Non-Relativistic Study of Mass Spectra, and Thermal Properties of a Quarkonium System with Eckart-Hellmann Potential
Abstract
In this present study, we model Eckart-Hellmann Potential (EHP) to interact in a quark-antiquark system. The solutions of the Schrödinger equation are obtained with EHP using the Nikiforov-Uvarov method. The energy equation and normalized wave function were obtained. The masses of the heavy mesons such as charmonium (cc-) and bottomonium (bb-) for different quantum numbers were predicted using the energy equation. Also, the partition function was calculated from the energy equation, thereafter other thermal properties such as mean energy, free energy, entropy, and specific heat capacity were obtained. The results obtained showed an improvement when compared with the work of other researchers and excellently agreed with experimental data.
Downloads
References
H. Ciftci, and H.F. Kisoglu, “Nonrelativistic-Arbitrary l-states of quarkonium through Asymptotic Iteration method”, Advances in High Energy Physics, 4549705 (2018). https://doi.org/10.1155/2018/4549705
H. Mutuk, “Mass Spectra and Decay constants of Heavy-light Mesons: A case study of QCD sum Rules and Quark model”, Advan. in High Energy Phys. 8095653 (2018). https://doi.org/10.1155/2018/8095653
M. Allosh, Y. Mustafa, N.K. Ahmed, and A.S. Mustafa, “Ground and Excited state mass spectra and properties of heavy-light mesons”, Few-Body Syst. 62, 26 (2021). https://doi.org/10.1007/s00601-021-01608-1
E. Omugbe, O.E. Osafile, I.B. Okon, E.P. Inyang, E.S. William, and A. Jahanshir, “Any L-state energy of the spinlessSalpeter equation under the Cornell potential by the WKB Approximation method: An Application to mass spectra of mesons”, Few-Body Systems, 63(1), 7 (2022). https://doi.org/10.1007/s00601-021-01705-1
E.S. William, E.P. Inyang, and E.A. Thompson, “Arbitrary -solutions of the Schrödinger equation interacting with Hulthén-Hellmann potential model”, Revista Mexicana Fisica, 66, 730 (2020). https://doi.org/10.31349/RevMexFis.66.730
E. Omugbe, “Non-relativistic eigensolutions of molecular and heavy quarkonia interacting potentials via the Nikiforov-Uvarov method”, Canadian Journal of Physics, 98, 1125 (2020). https://doi.org/10.1139/cjp-2020-0039
E.S. William, E.P. Inyang, I.O. Akpan, J.A. Obu, A.N. Nwachukwu, and E.P. Inyang, “Ro-vibrational energies and expectation values of selected diatomic molecules via Varshni plus modified Kratzer potential model”, Indian Journal of Physics, (2022). https://doi.org/10.1007/s12648-022-02308-0
E.P. Inyang, and E.O. Obisung, “The study of electronic states of NI and ScI molecules with screened Kratzer Potential”, East Eur. J. Phys. 3, 32 (2022). https://doi.org/10.26565/2312-4334-2022-3-04
E.P. Inyang, E.S. William, E. Omugbe, E.P. Inyang, E.A. Ibanga, F. Ayedun, I.O. Akpan, and J.E. Ntibi, “Application of Eckart-Hellmann potential to study selected diatomic molecules using Nikiforov-Uvarov-Functional analysis method”, Revista Mexicana de Fisica, 68, 14 (2022). https://doi.org/10.31349/RevMexFis.68.020401
E.P. Inyang, E.S.William , J.E.Ntibi, J.A.Obu, P.C.Iwuji, and E.P.Inyang “Approximate solutions of the Schrodinger equation with Hulthen plus screened Kratzer potential using the Nikiforov-Uvarov-Functional analysis method: An Application to diatomic molecules”., Canadian Journal of Physics (2022). https://doi.org/10.1139/cjp-2022-0030
E.P. Inyang, E.P. Inyang, I.O. Akpan, J.E. Ntibi, and E.S.William, “Analytical solutions of the Schrödinger equation with class of Yukawa potential for a quarkonium system via series expansion method”, European Journal of Applied Physics 2, 26 (2020). http://dx.doi.org/10.24018/ejphysics.2020.2.6.26
E.P. Inyang, P.C. Iwuji, J.E. Ntibi, E.S. William, and E.A. Ibanga, “Solutions of the Schrodinger equation with Hulthen –screened Kratzer potential: Application to diatomic molecules”, East. Eur. J. Phys. 1, 11 (2022). https://doi.org/10.26565/2312-4334-2022-2-02
M. Abu-Shady, T.A. Abdel-Karim, and E.M. Khokha, “Exact solution of the N-dimensional Radial Schrödinger Equation via Laplace Transformation method with the Generalized Cornell potential”, J. Quantum Phys. 45, 587 (2018). https://doi.org/10.48550/arXiv.1802.02092
E. Omugbe, O.E. Osafile, I.B. Okon, E.S. Eyube, E.P. Inyang, U.S. Okorie, A. Jahanshir, and C.A. Onate, “Non-relativistic bound state solutions with α-deformed Kratzer-type potential using the super-symmetric WKB method: application to theoretic-information measures”, European Physical Journal D, 76, 72 (2022), https://doi.org/10.1140/epjd/s10053-022-00395-6
E. Omugbe, O.E. Osafile, E.P. Inyang, and A. Jahanshir, “Bound state solutions of the hyper-radial Klein-Gordon equation under the Deng-Fan potential by WKB and SWKB methods”, Physica Scripta, 96(12), 125408 (2021). https://doi.org/10.1088/1402-4896/ac38d4
