# Atomic Concealment Due to Loss of Coherence of the Incident Beam of Projectiles in Collision Processes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definition of the Cross Section

## 3. Single Scattering Condition

## 4. Stationary Scattering Theory

## 5. Validity of the Stationary Scattering Theory

## 6. Coherence Length and Interference

_{2}molecule. For the description of the effects produced by the coherence length $\ell $, we employed the angle convolution described in Ref. [10]. We refer the reader to this article for further details of how this technique allows for the incorporation of the coherence effects in the description of the scattering process.

## 7. Atomic Concealment

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Geiger, H.; Marsden, E. The Laws of Deflexion of Alpha Particules through Large Angles. Phil. Mag. S. 6
**1913**, 25, 604–623. [Google Scholar] [CrossRef][Green Version] - Taylor, J.R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions; Wiley: New York, NY, USA, 1972. [Google Scholar]
- Schulz, M. The Role of Projectile Coherence in the Few-Body Dynamics of Simple Atomic Systems. Adv. Atoms. Mol. Opt. Phys.
**2017**, 66, 507–543. [Google Scholar] - Barrachina, R.O.; Navarrete, F.; Ciappina, M.F.; Schulz, M. Coherence and contextuality in scattering experiments. In Ion-Atom Collisions: The Few-Body Problem in Dynamic Systems; Schulz, M., Ed.; De Gruyter: Berlin, Germany, 2019; pp. 61–80. [Google Scholar]
- Schiff, L.I. Quantum Mechanics, 1st ed.; McGraw-Hill: New York, NY, USA, 1949; p. 116. [Google Scholar]
- Seitz, F. LI Schiff, Quantum Mechanics. Bull. Am. Math. Soc.
**1950**, 56, 191–192. [Google Scholar] [CrossRef] - Bacciagaluppi, G.; Valentini, A. Quantum Theory at the Crossroads. In Reconsidering the 1927 Solvay Conference; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Wichmann, E.H. Scattering of Wave Packets. Am. J. Phys.
**1965**, 33, 20–31. [Google Scholar] [CrossRef] - Gottfried, K.; Yan, T.-M. Quantum Mechanics: Fundamentals, 2nd ed.; Springer: New York, NY, USA, 2003. [Google Scholar]
- Sarkadi, L.; Fabre, I.; Navarrete, F.; Barrachina, R.O. Loss of wave-packet coherence in ion-atom collisions. Phys. Rev. A
**2016**, 93, 032702. [Google Scholar] [CrossRef][Green Version] - Fabre, I.; Navarrete, F.; Sarkadi, L.; Barrachina, R.O. Free Evolution of an Incoherent Mixture of States: A quantum mechanical approach to the Van Cittert—Zernike Theorem. Eur. J. Phys.
**2018**, 39, 015401. [Google Scholar] [CrossRef][Green Version] - Barrachina, R.O.; Navarrete, F.; Ciappina, M.F. Free evolution of a mixture of quantum states with classical space and momentum uncertainties. Eur. J. Phys.
**2019**, 40, 065402. [Google Scholar] [CrossRef] - Van Cittert, P.H. Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene. Physica
**1934**, 1, 201. [Google Scholar] [CrossRef] - Zernike, F. The concept of degree of coherence and its application to optical problems. Physica
**1938**, 5, 785. [Google Scholar] [CrossRef] - Egodapitiya, K.N.; Sharma, S.; Hasan, A.; Laforge, A.C.; Madison, D.H.; Moshammer, R.; Schulz, M. Manipulating Atomic Fragmentation Processes by Controlling the Projectile Coherence. Phys. Rev. Lett.
**2011**, 106, 153202. [Google Scholar] [CrossRef] [PubMed] - Chowdhury, U.; Schulz, M.; Madison, D.H. Differential cross sections for single ionization of H
_{2}by 75-keV proton impact. Phys. Rev. A**2011**, 83, 032712. [Google Scholar] [CrossRef][Green Version] - Feagin, J.M.; Hargreaves, L. Loss of wave-packet coherence in stationary scattering experiments. Phys. Rev. A
**2013**, 88, 032705. [Google Scholar] [CrossRef] - Kouzakov, K.A. Theoretical analysis of the projectile and target coherence in COLTRIMS experiments on atomic ionization by fast ions. Eur. Phys. J. D
**2017**, 71, 70613. [Google Scholar] [CrossRef] - Nagy, L.; Járai-Szabó, F.; Borbély, S. The effect of projectile wave packet width on the fully differential ionization cross-sections. J. Phys. B
**2018**, 51, 144005. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Doubly differential cross section for the ionization of ${H}_{2}$ by the impact of 75 keV protons as a function of the scattering angle $\theta $ and the energy loss $\epsilon $ of the projectiles. The energy loss is 30 eV. The experimental (circles) [15] and theoretical (dashed lines) [10] results are compared, for L = 6.5 cm (green) and 50 cm (red). In both cases, D = 0.15 mm. The solid green line incorporates the atomic concealment effect into the theoretical result (dashed green line) for L = 6.5 cm, as described in Section 7.

**Figure 2.**Doubly differential cross section for the ionization of ${H}_{2}$ by the impact of 75 keV protons as a function of the scattering angle $\theta $ and the energy loss $\epsilon $ of the projectiles. The energy loss is 30 eV. The theoretical [10] results are compared, for L = 50 cm (dashed red line) and 6.5 cm, with (solid green line) and without (dashed green line) the atomic concealment effect. In both cases, D = 0.15 mm. Note that the logarithmic scale of the vertical axis in Figure 1 has been changed to a linear one.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Barrachina, R.O.; Navarrete, F.; Ciappina, M.F. Atomic Concealment Due to Loss of Coherence of the Incident Beam of Projectiles in Collision Processes. *Atoms* **2021**, *9*, 5.
https://doi.org/10.3390/atoms9010005

**AMA Style**

Barrachina RO, Navarrete F, Ciappina MF. Atomic Concealment Due to Loss of Coherence of the Incident Beam of Projectiles in Collision Processes. *Atoms*. 2021; 9(1):5.
https://doi.org/10.3390/atoms9010005

**Chicago/Turabian Style**

Barrachina, Raúl O., Francisco Navarrete, and Marcelo F. Ciappina. 2021. "Atomic Concealment Due to Loss of Coherence of the Incident Beam of Projectiles in Collision Processes" *Atoms* 9, no. 1: 5.
https://doi.org/10.3390/atoms9010005