# Low-Energy Elastic Electron Scattering from Helium Atoms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Relativistic Polarized Orbital Approach

#### 2.2. Nonrelativistic B-Spline R-Matrix with Pseudostates

## 3. Results and Discussion

#### 3.1. Scattering Phase

#### 3.2. Scattering Length

`FARM`, but start to deviate from the latter and the almost straight line when k is smaller than about 0.07. This issue could likely be addressed by changing the default numerical parameters, but our goal is to illustrate that care is required when using these general codes in such relatively extreme situations.

#### 3.3. Cross Sections

**Figure 2.**Angle-integrated elastic and momentum-transfer (momtrans) cross section for e-He scattering. The lines show the present results obtained with RPO and the STGF asymptotic code for the BSR-498 model. The symbols show the same results for the FARM asymptotic package and the CCC results of Fursa and Bray [10] taken from the LXCat database [31].

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RPO | Relativistic Polarized Orbital |

BSR | B-Spline R-matrix |

## References

- Wan, J.Y.; Wu, M.S.; Zhang, J.Y.; Yan, Z.C. Confined variational calculations of low-energy electron-helium scattering. Phys. Rev. A
**2021**, 103, 042814. [Google Scholar] [CrossRef] - Atomic Spectra Database. Available online: https://www.nist.gov/pml/atomic-spectra-database (accessed on 12 September 2021).
- Brunger, M.J.; Buckman, S.J. Electron–molecule scattering cross-sections. I. Experimental techniques and data for diatomic molecules. Phys. Rep.
**2002**, 357, 215–458. [Google Scholar] [CrossRef] - Gopalan, A.; Bömmels, J.; Götte, S.; Landwehr, A.; Franz, K.; Ruf, M.W.; Hotop, H.; Bartschat, K. A novel electron scattering apparatus combining a laser photoelectron source and a triply differentially pumped supersonic beam target: Characterization and results for the He
^{−}(1s2s^{2}) resonance. Eur. Phys. J. D**2002**, 22, 17–29. [Google Scholar] [CrossRef] - Zatsarinny, O.; Bartschat, K. The B-spline R-matrix method for atomic processes: Application to atomic structure, electron collisions, and photoionization. J. Phys. B At. Mol. Opt. Phys.
**2013**, 46, 112001. [Google Scholar] [CrossRef] - Zatsarinny, O. BSR: B-Spline Atomic R-Matrix Codes. Comp. Phys. Commun.
**2006**, 174, 273. [Google Scholar] [CrossRef] - Oleg Zatsarinny’s GitHub Repository. Available online: https://github.com/zatsaroi (accessed on 12 September 2021).
- Atomic, Molecular, and Optical Sciences Gateway. Available online: https://ampgateway.org (accessed on 12 September 2021).
- Nesbet, R.K. Variational calculations of accurate e
^{-}-He cross sections below 19 eV. Phys. Rev. A**1979**, 20, 58–70. [Google Scholar] [CrossRef] - Fursa, D.V.; Bray, I. Calculation of electron-helium scattering. Phys. Rev. A
**1995**, 52, 1279. [Google Scholar] [CrossRef] [PubMed] - Hudson, E.T.; Bartschat, K.; Scott, M.P.; Burke, P.G.; Burke, V.M. Electron scattering from helium atoms. Phase shifts, resonance parameters and total cross sections. J. Phys. B At. Mol. Opt. Phys.
**1996**, 29, 5513. [Google Scholar] [CrossRef] - Chen, S.; McEachran, R.P.; Stauffer, A.D. Ab initio optical potentials for elastic electron and positron scattering from the heavy noble gases. J. Phys. B At. Mol. Opt. Phys.
