# Electron and Positron Scattering from Precious Metal Atoms in the eV to MeV Energy Range

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Potential Constituents and Their Influence on the DCS

## 4. Results

#### 4.1. Angle-Dependent DCS

**Figure 7.**Angular dependence of the differential cross-sections for 100 eV electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$. Experiment: Madison el al. [36] and Trajmar et al. [37]. Theory: Present OPM (—), Jablonski et al. [35], Czyzewski et al. [38], and Msezane and Henry [14].

**Figure 8.**Angular dependence of the differential cross-sections due to the OPM approach for electron scattering from Ni${}^{58}$, Cu${}^{63}$, Pd${}^{108}$, and Pt${}^{196}$ at impact energies of (

**a**) 500 eV and (

**b**) 10 keV.

**Figure 9.**Angular dependence of the differential cross-sections for electron scattering from Ni${}^{58}$ (—), Cu${}^{63}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $), Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(----)$, and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$ at impact energies of (

**a**) 1 MeV and (

**b**) 100 MeV.

#### 4.2. Angle-Dependent Sherman Function

#### 4.3. Energy-Dependent DCS and Sherman Function

**Figure 15.**Energy dependence of the differential cross-sections for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle of $\theta ={30}^{\xb0}$. Experiment: Madison el al. [36], Trajmar et al. [37], Ficenec et al. [41], Antonov et al. [42], and van der Laan [44]. Theory: Present OPM (—), Riley et al. [40], and Fink and Ingram [39].

**Figure 16.**Energy dependence of the differential cross-sections for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle of $\theta ={90}^{\xb0}$. Experiment: Madison el al. [36], Trajmar et al. [37], Ficenec et al. [41], van der Laan [44], and Shevchenko [43]. Theory: Present OPM (—), Riley et al. [40], and Fink and Ingram [39].

**Figure 17.**Energy-dependent Sherman function S for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle of $\theta ={90}^{\xb0}$, in comparison with the calculations of Fink and Ingram [39] for Cu${}^{63}$.

#### 4.4. Critical Minima and Total Polarization Points

#### 4.5. High-Energy Scaling Laws

- (a)
- Differential cross section

- (b)
- Sherman function

## 5. Positron Scattering and In-Plane Spin Asymmetries

#### 5.1. Angle-Dependent DCS

#### 5.2. Energy-Dependent DCS

#### 5.3. Sherman Function

#### 5.4. In-Plane Spin Polarization for Electrons and Positrons

- (a)
- Angular and energy dependence of U and T

- (b)
- High-energy scaling of U and T

## 6. Integrated Cross-Sections and Mean Free Paths

#### 6.1. Integrated Cross-Sections for Electron Scattering

**Figure 31.**Energy dependence of the integrated elastic cross-section (IECS) for electron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$. Experiment: (•) Trajmar et al. [37]. Theory: (—) present OPM approach, (◯) Mayol and Salvat [58], (⊖) Riley et al. [40], and (···) Zatsarinny and Bartschat [16].

**Figure 32.**Energy dependence of the momentum-transfer cross-section (MTCS) for electron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$. Experiment: (•) Trajmar et al. [37]. Theory: (—) present OPM approach, (◯) Mayol and Salvat [58], (⊖) Riley et al. [40], and (···) Zatsarinny and Bartschat [16].

**Figure 33.**The present OPM calculations of the viscosity cross-section (VCS) for electron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ in comparison with the Dirac–Hartree–Fock (DHF) calculations of Mayol and Salvat [58].

**Figure 34.**Energy dependence of the integrated inelastic cross-section (INCS) and total cross section (TCS) for electron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$. Theory: (—) present OPM approach and ($\cdots \cdots $) Gupta et al. [53].

**Figure 35.**Energy dependence of the total ionization cross-section (TICS) for electron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$. Experiment: (•) Koparnski reported in [51] for Ni, Freund et al. [55] and (▴) Bolorizadeh et al. [56] for Cu, and Nelson [57] for Au. Theory: (—) present OPM approach, ($\cdots \cdots $) Margreiter et al. [51], ($---$) Bartlett and Stelbovics [52] and Bartlett et al. [54], and (∘) Gupta et al. [53].

