# Frozen Core Approximation and Nuclear Screening Effects in Single Electron Capture Collisions

## Abstract

**:**

^{+}, He

^{2+}, Li

^{3+}, and C

^{6+}projectiles with velocities of 1 MeV/amu and 10 MeV/amu. In general, the DCS from the two models are found to differ by about one to two orders of magnitude with the active electron 4-body model showing better agreement with experiment. Comparison of the models reveals two possible sources of the magnitude difference: the inactive electron’s change of state and the projectile–target Coulomb interaction used in the different models. Detailed analysis indicates that the uncaptured electron’s change of state can safely be neglected in the frozen core approximation, but that care must be used in modeling the projectile–target interaction.

## 1. Introduction

^{+}+ He electron capture [37]. In our study of single ionization, we found that the initial state projectile–target interaction and the final state ionic potential were most influential on the magnitude and shape of the differential cross sections. The different treatments of these interactions in the three-body and four-body models represent different approximations for screening of the target nucleus and showed clear effects for both electron and heavy ion projectiles.

^{+}+ He single capture collision, our analysis also showed that the model used for screening either the projectile or the target nucleus could significantly affect the shape and magnitude of the DCS. In particular, much like the single ionization case, changes to the initial state projectile–target Coulomb interaction were the primary cause of the differences between the frozen core and active electron DCS. Based on the consistency of these prior studies, we expect the initial state projectile–target interaction to play an important role in the single capture process, as well. In order to focus solely on the effect of the frozen core approximation, we perform calculations with the first Born approximation (FBA), where the effect of the uncaptured electron can be studied in a straightforward manner. Clearly, more sophisticated models are available, but their use introduces additional complexities, making a one-to-one comparison of models more difficult. Our primary aim is to trace any differences in the DCS directly to specific aspects of the theoretical models and approximations. Using two simple models that are identical except for key features is crucial to limiting the possible causes of differences observed in the DCS. Atomic units are used throughout unless otherwise noted.

## 2. Theory

_{C}is the scattering angle of the projectile in the center of mass frame, which is related to the lab frame scattering angle by

^{+}, He

^{2+}, Li

^{3+}, and C

^{6+}projectiles. In the FBA, the motion of the incident and scattered projectiles is treated as a plane wave in both the initial and final state. The captured electron bound state is given by a hydrogenic wave function that is analytically known.

#### 2.1. Active Electron Four-Body Model

^{+}ion. We note that while the target electrons have been labeled here for clarity, their indistinguishability has been properly included by symmetrizing the total final state wave function with respect to the two electrons. The perturbation is given by the Coulomb interaction between the projectile and constituent particles of the helium atom

^{+}ion is simply a 1s hydrogenic wave function for charge $\beta =2$ given by

#### 2.2. Frozen Core Three-Body Model

_{i}and Z

_{i}can be found in [39].

## 3. Results and Discussion

#### 3.1. Screening and Correlation

^{+}, He

^{2+}, Li

^{3+}, and C

^{6+}projectiles colliding with helium at velocities of 1 Mev/amu and 10 MeV/amu. The validity of the Born approximation is restricted to energy regimes in which the perturbation parameter $\eta =\frac{{Z}_{p}}{{v}_{p}}$ is less than unity, and velocities chosen here ensure that this condition is met. A comparison of the three-body models shows that some differences appear, with the 5-Zeta wave function producing DCS that are uniformly about one order of magnitude larger than the hydrogenic wave function. The location of the minimum shifts to larger angles in the model with the 5-Zeta wave function, but generally the shape of the DCS does not change significantly with the choice of the target wave function.

#### 3.2. Frozen Core Effects

^{2}) atom to the ground state of a hydrogen-like atom with nuclear charge $\beta =2$. This electron is completely neglected in the three-body model and is therefore considered not to change state. Second, the perturbation potential is different for the three-body and four-body models. In particular, the four-body perturbation contains a sum of three terms, while the three-body perturbation is only two terms. Past work on the analysis of three-body and four-body models for single ionization has shown that the perturbation potential alters the shape of the fully differential cross sections [35,36]. Additionally, FBA models are known to predict a deep minimum in the electron capture DCS caused by a cancellation of terms in the perturbation [7,40,41,42].

