Markov Chain Monte Carlo Effective Range Analysis of Low-Energy Electron Elastic Scattering from Xenon

Kamil Fedus

doi:10.1007/s13538-015-0382-3

Abstract

Modified effective range theory formulated as a Bayesian statistical model through the combination with Markov Chain Monte Carlo integration and fitting tech- niques is used to check the compatibility of different e − −Xe scattering data such as the total cross-sections, the momentum transfer cross-sections, and the differential cross-sections that were determined experimentally in the region of Ramsauer-Townsend minimum. On the basis of this predictive approach, the most probable value of the scattering length, (−6.51 ± 0.05)a0, is proposed. The present analysis suggests that the non-relativistic spinless effective range theory is suitable for the description of angu- lar and energy dependencies of e − −Xe elastic scattering

Highlights

The modified effective range theory (MERT) for electron and positron scattering, originally proposed by O’Malley et al [1], is frequently used to extrapolate measured cross sections down to zero energy, the region hardly accessible experimentally
The f -wave phase shifts (η3) calculated using expressions introduced by Ali and Fraser [20] (3, 4) with and without (Born approximation) inclusion of quadrupole polarizability are shown for comparison. The latter results are multiply by 4 in order to match the scale on the vertical axis of d-wave plot role at high energies, E > 1 eV, and it has to be included within the MERT framework if we want to describe scattering cross-sections up to the threshold for the first inelastic process
This work shows that non-relativistic spinless model such as modified effective range theory (MERT) is able to describe cross-sections for low-energy electron (E < 10 eV) elastic scattering from Xenon atom at the same level of consistency with experiments as more advanced theoretical approaches

Summary

Introduction

The modified effective range theory (MERT) for electron and positron scattering, originally proposed by O’Malley et al [1], is frequently used to extrapolate measured cross sections down to zero energy, the region hardly accessible experimentally. The applicability of the original approach is limited to the very low energies To overcome this problem, Idziaszek and Karwasz proposed [3, 4] a different approach to MERT: the expression describing the scattering phase shifts of angular momentum partial waves as a function of incident electron energy is obtained exactly using Mathieu’s functions, i.e., analytical solutions of the radial Schrodinger equation with an adiabatic long-range dipole polarization potential (∼ r−4). It is proved that both the angular and the energy dependencies of scattering cross-sections can be well modeled within the framework of this non-relativistic model if the correct value of the static dipole polarizability is used an input parameter The latter quantity intrinsically contains the contribution of relativistic effects which are necessary in order to theoretically describe such a heavy atom as Xenon. MERT-derived dependence of the s-wave electron scattering length for rare-gas atoms on the dipole polarizability is discussed

MCMC–MERT

Total Cross Sections

Momentum Transfer Cross Sections

Differential Cross Sections

Scattering Length for Noble Gases

Conclusions