#### 6.2. Integrated Cross-Sections for Positron Scattering

#### 6.3. Mean Free Paths

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gargioni, E.; Grosswendt, B. Electron Scattering From Argon: Data Evaluation and Consistency. Rev. Mod. Phys.
**2008**, 80, 451. [Google Scholar] [CrossRef] - Horowitz, C. Parity violation in astrophysics. Eur. Phys. J. A
**2005**, 24, 167–170. [Google Scholar] [CrossRef] - Ichimura, S.; Shimizu, R. Backscattering correction for quantitative Auger analysis: I. Monte Carlo calculations of backscattering factors for standard materials. Surf. Sci.
**1981**, 112, 386. [Google Scholar] [CrossRef] - Mitroy, J.; Bromley, M.W.J.; Ryzhikh, G.G. Positron and positronium binding to atoms. J. Phys. B At. Mol. Opt. Phys.
**2002**, 35, R81. [Google Scholar] [CrossRef] - Breton, V.; Bricault, P.; Cardman, L.S.; Frois, B.; Goutte, D.; Isabelle, D.B.; Linzey, A.J.; Masson, G.; Maximon, L.C.; Offermann, E.A.J.M.; et al. High-accuracy comparison of electron and positron scattering from nuclei. Phys. Rev. Lett
**1991**, 66, 572–575. [Google Scholar] [CrossRef] [PubMed] - Shorifuddoza, M.; Das, P.K.; Kabir, R.; Haque, A.F.; Uddin, M.A. Angular distributions and critical minima in the elastic scattering of electrons by atomic copper. Int. J. Quantum Chem.
**2021**, 121, e26460. [Google Scholar] [CrossRef] - Motz, J.W.; Olsen, H.; Koch, H.W. Electron scattering without atomic or nuclear excitation. Rev. Mod. Phys.
**1964**, 36, 881–928. [Google Scholar] [CrossRef] - Kessler, J. Electron spin polarization by low-energy scattering from unpolarized targets. Rev. Mod. Phys.
**1969**, 41, 3–25. [Google Scholar] [CrossRef] - Tanuma, S.; Powell, C.J.; Penn, D.R. Calculations of electron inelastic mean free paths for 31 materials. Surf. Interface Anal.
**1988**, 11, 577–589. [Google Scholar] [CrossRef] - Powell, C.J.; Jablonski, A. Progress in quantitative surface analysis by X-ray photoelectron spectroscopy: Current status and perspectives. J. Electron. Spectrosc. Relat. Phenom.
**2010**, 178, 331–346. [Google Scholar] [CrossRef] - McCarthy, I.E.; Noble, C.J.; Phillips, B.A.; Turnbull, A.D. Optical model for electron scattering from inert gases. Phys. Rev. A
**1977**, 15, 2173. [Google Scholar] [CrossRef] - Berestetskii, V.B.; Lifshitz, E.M.; Pitaevskii, L.P. Quantum Electrodynamics; Elsevier: Oxford, UK, 1982; Volume 4. [Google Scholar]
- Saha, B.C.; Jakubassa-Amundsen, D.H.; Basak, A.K.; Haque, A.K.F.; Haque, M.M.; Khandker, M.H.; Uddin, M.A. Elastic scattering of electrons and positrons from alkali atoms. Adv. Quantum Chem. in press. [CrossRef]
- Msezane, A.Z.; Henry, R.J.W. Electron-impact excitation of atomic copper. Phys. Rev. A
**1986**, 33, 1631–1635. [Google Scholar] [CrossRef] - Zhou, Y.; Bray, I.; McCarthy, I. Model calculations of electron scattering from copper. J. Phys. B
**1999**, 32, 1033–1039. [Google Scholar] [CrossRef] - Zatsarinny, O.; Bartschat, K. Electron collisions with copper atoms: Elastic scattering and electron-impact excitation of the (3d
^{10}4s)^{2}S→(3d^{10}4p)^{2}P resonance transition. Phys. Rev. A**2010**, 82, 062703. [Google Scholar] [CrossRef] - Desclaux, J.P. A multiconfiguration relativistic Dirac-Fock program. Comput. Phys. Commun.
**1975**, 9, 31–45. [Google Scholar] [CrossRef] - Koga, T. Analytical Hartree-Fock electron densities for atoms He through Lr. Theor. Chim. Acta
**1997**, 95, 113–130. [Google Scholar] - Shorifuddoza, M.; Patoary, M.A.R.; Jakubassa-Amundsen, D.H.; Haque, A.K.F.; Uddin, M.A. Scattering of e
^{±}from ytterbium atoms. Eur. Phys. J. D**2019**, 73, 164. [Google Scholar] [CrossRef] - De Vries, H.; De Jager, C.W.; De Vries, C. Nuclear charge-density-distribution parameters from elastic electron scattering. At. Data Nucl. Data Tables
**1987**, 36, 495–530. [Google Scholar] [CrossRef] - Salvat, F.; Fernández-Varea, J.M.; Williamson, W., Jr. Accurate numerical solution of the radial Schrödinger and Dirac wave equations. Comput. Phys. Commun.
**1995**, 90, 151–168. [Google Scholar] [CrossRef] - Haque, A.K.F.; Haque, M.M.; Bhattacharjee, P.P.; Uddin, M.A.; Patoary, M.A.R.; Hossain, M.I.; Basak, A.K.; Mahbub, M.S.; Maaza, M.; Saha, B.C. Relativistic calculations for spin-polarization of elastic electron-mercury scattering. J. Phys. Commun.
**2017**, 1, 035014. [Google Scholar] [CrossRef] - Yennie, D.R.; Ravenhall, D.G.; Wilson, R.N. Phase-shift calculation of high-energy electron scattering. Phys. Rev.
**1954**, 95, 500. [Google Scholar] [CrossRef] - Joshipura, K.N.; Limbachiya, C.G. Theoretical total ionization cross-sections for electron impact on atomic and molecular halogens. Int. J. Mass Spectrom.
**2002**, 216, 239–247. [Google Scholar] [CrossRef] - Joshipura, K.N.; Vinodkumar, M.; Antony, B.K.; Mason, N.J. Theoretical total ionization cross-sections of CHx, CFx, SiHx, SiFx () and CCl4 targets by electron impact. Eur. Phys. J. D
**2003**, 23, 81–90. [Google Scholar] [CrossRef] - Guèye, P.; Kabir, A.A.; Giuliani, P.; Glister, J.; Lee, B.W.; Gilman, R.; Higinbotham, D.W.; Piasetzky, E.; Ron, G.; Sarty, A.J.; et al. Dispersive corrections in elastic electron-nucleus scattering: An investigation in the intermediate energy regime and their impact on the nuclear matter. Eur. Phys. J. A
**2020**, 56, 126. [Google Scholar] [CrossRef] - Überall, H. Electron Scattering from Complex Nuclei; Academic Press: New York, NY, USA, 1971. [Google Scholar]
- Donnelly, T.W.; Sick, I. Elastic magnetic electron scattering from nuclei. Rev. Mod. Phys.
**1984**, 56, 461–566. [Google Scholar] [CrossRef] - Beiser, B.A. Concepts of Modern Physics, 2nd ed.; McGraw-Hill Co.: New York, NY, USA, 1973. [Google Scholar]
- Bohr, N. XXXVII. On the constitution of atoms and molecules. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1913**, 26, 476–502. [Google Scholar] [CrossRef] - Sandor, R.K.J.; Blok, H.P.; Garg, U.; Harakeh, M.N.; De Jager, C.W.; Ponomarev, V.Y.; Vdovin, A.I.; De Vries, H. Interplay between single-particle and collective degrees of freedom in the excitation of the low-lying states in
^{142}Nd. Nucl. Phys. A**1991**, 535, 669–700. [Google Scholar] [CrossRef] - Segre, E. Nuclei and Particles: An Introduction to Nuclear and Subnuclear Physics; WA Benjamin: New York, NY, USA, 1974; p. 246. [Google Scholar]
- Ramsauer, C. Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen. i. fortsetzung. Ann. Phys.
**1921**, 66, 546–558. [Google Scholar] [CrossRef] - Townsend, J.S.; Bailey, V.A. LXX. The motion of electrons in argon. Philos. Mag.
**1922**, 43, 593–600. [Google Scholar] [CrossRef] - Jablonski, A.; Salvat, F.; Powell, C.J. Comparison of electron elastic-scattering cross sections calculated from two commonly used atomic potentials. J. Phys. Chem. Ref. Data
**2004**, 33, 409–451. [Google Scholar] [CrossRef] - Madison, D.H.; McEachran, R.P.; Ismail, M.; Teubner, P.J.O. Elastic scattering of electrons from copper at intermediate energies. J. Phys. B
**1998**, 31, 1127. [Google Scholar] [CrossRef] - Trajmar, S.; Williams, W.; Srivastava, S.K. Electron-impact cross sections for Cu atoms. J. Phys. B
**1977**, 10, 3323. [Google Scholar] [CrossRef] - Czyźewski, Z.; MacCallum, D.O.; Romig, A.; Joy, D.C. Calculations of Mott scattering cross section. J. Appl. Phys.
