Comparative study of single and double electron capture from atoms by fast bare ions

Abstract

Total cross sections for single and double electron capture in collisions of bare ion with helium-like atoms are calculated at energies ranging from 50 to 6,000 keV/amu. The present calculations are carried out using four-body formalism of Coulomb–Born distorted wave formalism and boundary corrected continuum intermediate state approximation. We have made a comparison between single and double electron capture cross sections using both the methods in wide range of energies at different charge state of the projectile. Total cross sections have been calculated by summing over all the contributions up to n = 2 shells and sub-shells. It is also noted that double electron capture cross sections are small thorough out the energy region compared with the single electron capture cross sections. However, the present computed results for asymmetric collisions have been compared with available theoretical and experimental results. We find that our numerical results for total cross sections show good agreement with the available experimental findings.

Appendix

Here A, B, C and D in Eq. (13) of the main text are given by

$$ A = A_{1} y^{2} + 2y\left( {\beta_{1} A_{1} + A_{12} + A_{22} } \right) + A_{3} ,\quad B = B_{1} y^{2} + 2y\left( {\beta_{1} B_{1} + B_{12} + A_{22} } \right) + B_{3} $$

$$ C = C_{1} y^{2} + 2y\left( {\beta_{1} C_{1} + C_{12} + C_{22} } \right) + C_{3} ,\quad D = D_{1} y^{2} + 2y\left( {\beta_{1} D_{1} + D_{12} + D_{22} } \right) + D_{3} $$

where

$$ A_{1} = \left( {\lambda_{1} + \omega + \varepsilon_{i} } \right)^{2} + q^{2} ,\quad B_{1} = - 2\vec{k}_{i} \cdot \vec{q} - 2ik_{i} \left( {\lambda_{1} + \omega + \varepsilon_{i} } \right), $$

$$ C_{1} = 2\vec{k}_{f} \cdot \vec{q} - 2ik_{f} \left( {\lambda_{1} + \omega + \varepsilon_{i} } \right),\quad D_{1} = - 2\left( {\vec{k}_{i} .\vec{k}_{f} + k_{i} k_{f} } \right), $$

$$ A_{12} = \lambda_{1} \left\{ {\left( {\omega + \varepsilon_{i} } \right)^{2} + \beta_{1}^{2} + q^{2} } \right\},\quad B_{12} = - 2\vec{k}_{i} \cdot \vec{q}\lambda_{1} - 2ik_{i} \lambda_{1} \left( {\omega + \varepsilon_{i} } \right), $$

$$ C_{12} = 2\vec{k}_{f} \cdot \vec{q}\lambda_{1} - 2ik_{f} \lambda_{1} \left( {\omega + \varepsilon_{i} } \right),\quad D_{12} = - 2\lambda_{1} \left( {\vec{k}_{i} \cdot \vec{k}_{f} + k_{i} k_{f} } \right), $$

$$ A_{22} = \left( {\omega + \varepsilon_{i} } \right)Q_{1} ,\quad B_{22} = - ik_{i} Q_{1} ,\quad C_{22} = - ik_{f} Q_{1} ,\quad D_{22} = 0, $$

$$ A_{3} = Q_{2} \left\{ {\left( {\omega + \varepsilon_{i} + \beta_{1} } \right)^{2} + q^{2} } \right\},\quad B_{3} = - 2\vec{k}_{i} \cdot \vec{q}Q_{2} - 2ik_{i} Q_{2} \left( {\omega + \varepsilon_{i} + \beta_{1} } \right), $$

$$ C_{3} = 2\vec{k}_{f} \cdot \vec{q}Q_{2} - 2ik_{f} Q_{2} \left( {\omega + \varepsilon_{i} + \beta_{1} } \right),\quad D_{3} = - 2Q_{2} \left( {\vec{k}_{i} \cdot \vec{k}_{f} + k_{i} k_{f} } \right). $$

The term Q₁ and Q₂ can be explicitly written as

$$ Q_{1} = \left( {\lambda_{1}^{2} + \beta_{1}^{2} } \right),\quad Q_{2} = \left( {\lambda_{1} + \beta_{1} } \right)^{2} . $$