68 © 2022 Adama Science & Technology University. All rights reserved Ethiopian Journal of Science and Sustainable Development e-ISSN 2663-3205 Volume 9 (2), 2022 Journal Home Page: www.ejssd.astu.edu.et ASTU Research Paper Estimation of Relativistic Mass Correction for Electronic and Muonic Hydrogen Atoms with Potential from Finite Size Source Eshetu Diriba Kena, Gashaw Bekele Adera Haramaya University, Department of Physics, P.O.Box 138, Dire Dawa, Ethiopia Article Info Abstract Article History: Received 25 April 2022 Received in revised form 26 July 2022 Accepted 28 July 2022 Hydrogen is the simplest atom in nature which helps us to study fundamental properties and structure of atoms. In this work, the perturbed Schrodinger equation is used to estimate relativistic mass correction with potentials from finite size sources. This study is done with the assumption that the changes in both energy eigenvalues and eigenfunctions are negligible when considering the finite size of the nuclei. The relativistic mass corrections to 1s 1/2 and 2s_1/2 states using potential from finite size source are obtained and compared with corrections using potential from point-like source. The results show that, for hydrogen-like atoms with light nuclei the relativistic mass corrections due to the finite size source roughly coincides with that of point-like source. However, for atoms with heavy nuclei the two corrections display strong disagreement in which the corrections with finite size nuclei are significantly smaller than that of point-like nuclei. Thus, this study offers the first attempt at including relativistic mass correction to energy eigenvalues of electronic and muonic hydrogen-like atoms with finite-size nuclei. When experimental data become available, comparison of these theoretical predictions may give a better insight into how relativistic corrections are affected by finite-size nuclei. Keywords: Atomic structure Charge distributions Finite size source Hydrogen-like atom Muonic hydrogen Point-like source Relativistic mass correction 1. Introduction The hydrogen atom is the simplest atom known to exist in nature, which makes it very suitable to study the fundamental properties and structure of atoms. The hydrogen atom consists of a single negatively charged electron that moves about a positively charged proton (Hudson and Nelson, 1990, Adamu et al., 2018). In 1911, Niels Bohr introduced the first model of hydrogen atom based on Planck’s hypothesis, which states that electromagnetic radiations can only be emitted or absorbed in quanta of energy. However, the complete understanding of the structure of atomic elements in the periodic table was made possible with the invention of new theory called quantum mechanics as of 1925 (Beiser, 2003) E-mail: eshetudiriba444@gmail.com https://doi.org/10.20372/ejssdastu:v9.i2.2022.474 In addition, recent studies show that it is possible to replace the electron in an atom by a muon to form muonic