**1990**, 68, 3066–3072. [Google Scholar] [CrossRef] - Fink, M.; Ingram, J. Theoretical electron scattering amplitudes and spin polarizations: Electron energies 100 to 1500 eV Part II. Be, N, O, Al, Cl, V, Co, Cu, As, Nb, Ag, Sn, Sb, I, and Ta targets. At. Data Nucl. Data Tables
**1972**, 4, 129–207. [Google Scholar] [CrossRef] - Riley, M.E.; MacCallum, C.J.; Biggs, F. Theoretical electron-atom elastic scattering cross sections: Selected elements, 1 keV to 256 keV. At. Data Nucl. Data Tables
**1975**, 15, 443–476. [Google Scholar] [CrossRef] - Ficenec, J.R.; Trower, W.P.; Heisenberg, J.; Sick, I. Elastic electron-nickel scattering. Phys. Lett. B
**1970**, 32, 460–462. [Google Scholar] [CrossRef] - Antonov, A.N.; Kadrev, D.N.; Gaidarov, M.K.; de Guerra, E.M.; Sarriguren, P.; Udias, J.M.; Lukyanov, V.K.; Zemlyanaya, E.V.; Krumova, G.Z. Charge and matter distributions and form factors of light, medium, and heavy neutron-rich nuclei. Phys. Rev. C
**2005**, 72, 044307. [Google Scholar] [CrossRef] - Shevchenko, N.G.; Polishchuk, V.N.; Kasatkin, Y.A.; Khomich, A.A.; Buki, A.Y.; Mazanko, B.V.; Shula, G.V. Charge-density distribution in the nuclei CR-50, CR-52, CR-53, CR-54 AND FE-54, FE-56. Sov. J. Nucl. Phys.
**1978**, 28, 139–142. [Google Scholar] - van der Laan, J.B. Electron Scattering Off Palladium Isotopes. Ph.D. Thesis, University of Amsterdam, Amsterdam, The Netherlands, 1986. [Google Scholar]
- Walker, D.W. Relativistic effects in low energy electron scattering from atoms. Adv. Phys.
**1971**, 20, 257–323. [Google Scholar] [CrossRef] - Kelemen, V.I.; Remeta, E.Y. Critical minima and spin polarization in the elastic electron scattering by the mercury atoms. J. Phys. B
**2012**, 45, 185202. [Google Scholar] [CrossRef] - Jakubassa-Amundsen, D.H. Equivalence of a tip bremsstrahlung quantum and an elastically scattered electron at ultrahigh energies. Phys. Rev. A
**2012**, 85, 042714. [Google Scholar] [CrossRef] - Jakubassa-Amundsen, D.H. An asymptotic DSM theory for high-energy near-tip bremsstrahlung. J. Phys. G
**2020**, 47, 075102, The angular scaling in (4.1) and (4.7) therein should be reversed. [Google Scholar] [CrossRef] - Bjorken, J.D.; Drell, S.D. Relativistic Quantum Mechanics; Mc Graw-Hill: New York, NY, USA, 1964. [Google Scholar]
- Dapor, M.; Miotello, A. Differential, total, and transport cross sections for elastic scattering of low energy positrons by neutral atoms (Z = 1–92, E = 500–4000 eV). At. Data Nucl. Data Tables
**1998**, 69, 1–100. [Google Scholar] [CrossRef] - Margreiter, D.; Deutsch, H.; Märk, T.D. A semiclassical approach to the calculation of electron impact ionization cross-sections of atoms: From hydrogen to uranium. Int. J. Mass Spectrom. Ion Process.
**1994**, 139, 127–139. [Google Scholar] [CrossRef] - Bartlett, P.L.; Stelbovics, A.T. Calculation of electron-impact total-ionization cross sections. Phys. Rev. A
**2002**, 66, 012707. [Google Scholar] [CrossRef] - Gupta, D.; Naghma, R.; Antony, B. Electron impact total and ionization cross sections for Sr, Y, Ru, Pd, and Ag atoms. Can. J. Phys.
**2013**, 91, 744–750. [Google Scholar] [CrossRef] - Bartlett, P.L.; Stelbovics, A.T. Electron-impact ionization cross sections for elements Z= 1 to Z= 54. At. Data Nucl. Data Tables
**2004**, 86, 235–265. [Google Scholar] [CrossRef] - Freund, R.S.; Wetzel, R.C.; Shul, R.J.; Hayes, T.R. Cross-section measurements for electron-impact ionization of atoms. Phys. Rev.
**1990**, 41, 3575. [Google Scholar] [CrossRef] - Bolorizadeh, M.A.; Patton, C.J.; Shah, M.B.; Gilbody, H.B. Multiple ionization of copper by electron impact. J. Phys. B
**1994**, 27, 175. [Google Scholar] [CrossRef] - Nelson, A.N. Technical Report AFML-TR-75-198; Massachusetts Inst of Tech.: Cambridge, MA, USA, 1975. [Google Scholar]
- Mayol, R.; Salvat, F. Total and transport cross sections for elastic scattering of electrons by atoms. At. Data Nucl. Data Tables
**1997**, 65, 55–154. [Google Scholar] [CrossRef] - Liljequist, D. A simple calculation of inelastic mean free path and stopping power for 50 eV–50 keV electrons in solids. J. Phys. D
**1983**, 16, 1567. [Google Scholar] [CrossRef] - Iakoubovskii, K.; Mitsuishi, K.; Nakayama, Y.; Furuya, K. Mean free path of inelastic electron scattering in elemental solids and oxides using transmission electron microscopy: Atomic number dependent oscillatory behavior. Phys. Rev. B
**2008**, 77, 104102. [Google Scholar] [CrossRef] - Pierce, D.T.; Siegmann, H.C. Attenuation Length of Hot Electrons in Ferromagnetic Ni. AIP Conf. Proc.
**1974**, 18, 1393. [Google Scholar] - Wooten, F.; Breen, W.M.; Stuart, R.N. Hot-electron scattering and the rigid-band model in ferromagnetic Ni and Ni-Al alloys. Phys. Rev.
**1968**, 165, 703–706. [Google Scholar] [CrossRef] - Jackson, D.C.; Gallon, T.E.; Chambers, A. A model for the Auger electron spectroscopy of systems exhibiting layer growth, and its application to the deposition of silver on nickel. Surf. Sci.
**1973**, 36, 381–394. [Google Scholar] [CrossRef] - Burke, M.A.; Schreurs, J.J. The inelastic mean free paths of auger electrons in thin films of copper and nickel. Surf. Interface Anal.
**1982**, 4, 42–46. [Google Scholar] [CrossRef] - Ridgway, J.W.; Haneman, D. Auger spectra and LEED patterns from vacuum cleaved silicon crystals with calibrated deposits of iron. Surf. Sci.
**1971**, 24, 451–458. [Google Scholar] [CrossRef] - Seah, M.P. Quantitative Auger electron spectroscopy and electron ranges. Surf. Sci.
**1972**, 32, 703–728. [Google Scholar] [CrossRef] - Brunner, J.; Zogg, H. Angular dependence of X-ray photoelectrons. J. Electron Spectrosc. Relat. Phenom.
**1974**, 5, 911–920. [Google Scholar] [CrossRef] - Mrozek, P.; Jablonski, A.; Sulyok, A. The inelastic mean free path of electrons in the ordered Al48Ni52 alloy. Surf. Interface Anal.
**1988**, 11, 499–501. [Google Scholar] [CrossRef] - Knapp, J.A.; Himpsel, F.J.; Eastman, D.E. Experimental energy band dispersions and lifetimes for valence and conduction bands of copper using angle-resolved photoemission. Phys. Rev. B
**1979**, 19, 4952. [Google Scholar] [CrossRef] - Palmberg, P.W.; Rhodin, T.N. Auger electron spectroscopy of fcc metal surfaces. J. Appl. Phys.
**1968**, 39, 2425–2432. [Google Scholar] [CrossRef] - Tanuma, S.; Shiratori, T.; Kimura, T.; Goto, K.; Ichimura, S.; Powell, C.J. Experimental determination of electron inelastic mean free paths in 13 elemental solids in the 50 to 5000 eV energy range by elastic-peak electron spectroscopy. Surf. Interface Anal.
**2005**, 37, 833–845. [Google Scholar] [CrossRef] - Shinotsuka, H.; Tanuma, S.; Powell, C.J.; Penn, D.R. Calculations of electron inelastic mean free paths. X. Data for 41 elemental solids over the 50 eV to 200 keV range with the relativistic full Penn algorithm. Surf. Interface Anal.
**2015**, 47, 871–888. [Google Scholar] [CrossRef] - Penn, D.R. Electron mean-free-path calculations using a model dielectric function. Phys. Rev. B
**1987**, 35, 482–486. [Google Scholar] [CrossRef] - Ashley, J.C. Energy loss rate and inelastic mean free path of low-energy electrons and positrons in condensed matter. J. Electron Spectrosc. Relat. Phenom.
**1990**, 50, 323–334. [Google Scholar] [CrossRef] - Tanuma, S.; Powell, C.J.; Penn, D.R. Calculations of electron inelastic mean free paths. III. Data for 15 inorganic compounds over the 50–2000 eV range. Surf. Interface Anal.
**1991**, 17, 927–939. [Google Scholar] [CrossRef] - Ghosh, V.J.; Aers, G.C. Positron stopping in elemental systems: Monte Carlo calculations and scaling properties. Phys. Rev. B
**1995**, 51, 45–59. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Coordinate system showing the $(x,z)$ scattering plane spanned by the lepton momenta ${\overrightarrow{k}}_{i}$ and ${\overrightarrow{k}}_{f}$, the spin polarization vectors ${\overrightarrow{\zeta}}_{i}$ and ${\overrightarrow{\zeta}}_{f}$, and the scattering angle $\theta $.

