Multiterm solution of a generalized Boltzmann kinetic equation for electron and positron transport in structured and soft condensed matter

Abstract

In this paper, we generalize the semiclassical Boltzmann kinetic equation for dilute gases to consider highly nonequilibrium electrons and positrons in soft condensed matter, accounting rigorously for all types of interactions, including positronium formation, and allowing for both coherent and incoherent scattering processes. The limitations inherent in the seminal paper of Cohen and Lekner [M. H. Cohen and J. Lekner, Phys. Rev. 158, 305 (1967); Y. Sakai, J. Phys. D 40, R441 (2007)] are avoided by solving the kinetic equation using a "multiterm" spherical harmonic representation of the velocity distribution function, as well as formulating a necessarily nonperturbative treatment of nonconservative collisional processes such as positronium formation. Numerical calculations of transport properties are carried out for a Percus-Yevick model of a hard-sphere system, and for positrons in liquid argon. New phenomena are predicted, including structure-induced negative conductivity and anisotropic diffusion.

On the approximation of transport properties in structured materials using momentum-transfer theory

“…The most general view of charged particle interactions with a medium is that of a wave representing the charged particle being scattered by the medium as a whole. A first approximation to the scattering is the 'single-scatterer approximation' [10] in which the scattered wave is the coherent sum of the contributions of many scattering centres in the molecule, which interfere to effectively produce a diffraction pattern of the medium [1]. The interference effects arising from coherent scattering processes can significantly affect the charged particle transport within the media.…”

“…From the definition of the double differential cross-section, d 2 σ/(dk dω), the rate of change of a property due to collisions can be found [1]:…”

“…The status of the field was detailed in a recent review by Sakai [8]. Even more recently, White et al [1,11] have formulated a new modification to the Boltzmann equation describing electron and positron transport in structured and soft-condensed matter. They solved the new kinetic equation using a multi-term solution technique that overcomes some of the limitations associated with previous solution techniques [8,9].…”

“…One aim of our work is to capitalize on this body of literature in the gas phase by adapting and applying it where possible. Our previous work focused on the implementation of gas phase cross-sections to consider such systems [1,11], and here we extend our aim to consider the calculation of transport properties in structured and soft-condensed materials from transport properties measured or calculated in the dilute gas phase limit.…”

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

“…The electron anisotropic nonequilibrium distribution functions are often expanded in the direct product of the spherical harmonics in (θ, φ) and the Sonine-Laguerre polynomials for the reduced speed or reduced energy dependence (Kumar et al 1980;Robson and Ness 1986;Robson 2006;White and Robson 2011;Abolhassani and Matte 2012). This methodology requires the matrix elements of the collision operator in the Boltzmann equation as discussed in Chap.…”