atom. In other words, Muonic hydrogen is an atom consisting of the proton and the muon which is very similar to the well-known electronic hydrogen (Pohl et al., 2013). As a result, theoretical descriptions of the two atomic types have many features in common. The main difference however, is the fact that the muon mass is about 207 times heavier than that of the electron. The massiveness of muon in turn results in a significant increase in the energy levels of the atoms (Pohl, 2014 and Patoary et al., 2018); and leads to the realization that muons tend to be close to the nucleus of an atom as http://www.ejssd.astu.edu/ https://doi.org/10.20372/ejssdastu:v9.i2.2022......... Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 69 compared to electrons in electronic hydrogen (Eshetu Diriba and Gashaw Bekele, 2021). In a simplest quantum description of hydrogen atom, its nucleus is assumed to be point-like in terms of its mass and charge distribution. Nevertheless, the more realistic model treats the nucleus as having finite size distributions. The shift in energy eigenvalues and eigenfunctions due to the finite size nucleus of muonic hydrogen atom has been studied non-relativistically using perturbation theory (Adamu and Ngadda, 2017). But in relativistic framework, the Dirac equation of bound muon can be solved exactly or using relativistic perturbation (Niri and Anjami, 2018, Firew Meka, 2020, Adamu and Ngadda, 2017, Deck et al., 2005 and Eshetu Diriba and Gashaw Bekele, 2021). It is worth noting that the relativistic effects play key role in the studies of the molecular properties such as electron affinities, ionization potentials, reaction, dissociation energies, spectroscopic and other properties (Iliaš et al., 2010 and Pyper, 2020). The relativistic effects can also be used to investigate the rotation curve of disk galaxies which is significant at large radii (Deur, 2021). In non-relativistic treatment of an atom, the standard Schrodinger equation conveys only a limited amount of information about properties of the atom. For instance, consideration of relativistic effects such as relativistic mass correction, spin-orbit coupling and Darwin terms, leads to a better description of atoms and their structure. Traditionally, those corrections are taken into account for hydrogen-like atoms with Coulomb potential from a point-like source (Schwabl, 2008 and Townsend, 2012). To the best of our knowledge, however, no work has been done that examines relativistic effects for those atoms with Coulomb potential from a finite size source. According to Ref. (Niri and Anjami, 2018 and Deck et al., 2005), the corrections to energy eigenvalues due to finite size nucleus for muonic hydrogen in non- relativistic treatment are negligibly small; and the same can be said for corrections to energy eigenfunctions. In this paper, the Schrodinger equation has been revisited as an attempt to address the relativistic mass correction to electronic and muonic hydrogen-like atoms by replacing the point-like potential source with Coulomb potential from finite size source. This is done so with the assumption that such replacement may not result in any significant shifts in both energy eigenvalues and eigenstates. Finally, the results of this work are compared with those obtained for hydrogen- like atoms with point-like source charge for light and heavy atomic nuclei. 