**Figure 2.**Electron charge density (${\varrho}_{e}$) for Ni${}^{58}$ (—, black), Cu${}^{63}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$, red), Pd${}^{108}$ (−-−-, blue), and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-$, green) as a function of the distance from the nucleus.

**Figure 3.**Nuclear charge density (${\varrho}_{N}$) for Ni${}^{58}$ (—), Cu${}^{63}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$, Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(----)$, and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$ as a function of the nuclear coordinate, ${r}_{N}$.

**Figure 4.**Energy dependence (

**top**: 1–10${}^{4}$ eV,

**bottom**: ${10}^{4}$–${10}^{9}$ eV) of the differential cross-sections for electron scattering from (

**a1**,

**a2**) Ni${}^{58}$, (

**b1**,

**b2**) Cu${}^{63}$, (

**c1**,

**c2**) Pd${}^{108}$, and (

**d1**,

**d2**) Pt${}^{196}$ at a fixed scattering angle $\theta ={90}^{\xb0}$. Shown in the top panels are the results from ${V}_{\mathrm{st}}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $, green), ${V}_{\mathrm{st}}+{V}_{\mathrm{ex}}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-$, blue), ${V}_{\mathrm{st}}+{V}_{\mathrm{ex}}+{V}_{\mathrm{cp}}\phantom{\rule{0.277778em}{0ex}}(---$, red), and ${V}_{\mathrm{OPM}}$ (—, black). The bottom panels show the results from ${V}_{\mathrm{st}}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $, green), ${V}_{\mathrm{OPM}}$ (—, black), ${V}_{\mathrm{c}}\phantom{\rule{0.277778em}{0ex}}(---$, red), and ${V}_{\mathrm{nuc}}\phantom{\rule{0.277778em}{0ex}}(\xb7\xb7\xb7\phantom{\rule{0.166667em}{0ex}}\xb7$, blue).