^{2}) atom to the ground state of a He

^{+}ion. Second, the Coulomb interaction in the perturbation potential of the four-body model contains three terms, whereas in the three-body model, it only contains two terms. The four-body model perturbation contains the Coulomb interaction of the projectile with each constituent particle of the He

^{+}ion core, but the three-body model perturbation contains only an interaction for the core as a single point particle. Both of these differences between the three-body and four-body models can be individually explored through modifications to the four-body model.

^{+}ion wave function is hydrogenic with nuclear charge 27/16 (the same as the initial state orbital). All other features of the four-body model are unaltered. Results of the three-body, four-body, and four-body with $\beta =27/16$ models are shown in Figure 2. For all projectiles and projectile velocities, the final state He

^{+}ion’s nuclear charge has no discernable effect on the DCS. Closer examination of the two calculations reveals that they differ by at most 10%, with the results for $\beta =\frac{27}{16}$ being larger than those for $\beta =2$. Thus, we conclude that the change of state of the inactive electron is not the primary cause of the differences between the three-body and four-body DCS. This is consistent with our previous work for the single ionization process, in which the change of state of the inactive electron had a negligible effect on the DCS [35,36].

^{+}core consists of only one term in which the core is fully screened by the inactive electron and therefore modeled as a point particle with charge +1. To test the effect of this approximation, we perform a calculation using the four-body model, but replace the four-body perturbation of Equation (7) with the three-body perturbation of Equation (17). Results are shown in Figure 3, and it is apparent that the choice of perturbation significantly affects the magnitude of the DCS. In particular, the DCS using the four-body model with the three-body perturbation are generally smaller than those of the complete four-body calculation, although not as small as the three-body model. For all projectile energies, the DCS calculated using the four-body model with the three-body perturbation fall between the three-body and four-body models. This indicates that full target nuclear screening in the three-body model results in lower capture cross sections and points to the important role of nuclear–nuclear interactions in the capture process. Clearly the model chosen for the projectile–target interaction significantly alters the DCS much more than the inactive electron changing state. We can then conclude that the concept of the frozen core approximation, in which the inactive electron does not change state, is valid for single electron capture, but that care must be used in modeling the projectile–target interaction. These results are consistent with our past studies of the frozen core approximation and nuclear screening [35,36,37].

#### 3.3. Comparison with Experiment

^{+}+ He single capture collision system [37] and can be attributed to nuclear screening being less important in grazing collisions. In general, both models do a surprisingly good job of predicting experiment given their simplicity. However, no clear conclusion can be reached for which model best predicts experiment. In some cases, the four-body model is in better agreement, while in others, three-body model more closely matches the data. For H

^{+}projectiles (Figure 4), both the three-body and four-body models do a reasonable job predicting experiment at small and large scattering angles for the larger projectile velocities. However, both models underestimate the small angle DCS at 100 keV, which may be pushing the limits of the applicability of the FBA model. The deep minimum occurring in the three-body and four-body DCS typically appears near the elbow in the experimental data and causes a clear mismatch between theory and experiment in this region. In Figure 4a, the captured electron is known to be in the ground state, while in Figure 4b,c, the bound state of the captured electron is unknown. From Figure 7 and the Oppenheimer ${n}^{-3}$ rule, we expect that contributions from excited states will increase the three-body and four-body DCS by about 20%. For each energy, we include predictions of other available theoretical models, which are generally in better agreement with experiment. This is expected since each of the additional models includes better treatments of screening and correlation.

^{3}He

^{2+}projectiles (Figure 5), similar features are observed. However, in this case the four-body model better predicts experiment at small scattering angle, while at large scattering angles, the three-body model produces better agreement. This indicates that large angle scattering is overestimated by the use of a bare nucleus and that the nuclear–nuclear interaction in the four-body model is too strong. The non-perturbative two center-basis generator method (TC-BGM) model very accurately predicts experiment, while the perturbative Coulomb distorted wave–Born initial state (CDW-BIS) model predicts a similar deep minimum due to interference from the perturbative terms.