2. Mathematical methodology In this chapter we are going to introduce the mathematical methodology and methods needed to achieve our research objectives 2.1. Dirac Equation: Minimal Coupling Dirac equation for a particles in an electromagnetic field, 𝐴𝜇, is modified as [𝛾𝜇𝑖𝜕𝜇 − 𝑚𝐼]𝛹 = 0 ⟶ [𝛾 𝜇(𝑖𝜕𝜇 − 𝑒𝐴𝜇) − 𝑚𝐼]𝛹 = 0 (1) More explicitly Eq. (1) can be written as [𝛾0𝑖𝜕0 − 𝑒𝐴0 + 𝛾 ⋅ (𝑖�⃗⃗� + 𝑒𝐴)− 𝑚𝐼]𝛹 = 0 (2) where, 𝛾0 = ( 𝐼 0 0 −𝐼 ), �⃗� = ( 𝑜 �⃗� −�⃗� 0 ), 𝐼 = ( 1 0 0 1 ) 2.2. Algebra of Gamma Matrices 2.2.1. Pauli matrices The Pauli matrices are a set of 2 × 2 complex, Hermitian, and unitary matrices 𝜎𝑥 = ( 0 1 1 0 ), 𝜎𝑦 = ( 0 −𝑖 𝑖 0 ), 𝜎𝑧 = ( 1 0 0 −1 ) (3) Using the commutation and anti-commutation relation of Pauli matrices we obtain 𝜎𝑖𝜎𝑗 = 𝛿𝑖𝑗 + 𝑖∑𝜖𝑖𝑗𝑘𝜎𝑘 (4) Equation 4 helps us to prove the relation (�⃗� ⋅ A⃗⃗⃗)(�⃗� ⋅ �⃗⃗�) = 𝐴 ⋅ �⃗⃗� + 𝑖�⃗� ⋅ (𝐴 × �⃗⃗�) (5) 2.3. Numerical Calculation of Calculation of Relativistic Mass correction We employ MATLAB 2013a software to numerically generate relativistic mass correction. Finally, we plot the graph of electronic and muonic hydrogen atoms as a function of proton number. 3. Formalism In this section, we cover the mathematical origin of relativistic effects and then derive relativistic mass corrections for hydrogen-like atoms with potential from point-like and finite size sources. In this work, we invoke the natural unit whereby we set ℏ = 𝑐 = 1. Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 70 3.1. Reduction of Dirac Equation into Non-relativistic Equation In the presence of a spherically symmetric electromagnetic potential the Dirac equation takes the form 𝐻𝜓 = (�⃗� ⋅ �⃗⃗� + 𝛽𝑚 + 𝑉(𝑟))𝜓, (6) where, the kinematic momentum operator is given in terms of vector potential 𝐴 as �⃗⃗� = 𝑝 − 𝑒𝐴 (7) The four parameters �⃗� and 𝛽 are written in block matrix format using Pauli and identity matrices: �⃗� = ( 0 �⃗� �⃗� 0 ); 𝛽 = ( 1 0 0 −1 ) (8) Let’s redefine the Dirac spinor 𝜓, which is a four dimensional column vector as 𝜓 = ( 𝜙 𝜒 ) (9) Substituting Eqs. (6) - (9) into Eq. (6) gives (𝐸 − 𝑚 − 𝑉)𝜙 = �⃗� ⋅ �⃗⃗�𝜒 (10) (𝐸 + 𝑚 − 𝑉)𝜒 = �⃗� ⋅ �⃗⃗�𝜙 (11) In non-relativistic limit, we relate the relativistic and non-relativistic energies as (Townsend, 2012 and Maggiore, 2005) 𝐸𝑁𝑅𝑇 = 𝐸 − 𝑚 (12) Plugging Eq. (12) into Eqs. (10) and (11) yields (𝐸𝑁𝑅𝑇 − 𝑉)𝜙 = 𝑖�⃗� ⋅ �⃗⃗�𝜒 (13) (𝐸𝑁𝑅𝑇 + 2𝑚 − 𝑉)𝜒 = �⃗� ⋅ �⃗⃗�𝜙 (14) The non-relativistic energy 𝐸𝑁𝑅𝑇 and the potential 𝑉 terms are very small in comparison with 2𝑚 and are therefore neglected (Messiah, 1966). Hence Eq. (14) is approximated as 𝜒 = �⃗� ⋅ �⃗⃗� 