**Figure 5.**Angular dependence of the differential cross sections for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a fixed energy ${E}_{i}$ = 40 eV. Shown are the results from ${V}_{\mathrm{st}}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $, green); ${V}_{\mathrm{st}}+{V}_{\mathrm{ex}}+{V}_{\mathrm{cp}}\phantom{\rule{0.277778em}{0ex}}(---$, red); and ${V}_{\mathrm{OPM}}$ (—, black), along with the results for the Coulomb field, ${V}_{\mathrm{c}}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-$, blue).

**Figure 10.**Angular variation of the Sherman function predicted by the OPM approach for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at impact energy ${E}_{i}$ = 40 eV.

**Figure 11.**Angular variation of the Sherman function for electron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at impact energy ${E}_{i}$ = 100 eV as computed by the present OPM (—) and by Fink and Ingram $(\xb0\xb0\xb0)$ [39].

**Figure 12.**Angular variation of the Sherman function predicted by the OPM approach for electron scattering from Ni${}^{58}$, Cu${}^{63}$, Pd${}^{108}$, and Pt${}^{196}$ at impact energies of (

**a**) 500 eV and (

**b**) 1 keV.

**Figure 13.**Angular variation of the Sherman function due to the present OPM approach for electron scattering from Ni${}^{58}$, Cu${}^{63}$, Pd${}^{108}$, and Pt${}^{196}$ at impact energies of 10 keV (

**a**) and 100 keV (

**b**).

**Figure 14.**Angular variation of the Sherman function S for electron scattering from Ni${}^{58}$ (—), Cu${}^{63}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$, Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(----)$, and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$ at impact energies of (

**a**) 1 MeV and (

**b**) 100 MeV.

**Figure 18.**Angular dependence of the DCS for three incident energies in the vicinity of the critical minima: (

**a**) (${E}_{\mathrm{c}}=6.8;{\theta}_{\mathrm{c}}=$ 104.5°) for Ni, (

**b**) (${E}_{\mathrm{c}}=3.57;{\theta}_{\mathrm{c}}=$ 73.0°) for Cu, (

**c**) (${E}_{\mathrm{c}}=2.93;$${\theta}_{\mathrm{c}}=$ 116.6°) for Pd, and (

**d**) (${E}_{\mathrm{c}}=12.65;{\theta}_{\mathrm{c}}=$ 43.0°) for Pt.

**Figure 19.**Differential cross section for 140 MeV electron impact (—) on Ni${}^{58}$ (lower line) and Pt${}^{196}$ (upper line) as function of the scattering angle $\theta $. Included are the results at 70 MeV impact for Ni${}^{58}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$ and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$ (scaled to 140 MeV) as well as the scaled results at 280 MeV impact for Ni${}^{58}\phantom{\rule{0.277778em}{0ex}}(----)$ and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\cdots -\cdots -)$ according to (15), with ${E}_{i}^{\left(1\right)}$ = 140 MeV.

**Figure 20.**Sherman function from 140 MeV electron impact (—) on (

**a**) Pd${}^{108}$ (upper line near ${40}^{\xb0}$) and Pt${}^{196}$ (lower line near ${40}^{\xb0}$) and (

**b**) Ni${}^{58}$ as function of the scattering angle $\theta $. Included in (a) are the scaled results (to 140 MeV, according to (19)) for 70 MeV impact on Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$ and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$ and for 280 MeV impact on Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(----)$ and Pt${}^{196}$ (short-dashed line). Included in (

**b**) are the scaled results according to (19) for 70 MeV $(-\xb7-\xb7-)$ and 280 MeV $(----)$ impact. The scaled results for 70 MeV impact according to Born scaling of the magnitude of $S\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots )$ are included as well.

**Figure 21.**Angular dependence of the differential cross-sections calculated using our OPM approach for 100 eV positron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$.

**Figure 22.**Angular dependence of the differential cross-sections for 500 eV positron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ calculated from our OPM approach in comparison with the results of Dapor and Miotello [50].

**Figure 23.**Energy-dependent differential cross-sections for positron scattering (—) from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle $\theta ={30}^{\xb0}$. The calculations of Dapor and Miotello [50] are shown as well. The electron scattering results ($\cdots \cdots $) from Figure 15 are included here for comparison.

**Figure 24.**Energy dependent differential cross-sections for positron (—) scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle $\theta ={90}^{\xb0}$. The calculations of Dapor and Miotello [50] are shown as well. The electron scattering results ($\cdots \cdots $) from Figure 16 are included for comparison.

**Figure 25.**Energy-dependent Sherman function S for positron scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a scattering angle $\theta ={90}^{\xb0}$.

**Figure 26.**Energy variation of the spin asymmetry parameter U for both electron (—) and positron ($\cdots \cdots $) impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at scattering angle $\theta ={90}^{\xb0}$.

**Figure 27.**Energy variation of the spin asymmetry parameter T for both electron and positron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at scattering angle $\theta ={90}^{\xb0}$.

**Figure 28.**Angular variation of the spin asymmetry parameter U for e${}^{\pm}$ scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a collision energy of 100 eV. (—, red), electrons; ($\cdots \cdots $, blue), positrons.

**Figure 29.**Angular variation of the spin asymmetry parameter T for e${}^{\pm}$ impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ at a collision energy of 100 eV. (—, red), electrons; ($\cdots \cdots $, blue), positrons.

**Figure 30.**(

**a**) Spin asymmetry U and (

**b**) spin asymmetry T as functions of the scattering angle $\theta $. Shown are results for 70 MeV electron scattering from Ni${}^{58}\phantom{\rule{0.277778em}{0ex}}(----)$, Pd${}^{108}\phantom{\rule{0.277778em}{0ex}}(-\xb7-\xb7-)$, and Pt${}^{196}$ (—), as well as for 70 MeV positron scattering from Ni${}^{58}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $, upper line) and Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(-\cdots -\cdots -)$. Included are results for 140 MeV electron scattering from Pt${}^{196}\phantom{\rule{0.277778em}{0ex}}(\cdots \cdots $, lower line).

