Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Abstract

The hyperspherical hidden crossing method with the correction term (HHCM +cor ) has been used to calculate partial wave Ps(1s)-formation cross sections for low-energy e + − H, e + − Li and e + − Na collisions. The Stückelberg phase varies in a systematic way as a function of atomic number Z , incident positron momentum k + , and total orbital angular momentum L and provides an explanation for the small S-wave cross section for all three systems. 13 References 13 Including the correction term does not compromise the simplicity of the method; it simply provides an improved value for the wave vector K (R). However, there is one undesirable consequence of using the correction term. The correction term diverges at the branch point, and therefore we include it only on the real axis. As a result, the cross section becomes path dependent. The correction term can be included in a consistent way by following the prescription for the paths given in [6]; we refer to these calculations as HHCM +cor .The HHCM has previously been applied with success to three-body collisions, including electron excitation [1] and impact ionization [1,[7][8][9][10][11][12]. Earlier HHCM calculations for Ps(1s) formation in e + − H collisions included an asymptotic approximation to the correction term (for the P-and D-waves) and provided an explanation for the small S-wave Ps(1s)-formation cross section in terms of the Stückelberg phase [2]. The HHCM was also used to treat near-threshold positron impact ionization of hydrogen [13]. The HHCM enables one to understand why the S-wave cross section for any process for e + − H collisions (including ionization) that proceeds via the Ps(1s) formation channel is small. Although the HHCM was derived for three-body Coulomb systems, it has been successfully used to study the recombination of three identical bosons with zero range two-body potentials [14]. We previously used the HHCM to calculate partial wave Ps(1s)-formation cross sections for e + − Li collisions [15]. Despite the success of these calculations, the application of the HHCM has been somewhat limited.We have carried out a systematic study of Ps(1s) formation in positron collisions with H, Li and Na using the HHCM +cor . The HHCM +cor results for e + − H and e + − Na reported here are new; we also include the previously published HHCM +cor e + − Li results [6] for completeness. We examine the behavior of the Stückelberg phase as a function of atomic number Z , incident positron momentum k + , and total orbital angular momentum L for the three atoms with a single l = 0 valence electron. For the alkalis, we use a model potential to describe the interaction of the valence electron and the positron with the ion core.

Threshold behavior of positronium formation in positron–alkali-metal scattering

“…Recently, Ward and Shertzer [20] and Ward et al [21] applied the hyperspherical hidden crossing method to Ps formation in positronlithium scattering and found a rise in the s-wave cross section for Ps formation as the momentum of the incident positron is decreased. A similar result was also obtained for positron-sodium scattering [26].…”

Solving the Faddeev-Merkuriev Equations in Total Orbital Momentum Representation via Spline Collocation and Tensor Product Preconditioning

“…Even though the basic mathematical model for such a broad spectrum of physical systems is the Schrödinger equation, the diversity of model interactions and particular physical states leads to a variety of employed computational methods [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Thus, our ability to perform direct model-free calculations for such wide range of systems is of utmost importance for many branches of physics.…”

Solving the Faddeev-Merkuriev equations in total orbital momentum representation via spline collocation and tensor product preconditioning

“…Even though the basic mathematical model for such a broad spectrum of physical systems is the Schrödinger equation, the diversity of model interactions and particular physical states leads to a variety of employed computational methods [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Thus, our ability to perform direct model-free calculations for such wide range of systems is of utmost importance for many branches of physics.…”