Hyperspherical partial-wave theory applied to electron–hydrogen-atom ionization calculation for equal-energy-sharing kinematics

Abstract

Hyperspherical partial wave theory has been applied here in a new way in the calculation of the triple differential cross sections for the ionization of hydrogen atoms by electron impact at low energies for various equal-energy-sharing kinematic conditions. The agreement of the cross section results with the recent absolute measurements of Röder et al [51] and with the latest theoretical results of the ECS and CCC calculations [29] for different kinematic conditions at 17.6 eV is very encouraging. The other calculated results, for relatively higher energies, are also generally satisfactory, particularly for large Θ ab geometries. In view of the present results, together with the fact that it is capable of describing unequal-energy-sharing kinematics [35], it may be said that the hyperspherical partial wave theory is quite appropriate for the description of ionization events of electron-hydrogen type systems. It is also clear that the present approach in the implementation of the hyperspherical partial wave theory is very appropriate.

Hyperspherical partial wave calculation for double photoionization of the helium atom at 20 eV excess energy

“…With judicial choice of the parameter R ∞ and possibly with the availability of better computational facilities, the method may be applied from very low energy to high energy cases. At this point if we recall the capability of the hyperspherical partial wave approach in representing electron-hydrogen-atom ionization collisions [33,34,35] at low energies (and also consider situations of very low energy cases, with excess energy 1 eV and below for which we had to take R ∞ about 4000 -5000 a.u. and get reliable results for ionization cross sections [42]), consider the present success, then we may expect the hyperspherical partial wave theory to have a very good prospect.…”

“…But very often, this invites pseudo resonance type behaviour causing undesirable oscillations in the cross sections. Recently [35] for (e, 2e) problem, we applied the finite difference method (a five-point scheme) for solutions in the interval (0, ∆) and thereby get rid of undesirable oscillations. So we adopted here the same approach but with a seven-point scheme in place of the five-point scheme.…”

“…This is the Hyperspherical partial wave approach of one of the authors (Das [31,32,33,34], Das et al [35]) which is very successful in representing the three-particle continuum wave function in the final channel very accurately.…”

Hyperspherical partial wave theory with two-term error correction

“…Also we set P = p 2 1 + p 2 2 , α 0 = arctan(p 2 /p 1 ), p1 = (θ p1 , φ p1 ), p2 = (θ p2 , φ p2 ) and ω 0 = (α 0 , p1 , p2 ) where r i and p i (i = 1, 2) are the coordinates and momenta of i th charged particles. Ψ (−) f s is then expanded in symmetrized hyperspherical harmonics [1] that are functions of five angular variables and l 1 , l 2 , n, L, M , which are, respectively, the angular momenta of two electrons, the order of the Jacobi polynomial in hyperspherical harmonics, the total angular momentum and its projection. For a given symmetry s (s = 0 for singlet and s = 1 for triplet), we decompose the final state as…”

“…Another promising approach for the electron-hydrogen atom ionization problem is the Hyperspherical Partial Wave (HPW) approach. After the successful applications of HPW theory to compute triple differential equal-energy-sharing cross-section results [1,2,3,4,5,6], we aspire to calculate SDCS results. Before considering the full electron-hydrogen ionization problem, here, we consider Coulomb three-body system within Temkin-Poet (TP) model [7,8].…”