**Figure 36.**The present OPM calculations of the IECS for positron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ in comparison with the results of Dapor and Miotello [50].

**Figure 37.**The present OPM calculations of the MTCS for positron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ in comparison with the results of Dapor and Miotello [50].

**Figure 38.**The present OPM calculations of the VCS for positron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$ in comparison with the results from Dapor and Miotello [50].

**Figure 39.**The present OPM calculations of the INCS, TCS, and TICS for positron impact scattering from (

**a**) Ni${}^{58}$, (

**b**) Cu${}^{63}$, (

**c**) Pd${}^{108}$, and (

**d**) Pt${}^{196}$.

**Figure 40.**Inelastic mean free path ${\lambda}_{\mathrm{in}}$ (IMFP) in $\dot{\mathrm{A}}$ for electrons colliding with (

**a**) Ni, (

**b**) Cu, (

**c**) Pd, and (

**d**) Pt as a function of collision energy, ${E}_{i}$. Shown are the present results (—, red) and the theoretical results from Shinotsuka et al. ($----$, blue [72]), Ashley $(-\xb7-\xb7-,$ green [74]), and Penn (in (

**b**): upper full line, black [73]). Included are the experimental data from Iakoubovskii et al. [60], Pierce and Siegmann [61], Wooten et al. [62], Jackson et al. [63], Burke and Schreurs [64], Seah [66], Ridgway and Haneman [65], Brunner and Zogg [67], Mrozek et al. [68], Knapp et al. [69], Palmberg and Rhodin [70], and Tanuma et al. [71].

**Figure 41.**Inelastic mean free path ${\lambda}_{\mathrm{in}}$ (IMFP) in $\dot{\mathrm{A}}$ for positrons colliding with (

**a**) Ni, (

**b**) Cu, (

**c**) Pd, and (

**d**) Pt as function of collision energy, ${E}_{i}$. Shown are the present results (—, red) and the theoretical results from Ashley (—, upper line in (

**b**), black [74]) and Ghosh and Aers ($----$, blue [76]).

**Figure 42.**(

**a**,

**b**) Elastic mean free path ${\lambda}_{\mathrm{el}}$ (EMFP) and (

**c**,

**d**) total mean free path ${\lambda}_{\mathrm{tot}}$ (TMFP) in $\dot{\mathrm{A}}$ for (

**a**,

**c**) electrons and (

**b**,

**d**) positrons colliding with Ni ($-\xb7-\xb7-$, green), Cu ($----$, black), Pd ($\cdots \cdots $, blue), and Pt (—, red) as function of collision energy ${E}_{i}$.

**Table 1.**The energy and angular positions and the values of direct and spin-flip scattering amplitudes of the deep minima (DM) in the DCS for elastic e${}^{-}$-Ni scattering.

${\mathit{E}}_{\mathit{c}}$ (eV) | ${\mathit{\theta}}_{\mathit{c}}$ (deg.) | $\mid \mathit{A}\left(\mathit{\theta}\right)\mid $ (cm) | $\mid \mathit{B}\left(\mathit{\theta}\right)\mid $ (cm) |
---|---|---|---|

6.8 | 104.5 | 1.55 × 10${}^{-10}$ | 2.11 × 10${}^{-10}$ |

11.2 | 97.0 | 1.31 × 10${}^{-11}$ | 1.62 × 10${}^{-10}$ |

15.5 | 54.5 | 2.96 × 10${}^{-11}$ | 1.23 × 10${}^{-10}$ |

29.5 | 34.5 | 2.86 × 10${}^{-10}$ | 9.64 × 10${}^{-11}$ |

74.5 | 146.5 | 5.07 × 10${}^{-11}$ | 2.72 × 10${}^{-11}$ |

135.0 | 73.5 | 7.10 × 10${}^{-12}$ | 5.01 × 10${}^{-11}$ |

297.1 | 123.5 | 1.63 × 10${}^{-11}$ | 3.33 × 10${}^{-11}$ |

**Table 2.**Maximum spin polarization points, along with their positions and deviations in energy $\Delta E$ and angle $\Delta \theta $ from their respective critical minima positions, for e${}^{-}$-Ni elastic scattering.

${\mathit{S}}_{\mathbf{m}}$ | ${\mathit{E}}_{\mathbf{m}}$ (eV) | $\pm \mathbf{\Delta}\mathit{E}$ (eV) | ${\mathit{\theta}}_{\mathbf{m}}$ (deg) | $\pm \mathbf{\Delta}\mathit{\theta}$ (deg) |
---|---|---|---|---|

+0.96 | 6.84 | 0.04 | 104.0 | 0.5 |

−1.00 | 5.80 | 1.00 | 106.5 | 2.0 |

+0.98 | 10.00 | 1.20 | 98.5 | 1.5 |

−1.00 | 12.05 | 0.85 | 96.0 | 1.0 |

+0.98 | 15.60 | 0.10 | 53.0 | 1.5 |

−0.99 | 15.45 | 0.05 | 55.5 | 1.0 |

+0.53 | 29.20 | 0.30 | 34.5 | 0.0 |

−0.69 | 32.00 | 2.50 | 33.0 | 1.5 |

+0.68 | 73.70 | 0.80 | 147.0 | 0.5 |

−0.78 | 74.60 | 0.10 | 146.5 | 0.0 |

+0.84 | 133.20 | 1.80 | 73.5 | 0.0 |

−0.98 | 136.60 | 1.60 | 73.5 | 0.0 |

+0.84 | 296.00 | 1.10 | 123.5 | 0.0 |

−0.99 | 294.60 | 2.50 | 124.0 | 0.5 |

**Table 3.**The energy and angular positions and the values of direct and spin-flip scattering amplitudes of the deep minima (DM) in the DCS for elastic e${}^{-}$-Cu scattering [6].

