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Physical Review A

Virtual states in electron-molecule scattering via modified effective-range theory

Kamil Fedus, Grzegorz P. Karwasz

Published Feb 1, 2024• 3 citations
Publisher

Abstract

The numerical approach to calculating the $S$-matrix poles associated with virtual states in electron-molecule collisions is proposed within the modified effective-range theory frame. The $S$-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 ${\mathrm{N}}_{2}, {\mathrm{CH}}_{4}, {\mathrm{CO}}_{2}$, and ${\mathrm{SF}}_{6}$. The relation between the $S$-matrix pole position and the $s$-wave scattering length is discussed.

DOI
10.1103/physreva.109.022801

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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.
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.
Electron scattering by rare-gas dimers is studied for very low incident energies using the zero range potential (ZRP) method. Beyond the traditional ZRP, we consider an alternative formulation accounting for the atomic polarization, inspired in the modified effective range theory. The scattering calculations are reported in fixed nuclei, rigid rotor, and rovibrational approximations, the first two being analytical. An expression for the electron-molecule scattering length is obtained. We find that short-range interactions are the dominant mechanism for rotational transitions in electron scattering by rare-gas dimers, while the long-range interactions may be neglected. Our results show how the elastic, rotational, and rovibrational cross sections depend on the electron-atom scattering length, on the molecular parameters, and also on the inclusion of polarization effects. The principle of detailed balance is discussed in the context of the ZRP method. Finally, we show that the rovibrational coupling considerably affects the rotational cross sections when the rotational constant becomes comparable to the vibrational constant.
Quantum scattering calculations are carried out down to very low energy for collisions of electrons with CF4{\mathrm{CF}}_{4}CF4​ molecules in the gas phase. The interaction forces are obtained from ab initio data for the static interaction and through model potentials for the exchange and correlation-polarization effects. A numerical analysis is performed to evaluate the partial wave convergence behavior over the whole energy range, to assess the reliability of modified effective range theory close to zero energy, and to extract the value and the sign of the scattering length. A comparison with experiments and earlier calculations in the low-energy domain is also presented.
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.
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.