Elastic Scattering of Slow Electrons by Noble Gases—The Effective Range Theory and the Rigid Sphere Model

Abstract

We report on an extensive semi-empirical analysis of scattering cross-sections for electron elastic collision with noble gases via the Markov Chain Monte Carlo-Modified Effective Range Theory (MCMC−MERT). In this approach, the contribution of the long-range polarization potential (∼r−4) to the scattering phase shifts is precisely expressed, while the effect of the complex short-range interaction is modeled by simple quadratic expression (the so-called effective range expansion with several adjustable parameters). Additionally, we test a simple potential model of a rigid sphere combined with r−4 interaction. Both models, the MERT and the rigid sphere are based on the analytical properties of Mathieu functions, i.e., the solutions of radial Schrödinger equation with pure polarization potential. However, in contrast to MERT, the rigid sphere model depends entirely upon one adjustable parameter—the radius of a hard-core. The model’s validity is assessed by a comparative study against numerous experimental cross-sections and theoretical phase shifts. We show that this simple approach can successfully describe the electron elastic collisions with helium and neon for energies below 1 eV. The purpose of the present analysis is to give insight into the relations between the parameters of both models (that translate into the cross-sections in the very low energy range) and some “macroscopic” features of atoms such as the polarizability and atomic “radii”.

Calculation of electron scattering lengths on Ar, Kr, Xe, Rn and Og atoms

“…The theoretical results that we are comparing were obtained by different methods, with different degrees of sophistication, and therefore differ in the accuracy of the results. In the simplest methods, the interaction was described as a combination of the hard-sphere and the long-range polarization potential (rigid sphere approach, [27]). From the other side, we compare to one of the most advanced models, based on the R-matrix method [28].…”

“…All data are given in atomic units. A calculated from phase shifts and extrapolating results for k → 0 (method A) B B calculated from extrapolation of the linear part of the wave function at zero energy (method B) * no polarization a Kurokawa et al [3] b Haddad and O'Malley [7] c Ferch et al [8] d Weyhreter et al [30] e Buckman and Mitroy [31] f Petrovic et al [32] g Buckman and Lohmann [33] h Milloy et al [34] i Cheng et al [9] j Saha [10] k McEachran and Stauffer [35] l Mimnagh et al [36] m Bell et al [37] n Fedus [27] o Buckman and Lohmann [38] p Hunter et al [39] q England and Elford [40] r Brennan and Ness [41] s McEachran and Stauffer [42] t Zatsarinny et al [28] u Cheng et al [43] w Meshkov et al [44] Table 2. Values of the fitting parameters and the reciprocals of the fine structure constant obtained from them.…”

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

“…We have already shown in a series of papers that the approximation in equation ( 9) is valid over a broad range of energies (even up to the ionization threshold for atomic targets), e.g. see [39,40,47]. Importantly, Mathieu functions used to derive equations (6) -(9) are valid not only for real and positive k but in the sector −π < Arg k <π [33].…”

“…Therefore, our current approach is entirely based on equation 6, which was semi-empirically proved to be valid also for kR * > 1. (see [39,40,47]).…”

“Atoms” Special Issue (Electron Scattering in Gases—From Cross Sections to Plasma Modeling)

“…In this volume, we present a "cross-section" of different approaches, applications and processes, from industrial plasmas [3,4] to molecules of biological interest [5][6][7]; from experiments at ultra-low temperatures [8,9] to processes relevant for thermonuclear synthesis [10]. Thanks to a global response to the invitation, this volume hosts in an equilibrated manner the experimental aspects [6,8,9], ab-initio theories [7,[10][11][12] and semi-empirical approaches [13][14][15][16], both for electron and positron scattering [11,[15][16][17].…”