${\mathit{E}}_{\mathit{c}}$ (eV) | ${\mathit{\theta}}_{\mathit{c}}$ (deg.) | $\mid \mathit{A}\left(\mathit{\theta}\right)\mid $ (cm) | $\mid \mathit{B}\left(\mathit{\theta}\right)\mid $ (cm) |
---|---|---|---|

3.26 | 135.0 | 6.83$\times {10}^{-10}$ | 3.00$\times {10}^{-09}$ |

3.57 | 73.0 | 1.43$\times {10}^{-10}$ | 2.26$\times {10}^{-09}$ |

13.1 | 50.5 | 5.48$\times {10}^{-11}$ | 1.85$\times {10}^{-10}$ |

19.8 | 117.5 | 2.39$\times {10}^{-11}$ | 1.07$\times {10}^{-10}$ |

28.9 | 76.5 | 4.16$\times {10}^{-11}$ | 1.05$\times {10}^{-10}$ |

86.0 | 145.0 | 4.60$\times {10}^{-12}$ | 2.07$\times {10}^{-11}$ |

158.2 | 70.5 | 1.74$\times {10}^{-12}$ | 4.29$\times {10}^{-11}$ |

337.4 | 122.5 | 1.20$\times {10}^{-11}$ | 3.26$\times {10}^{-11}$ |

**Table 4.**Maximum spin polarization points, along with their positions and deviations in energy ($\Delta E$) and angle ($\Delta \theta $) from the respective critical minimum positions, for e${}^{-}$-Cu scattering [6].

${\mathit{S}}_{\mathbf{m}}$ | ${\mathit{E}}_{\mathbf{m}}$ (eV) | $\pm \mathbf{\Delta}\mathit{E}$ (eV) | ${\mathit{\theta}}_{\mathbf{m}}$ (deg) | $\pm \mathbf{\Delta}\mathit{\theta}$ (deg) |
---|---|---|---|---|

+1.00 | 3.56 | 0.30 | 129.0 | 6.0 |

−1.00 | 3.15 | 0.11 | 139.0 | 4.0 |

+1.00 | 3.63 | 0.06 | 70.5 | 2.5 |

−1.00 | 3.53 | 0.04 | 75.0 | 2.0 |

+1.00 | 13.20 | 0.10 | 51.0 | 0.5 |

−0.96 | 12.86 | 0.24 | 50.5 | 0.0 |

+0.96 | 19.73 | 0.07 | 117.0 | 0.5 |

−0.99 | 19.73 | 0.07 | 118.0 | 0.5 |

+1.00 | 28.05 | 0.85 | 76.0 | 0.5 |

−0.97 | 30.37 | 1.47 | 77.5 | 1.0 |

+0.86 | 85.69 | 0.31 | 145.0 | 0.0 |

−0.61 | 86.56 | 0.56 | 145.0 | 0.0 |

+0.92 | 155.80 | 2.40 | 70.5 | 0.0 |

−0.90 | 160.10 | 1.90 | 70.5 | 0.0 |

+0.83 | 340.00 | 2.60 | 122.0 | 0.5 |

−0.99 | 334.50 | 2.90 | 123.0 | 0.5 |

**Table 5.**The energy and angular positions and the values of direct and spin-flip scattering amplitudes of the deep minima (DM) in the DCS for elastic e${}^{-}$-Pd scattering.

${\mathit{E}}_{\mathit{c}}$ (eV) | ${\mathit{\theta}}_{\mathit{c}}$ (deg.) | $\mid \mathit{A}\left(\mathit{\theta}\right)\mid $ (cm) | $\mid \mathit{B}\left(\mathit{\theta}\right)\mid $ (cm) |
---|---|---|---|

2.93 | 116.60 | 1.92$\times {10}^{-10}$ | 1.04$\times {10}^{-9}$ |

10.28 | 49.80 | 1.18$\times {10}^{-11}$ | 4.40$\times {10}^{-10}$ |

10.69 | 116.60 | 3.35$\times {10}^{-11}$ | 3.15$\times {10}^{-10}$ |

38.40 | 79.00 | 5.01$\times {10}^{-11}$ | 2.48$\times {10}^{-10}$ |

68.56 | 141.40 | 3.87$\times {10}^{-11}$ | 5.79$\times {10}^{-11}$ |

129.00 | 51.40 | 9.62$\times {10}^{-10}$ | 1.33$\times {10}^{-10}$ |

196.20 | 151.00 | 2.17$\times {10}^{-11}$ | 4.19$\times {10}^{-11}$ |

289.10 | 90.60 | 4.81$\times {10}^{-12}$ | 9.19$\times {10}^{-11}$ |

638.90 | 129.40 | 1.18$\times {10}^{-11}$ | 6.56$\times {10}^{-11}$ |

**Table 6.**Maximum spin polarization points with their positions and deviations in energy ($\Delta E$) and angle ($\Delta \theta $) from the respective critical minimum positions for e${}^{-}$-Pd scattering.

${\mathit{S}}_{\mathbf{m}}$ | ${\mathit{E}}_{\mathbf{m}}$ (eV) | $\pm \mathbf{\Delta}\mathit{E}$ (eV) | ${\mathit{\theta}}_{\mathbf{m}}$ (deg) | $\pm \mathbf{\Delta}\mathit{\theta}$ (deg) |
---|---|---|---|---|

+1.00 | 2.84 | 0.09 | 114.2 | 2.4 |

−1.00 | 2.92 | 0.01 | 119.0 | 2.4 |

+1.00 | 10.14 | 0.14 | 52.2 | 2.4 |

−1.00 | 10.42 | 0.14 | 47.8 | 2.0 |

+1.00 | 10.20 | 0.49 | 115.0 | 1.6 |

−1.00 | 11.11 | 0.42 | 117.8 | 1.2 |

+0.99 | 39.12 | 0.72 | 78.2 | 0.8 |

−0.99 | 36.35 | 2.05 | 80.2 | 1.2 |

+0.95 | 69.90 | 1.34 | 141.0 | 0.4 |

−0.53 | 72.31 | 3.75 | 141.0 | 0.4 |

−0.22 | 130.50 | 1.50 | 52.5 | 1.1 |

+0.82 | 196.00 | 0.20 | 151.0 | 0.0 |

−0.88 | 197.60 | 1.40 | 150.6 | 0.4 |

+0.99 | 288.00 | 1.10 | 90.2 | 0.4 |

−0.97 | 290.30 | 1.20 | 91.0 | 0.4 |

+0.97 | 640.00 | 1.10 | 129.0 | 0.4 |

−1.00 | 628.00 | 10.90 | 130.2 | 0.8 |

**Table 7.**The energy and angular positions and the values of direct and spin-flip scattering amplitudes of the deep minima (DM) in the DCS for elastic e${}^{-}$-Pt scattering.