^{5+}projectiles. While the experimental results that were used here for comparison were able to distinguish capture to the ground state from capture to excited states, this is not always the case. Our calculations showed that capture to excited states is more important for higher charge projectiles, as expected.

## Funding

## Conflicts of Interest

## References

- Ullrich, J.; Moshammer, R.; Dorn, A.; Dörner, R.; Schmidt, L.P.H.; Schmidt-Böcking, H. Recoil-ion and electron momentum spectroscopy: reaction-microscopes. Rep. Prog. Phys.
**2003**, 66, 1463–1545. [Google Scholar] [CrossRef][Green Version] - Kim, H.-K.; Schöffler, M.S.; Houamer, S.; Chuluunbaatar, O.; Titze, J.N.; Schmidt, L.P.H.; Jahnke, T.; Schmidt-Böcking, H.; Galstyan, A.; Popov, Y.V.; et al. Electron transfer in fast proton-helium collisions. Phys. Rev. A
**2012**, 85, 022707. [Google Scholar] [CrossRef] - Mergel, V.; Dörner, R.; Khayyat, K.; Achler, M.; Weber, T.; Jagutzki, O.; Lüdde, H.J.; Cocke, C.L.; Schmidt-Böcking, H. Strong Correlations in the He Ground State Momentum Wave Function Observed in the Fully Differential Momentum Distributions for the p + He Transfer Ionization Process. Phys. Rev. Lett.
**2001**, 86, 2257–2260. [Google Scholar] [CrossRef] - Zapukhlyak, M.; Kirchner, T.; Hasan, A.; Tooke, B.; Schulz, M. Projectile angular-differential cross sections for transfer and transfer excitation in proton collisions with helium. Phys. Rev. A
**2008**, 77, 012720. [Google Scholar] [CrossRef] - Fischer, D.; Støchkel, K.; Cederquist, H.; Zettergren, H.; Reinhed, P.; Schuch, R.; Källberg, A.; Simonsson, A.; Schmidt, H.T. Experimental separation of the Thomas charge-transfer process in high-velocity p-He collisions. Phys. Rev. A
**2006**, 73, 052713. [Google Scholar] [CrossRef] - Fischer, D.; Gudmundsson, M.; Berényi, Z.; Haag, N.; Johansson, H.A.B.; Misra, D.; Reinhed, P.; Källberg, A.; Simonsson, A.; Støchkel, K.; et al. Importance of Thomas single-electron transfer in fast p-He collisions. Phys. Rev. A
**2010**, 81, 012714. [Google Scholar] [CrossRef] - Schöffler, M.S.; Titze, J.; Schmidt, L.P.H.; Jahnke, T.; Neumann, N.; Jagutzki, O.; Schmidt-Böcking, H.; Dörner, R.; Mančev, I. State-selective differential cross sections for single and double electron capture in He
^{+,2+}-He and p-He collisions. Phys. Rev. A**2009**, 79, 064701. [Google Scholar] [CrossRef] - Zapukhlyak, M.; Kirchner, T. Projectile angular-differential cross sections for electron transfer processes in ion-helium collisions: Evidence for the applicability of the independent electron model. Phys. Rev. A
**2009**, 80, 062705. [Google Scholar] [CrossRef] - Simony, P.R.; McGuire, J.H.; Eichler, J. Exact second Born electron capture for p + He. Phys. Rev. A
**1982**, 26, 1337–1343. [Google Scholar] [CrossRef] - Guo, D.L.; Ma, X.; Zhang, R.T.; Zhang, S.F.; Zhu, X.L.; Feng, W.T.; Gao, Y.; Hai, B.; Zhang, M.; Wang, H.B.; et al. State-selective electron capture in 30- and 100-keV He
^{+}+ He collisions. Phys. Rev. A**2017**, 95, 012707. [Google Scholar] [CrossRef] - Gao, J.W.; Wu, Y.; Wang, J.G.; Sisourat, N.; Dubois, A. State-selective electron transfer in He
^{+}+ He collisions at intermediate energies. Phys. Rev. A**2018**, 97, 052709. [Google Scholar] [CrossRef] - Adivi, E.G.; Bolorizadeh, M.A. Faddeev treatment of single-electron capture by protons in collision with many-electron atoms. J. Phys. B At. Mol. Opt. Phys.
