# A Complete CDW Theory for the Single Ionization of Multielectronic Atoms by Bare Ion Impact

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Prior Version of the CDW Approximation

#### 2.2. Post Version of the CDW Approximation

#### 2.3. Complete Hybrid Post CDW

#### 2.4. Study of the Divergences in the CDW Theory

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CDW | Continuum distorted wave |

CDW-EIS | Continuum distorted wave-eikonal initial state |

GSZ | Green–Sellin–Zachor |

## Appendix A. Fourier Transforms Depending on ϕ_{i} and ϕ_{f}

#### Appendix A.1. Fourier Transform in the Prior Versions of CDW

#### Appendix A.2. Fourier Transforms in the Post Versions of CDW

**Table A1.**Coefficients C${}_{l}$ and exponents Z${}_{l}$ for the expansion of the $\Omega $ function as described in Equation (A18) for each target considered.

He | Ne | Ar | |||
---|---|---|---|---|---|

C${}_{\mathit{l}}$ | Z${}_{\mathit{l}}$ | C${}_{\mathit{l}}$ | Z${}_{\mathit{l}}$ | C${}_{\mathit{l}}$ | Z${}_{\mathit{l}}$ |

1.77350 | 2.07594 | 0.67957 | 1.80073 | −2.88392 | 2.79017 |

−2.02508 | 3.40934 | −0.49762 | 3.35937 | 2.29974 | 3.13317 |

−1.62769 | 2.31277 | 1.11389 | 3.39575 | 0.22797 | 0.73040 |

4.98588 | 2.68460 | 0.07557 | 7.26484 | 0.14760 | 9.88781 |

−2.10744 | 2.21215 | −0.37146 | 3.17105 | 1.20838 | 2.22245 |

## References

- Belkić, D. A quantum theory of ionisation in fast collisions between ions and atomic systems. J. Phys. B At. Mol. Phys.
**1978**, 11, 3529–3552. [Google Scholar] [CrossRef] - Crothers, D.S.F.; McCann, J.F. Ionisation of atoms by ion impact. J. Phys. B At. Mol. Phys.
**1983**, 16, 3229–3242. [Google Scholar] [CrossRef] - Fainstein, P.D.; Ponce, V.H.; Rivarola, R.D. A theoretical model for ionisation in ion-atom collisions. Application for the impact of multicharged projectiles on helium. J. Phys. B At. Mol. Phys.
**1988**, 21, 287–299. [Google Scholar] [CrossRef] - Fainstein, P.D.; Ponce, V.H.; Rivarola, R.D. Two-centre effects in ionization by ion impact. J. Phys. B At. Mol. Phys.
**1991**, 24, 3091–3119. [Google Scholar] [CrossRef] - Galassi, M.E.; Champion, C.; Weck, P.F.; Rivarola, R.D.; Fojón, O.; Hanssen, J. Quantum-mechanical predictions of DNA and RNA ionization by energetic proton beams. Phys. Med. Biol.
**2012**, 57, 2081–2099. [Google Scholar] [CrossRef] - Gulyás, L.; Tóth, I.; Nagy, L. CDW-EIS calculation for ionization and fragmentation of methane impacted by fast protons. J. Phys. B At. Mol. Phys.
**2013**, 46, 075201. [Google Scholar] [CrossRef] - Gulyás, L.; Egri, S.; Ghavaminia, H.; Igarashi, A. Single and multiple electron removal and fragmentation in collisions of protons with water molecules. Phys. Rev. A
**2016**, 93, 032704. [Google Scholar] [CrossRef] [Green Version] - Mendez, A.M.P.; Montanari, C.C.; Miraglia, J.E. Ionization of biological molecules by multicharged ions using the stoichiometric model. J. Phys. B At. Mol. Phys.
**2020**, 53, 055201. [Google Scholar] [CrossRef] [Green Version] - Mendez, A.M.P.; Montanari, C.C.; Miraglia, J.E. Scaling rules for the ionization of biological molecules by highly charged ions. J. Phys. B At. Mol. Phys.
**2020**, 53, 175202. [Google Scholar] [CrossRef] - Gulyás, L.; Fainstein, P.D. CDW theory of ionization by ion impact with a Hartree-Fock-Slater description of the target. J. Phys. B At. Mol. Phys.
**1998**, 31, 3297–3305. [Google Scholar] [CrossRef] - Monti, J.M.; Fojón, O.A.; Hanssen, J.; Rivarola, R.D. Single electron ionization of multishell atoms: Dynamic screening and post-prior discrepancies in the CDW-EIS model. J. Phys. B At. Mol. Phys.
**2013**, 46, 145201. [Google Scholar] [CrossRef] - Ciappina, M.F.; Cravero, W.R.; Garibotti, C.R. Post–prior discrepancies in the continuum distorted wave–eikonal initial state approximation for ion–helium ionization. J. Phys. At. Mol. Opt. Phys.
**2003**, 36, 3775–3786. [Google Scholar] [CrossRef] - Ciappina, M.F.; Cravero, W.R. Post–prior discrepancies in CDW–EIS calculations for ion impact ionization fully differential cross-sections. J. Phys. B At. Mol. Opt. Phys.
**2006**, 39, 1091–1100. [Google Scholar] [CrossRef] [Green Version] - Monti, J.M.; Fojón, O.A.; Hanssen, J.; Rivarola, R.D. Influence of the dynamic screening on single-electron ionization of multi-electron atoms. J. Phys. B At. Mol. Phys.
**2010**, 43, 205203. [Google Scholar] [CrossRef] - Brauner, M.; Macek, J.H. Ion-impact ionization of He targets. Phys. Rev. A
**1992**, 46, 2519–2531. [Google Scholar] [CrossRef] [PubMed] - Dubeé, L.J.; Dewangan, D.P. Reinstating an Ionization Theory beyond Reasonable Doubt. In Proceedings of the ICPEAC XIX Conference, Whistler, BC, Canada, 26 July–1 August 1995; p. 62. [Google Scholar]
- Cheshire, I.M. Continuum distorted wave approximation; resonant charge transfer by fast protons in atomic hydrogen. Proc. Phys. Soc.
**1964**, 84, 89–98. [Google Scholar] [CrossRef] - Belkić, D.; Gayet, R.; Salin, A. Electron capture in high-energy ion-atom collisions. Phys. Rep.
**1979**, 56, 279–369. [Google Scholar] [CrossRef] - Stolterfoht, N.; DuBois, R.D.; Rivarola, R.D. Electron Emission in Heavy Ion-Atom Collisions; Springer: Berlin, Germany, 1997. [Google Scholar]
- Clementi, E.; Roetti, C. Roothaan-Hartree-Fock Atomic Wavefunctions: Basis Functions and Their Coefficients for Ground and Certain Excited States of Neutral and Ionized Atoms, Z <= 54. At. Data Nucl. Data Tables
**1974**, 14, 177. [Google Scholar] [CrossRef] - Gravielle, M.S.; Miraglia, J.E. Some Nordsieck integrals of interest in radiation and atomic collision theories. Comput. Phys. Commun.
**1992**, 69, 53–58. [Google Scholar] [CrossRef] - Szydlik, P.P.; Green, A.E. Independent-particle-model potentials for ions and neutral atoms with Z <= 18. Phys. Rev. A
**1974**, 9, 1885–1894. [Google Scholar] [CrossRef] - Piessens, R.; De Doncker-Kapenga, E.; Überhuber, C.W. Quadpack: A Subroutine Package for Automatic Integration; Springer: New York, NY, USA, 1983. [Google Scholar]
- Lee, D.H.; Richard, P.; Zouros, T.J.M.; Sanders, J.M.; Shinpaugh, J.L.; Hidmi, H. Binary-encounter electrons observed at 0° in collisions of 1-2-MeV/amu H
^{+}, C^{6+}, N^{7+}, O^{8+}, and F^{9+}ions with H_{2}and He targets. Phys. Rev. A**1990**, 41, 4816–4823. [Google Scholar] [CrossRef] [PubMed] - Biswas, S.; Misra, D.; Monti, J.M.; Tachino, C.A.; Rivarola, R.D.; Tribedi, L.C. Energy and angular distribution of electrons in ionization of He and Ne by 6-MeV/u bare carbon ions: Comparison with continuum-distorted wave eikonal-initial-state calculations in prior and post forms. Phys. Rev. A
**2014**, 90, 052714. [Google Scholar] [CrossRef] - Rudd, M.E.; Toburen, L.H.; Stolterfoht, N. Differential Cross Sections for Ejection of Electrons from Argon by Protons. At. Data Nucl. Data Tables
**1979**, 23, 405. [Google Scholar] [CrossRef]