${\mathit{E}}_{\mathit{c}}$ (eV) | ${\mathit{\theta}}_{\mathit{c}}$ (deg.) | $\mid \mathit{A}\left(\mathit{\theta}\right)\mid $ (cm) | $\mid \mathit{B}\left(\mathit{\theta}\right)\mid $ (cm) |
---|---|---|---|

12.65 | 43.0 | 2.33$\times {10}^{-10}$ | 1.49$\times {10}^{-09}$ |

14.75 | 123.5 | 2.57$\times {10}^{-10}$ | 1.04$\times {10}^{-09}$ |

27.26 | 81.0 | 1.46$\times {10}^{-09}$ | 9.48$\times {10}^{-10}$ |

100.06 | 128.5 | 2.94$\times {10}^{-11}$ | 2.89$\times {10}^{-10}$ |

156.01 | 107.0 | 1.02$\times {10}^{-11}$ | 3.76$\times {10}^{-10}$ |

200.01 | 85.5 | 1.62$\times {10}^{-11}$ | 4.29$\times {10}^{-10}$ |

230.68 | 36.0 | 3.01$\times {10}^{-09}$ | 4.26$\times {10}^{-10}$ |

252.17 | 145.5 | 3.27$\times {10}^{-11}$ | 1.64$\times {10}^{-10}$ |

307.74 | 118.5 | 8.54$\times {10}^{-12}$ | 2.94$\times {10}^{-10}$ |

453.76 | 66.5 | 1.14$\times {10}^{-11}$ | 3.00$\times {10}^{-10}$ |

526.09 | 153.0 | 2.36$\times {10}^{-11}$ | 1.00$\times {10}^{-10}$ |

882.63 | 98.5 | 3.85$\times {10}^{-11}$ | 2.21$\times {10}^{-10}$ |

1594.10 | 136.5 | 1.80$\times {10}^{-11}$ | 1.93$\times {10}^{-10}$ |

**Table 8.**Maximum spin polarization points with their positions and deviations in energy ($\Delta E$) and angle ($\Delta \theta $) from the respective critical minimum positions for e${}^{-}$-Pt scattering.

${\mathit{S}}_{\mathbf{m}}$ | ${\mathit{E}}_{\mathbf{m}}$ (eV) | $\pm \mathbf{\Delta}\mathit{E}$ (eV) | ${\mathit{\theta}}_{\mathbf{m}}$ (deg) | $\pm \mathbf{\Delta}\mathit{\theta}$ (deg) |
---|---|---|---|---|

−1.00 | 13.67 | 1.02 | 39.5 | 3.5 |

+1.00 | 10.9 | 1.75 | 52.7 | 9.7 |

−1.00 | 15.66 | 0.91 | 126.5 | 3.0 |

+1.00 | 11.51 | 3.24 | 117.0 | 6.5 |

−0.40 | 44.00 | 16.74 | 85.0 | 4.0 |

+1.00 | 19.90 | 7.36 | 77.5 | 3.5 |

−1.00 | 110.03 | 9.97 | 125.0 | 3.5 |

+1.00 | 92.30 | 7.76 | 131.5 | 3.0 |

−0.99 | 157.08 | 1.07 | 109.5 | 2.5 |

+1.00 | 157.27 | 1.26 | 103.5 | 3.5 |

−1.00 | 215.93 | 15.92 | 83.5 | 2.0 |

+1.00 | 185.00 | 15.01 | 88.5 | 3.0 |

−0.26 | 203.92 | 26.76 | 39.0 | 3.0 |

−0.96 | 250.39 | 1.78 | 146.5 | 1.0 |

+0.98 | 253.28 | 1.11 | 144.5 | 1.0 |

−1.00 | 324.61 | 16.87 | 117.0 | 1.5 |

+0.99 | 296.19 | 11.55 | 120.0 | 1.5 |

−0.97 | 474.09 | 20.33 | 66.5 | 0.0 |

+1.00 | 432.71 | 21.05 | 66.5 | 0.0 |

−0.87 | 529.35 | 3.26 | 152.5 | 0.5 |

+1.00 | 523.37 | 2.72 | 153.5 | 0.5 |

−0.97 | 879.02 | 3.61 | 100.0 | 1.5 |

+1.00 | 854.77 | 27.86 | 98.0 | 0.5 |

−1.00 | 1506.30 | 87.80 | 138.5 | 2.0 |

+1.00 | 1643.70 | 49.60 | 135.0 | 1.5 |

Element | ${\mathit{Z}}_{\mathit{T}}$ | n | ${\mathit{N}}_{\mathit{c}}$ |
---|---|---|---|

Na | 11 | 1 | – |

K | 19 | 2 | 3 |

Ni | 28 | 2 | 5 |

Cu | 29 | 2 | 7 |

Rb | 37 | 3 | 6 |

Pd | 46 | 3 | 7 |

Cs | 55 | 4 | 14 |

Yb | 70 | 4 | 11 |

Pt | 78 | 4 | 11 |

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

© 2022 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**

Jakubassa-Amundsen, D.H.; Haque, A.K.F.; Haque, M.M.; Billah, M.M.; Basak, A.K.; Saha, B.C.; Uddin, M.A.
Electron and Positron Scattering from Precious Metal Atoms in the eV to MeV Energy Range. *Atoms* **2022**, *10*, 82.
https://doi.org/10.3390/atoms10030082

**AMA Style**

Jakubassa-Amundsen DH, Haque AKF, Haque MM, Billah MM, Basak AK, Saha BC, Uddin MA.
Electron and Positron Scattering from Precious Metal Atoms in the eV to MeV Energy Range. *Atoms*. 2022; 10(3):82.
https://doi.org/10.3390/atoms10030082

**Chicago/Turabian Style**

Jakubassa-Amundsen, Doris H., Abul Kalam Fazlul Haque, Md. Monirul Haque, Md. Masum Billah, Arun Kumar Basak, Bidhan Chandra Saha, and Md. Alfaz Uddin.
2022. "Electron and Positron Scattering from Precious Metal Atoms in the eV to MeV Energy Range" *Atoms* 10, no. 3: 82.
https://doi.org/10.3390/atoms10030082