American Journal of Modern Physics

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Atomic Form Factor Calculations of S-states of Helium

Received: 12 September 2019    Accepted: 04 October 2019    Published: 15 October 2019
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Abstract

Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S2 11S and 1S2S 21, 3S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition. The results presented here, indicate that the highly compact twenty-six variational parameters model will have the quantitative and qualitative applicability for the study of electronic correlation. The correlated wave functions are used to calculate the atomic form factor for the diffusion of electrons by the helium atom. The atomic form factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical correlated wave functions for free atoms. Finally, suggestions are made as to the way the atomic form factor of the helium atom may be approximated by a sum of Gaussians for efficiency use.

DOI 10.11648/j.ajmp.20190804.12
Published in American Journal of Modern Physics (Volume 8, Issue 4, July 2019)
Page(s) 66-71
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Helium, Variational Method, Correlation Effect, Charge Distribution, Scattering

References
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[7] K. D. Sen, Jacob Katriel, H. E. Montgomery, (2018) A comparative study of two-electron systems with screened Coulomb potentials. Annals of Physics 397, pp 192-212.
[8] Li Guang Jiao and Yew Kam Ho, (2014) Calculation of screened Coulomb potential matrices and its application to He bound and resonant states. Physical Review A 90, 012521.
[9] Freund, David E. and Huxtable, Barton D. and Morgan, John D. (1984) Variational calculations on the helium isoelectronic sequence. Phys. Rev. A, 29 (2): 980—982.
[10] Thakkar, Ajit J. and Koga, Toshikatsu. (1994) Ground-state energies for the helium isoelectronic series. Phys. Rev. A, 50 (1): 854-856.
[11] G. W. F. Drake. (1996) Atomic, molecular and optical physics handbook. AIP Press, Woodbury, NY}, pp 154-171.
[12] Accad, Y. and Pekeris, C. L. and Schiff, B. (1975) Two-electron S and P term values with smooth Z dependence. Phys. Rev. A, 11 (4): 1479-1481.
[13] Green, Louis C. and Mulder, Marjorie M. and Milner, Paul C. (1953) Correlation Energy in the Ground State of He I, Phys. Rev. 91 (1): 35-39.
[14] Chandrasekhar, S. and Herzberg, G. (1955) Energies of the Ground States of He, Li+, and O6+. Phys. Rev. 98 (4): 1050—1054.
[15] Hart, J. F. and Herzberg, G. (1957) Twenty-Parameter Eigen functions and Energy Values of the Ground States of He and He-Like Ions. Phys. Rev. 106 (1): 79-82.
[16] S Bhattacharyya and A Bhattacharyya and B Talukdar and N C Deb, (1996) Analytical approach to the helium-atom ground state using correlated wave functions. Journal of Physics B: Atomic, Molecular and Optical Physics 29 (5): L147.
[17] Hans A. Bethe and Edwin E. Salpeter, (1957) Quantum mechanics of one and two electron atoms. Springer Berlin pp 155.
[18] Korobov and Vladimir I. (2004) Bethe logarithm for the helium atom. Phys. Rev. A, 69 (5): 054501.
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[20] Pekeris, C. L. (1959) 11S and 23S States of Helium. Phys. Rev. 115 (5): 1216-1221.
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Author Information
  • Department of Physics, Faculty of Sciences, University Cheikh Anta Diop, Dakar, Senegal

  • Department of Physics, Faculty of Sciences, University Cheikh Anta Diop, Dakar, Senegal

  • Department of Physics, Faculty of Sciences, University Cheikh Anta Diop, Dakar, Senegal

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    Saïdou Diallo, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou. (2019). Atomic Form Factor Calculations of S-states of Helium. American Journal of Modern Physics, 8(4), 66-71. https://doi.org/10.11648/j.ajmp.20190804.12

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    Saïdou Diallo; Ibrahima Gueye Faye; Louis Gomis; Moustapha Sadibou Tall; Ismaïla Diédhiou. Atomic Form Factor Calculations of S-states of Helium. Am. J. Mod. Phys. 2019, 8(4), 66-71. doi: 10.11648/j.ajmp.20190804.12

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    AMA Style

    Saïdou Diallo, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou. Atomic Form Factor Calculations of S-states of Helium. Am J Mod Phys. 2019;8(4):66-71. doi: 10.11648/j.ajmp.20190804.12

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  • @article{10.11648/j.ajmp.20190804.12,
      author = {Saïdou Diallo and Ibrahima Gueye Faye and Louis Gomis and Moustapha Sadibou Tall and Ismaïla Diédhiou},
      title = {Atomic Form Factor Calculations of S-states of Helium},
      journal = {American Journal of Modern Physics},
      volume = {8},
      number = {4},
      pages = {66-71},
      doi = {10.11648/j.ajmp.20190804.12},
      url = {https://doi.org/10.11648/j.ajmp.20190804.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajmp.20190804.12},
      abstract = {Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S2 11S and 1S2S 21, 3S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition. The results presented here, indicate that the highly compact twenty-six variational parameters model will have the quantitative and qualitative applicability for the study of electronic correlation. The correlated wave functions are used to calculate the atomic form factor for the diffusion of electrons by the helium atom. The atomic form factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical correlated wave functions for free atoms. Finally, suggestions are made as to the way the atomic form factor of the helium atom may be approximated by a sum of Gaussians for efficiency use.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Atomic Form Factor Calculations of S-states of Helium
    AU  - Saïdou Diallo
    AU  - Ibrahima Gueye Faye
    AU  - Louis Gomis
    AU  - Moustapha Sadibou Tall
    AU  - Ismaïla Diédhiou
    Y1  - 2019/10/15
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajmp.20190804.12
    DO  - 10.11648/j.ajmp.20190804.12
    T2  - American Journal of Modern Physics
    JF  - American Journal of Modern Physics
    JO  - American Journal of Modern Physics
    SP  - 66
    EP  - 71
    PB  - Science Publishing Group
    SN  - 2326-8891
    UR  - https://doi.org/10.11648/j.ajmp.20190804.12
    AB  - Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S2 11S and 1S2S 21, 3S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition. The results presented here, indicate that the highly compact twenty-six variational parameters model will have the quantitative and qualitative applicability for the study of electronic correlation. The correlated wave functions are used to calculate the atomic form factor for the diffusion of electrons by the helium atom. The atomic form factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical correlated wave functions for free atoms. Finally, suggestions are made as to the way the atomic form factor of the helium atom may be approximated by a sum of Gaussians for efficiency use.
    VL  - 8
    IS  - 4
    ER  - 

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