2𝑚 𝜙 (15) Inserting Eq. (15) back into Eq. (13) and we get 𝐸𝑁𝑅𝑇𝜙 = [ 1 2𝑚 (�⃗� ⋅ �⃗⃗�)(�⃗� ⋅ �⃗⃗�) + 𝑒𝐴0]𝜙, (16) where, the potential energy V has been expressed in terms of scalar potential𝐴0(𝑉 = 𝑒𝐴0). By using the property of Pauli matrices, we may write (�⃗� ⋅ �⃗⃗�)(�⃗� ⋅ �⃗⃗�) = �⃗⃗�2 + 𝑖�⃗� ⋅ (�⃗⃗� × �⃗⃗�) = �⃗⃗�2 − 𝑒�⃗� ⋅ �⃗⃗�, which in turn allows us to re-express Eq. (16) as 𝐸𝑁𝑅𝑇𝜙 = [ (�⃗� − 𝑒𝐴) 2 2𝑚 − 𝑒�⃗� ⋅ �⃗⃗� 2𝑚 + 𝑒𝐴0]𝜙. (17) The expression in Eq. (17) is famously known as the Pauli equation. This Pauli equation can further be manipulated to become the non-relativistic Schrodinger equation which naturally incorporate all relativistic corrections and appropriate for spin-half particles (Maggiore, 2005): 𝐸𝑁𝑅𝑇𝜙 = [ 𝑝2 2𝑚 + 𝑉 − 𝑝4 8𝑚3 + 1 2𝑚2 1 𝑟 𝑑𝑉 𝑑𝑟 𝑆 ⋅ �⃗⃗� − 𝑒 8𝑚2 ∇⃗⃗⃗ ⋅ �⃗⃗� ]𝜙, (18) where, −𝑝4/8𝑚3 is the relativistic mass correction, the term 𝑆 ⋅ �⃗⃗� is spin-orbit coupling and ∇⃗⃗⃗ ⋅ �⃗⃗� is the Darwin term. 3.2. Relativistic mass correction for point-like source Relativistic mass correction is often referred as kinetic energy correction, which can also be obtained from energy-momentum relation for relativistic particle in the limit |𝑝| ≪ 𝑚. As we can see from Eq. (13) the perturbed Hamiltonian due to relativistic mass correction is given by �̂�1 = − 𝑝4 8𝑚3 (19) The Hamiltonian of unperturbed system is given by 𝐻0 = 𝑝2 2𝑚 + 𝑉 Squaring both sides and rearranging give rise to 𝑝4 8𝑚3 = 1 2𝑚 (𝐻0 − 𝑉) 2 (20) By using Eq. (20), we may rewrite Eq. (19) as �̂�1 = − 1 2𝑚 (𝐻0 − 𝑉) 2 (21) The relativistic mass correction to energy levels of hydrogen atom can be written as (Griffiths, 1995) ∆𝐸1 (𝑛) = ⟨𝜓𝑛𝑙𝑚|�̂�1|𝜓𝑛𝑙𝑚⟩ (22) Putting Eq. (21) into the above equation we obtain ∆𝐸1 (𝑛) = − 1 2𝑚 ⟨𝜓𝑛𝑙𝑚|(𝐻0 − 𝑉) 2|𝜓𝑛𝑙𝑚⟩ = − 1 2𝑚 [𝐸𝑛 2 − 2𝐸𝑛⟨𝜓𝑛𝑙𝑚|𝑉|𝜓𝑛𝑙𝑚⟩+ ⟨𝜓𝑛𝑙𝑚|𝑉 2|𝜓𝑛𝑙𝑚⟩] (23) Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 71 For a point-like nucleus the Coulomb potential is given by 𝑉 = − 𝑍𝑒2 4𝜋 0 1 𝑟 = − 𝑍𝛼 𝑟 where, the fine structure constant is given by 𝛼 = 𝑒2 4𝜋 0 = 1 137 With such potential, Eq. (23) becomes ∆𝐸1 (𝑛) = − 1 2𝑚 [𝐸𝑛 2 + 2𝐸𝑛 ⟨𝜓𝑛𝑙𝑚 | (𝑍𝛼) 𝑟 |𝜓𝑛𝑙𝑚⟩ + ⟨𝜓𝑛𝑙𝑚 | (𝑍𝛼)2 𝑟2 |𝜓𝑛𝑙𝑚⟩] (24) The expectation values of 1/𝑟 and 1/𝑟2 in the Eq. (24) are given by (Sakurai and Napolitano, 2011) ⟨𝜓𝑛𝑙𝑚 | 𝑍𝛼 𝑟 |𝜓𝑛𝑙𝑚⟩ = 𝑚(𝑍𝛼) 2 1 𝑛2 = −2𝐸𝑛, (25) and ⟨𝜓𝑛𝑙𝑚 | (𝑍𝛼)2 𝑟2 |𝜓𝑛𝑙𝑚⟩ = 𝑍2𝛼2 𝑎0 2𝑛3 (𝑙 + 1 2 ) = 4𝐸𝑛 2 𝑛 (𝑙 + 1 2 ) , (26) where, 𝐸𝑛 = − 𝑚(𝑍𝛼)2 2 1 𝑛2 = − 𝑍2𝛼 2𝑎0 1 𝑛2 , (27) The Bohr’s radius 𝑎0 for electronic hydrogen is given by 𝑎0 = 1 𝑚𝛼 By substituting the result in Eqs. (25) - (27) into Eq. (24) produces ∆𝐸1 (𝑛) = − 𝑍2𝛼𝐸𝑛 𝑚𝑎0 1 𝑛2 [ 3 4 − 𝑛 (𝑙 + 1 2 ) ] (28) For 1𝑠1/2 state, Eq. (28) reduces to ∆𝐸 1 (1𝑠1/2) = 5 4 𝑍2𝛼𝐸1 𝑚𝑎0 (29) Similarly, for 2𝑠1/2 state Eq. (28) becomes ∆𝐸 1 (2𝑠1/2) = 5 16 𝑍2𝛼𝐸2 𝑚𝑎0 , or ∆𝐸 1 (2𝑠1/2) = 5 64 𝐸1 (30) Note that for muonic hydrogen, the relativistic mass corrections for 1𝑠1/2 and 2𝑠1/2 states are obtained from Eqs. (29) and (30) by substituting the Bohr radius 𝑎0 with 𝑎𝜇 ⋍ 1 207 𝑎0. 