**Aliyu Adamu, Adamu Hassan, Ahmad Abdulkadir and Adamu Z. Ngari **

**Abstract**

The effects of relativistic motion, the spin-orbit interaction and Zitterbewegung of an electron, which are of the same order of magnitude, defined the fine structure correction to hydrogen spectra. In this work, perturbation theory, as an approximation method is applied to examine the effects of finite sized of atomic nucleus on relativistic motion of electron in hydrogen atom. The nuclear finite corrections to 1s, 2s, 3s, 4s and 5s energy states in hydrogen atom due to the finite size of nucleus were computed and the results showed that the nuclear size effects, which is of order of 10-6 eV, depends on the size, A(N, Z) of the nucleus and energy states, n of relativistic electron. This suggested that the nuclear size is more effective on relativistic electron in the lower energy levels of heavy nuclei, as the effect varied directly with the nuclear size and inversely with the electron states, n. Thus, the finite size of atomic nucleus has an impact on relativistic motion of an electron. Moreover, a simple model was developed to predict the energy level variation as a function of the size of the nucleus. Therefore, this study justifies the effects of nuclear size on relativistic electron and the measured values are of greatest interest since it will reveal significant changes of the nuclear structure and may also improve the knowledge of fine structure correction.

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**Keywords**: relativistic motion fine structure electron finite-size nucleus perturbation energy states

**References **

[1] Maclay, G. J. (2019). History and Some Aspects of the Lamb Shift. Physics, Volume 2, Number 8, page 105 – 149.

[2] Meka, F. (2020). Investigating Muonic Hydrogen Atom Energy Spectrum Using Perturbation Theory in Lowest Order. Advances in Physics Theories and Applications. Volume 83, page 7 – 21.

[3] Schrödinger, E. (1926).QuantisierungalsEigenwert-Problem.Annalen der Physik Volume 80, page 437.

[4] Landau, L. D. and Lifshitz, E. M., (1991). Quantum Mechanics, Non-relativistic Theory, Volume 3 of Course of Theoretical Physics. Third edition, Pergamon Press, Oxford, England. page 119.

[5] Greiner, W. (2001). Quantum Mechanics: An Introduction. Fourth Edition. Springer, Berlin, Germany, page 181, 220 – 227.

[6] Herzberg, G. (1944). Atomic Spectra and Atomic Structure. Dover: Mineola, New York.

[7] White, H. E. (1934). Introduction to Atomic Spectra. (McGraw-Hill Book Company, New York), Chap. 8. New York, 1934.

[8] Sala, O., Araki, K. and Noda, L. K. (1999). A Procedure to Obtain the Effective Nuclear Charge from the Atomic Spectrum of Sodium. Journal of Chemical Education. Volume 76, Number 9, page 1269–1271.

[9] Adamu, A., Tartius, P.and Amshi, S. A., (2016).The Spectroscopy of Single Electron and Muonic Atoms. Journal of Physical Science and Innovations, Vol. 8, No. 2, pp 1 – 13.

[10]Adamu A., (2016): Corrections to the Energy Levels of Finite – Size Nuclei due to Fluctuating Electromagnetic Fields in Vacuum. J – NAMP. Vol. 36, (July Issue), pp 215 – 222.

[11]Adamu, A. andNgadda, Y. H., (2015): The Nuclear Finite–Size Corrections to Energies of n = 1, n = 2 and n = 3 States of Hydrogen Atom.J – NAMP. Vol. 30, pp 137 – 137.

[12]Hernandez, O. J., Ji, C., Bacca, S., Dinur, N. N. andBarnea, N. (2014). Improved estimates of the nuclear structure corrections in μD. arXiv:1406.5230v1 [nucl-th] 19 Jun 2014.

[13]Antognini, A., Kottmann, F., Biraben, F., Indelicato, P., Nez, F. and Pohl, R. (2013). Theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen. Annals of Physics. Volume 331, page 127–145.

[14]Bruch, R., Heilig, K., Kaletta, D., Steudel, A. and Wendlandt, D. (1969). Nuclear Volume and Mass Effect in the Optical Isotope Shift of Light Elements. Journal de Physique Colloques, Volume 30 (C1), page C1-51-C1-58.

[15]Mohammadi, S., Giv, B. N. and Shakib, N. S. (2017). Energy Levels Calculations of 24Al and 25Al Isotopes. Nuclear Science. Volume 2, Number 1, page 1-4. [16]Mohr, P. J., Taylor, B. N. and Newel, D. B. (2012). CODATA recommended values of the fundamental physical constants: 2010. Reviews of Modern Physics. Volume 84, page 1527 – 1605.

[17]Galván, A. P., Zhao, Y. and Orozco, L. A. (2008). Measurement of the hyperfine splitting of the 6S½ level in rubidium. Physical Review A78, 012502 2008.