**2008**, 41, 025201. [Google Scholar] [CrossRef] - McEachran, R.P.; Blanco, F.; García, G.; Brunger, M.J. A Relativistic Complex Optical Potential Calculation for Electron–Beryllium Scattering: Recommended Cross Sections. J. Phys. Chem. Ref. Data
**2018**, 47, 033103. [Google Scholar] [CrossRef] - McEachran, R.P.; Blanco, F.; García, G.; Stokes, P.W.; White, R.D.; Brunger, M.J. Integral Cross Sections for Electron–Magnesium Scattering Over a Broad Energy Range (0–5000 eV). J. Phys. Chem. Ref. Data
**2018**, 47, 043104. [Google Scholar] [CrossRef] - Grant, I.; McKenzie, B.; Norrington, P.; Mayers, D.; Pyper, N. An atomic multiconfigurational Dirac-Fock package. Comput. Phys. Commun.
**1980**, 21, 207–231. [Google Scholar] [CrossRef] - Hibbert, A. Energies and Oscillator Strengths Using Configuration Interaction Wavefunctions. In Computational Atomc Physics; Bartschat, K., Ed.; Springer: Heidelberg, Germany; New York, NY, USA, 1996; Chapter 3; pp. 27–64. [Google Scholar]
- McEachran, R.P.; Morgan, D.L.; Ryman, A.G.; Stauffer, A.D. Positron scattering from noble gases: Corrected results for helium. J. Phys. B At. Mol. Phys.
**1978**, 11, 951–953. [Google Scholar] [CrossRef] - McEachran, R.P.; Stauffer, A.D. Dynamic distortion effects in electron-atom scattering. J. Phys. B At. Mol. Opt. Phys.
**1990**, 23, 4605–4614. [Google Scholar] [CrossRef] - McEachran, R.P.; Stauffer, A.D. Relativistic Effects in Low-Energy Electron–Argon Scattering. Aust. J. Phys.
**1997**, 50, 511–5244. [Google Scholar] [CrossRef] [Green Version] - Puchalski, M.; Szalewicz, K.; Lesiuk, M.; Jeziorski, B. QED calculation of the dipole polarizability of helium atom. Phys. Rev. A
**2020**, 101, 022505. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.S.; Kar, S.; Ho, Y.K. Dynamic Multipole Polarizabilities of Helium and Screened-Helium Atoms. Atoms
**2020**, 8, 90. [Google Scholar] [CrossRef] - Zatsarinny, O.; Bartschat, K. Nonperturbative B-spline R-matrix-with-pseudostates calculations for electron-impact ionization of helium. Phys. Rev. A
**2012**, 85, 062709. [Google Scholar] [CrossRef] - Pekeris, C.L. 1
^{1}S and 2^{3}S States of Helium. Phys. Rev.**1959**, 115, 1216–1221. [Google Scholar] [CrossRef] - Bartschat, K.; Hudson, E.T.; Scott, M.P.; Burke, P.G.; Burke, V.M. Electron-atom scattering at low and intermediate energies using a pseudo-state/R-matrix basis. J. Phys. B At. Mol. Phys.
**1996**, 29, 115. [Google Scholar] [CrossRef] - Berrington, K.A.; Eissner, W.; Norrington, P.H. RMATRIX-I: Belfast Atomic R-Matrix Codes. Comput. Phys. Commun.
**1995**, 92, 290. [Google Scholar] [CrossRef] - O’Malley, T.F.; Spruch, L.; Rosenberg, L. Modification of Effective-Range Theory in the Presence of a Long-Range (r
^{−4}) Potential. J. Math. Phys.**1961**, 2, 491. [Google Scholar] [CrossRef] - Fernández-Menchero, L.; Conroy, A.; Ballance, C.; Badnell, N.; Mitnik, D.; Gorczyca, T.; Seaton, M. PSTGF: Time-independent R-matrix atomic electron-impact code. Comput. Phys. Commun.
**2020**, 256, 107489. [Google Scholar] [CrossRef] - Burke, V.; Noble, C. Farm—A flexible asymptotic R-matrix package. Comput. Phys. Commun.
**1995**, 85, 471–500. [Google Scholar] [CrossRef] - Ali, M.K.; Fraser, P.A. The contribution of long-range forces to low-energy phaseshifts. J. Phys. B At. Mol. Phys.
**1977**, 10, 3091–3104. [Google Scholar] [CrossRef] - Alves, L.L.; Bartschat, K.; Biagi, S.F.; Bordage, M.C.; Pitchford, L.C.; Ferreira, C.M.; Hagelaar, G.J.M.; Morgan, W.L.; Pancheshnyi, S.; Phelps, A.V.; et al. Comparisons of sets of electron–neutral scattering cross sections and swarm parameters in noble gases: II. Helium and neon. J. Phys. D Appl. Phys.
**2013**, 46, 334002. [Google Scholar] [CrossRef] - LXCat Database. Available online: https://nl.lxcat.net (accessed on 14 September 2021).