3.3. Relativistic mass correction for finite size source Evaluation of the first-order relativistic mass correction for hydrogen-like atoms with Coulomb potential due to a finite size source also starts with Eq. (23). So for the ground state one may write ∆𝐸 1 (1𝑠1/2) = − 1 2𝑚 [𝐸1 2 − 2𝐸1⟨𝜓100|𝑉|𝜓100⟩ + ⟨𝜓100|𝑉 2|𝜓100⟩] (31) where, 𝑉(𝑟) is the potential induced by a uniformly charged spherical nucleus and given by ( Adamu and Ngadda, 2017) 𝑉(𝑟) = { 𝑍𝛼(− 3 2𝑅 + 𝑟2 2𝑅3 ), for 𝑟 ≤ 𝑅 −𝑍𝛼 𝑟 , for 𝑟 > 𝑅 (32) By employing this potential, we can evaluate the expectation values of 𝑉(𝑟) and 𝑉(𝑟)2. Let’s start with 〈𝑉(𝑟)〉 with respect to the 1𝑠1/2 state. That is ⟨𝜓100|𝑉|𝜓100⟩ = ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = ∫𝑅10 ∗ 𝑌00 ∗ 𝑉𝑅10𝑌00d𝜏 where, d𝜏 = 𝑟2d𝑟sin𝜃d𝜃d𝜙 = 𝑟2d𝑟dΩ Let’s now rewrite the above integral as ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = ∫𝑅10 ∗ 𝑉𝑅10𝑟 2d𝑟∫𝑌00 ∗ 𝑌00dΩ (33) The spherical harmonics is normalized, so that the second integral becomes unity (Krane, 988); and hence ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = ∫𝑅10 ∗ 𝑉𝑅10𝑟 2d𝑟. (34) In order to evaluate Eq. (24), we partition the integration into two subintervals such that ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = ∫ 𝑅10 ∗ 𝑉𝑟≤𝑅𝑅10𝑟 2d𝑟 𝑅 0 + ∫ 𝑅10 ∗ 𝑉𝑟≥𝑅𝑅10𝑟 2d𝑟 ∞ 𝑅 (35) If we insert Eq. (32) into Eq. (35), then we may obtain Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 72 ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = 𝑍𝛼{∫ 𝑅10 ∗ (− 3 2𝑅 + 𝑟2 2𝑅3 )𝑅10𝑟 2d𝑟 𝑅 0 − ∫ 𝑅10 ∗ 1 𝑟 𝑅10𝑟 2d𝑟 ∞ 𝑅 } (37) The radial wave function of electronic hydrogen in 1𝑠1/2 state is given by (Beiser, 2003) 𝑅10 = 2 𝑎0 3/2 𝑒 −𝑍𝑟 𝑎0 (38) where, 𝑎0 is the Bohr radius of electronic hydrogen whose value is given by 𝑎0 = 5.29 × 10 −11m. Substituting Eq. (38) into Eq. (37) we obtain ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = 4𝑍𝛼 𝑎0 3 {∫ ( −3 2𝑅 + 𝑟2 2𝑅3 )𝑟2𝑒 −2𝑍𝑟 𝑎0 d𝑟 𝑅 0 − ∫ 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 ∞ 𝑅 } (39) We now apply Taylor expansion of the exponential factor in the first term on the right hand side of Eq. (33). Since 𝑟/𝑎0 ≪ 1, we can approximate as (Arfken, and Weber, 2005) 𝑒−2𝑍𝑟/𝑎0 ≅ 1 − 2𝑍𝑟 𝑎0 (40) So that ∫𝜓100 ∗ 𝑉𝜓100d𝜏 = 4𝑍𝛼 𝑎0 3 {∫ ( −3 2𝑅 + 𝑟2 2𝑅3 ) (1 − 2𝑍𝑟 𝑎0 )𝑟2d𝑟 𝑅 0 − ∫ 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 ∞ 𝑅 } = 4𝑍𝛼 𝑎0 3 {− 4𝑅2 10 − ∫ 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 ∞ 𝑅 } (41) By using product rule of differentiation we have d(𝑟𝑒 −2𝑍𝑟 𝑎0 ) = 𝑒 −2𝑍𝑟 𝑎0 d𝑟 − 2𝑍 𝑎0 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 Rearranging, we get 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 = 𝑎0 2𝑍 𝑒 −2𝑍𝑟 𝑎0 d𝑟 − 𝑎0 2𝑍 d(𝑟𝑒 −2𝑍𝑟 𝑎0 ) (42) Integrating the above equation, we obtain ∫ 𝑟𝑒 −2𝑍𝑟 𝑎0 d𝑟 ∞ 𝑅 = − 𝑎0 2 4𝑍2 𝑒 −2𝑍𝑟 𝑎0 | 𝑅 ∞ − 𝑎0 2𝑍 𝑟𝑒 −2𝑍𝑟 𝑎0 | 𝑅 ∞ = 𝑎0 2 4𝑍2 (1 + 2𝑍𝑅 𝑎0 )𝑒 −2𝑍𝑅 𝑎0 (43) Substituting Eq. (43) into Eq. (41) we have ∫𝜓100 ∗ 𝑉𝜓100𝑑𝜏 = − 8𝑍𝛼𝑅2 5𝑎0 3 − 𝛼 𝑍𝑎0 (1 + 2𝑍𝑅 𝑎0 )𝑒 −2𝑍𝑅 𝑎0 (44) And also the expectation value of 𝑉2 can be