-energy electrons by helium using a variational procedure which takes polarization and correlation effects into account by means of continuum Bethe-Goldstone equations. The convergence of the results is examined using two methods to obtain a complete set of basis functions. Results for the $s$, $p$, $d$, and $f$ waves are presented. The width of the $^{2}S$ closed-channel resonance below the $n=2$ threshold is found to be 0.015 eV, in comparison with the 0.015-0.020-eV estimates of Andrick and Ehrhardt from experiment and the 0.01-eV estimate of Simpson and Fano.

DOI
10.1103/physreva.6.2118

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We present a variational basis-set calculational scheme for elastic scattering of positronium atom by helium atom in S wave and apply it to the calculation of the scattering length. Highly correlated trial functions with appropriate symmetry are used in this calculation. We report numerical result for the scattering length in atomic unit: $(1.0\ifmmode\pm\else\textpm\fi{}{0.1)a}_{0}.$ This corresponds to a zero-energy elastic cross section of $(4.0\ifmmode\pm\else\textpm\fi{}0.8)\ensuremath{\pi}{a}_{0}^{2}.$
Low-energy properties of spin-polarized electron-helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) and spin-polarized positronium-helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) scattering are calculated using the modified confined variational method. To take the van der Waals interaction into consideration, the van der Waals coefficient between positronium and helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) is calculated using the explicitly correlated Gaussian basis. Compared with the ground state noble gases, the larger scattering length and zero pickoff annihilation rate indicates that He(23S)\mathrm{He}(2\phantom{\rule{0.16em}{0ex}}^{3}S)He(23S) may be a more suitable cooling gas. In the energy range we considered, no similarity is found for the SSS-wave cross sections between these two scatterings. A Ramsauer-Townsend minimum is observed in spin-polarized electron-helium(23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) scattering due to the large polarization of metastable helium. This may open a new way to find and study the Ramsauer-Townsend effect.
Elastic cross-sections for electron scattering on neon from 0 energy up to 16 eV are analyzed by an analytical approach to the modified effective range theory (MERT). It is shown that energy and angular variations of elastic differential, integral and momentum transfer cross-sections can be accurately parameterized by six MERT coefficients up to the energy threshold for the first Feshbach resonance. MERT parameters are determined empirically by numerical comparison with large collection of available experimental data of elastic total (integral) cross-sections. The present analysis is validated against numerous electron beams and swarm experiments. The comparison of derived MERT parameters with those found for other noble gases, helium, argon and krypton, is done. The derived scattering length (for the s-partial wave) in neon, 0.227 a 0, agrees well with recent theories; it is small but, differently from Ar and Kr, still positive. Analogue parameters for the p-wave and the d-wave are negative and positive respectively for all the four gases compared.
In this study, we investigate the structural properties of the three-body system $\textmu{}{\text{He}}^{+}$ and the four-body system $\textmu{}\text{He}$ using explicitly correlated Gaussian basis functions with global vectors. Our results quantitatively demonstrate the pronounced shielding effect of the muon on the nucleus and reveal structural characteristics of $\textmu{}{\text{He}}^{+}$ and $\textmu{}\text{He}$, analogous to those of H and ${\mathrm{H}}^{\ensuremath{-}}$, respectively. Using the confined variational method, we examine sss-wave electron scattering from $\textmu{}{\text{He}}^{+}$ at incident energies below 8.7 eV, considering both 1S^{1}S1S and 3S^{3}S3S symmetries of the two electrons. By fitting near-threshold phase shifts to the modified effective range theory, we determine the scattering lengths to be 5.95 a0{a}_{0}a0​ for 1S^{1}S1S symmetry and 1.76 a0{a}_{0}a0​ for 3S^{3}S3S symmetry. Finally, we explore the distortion effects on $\textmu{}{\text{He}}^{+}$ induced by electron impact.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.