[18]Das, A. and Sidharth, B. G. (2015). Revisiting the Lamb Shift.Electronic Journal of Theoretical Physics. Volume 12, Number IYL15-34, page 139–152.

[19]Adamu, A., Hassan, M., Dikwa, M. K. and Amshi, S. A., (2018). Determination of Nuclear Structure Effects on Atomic Spectra by Applying Rayleigh–Schrödinger Perturbation Theory. American Journal of Quantum Chemistry and Molecular Spectroscopy. Volume 2, Number 2, page 39-51.

[20]Deck, R. T., Amar, J. G. and Fralick, G. (2005). Nuclear Size Corrections to the Energy Levels of Single-electron and -Muon Atoms. Journal of Physics B: Atomic Molecular and Optical Physics, Volume 38, page 2173 – 2186.

[21]Niri, B. N. and Anjami, A. (2018). Nuclear Size Corrections to the Energy Levels of Single-Electron Atoms. Nuclear Science. Volume 3, Number 1, page 1 – 8. [22]Hofstadter, R. and McAllister, R. W. (1955). Electron Scattering from the Proton. Physical Review. Volume 98, page 217.

[23]McAllister, R. W. and Hofstadter, R. (1956). Elastic Scattering of 188-Mev Electrons from the Proton and the Alpha Particle. Physical Review. Volume 102, page 851.

[24]Yearian, M. R. and Hofstadter, R. (1958). Magnetic Form Factor of Neutron. Physical Review. Volume 110, page 552.

[25]Ohmura, T. (1959). Effect of the Finite Size of the Proton on the Coulomb Energy of He3 . Progress of Theoretical Physics. Volume 22 Issue 1, page 148 – 150.

[26]Adamu, A. and Ngadda, Y. H. (2017). Determination of Nuclear Potential Radii and its Parameter from Finite – Size Nuclear Model.International Journal of Theoretical and Mathematical Physics. Volume 7, Number 1, page 9 – 13.

[27]Bayram, T., Akkoyun, S., Kara, S. O. and Sinan, A. (2013). New parameters for nuclear charge radius formulas.ACTA PHYSICA POLONICA B. Volume 44, number 8, page 1791 – 1799.

[28]Angeli, I. (2013). Manifestation of Non-Traditional Magic Nucleon Numbers in Nuclear Charge Radii. ACTA PHYSICA DEBRECINA. Volume 47, Number 7.

[29]Merino, C., Novikov, I. S. and ShabelskiY. M. (2009). Nuclear Radii Calculations in Various Theoretical Approaches for Nucleus-Nucleus Interactions. arXiv:0907.1697v1 [nucl-th] 10 Jul 2009. Aliyu Adamu etal./ NIPES Journal of Science and Technology Research 2(3) 2020 pp. 36-44 43

[30]Patoary, A. M. and Oreshkina, N. S. (2018). Finite nuclear size effect to the fine structure of heavy muonic atoms. The European Physical Journal D. Volume 72, page 54.

[31]Martensson-Pendrill, A. M. and Gustavsson, M. G. H. (2003). Handbook of Molecular Physics and Quantum Chemistry. John Wiley & Sons, Ltd, Chichester, Volume 1, Part 6, Chapter 30, page 477 – 484. Edited by Stephen Wilson.

[32]Adamu, A. andNgadda, Y. H., (2014).The Effect of 1st Order Time Independent Perturbation on the Finite Size of the Nuclei of Atoms.International Journal of Theoretical and Mathematical Physics. Volume 28, Number 1, page 333 – 339.

[33]Adamu A. and Ngadda Y. H., (2017): Determination of Nuclear Potential Radii and Its Parameter from Finite – Size Nuclear Model. International Journal of Theoretical and Mathematical Physics. Volume 7, Number 1, page 9 – 13.

[34]Borie, E. (2012). Lamb Shift in Light Muonic Atoms – Revisited. Annals of Physics. Volume 327, page 733 – 763. [35]Godunov, S .I. and M. I. Vysotsky, (2013). The Dependence of the Atomic Energy Levels on a Super Strong Magnetic Field with Account of a Finite Nucleus Radius and Mass, arXiv:1304.7940v1 [hep-ph].

[36]El Shabshiry, M., Ismaeel, S. M. E. and Abdel-Mageed, M. M. (2015). Finite Size Uehling Corrections in Energy Levels of Hydrogen and Muonic Hydrogen Atom. IOSR Journal of Applied Physics (IOSR-JAP), Volume 7, Issue 5 Ver. I, page 60 – 66.

[37]Krane, K. S. (1988). Introductory Nuclear Physics. John Wiley and Sons Inc., New York. Page 49. [38]Neznamov, V. P. and Safronov, I. I. (2015). A new Method for Solving the Z > 137 Problem and for Determination of Energy Levels of Hydrogen-Like Atoms. arXiv: 1307.0209 V3.