given as ∫𝜓100 ∗ 𝑉2𝜓100d𝜏 = ∫ 𝑅10 ∗ 𝑉2𝑅10𝑟 2d𝑟 ∞ 0 = ∫ 𝑅10 ∗ [𝑉𝑟≤𝑅] 2 𝑅10𝑟 2d𝑟 𝑅 0 + ∫ 𝑅10 ∗ [𝑉𝑟≥𝑅] 2𝑅10𝑟 2d𝑟 ∞ 𝑅 = 4(𝑍𝛼)2 𝑎0 3 {∫ ( 9 4𝑅2 − 3𝑟2 2𝑅4 + 𝑟4 4𝑅6 )𝑟2𝑒 −2𝑍𝑟 𝑎0 d𝑟 𝑅 0 + ∫ 𝑒 −2𝑍𝑟 𝑎0 d𝑟 ∞ 𝑅 } (45) Applying the Taylor expansion on the exponential factor in the first integral term of Eq. (45) and then carrying out the integration give rise to ∫𝜓100 ∗ 𝑉2𝜓100d𝜏 = 68(𝑍𝛼)2𝑅 35𝑎0 3 + 2𝑍𝛼2 𝑎0 2 𝑒 −2𝑍𝑅 𝑎0 (46) Using Eqs. (44) and (46), Eq. (31) becomes ∆𝐸 1 (1𝑠1/2) = − 1 2𝑚 [𝐸1 2 + 16𝑍𝛼𝑅2𝐸1 5𝑎0 3 + 2𝐸1𝛼 𝑍𝑎0 (1 + 2𝑍𝑅 𝑎0 )𝑒 −2𝑍𝑅 𝑎0 + 68(𝑍𝛼)2𝑅 35𝑎0 3 + 2𝑍𝛼2 𝑎0 2 𝑒 −2𝑍𝑅 𝑎0 ] = − 1 2𝑚 {𝐸1 2 + 16𝑍𝛼𝑅2𝐸1 5𝑎0 3 + 68(𝑍𝛼)2𝑅 35𝑎0 3 + [ 2𝐸1𝛼 𝑍𝑎0 (1 + 2𝑍𝑅 𝑎0 ) + 2𝑍𝛼2 𝑎0 2 ]𝑒 −2𝑍𝑅 𝑎0 } (47) Then for the 1𝑠1/2 state, the first-order relativistic mass correction for finite size nuclei becomes ∆𝐸 1 (1𝑠1/2) = − 1 2𝑚 {𝐸1 2 + 4𝑍𝛼𝑅(28𝑅𝐸1 + 17𝑍𝛼) 35𝑎0 3 + [ 2𝐸1𝛼 𝑍𝑎0 (1 + 2𝑍𝑅 𝑎0 ) + 2𝑍𝛼2 𝑎0 2 ]𝑒 −2𝑍𝑅 𝑎0 } (48) Similarly, the first-order relativistic mass correction for the 2𝑠1/2 state is obtained by rewriting Eq. (31) as ∆𝐸 1 (2𝑠1/2) = − 1 2𝑚 [𝐸2 2 − 2𝐸2⟨𝜓200|𝑉|𝜓200⟩ + ⟨𝜓200|𝑉 2|𝜓200⟩] (49) The expectation values of 𝑉 and 𝑉2 with respect to the 2𝑠1/2 state can now be evaluated in what follows. Starting with that of 𝑉 we have ∫𝜓200 ∗ 𝑉𝜓200d𝜏 = ∫ 𝑅20 ∗ 𝑉𝑅20𝑟 2d𝑟 ∞ 0 Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 73 = ∫ 𝑅20 ∗ 𝑉𝑟≤𝑅𝑅20𝑟 2d𝑟 𝑅 0 + ∫ 𝑅20 ∗ 𝑉𝑟≥𝑅𝑅20𝑟 2d𝑟 ∞ 𝑅 (50) where, 𝑅20 = 1 2√2𝑎0 3 2 (2 − 𝑍𝑟 𝑎0 )𝑒 −𝑍𝑟 2𝑎0 (51) Substituting Eq. (51) into Eq. (50) we obtain ∫𝜓200 ∗ 𝑉𝜓200d𝜏 = 𝑍𝛼 8𝑎0 3 {∫ 𝑉𝑟≤𝑅 (2 − 𝑍𝑟 𝑎0 ) 2 𝑒 −𝑍𝑟 𝑎0 𝑟2d𝑟 𝑅 0 + ∫ 𝑉𝑟≥𝑅 (2 − 𝑍𝑟 𝑎0 ) 2 𝑒 −𝑍𝑟 𝑎0 𝑟2d𝑟 ∞ 𝑅 } (52) In the limit 𝑟 ≪ 𝑎0, we can employ the binomial expansion for the one of the factors in the first integral becomes (2 − 𝑍𝑟 𝑎0 ) 2 = 4(1 − 𝑍𝑟 2𝑎0 ) 2 ≃ 4[1 − 𝑍𝑟 2𝑎0 ] (53) Once again, we apply the Taylor expansion to the exponential factor for the first integration and using Eqs. (32) and (53), we have ∫𝜓200 ∗ 𝑉𝜓200d𝜏 = 𝑍𝛼 2𝑎0 3 ∫ (− 3𝑟2 2𝑅 + 𝑟4 2𝑅3 )d𝑟 𝑅 0 − 𝑍𝛼 8𝑎0 3 ∫ (4𝑟 − 4𝑍𝑟2 𝑎0 + 𝑍2𝑟3 𝑎0 2 )𝑒 −𝑍𝑟 𝑎0 𝑟d𝑟 ∞ 𝑅 By following similar steps as in Eqs. (42) and (43) we have ∫𝜓200 ∗ 𝑉𝜓200d𝜏 = −𝑍𝛼𝑅2 5𝑎0 3 − 𝛼[ 1 4𝑍𝑎0 (1 + 𝑍𝑅 𝑎0 ) − 𝑍𝑅2 8𝑎0 3 + 𝑍2𝑅3 8𝑎0 4 ]𝑒 −𝑍𝑅 𝑎0 (54) and ∫𝜓200 ∗ 𝑉2𝜓200d𝜏 = ∫ 𝑅20 ∗ (𝑉𝑟≤𝑅) 2𝑅20𝑟 2d𝑟 𝑅 0 + ∫ 𝑅20 ∗ (𝑉𝑟≥𝑅) 2𝑅20𝑟 2d𝑟 ∞ 𝑅 = 17(𝑍𝛼)2𝑅 70𝑎0 3 + 𝛼2 [ 𝑍 2𝑎0 2 − 𝑍 4𝑎0 2 (1 + 𝑍𝑅 𝑎0 ) + 𝑍3𝑅2 8𝑎0 4 ]𝑒 −𝑍𝑅 𝑎0 (55) Substituting Eqs. (54) and (55) into Eq. (49) we obtain ∆𝐸 1 (2𝑠1/2) = − 1 2𝑚 {𝐸2 2 + 2𝛼𝐸2𝑅 2 5𝑎0 3 + 17𝑍2𝛼2𝑅 70𝑎0 3 + [ 𝑍𝛼2 2𝑎0 2 + 𝛼𝐸2 2𝑍𝑎0 + ( 𝐸2 2𝑎0 − 𝑍2𝛼 4𝑎0 2 )𝛼𝑅 + ( 𝑍2𝛼 8𝑎0 4 − 𝐸2 4𝑎0 3 )𝑍𝛼𝑅2 + 𝑍2𝛼𝐸2 4𝑎0 4 𝑅3]𝑒 −𝑍𝑅 𝑎0 } (56) In case of muonic hydrogen-like atoms, we may use Eqs. (48) and (56) to estimate the relativistic mass corrections to 1𝑠1/2 and 2𝑠1/2 states, respectively, by substituting 𝑎0 with 𝑎𝜇 whose value is given by 𝑎𝜇 = 2.56 × 10−13 m. 4. Results and Discussion The corrections in Eqs. (28), (48) and (56) are converted into Matlab source code to produce numerical results. We have used MATLAB software to generate the graph of