Creation, destruction, and transfer of atomic multipole moments by electron scattering: Liouville-space formulation

Abstract

In previous works, expressions for the atomic multipole moment cross sections were derived from a traditional collision approach. In the present work, we have derived the fundamental formula (see equation ( 35)) from which all of the atomic multipole moment cross sections can be obtained by using Liouville-space methods introduced by Fano (1963 Phys. Rev. 131 259). This simple, elegant formula is an expression for the multipole cross sections in terms of the Liouville-space transition operator (sometimes referred to as the tetradic transition matrix or the transition superoperator). The transition superoperator, in turn, can be expressed in terms of the traditional quantum mechanical transition operators via a formula which is sometimes referred to as 'Fano's convolution formula'. Upon application of this formula to our Liouville-space expression for the multipole cross sections, the resulting cross section formulae are identical to those obtained in previous works. Establishing this connection with the Liouville-space formalism allows us to apply powerful group theoretical techniques in order to obtain expressions of practical interest. As a specific example, we consider the transition rate for the final-state multipole moment which can be obtained via the use of a 'connecting factor' from the initial values of the multipole moments. The 'connecting factor', in turn, is expressed in this work as a Liouville-space matrix element of the tetradic transition matrix. Based on this expression and the symmetry properties of the electron-atom collisional system, certain symmetry relations are obtained for the 'connecting factors'. Since these factors are proportional to the multipole cross sections, corresponding relations are also obtained for those cross sections, which results in a reduction in the number of values that needs to be calculated for plasma modelling applications. An additional corollary of practical importance is that, in the case of cylindrically symmetric plasmas, the same symmetry relations also hold for the multipole rate coefficients. We provide an explicit derivation of this new, important result.

The creation, destruction and transfer of multipole moments in electron scattering by ions

“…The definition of cross sections associated with collision processes in Liouville space has been discussed by Snider and collaborators [46,47]. It has been recently shown [48] that such Liouville-space techniques can also be used for obtaining the fundamental formulae that lead to the atomic multipole cross sections, but, to the best of our knowledge, such techniques have not been used in the past for electronion scattering processes. This work in effect constitutes such an undertaking.…”

The creation, destruction, and transfer of multipole moments in electron–ion three-body recombination using the Gell-Mann–Goldberger–Watson method

“…In an earlier work [16] we have discussed electron scattering by targets with short-range interaction potentials in a Liouville-space description. In that work both initial and final channels belonged to the same arrangement channel (partition).…”

“…For conventional rate coefficients such a Liouville-space description has been given by Snider and collaborators [21][22][23]. Here we briefly discuss the generalization of their formalism for density matrices or the generalization of our earlier formalism in [16] for multiarrangement-channel processes.…”

“…Following our earlier work [16], we denote here by R z the rotational operator of the whole collisional system by the angle f around the z axis, which can then be written in the form R e ,…”

“…This approach is advantageous because it allows the assumption of an infinitely heavy nucleus at the start, and lends itself to a straightforward inclusion of exchange effects, thereby extending the applicability of the formula obtained to lower incident electron energies. Another advantage of this approach is the fact that it leads to a convenient formulation of the problem in Liouville space, thereby generalizing our earlier treatment [16]. The Liouville-space formulation, in turn, allows us, via the use of some symmetry arguments, to derive a selection rule for cylindrically symmetric plasmas.…”