Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elastic and charge transfer cross sections for low to ultralow $\rm{H}(1s) + \rm{H}^{+}$ collisions. Quantal and semiclassical calculations

Mykhaylo Khoma

TL;DR

This work addresses elastic and resonant charge-transfer cross sections in $\rm{H}(1s)+\rm{H}^+$ collisions over $E_{ m cm}$ from $10^{-10}$ to $10$ eV. It combines fully quantum radial-equation solutions with Born–Oppenheimer potentials $E^{\rm BO}_{g,u}(R)$ and a semiclassical JWKB treatment (including a three-turning-point extension) to obtain phase shifts $\delta_{g}(l)$, $\delta_{u}(l)$ and the corresponding cross sections $\sigma^{(i)}_{\rm EL}$, $\sigma^{(d)}_{ m EL}$, and $\sigma_{ m CT}$. The results show excellent agreement (∼0.1%) with prior data across most energies, reveal new oscillations in the charge-transfer cross section around $E_{ m cm}\approx 0.127$ eV and $0.363$ eV, and demonstrate that the semiclassical approach remains reliable down to $E_{ m cm}\sim 0.01$ eV while highlighting the need for three-turning-point corrections at lower energies. The study thus extends the practical applicability of semiclassical methods to ultra-low-energy ion–atom collisions and provides detailed phase-shift data for $\rm{H}(1s)+\rm{H}^+$.

Abstract

The elastic scattering and resonant charge transfer integral cross sections in $\rm{H}(1s) + \rm{H^+}$ collisions are computed for the center-of-mass energy range of $10^{-10}-10$ eV. Fully quantal and semiclassical approaches are utilized in these calculations. The reliability of the semiclassical approximation for very low collision energies is discussed. The results are compared with available data from the literature.

Elastic and charge transfer cross sections for low to ultralow $\rm{H}(1s) + \rm{H}^{+}$ collisions. Quantal and semiclassical calculations

TL;DR

This work addresses elastic and resonant charge-transfer cross sections in collisions over from to eV. It combines fully quantum radial-equation solutions with Born–Oppenheimer potentials and a semiclassical JWKB treatment (including a three-turning-point extension) to obtain phase shifts , and the corresponding cross sections , , and . The results show excellent agreement (∼0.1%) with prior data across most energies, reveal new oscillations in the charge-transfer cross section around eV and eV, and demonstrate that the semiclassical approach remains reliable down to eV while highlighting the need for three-turning-point corrections at lower energies. The study thus extends the practical applicability of semiclassical methods to ultra-low-energy ion–atom collisions and provides detailed phase-shift data for .

Abstract

The elastic scattering and resonant charge transfer integral cross sections in collisions are computed for the center-of-mass energy range of eV. Fully quantal and semiclassical approaches are utilized in these calculations. The reliability of the semiclassical approximation for very low collision energies is discussed. The results are compared with available data from the literature.

Paper Structure

This paper contains 6 sections, 17 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Semiclassical calculation
  4. Quantal calculation
  5. Results
  6. Discussion and concluding remarks

Figures (4)

  • Figure 1: Charge transfer (CT), elastic distinguishable (ELd), and elastic indistinguishable (ELi) integral cross sections from the present fully quantal (solid lines) and JWKB (symbols $\bigcirc$, $\square$, and $\triangle$) calculations for center-of-mass collision energies within the range of $10^{-4}$--$10^{-2}$ eV. Other theoretical results: $-\cdot-$, the $\sigma_{\rm{CT}}$ and $\sigma^{\rm (i)}_{\rm{EL}}$ calculations of Krstić et al.Krstic-PRA-70; $-~-$, the $\sigma_{\rm{CT}}$ and $\sigma^{\rm (d)}_{\rm{EL}}$ calculations of Schultz et al.Schultz-Stancil; $\times$, the $\sigma_{\rm{CT}}$ calculations of Hunter and Kuriyan Hunter-77
  • Figure 2: (a) Charge transfer integral cross section from the present fully quantal (solid line) and JWKB (symbols $\bigcirc$) calculations within the range of $0.01$--$0.15$ eV. Other theoretical results: $\times$, Hunter-77; $-\cdot-$, Krstic-PRA-70; $-~-$, Kato-PRA. The inset shows a close-up of the sharp oscillation at $\sim$ 0.127 eV. (b) The same as in (a), except for the energy range of $0.15$--$0.4$ eV
  • Figure 3: Charge transfer (CT), elastic distinguishable (ELd), and elastic indistinguishable (ELi) integral cross sections for center-of-mass collision energies within the range of $0.01$--$10$ eV. Solid lines, the present quantal calculations; $-~-$, the quantal calculations from Schultz-Stancil; $-\cdot-$, the quantal calculations from Krstic-PRA-70; $\cdots$, the quantal calculations from Kato-PRA; $\bigtriangleup$ and $\square$, the JWKB calculations for ELi and CT from Krstic-jpb-99
  • Figure 4: Integral elastic cross sections for the energy range of $10^{-10}$--$10^{-4}$ eV. Lines, the present calculations of $\sigma^{\rm (d)}_{\rm{EL}}$ and $\sigma^{\rm (i)}_{\rm{EL}}$; symbols $\bigcirc$, the $\sigma^{\rm (i)}_{\rm{EL}}$ calculations from Schultz-astro-05