**2004**, 37, 3321–3338. [Google Scholar] [CrossRef] - Slim, H.A.; Heck, E.L.; Bransden, B.H.; Flower, D.R. Theoretical differential cross sections for proton-helium scattering at intermediate energies. J. Phys. B At. Mol. Opt. Phys.
**1991**, 24, 2353–2358. [Google Scholar] [CrossRef] - Leigh, T.H. Fowler Ralph Howard on the capture of electrons by moving electrified particles. Proc. R. Soc. Lond. Ser. A
**1927**, 114, 561–576. [Google Scholar] - Oppenheimer, J.R. On the Quantum Theory of the Capture of Electrons. Phys. Rev.
**1928**, 31, 349–356. [Google Scholar] [CrossRef] - Brinkman, H.C.; Kramers, H.A. Zur Theorie der Einfangung von Elektronen durch α-Teilchen. Proc. Acad. Sci. Amst.
**1930**, 33, 973. [Google Scholar] - Belkić, D.; Mančev, I.; Hanssen, J. Four-body methods for high-energy ion-atom collisions. Rev. Mod. Phys.
**2008**, 80, 249–314. [Google Scholar] [CrossRef] - Martínez, A.E.; Deco, G.R.; Rivarola, R.D.; Fainstein, P.D. K-Shell vacancy production in asymmetric collisions. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At.
**1988**, 34, 32–36. [Google Scholar] [CrossRef] - Ghanbari-Adivi, E. Coulomb–Born distorted wave approximation applied to the proton–helium single-electron capture process. J. Phys. B At. Mol. Opt. Phys.
**2011**, 44, 165204. [Google Scholar] [CrossRef] - Dewangan, D.P.; Eichler, J. Electron capture and the long range of the Coulomb interaction. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At.
**1987**, 23, 160–163. [Google Scholar] [CrossRef] - Ghanbari-Adivi, E.; Ghavaminia, H. Four-body treatment of the single-electron capture in energetic proton–helium collisions. Phys. Scr.
**2014**, 89, 105402. [Google Scholar] [CrossRef] - Halder, S.; Mondal, A.; Samaddar, S.; Mandal, C.R.; Purkait, M. Differential and total cross sections for charge transfer and transfer-excitation in ion-helium collisions. Phys. Rev. A
**2017**, 96, 032717. [Google Scholar] [CrossRef] - Belkić, D.; Mančev, I. Formation of H-by double charge exchange in fast proton-helium collisions. Phys. Scr.
**1992**, 45, 35–42. [Google Scholar] [CrossRef] - Belkić, D.; Mančev, I. Four-body CDW approximation: dependence of prior and post total cross sections for double charge exchange upon bound-state wave-functions. Phys. Scr.
**1993**, 47, 18–23. [Google Scholar] [CrossRef] - Belkić, D. Symmetric double charge exchange in fast collisions of bare nuclei with heliumlike atomic systems. Phys. Rev. A
**1993**, 47, 189–200. [Google Scholar] [CrossRef] - Belkic, D. Two-electron capture from helium-like atomic systems by completely stripped projectiles. J. Phys. B At. Mol. Opt. Phys.
**1993**, 26, 497–508. [Google Scholar] [CrossRef] - Harris, A.L.; Madison, D.H. Effect of the center-of-mass approximation on the scaling of electron-capture fully differential cross sections. Phys. Rev. A
**2014**, 90, 022701. [Google Scholar] [CrossRef] - Mancev, I.; Mergel, V.; Schmidt, L. Electron capture from helium atoms by fast protons. J. Phys. B At. Mol. Opt. Phys.
**2003**, 36, 2733–2746. [Google Scholar] [CrossRef][Green Version] - Samanta, R.; Purkait, M.; Mandal, C.R. Single-electron-capture processes in collisions of He
^{2+}, Li^{q+}(q = 1, 2, 3), C^{6+}, and O^{8+}ions with helium. Phys. Rev. A**2011**, 83, 032706. [Google Scholar] [CrossRef] - Samaddar, S.; Halder, S.; Mondal, A.; Mandal, C.R.; Purkait, M.; Das, T.K. Single and double electron capture in p-He and α-He collisions. J. Phys. B At. Mol. Opt. Phys.