relativistic mass correction versus proton number. The mass of muon and electron are obtained from CODATA and the nuclear radii for each nucleus is obtained from International Atomic Energy Agency. Table 1 contains the experimental values of important parameters such as rest masses of parti. By employing the experimental values of the parameters given in Table 1, we can estimate the numerical values of first-order relativistic mass correction to 1𝑠1/2 and 2𝑠1/2 states for both types of hydrogen-like atoms in the unit of electronvolt (eV). Table 2 displays those corrections using point-like and finite size charge distributions, separately, for atoms with light and heavy nuclei. It is evident that for electronic hydrogen-like atom, the first-order relativistic mass correction is more prominent in 1s_(1/2) state than 2s_(1/2) state for both types of charge distributions. It was indicated by the authors (Xie et al., 2021) that the relativistic mass correction takes the negative values and decrease continuously. For point-like source, the relativistic mass correction of ground state has been calculated and the result have strong agreement with the present work (Wikipedia, 2022). Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 74 Table 1: Experimental values of some parameters used in this numerical analysis. Nucleus Electron mass (𝐌𝐞𝐕) Muon mass (𝐌𝐞𝐕) Bohr Radius of electronic atom (𝐌𝐞𝐕−𝟏) Bohr Radius muonic atom (𝐌𝐞𝐕−𝟏) Nuclear charge Radius (𝐌𝐞𝐕−𝟏) 0.5110 105.65836 268.21 1.2975 𝐻1 1 − − − − 0.00445 𝐻4 𝑒2 − − − − 0.00997 𝐿𝑖6 3 − − − − 0.01312 𝐵𝑒7 4 − − − − 0.01341 𝐶12 6 − − − − 0.01252 𝑍𝑛64 30 − − − − 0.01991 Table 2: The first-order relativistic mass correction for electronic hydrogen-like atoms. Nucleus ∆𝑬 𝟏 (𝟏𝒔𝟏/𝟐) with point-like source (eV) ∆𝑬 𝟏 (𝟏𝒔𝟏/𝟐) with finite size source (eV) ∆𝑬 𝟏 (𝟐𝒔𝟏/𝟐) with point-like source (eV) ∆𝑬 𝟏 (𝟐𝒔𝟏/𝟐) with finite size source (eV) 𝐻1 1 −9.0569 × 10 −4 −9.0655 × 10−4 −1.4717 × 10−4 −1.4772 × 10−4 𝐻3 𝑒2 −3.6227 × 10 −3 −2.7192 × 10−3 −5.8870 × 10−4 −3.5482 × 10−4 𝐿𝑖6 3 −8.1551 × 10 −3 −4.2899 × 10−3 −1.3246 × 10−3 −5.5065 × 10−4 𝐵𝑒7 4 −1.4500 × 10 −2 −5.8000 × 10−3 −2.3548 × 10−3 −7.4419 × 10−4 𝐶12 6 −3.2605 × 10 −2 −8.7599 × 10−3 −5.2983 × 10−3 −1.1314 × 10−3 𝑍𝑛64 30 −8.1511 × 10 −1 −4.3569 × 10−2 −1.3246 × 10−1 −7.0491 × 10−3 Table 3 shows that the corrections due to point-like and finite size charge distributions, separately, for light and heavy nuclei muonic hydrogen. For muonic hydrogen-like atoms, the first-order relativistic mass correction, when considering point-like nuclei, has a better contribution in the 1𝑠1/2 state than the 2𝑠1/2 state over the whole range of 𝑍. However, the same cannot be said when taking into account finite size nuclei as the relativistic mass correction is more prominent in 1𝑠1/2 state than 2𝑠1/2 state for muonic hydrogen-like atoms with 𝑍 < 15. But for muonic atoms with heavier nuclei the situation changes in such away that the relativistic mass correction