**Figure 1.**Extraction of the scattering length. Due to the limitations of the asymptotic FARM and STGF codes (see text for details), the smallest k-value in the BSR-498 calculation is 0.01, while the RPO code allows the calculation at zero collision energy.

k | Nesbet [9] | RMPS [11] | Wan et al. [1] | Present RPO | Present BSR-498 |
---|---|---|---|---|---|

0.1 | −0.1282 | −0.1260 | −0.1281 | −0.1262 | −0.1286 |

0.2 | −0.2655 | −0.2607 | −0.2650 | −0.2614 | −0.2560 |

0.3 | −0.4021 | −0.3979 | −0.4030 | −0.3978 | −0.4043 |

0.4 | −0.5388 | −0.5380 | −0.5319 | −0.5398 | |

0.5 | −0.6684 | −0.6617 | −0.6678 | −0.6609 | −0.6699 |

0.6 | −0.7930 | −0.7848 | −0.7907 | −0.7835 | −0.7932 |

0.7 | −0.9067 | −0.9062 | −0.8990 | −0.9088 | |

0.8 | −1.0155 | −1.0085 | −1.103 | −1.0069 | −1.0162 |

0.9 | −1.1163 | −1.1077 | −1.112 | −1.1071 | −1.1151 |

1.0 | −1.2056 | −1.202 | −1.1998 | −1.2055 | |

1.1 | −1.2848 | −1.2789 | −1.291 | −1.2854 | −1.2863 |

k | Nesbet [9] | RMPS [11] | Wan et al. [1] | Present RPO | Present BSR-498 |
---|---|---|---|---|---|

0.1 | 0.00308 | 0.0026 | 0.003029 | 0.00304 | 0.00305 |

0.2 | 0.01311 | 0.0129 | 0.01269 | 0.01264 | 0.01261 |

0.3 | 0.03063 | 0.0306 | 0.02964 | 0.02944 | 0.02945 |

0.4 | 0.05519 | 0.05390 | 0.05348 | 0.05352 | |

0.5 | 0.08605 | 0.0880 | 0.08449 | 0.08380 | 0.08386 |

0.6 | 0.1209 | 0.1244 | 0.1209 | 0.11856 | 0.11865 |

0.7 | 0.1588 | 0.1588 | 0.15535 | 0.15556 | |

0.8 | 0.1960 | 0.2000 | 0.1960 | 0.19175 | 0.19211 |

0.9 | 0.2305 | 0.2349 | 0.2305 | 0.22576 | 0.22657 |

1.0 | 0.2626 | 0.2626 | 0.25608 | 0.25772 | |

1.1 | 0.2932 | 0.2934 | 0.2932 | 0.28211 | 0.28524 |

k | Nesbet [9] | Wan et al. [1] | Present RPO | Present BSR-498 |
---|---|---|---|---|

0.1 | 0.0004 | 0.0004693 | 0.00042 | 0.00079 |

0.2 | 0.0017 | 0.001658 | 0.00165 | 0.00178 |

0.3 | 0.0037 | 0.003658 | 0.00367 | 0.00366 |

0.4 | 0.0066 | 0.006497 | 0.00650 | 0.00651 |

0.5 | 0.0104 | 0.01006 | 0.01017 | 0.01014 |

0.6 | 0.0149 | 0.01448 | 0.01471 | 0.01462 |

0.7 | 0.0203 | 0.01970 | 0.02010 | 0.02002 |

0.8 | 0.0265 | 0.02567 | 0.02630 | 0.02629 |

0.9 | 0.0335 | 0.0323 | 0.03318 | 0.03326 |

1.0 | 0.0414 | 0.0396 | 0.04060 | 0.04112 |

1.1 | 0.0501 | 0.0450 | 0.04837 | 0.04969 |

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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

McEachran, R.P.; Hamilton, K.R.; Bartschat, K.
Low-Energy Elastic Electron Scattering from Helium Atoms. *Atoms* **2021**, *9*, 82.
https://doi.org/10.3390/atoms9040082

**AMA Style**

McEachran RP, Hamilton KR, Bartschat K.
Low-Energy Elastic Electron Scattering from Helium Atoms. *Atoms*. 2021; 9(4):82.
https://doi.org/10.3390/atoms9040082

**Chicago/Turabian Style**

McEachran, Robert P., Kathryn R. Hamilton, and Klaus Bartschat.
2021. "Low-Energy Elastic Electron Scattering from Helium Atoms" *Atoms* 9, no. 4: 82.
https://doi.org/10.3390/atoms9040082