**Figure 1.**Double differential cross-section for the single ionization of He atoms by $1.5$ MeV/u F${}^{9+}$ and H${}^{+}$ impact as a function of the electron energy for a ${0}^{\circ}$ fixed emission angle. Theory: full-line prior CDW integrated with a 20-point Gauss quadrature; dashed-double-dotted-line, complete post CDW integrated with a 20-point Gauss quadrature; dashed-line, complete post CDW integrated with a 60-point Gauss quadrature; dashed-dotted-line, complete post CDW integrated with QUADPACK library; experiments, circles [24].

**Figure 2.**Double differential cross-section for the single ionization of He atoms by $1.5$ MeV/u F${}^{9+}$ and H${}^{+}$ impact as a function of the electron energy for a ${0}^{\circ}$ fixed emission angle. Theory: dotted-line, post CDW; full-line, complete hybrid post CDW; dashed-line, CDW theory from [10]; experiments, circles [24].

**Figure 3.**Double differential cross-section for single ionization of He atoms by 6 MeV/u C${}^{6+}$ impact as a function of the electron energy for a fixed emission angle. Theory: dotted-line, post CDW; full-line, complete hybrid post CDW; experiments, circles [25].

**Figure 4.**Same as Figure 3, but for the ionization of Ne atoms.

**Figure 5.**Ionization DDCS of He (left panel) and Ne (right panel) by 6 MeV/u C${}^{6+}$ impact for a fixed ${60}^{\circ}$ emission angle as shown in Figure 3 and Figure 4. Dashed lines show the percentage of the relative difference (right axis) between post and complete hybrid post CDW calculations.

**Table 1.**Parameters d and K of the GSZ potential for the different targets considered, extracted from [22].

Target | d (a.u.) | K (a.u.) |
---|---|---|

He | 0.381 | 1.77 |

Ne | 0.558 | 2.71 |

Ar | 1.045 | 3.50 |

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**MDPI and ACS Style**

Monti, J.M.; Quinto, M.A.; Rivarola, R.D.
A Complete CDW Theory for the Single Ionization of Multielectronic Atoms by Bare Ion Impact. *Atoms* **2021**, *9*, 3.
https://doi.org/10.3390/atoms9010003

**AMA Style**

Monti JM, Quinto MA, Rivarola RD.
A Complete CDW Theory for the Single Ionization of Multielectronic Atoms by Bare Ion Impact. *Atoms*. 2021; 9(1):3.
https://doi.org/10.3390/atoms9010003

**Chicago/Turabian Style**

Monti, Juan M., Michele A. Quinto, and Roberto D. Rivarola.
2021. "A Complete CDW Theory for the Single Ionization of Multielectronic Atoms by Bare Ion Impact" *Atoms* 9, no. 1: 3.
https://doi.org/10.3390/atoms9010003