**2017**, 50, 065202. [Google Scholar] [CrossRef] - Mančev, I.; Milojević, N.; Belkić, D. Boundary-corrected four-body continuum-intermediate-state method: Single-electron capture from heliumlike atomic systems by fast nuclei. Phys. Rev. A
**2015**, 91, 062705. [Google Scholar] [CrossRef] - Mančev, I. Single-electron capture from helium-like atomic systems by bare projectiles. EPL
**2004**, 69, 200. [Google Scholar] [CrossRef] - Mancev, I. Single charge exchange in fast collisions of alpha particles with helium. J. Phys. B At. Mol. Opt. Phys.
**2002**, 36, 93–104. [Google Scholar] [CrossRef] - Mančev, I. Electron correlations in single-electron capture from heliumlike atomic systems. Phys. Rev. A
**1999**, 60, 351–358. [Google Scholar] [CrossRef] - Harris, A.L.; Morrison, K. Comprehensive study of 3-body and 4-body models of single ionization of helium. J. Phys. B At. Mol. Opt. Phys.
**2013**, 46, 145202. [Google Scholar] [CrossRef] - Harris, A.L. Effect of frozen core approximation in heavy-ion impact ionization of helium. J. Phys. B At. Mol. Opt. Phys.
**2015**, 48, 115203. [Google Scholar] [CrossRef] - Harris, A.L.; Plumadore, A. Quantum mechanical potentials and inactive electron effects in charge exchange collisions. J. Phys. B At. Mol. Opt. Phys.
**2019**, 52, 055203. [Google Scholar] [CrossRef] - McDowell, M.R.C.; Coleman, J.P. Introduction to the Theory of Ion-Atom Collisions; Elsevier Science Publishing Co Inc.: New York, NY, USA, 1970; ISBN 978-0-7204-0166-0. [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, 177–478. [Google Scholar] [CrossRef] - Ghanbari-Adivi, E.; Ghavaminia, H. Projectile angular-differential cross sections for single electron transfer in fast He
^{+}–He collisions. Chin. Phys. B**2015**, 24, 033401. [Google Scholar] [CrossRef] - Belkic, D.; Salin, A. Differential cross sections for charge exchange at high energies. J. Phys. B At. Mol. Phys.
**1978**, 11, 3905–3911. [Google Scholar] [CrossRef] - Omidvar, K. Asymptotic form of the charge-exchange cross section in three-body rearrangement collisions. Phys. Rev. A
**1975**, 12, 911–926. [Google Scholar] [CrossRef][Green Version] - Kambara, T.; Igarashi, A.; Watanabe, N.; Nakai, Y.; Kojima, T.M.; Awaya, Y. Recoil-ion momentum distribution of single-electron capture to the ground and excited states in 0.5–He collisions. J. Phys. B At. Mol. Opt. Phys.
**1997**, 30, 1251–1260. [Google Scholar] [CrossRef] - Mergel, V.; Dörner, R.; Ullrich, J.; Jagutzki, O.; Lencinas, S.; Nüttgens, S.; Spielberger, L.; Unverzagt, M.; Cocke, C.L.; Olson, R.E.; et al. State Selective Scattering Angle Dependent Capture Cross Sections Measured by Cold Target Recoil Ion Momentum Spectroscopy. Phys. Rev. Lett.
**1995**, 74, 2200–2203. [Google Scholar] [CrossRef] - Kamber, E.Y.; Cocke, C.L.; Cheng, S.; Varghese, S.L. Angular Distribution of Fast Protons from Singly and Doubly Ionizing Collisions with He. Phys. Rev. Lett.
**1988**, 60, 2026–2029. [Google Scholar] [CrossRef] - Dörner, R.; Ullrich, J.; Schmidt-Böcking, H.; Olson, R.E. Three-body interactions in proton-helium angular scattering. Phys. Rev. Lett.
**1989**, 63, 147–150. [Google Scholar] [CrossRef] - Gulyás, L.; Igarashi, A.; Kirchner, T. Projectile scattering in one- and two-electron transitions. J. Phys. B At. Mol. Opt. Phys.
**2012**, 45, 085205. [Google Scholar] [CrossRef]

