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Journal of Physics Communications

Virtual and bound states in low-energy positron scattering by atoms and molecules via modified effective range theory

Kamil Fedus, Grzegorz P. Karwasz

Published Apr 1, 2024• 2 citations
Publisher

Abstract

Abstract Modified effective range theory is applied as a tool to determine bound and virtual state energies in low-energy positron elastic scattering by atoms and molecules. This is achieved by the S-matrix continuation into the complex momentum plane, allowing to identify poles related to shallow energy states. The influence of the long-range polarization potential (∼r −4) on the bound and virtual-state pole positions is analyzed for noble gases and nonpolar molecules such as H2, N2, and CH4. The quantitative relations between the S-matrix poles and the s-wave scattering length accounting for dipole polarization are introduced.

DOI
10.1088/2399-6528/ad4001

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Abstract Modified effective range theory is applied as a tool to determine bound and virtual state energies in low-energy positron elastic scattering by atoms and molecules. This is achieved by the S-matrix continuation into the complex momentum plane, allowing to identify poles related to shallow energy states. The influence of the long-range polarization potential (∼r −4) on the bound and virtual-state pole positions is analyzed for noble gases and nonpolar molecules such as H2, N2, and CH4. The quantitative relations between the S-matrix poles and the s-wave scattering length accounting for dipole polarization are introduced.
The numerical approach to calculating the SSS-matrix poles associated with virtual states in electron-molecule collisions is proposed within the modified effective-range theory frame. The SSS-matrix continuation into the complex momentum plane is possible thanks to the analytical properties of modified Mathieu functions, that is, the exact solutions of the Schr\"odinger equation with the long-range polarization potential ($\ensuremath{\sim}{r}^{\ensuremath{-}4}$). The short-range electron-molecule interaction is included in the introduced numerical approach by model comparison with experimental integral, differential, and momentum-transfer elastic cross sections. The influence of the polarization potential on the virtual-state pole positions is analyzed for nonpolar targets such as N2,CH4,CO2{\mathrm{N}}_{2}, {\mathrm{CH}}_{4}, {\mathrm{CO}}_{2}N2​,CH4​,CO2​, and SF6{\mathrm{SF}}_{6}SF6​. The relation between the SSS-matrix pole position and the sss-wave scattering length is discussed.
Recent results for positron scattering from noble gases over an energy range from 0.5 to 60eV are presented. Measurements include the grand total ( σ GT ), Ps formation ( σ Ps ) and Grand total – Ps formation ( (σ GT -P s ) cross sections. Some preliminary DCS results will also be presented. Work on a formulation of modified effective range theory (MERT) is being undertaken to determine the value of the scattering length [1] which may be useful for identifying a bound state. Plans for experiments on metal atoms will be outlined.
The parameters for SSS-wave elastic scattering of near-zero-energy electrons and positrons by H2{H}_{2}H2​ molecules are calculated using the stabilization method with explicitly correlated Gaussians. The confined variational method is applied to optimize the Gaussians to describe the short-range interaction of incident ${e}^{\ifmmode\pm\else\textpm\fi{}}$ with H2{H}_{2}H2​ in the fixed-nuclei approximation. For e+{e}^{+}e+-H2{H}_{2}H2​ scattering the scattering length of previous work [Phys. Rev. Lett. 103, 223202 (2009)] is substantially improved. More importantly, for ${e}^{\ensuremath{-}}$-H2{H}_{2}H2​ scattering, from first principles, the scattering length is computed as a function of the internuclear distance. In the case that the two nuclei are at the equilibrium distance the results are in a good agreement with values derived from fitting experimental total and diffusion cross sections to the modified effective range theory.
A simple expansion of the scattering length in terms of SSS-matrix poles is derived for the angular momenta l>0l>0l>0. This expression shows that the dominant role in the low-energy collisions is played by the poles lying close to the origin of the complex momentum plane. Among these poles a peculiar class of virtual states is found to respond very weakly to the scaled perturbation potentials and thus resistant to producing trajectories in the complex momentum plane. Properties and an impact on low-energy collisions of these resistant virtual states are discussed.
Scattering processes are a fundamental way of experimentally probing distributions and properties of systems in several areas of physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than the range of the potential, partial waves with high angular momentum are typically unimportant. The dominant contribution comes from l=0l=0l=0 partial waves, commonly known as sss-wave scattering. This situation is very relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g. nuclear structure and matter. This manuscript is intended as a pedagogical introduction to the topic while covering a numerical approach to compute the desired quantities. We introduce low-energy scattering with particular attention to the concepts of scattering length and effective range. These two quantities appear in the effective-range approximation, which universally describes low-energy processes. We outline a numerical procedure for calculating the scattering length and effective range of spherically symmetric two-body potentials. As examples, we apply the method to the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to provide the tools so students can implement similar calculations and extend them to other potentials.
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.