begins to exhibit strong influence over the 2𝑠1/2 state as compared to the 1𝑠1/2 state. Table 3: The first-order relativistic mass correction for muonic hydrogen-like atoms. Nucleus ∆𝑬 𝟏 (𝟏𝒔𝟏/𝟐) with point- like source (eV) ∆𝑬 𝟏 (𝟏𝒔𝟏/𝟐) with finite size source (eV) ∆𝑬 𝟏 (𝟐𝒔𝟏/𝟐) with point-like source (eV) ∆𝑬 𝟏 (𝟐𝒔𝟏/𝟐) with finite size source (eV) 𝐻1 1 −1.8736 × 10 −1 −1.8609 × 10−1 −3.0447 × 10−2 −4.7251 × 10−2 𝐻3 𝑒2 −7.4946 × 10 −1 −5.5244 × 10−1 −1.2179 × 10−1 −2.2342 × 10−1 𝐿𝑖6 3 −1.68630 × 10 0 −8.5973 × 10−1 −1.9673 × 10−1 −5.5836 × 10−1 𝐵𝑒7 4 −2.99783 × 10 0 −1.15121 × 100 −4.8715 × 10−1 −9.6176 × 10−1 𝐶12 6 −6.74512 × 10 0 −1.71430 × 100 −1.09610 × 100 −1.93145 × 100 𝑍𝑛64 30 −1.68640 × 10 2 −7.62900 × 100 −2.74020 × 101 −6.92000 × 101 Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 75 Figure 1: Relativistic mass correction versus proton number for 1𝑠1/2 state electronic atom. Figure 2: Relativistic mass correction versus proton number for 2𝑠1/2 sate electronic atom. Figures 1 and 2 depict that the results of both present work (finite size source) and point-like source for electronic hydrogen atom. The results show that as the atomic number (𝑍) increases, the relativistic mass corrections decrease for the two types of hydrogen-like atoms. The plots also clearly indicate that the correction with finite size charge distribution agrees with that of the point-like source in case of hydrogen-like atoms with light nuclei. Nevertheless, as 𝑍 increases, there is strong deviation of the corrections with finite size distributions from the ones with point-like sources. Eshetu Diriba &Gashaw Bekele. Ethiop.J.Sci.Sustain.Dev., Vol. 9 (2), 2022 76 Figure 3: Relativistic mass correction versus proton number for 1𝑠1/2 state muonic atom. Figure 4: Relativistic mass correction versus proton number for 2𝑠1/2 state muonic atom. Figures 3 and 4 illustrate the dependence of the relativistic mass correction on the proton number 𝑍 for 1𝑠1/2 and 2𝑠1/2 states of muonic hydrogen-like atoms. According to Figure 3, as the atomic number increases so is the deviation of the relativistic mass correction to 1𝑠1/2 state with finite size nuclei from the one with point-like charge distribution. Based on Figure 4 the relativistic mass corrections to 2𝑠1/2 state of muonic hydrogen-like atoms with finite size and point-like nuclei show slightly better agreement as opposed to the case of the 1𝑠1/2 states of similar atoms. 5. Conclusion In this paper, we have shown how, in non-relativistic limit, Dirac equation can be reduced into Schrodinger equation and we have presented the mathematical approach used to calculate the relativistic mass correction by using potentials from point-like and finite size source charge distributions. The results show that, for hydrogen-like atoms with light nuclei the relativistic mass corrections due to the finite size source roughly coincides with that of point-like source. 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