**Figure 1.**Differential cross sections for single electron capture to the ground state with the residual He

^{+}ion also in the ground state. The columns contain results for incident projectile velocities of 1 MeV/amu ((

**a–d**)) and 10 MeV/amu ((

**e–h**)). Results are presented for projectiles H

^{+}(

**a**,

**e**), He

^{2+}(

**b**,

**f**), Li

^{3+}(

**c**,

**g**), and C

^{6+}(

**d**,

**h**) the three-body model with either the hydrogenic wave function of Equation (15) or the 5-Zeta wave function of Equation (18) as well as the four-body model with the variational wave function of Equation (13) or the correlated wave function of Equation (14).

**Figure 2.**Same as Figure 1, with DCS calculated using the four-body model with the uncorrelated wave function and the three-body model with the hydrogenic wave function. A calculation using the four-body model (uncorrelated) with $\beta =27/16$ is also shown.

**Figure 3.**Same as Figure 3, but results are included for the four-body model using the three-body perturbation of Equation (17), as described in the text.

**Figure 4.**Differential cross sections for p + He single electron capture to the ground state with the residual He

^{+}ion also in the ground state. Results from the three-body and four-body models are compared to experimental results from (

**a**) [7], (

**b**,

**c**) [6] and the theoretical models from (

**a**) [7], (

**b**) [31], and (

**c**) [47]. For (

**a**), the captured electron is known to be in the ground state in, however for (

**b**,

**c**), the final captured state of the electron is not specified. Incident projectile energies are (

**a**) 100 keV, (

**b**) 1.3 MeV, and (

**c**) 7.5 MeV.

**Figure 5.**Differential cross sections for

^{3}He

^{2+}+ He single electron capture to the ground state with the residual He

^{+}ion also in the ground state. Results from the three-body and four-body models are compared to experiment [7] and other theoretical models [7,8] Incident projectile energies are (

**a**) 300 keV/amu, (

**b**) 450 keV/amu, and (

**c**) 630 keV/amu. In all cases, the captured electron is known to be in the ground state.

**Figure 6.**Differential cross sections for B

^{5+}+ He single electron capture to the ground state with the residual He

^{+}ion also in the ground state. Experimental results are from [43]. Incident projectile energies are (

**a**) 750 keV/amu and (

**b**) 1 MeV/amu. In all cases, the captured electron is known to be in the ground state.

**Figure 7.**Differential cross sections using the four-body model (uncorrelated) for capture to the ground state and capture to all excited states up to and including $n=6$.

© 2019 by the author. 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**

Harris, A. Frozen Core Approximation and Nuclear Screening Effects in Single Electron Capture Collisions. *Atoms* **2019**, *7*, 44.
https://doi.org/10.3390/atoms7020044

**AMA Style**

Harris A. Frozen Core Approximation and Nuclear Screening Effects in Single Electron Capture Collisions. *Atoms*. 2019; 7(2):44.
https://doi.org/10.3390/atoms7020044

**Chicago/Turabian Style**

Harris, Allison. 2019. "Frozen Core Approximation and Nuclear Screening Effects in Single Electron Capture Collisions" *Atoms* 7, no. 2: 44.
https://doi.org/10.3390/atoms7020044