C.O. Edet, S. Mahmoud, E.P. Inyang, N. Ali, S.A. Aljunid, R. Endut, A.N. Ikot, and M. Asjad, “Non-Relativistic Treatment of the 2D Electron System Interacting via Varshni-Shukla Potential Using the Asymptoptic Iteration Method”, Mathematics, 10, 2824 (2022). https://doi.org/10.3390/math10152824
M. Abu-shady, C.O. Edet, and A.N. Ikot, “Non-relativistic Quark model under external magnetic and Aharanov-Bohm (AB) fields in the presence of Temperature-Dependent confined Cornell potential”, Canadian Journal of Physics, (2021) https://doi.org/10.11139/cjp-2020-0101
R. Kumar, and F. Chand, “Asymptotic study to the N-dimensional Radial Schrodinder Equation for the quark-antiquark system”, Commun. In Theor. Phys .59, 467 (2013). https://doi.org/10.1088/0253-6102/59/5/02
J.P. Prasanth, K. Sebastian, and V.M. Bannur, “Revisiting Cornell potential model of the Quark-Gluon plasma”, Physica A, 558,124921 (2020). https://doi.org/10.1016/j.physa.2020.124921
H. S. Chung, J. Lee, and D. Kang, “Cornel potential parameters for S-wave heavy quarkonia”, Journal of the Korean physical society, 52, 1151 (2008). https://doi.org/10.3938/jkps.52.1151
R. Kumar, R.M. Singh, S.B. Bhahardivaj, R. Rani, and F. Chand, “Analytical solutions to the Schrodinger equation for generalized Cornell potential and its application to diatomic molecules and heavy mesons”, Mod. Phys. Lett. A. 37, 2250010 (2022). https://doi.org/10.1142/S0217732322500109
A. Vega, and J. Flores, “Heavy quarkonium properties from Cornell potential using variational method and supersymmetric quantum mechanics”, Pramana – J. Phys. 87, 73 (2016). https://doi.org/10.1007/s12043-016-1278-7
H. Mutuk, “Cornell Potential: A Neural Network Approach”, Advan. in High Energy Phys. 22, 3105373 (2019). https://doi.org/10.1155/2019/3105373
H. Hassanabadi, M. Ghafourian, and S. Rahmani, “Study of the Heavy-Light mesons properties via the Variational method for Cornell interaction”, Few-Body Syst. 57, 249 (2016). https://doi.org/10.1007/s00601-015-1040-6
M. Abu-Shady, and S.Y. Ezz-Alarab, “Trigonometric Rosen–Morse Potential as a Quark–Antiquark Interaction Potential for Meson Properties in the Non-relativistic Quark Model Using EAIM”. Few-Body Systems, 60, 66 (2019). https://doi.org/10.1007/s00601-019-1531-y
M. Abu-shady, H.M. Mansour, and A.I. Ahmador, “Dissociation of Quarkonium in Hot and Dense media in an Anisotropic plasma in the Non-relativistic Quark model”, Advances in High Energy Physics, 4785615 (2019). https://doi.org/10.1155/2019/4785615
E.P. Inyang, E.P. Inyang, J.E. Ntibi, E.E. Ibekwe, and E.S. William, “Approximate solutions of D-dimensional Klein-Gordon equation with Yukawa potential via Nikiforov-Uvarov method”, Indian Journal of Physics, 95, 2733 (2021). https://doi.org/10.1007/s12648-020-01933-x
E.P. Inyang, E.P. Inyang, E.S. William, and E.E. Ibekwe, “Study on the applicability of Varshni potential to predict the mass-spectra of the Quark-Antiquark systems in a non-relativistic framework”, Jordan Journal of Physics, 14, 337 (2021). https://journals.yu.edu.jo/jjp/Draft/Vol14No4/8-%20Study%20on%20the%20Applicability,%20Inyang%20et%20al.pdf
E.E. Ibekwe, U.S. Okorie, J.B. Emah, E.P. Inyang, and S.A. Ekong, “Mass spectrum of heavy quarkonium for screened Kratzer potential (SKP) using series expansion method”, Eur. Phys. J. Plus, 87, 136 (2021). https://doi.org/10.1140/epjp/s13360-021-01090-y
I.O. Akpan, E.P. Inyang, E.P. Inyang, and E.S. William, “Approximate solutions of the Schrödinger equation with Hulthen-Hellmann Potentials for a Quarkonium system”, Revista Mexica De Fisica, 67, 482 (2021). https://doi.org/10.31349/RevMexFis.67.482
K.R. Purohit, P. Jakhad, and A.K. Rai, “Quarkonium spectroscopy of the linear plus modified Yukawa potential”, Phys. Scripta, 97, 044002 (2022). https://doi.org/10.1088/1402-4896/ac5bc2
S. Patel, P.C. Vinodkumar, and S. Bhatnagar, ”Decay rates of charmonia within a quark-antiquark confining potential”, Chinese Physics C, 40, 053102 (2016). https://doi.org/10.1088/1674-1137/40/5/053102
V. Mateu, P.G. Ortega, D.R. Entem, and F. Fernadez, “Calibrating the nave Cornell model with NRQCD”, The European Physical Journal C, 79, 323 (2019). https://doi.org/10.1140/epjc/s10052-019-6808-2
F. Brau, and C. Sernay, “The three-dimensional Fourier grid Hamiltonian method”, Journal of computational physics, 139, 136 (1998). https://doi.org/10.1006/jcph.1997.5866
A. Bhaghyesh, “Charmonium properties using the Discrete variable representation (DVR) method”, Advances in High Energy Physics, 9991152 (2021). https://doi.org/10.1155/2021/9991152
S. Jacobs, M.G. Olsson, and C. Suchyta, “Comparing the Schrodinger and Spinless Salpeter equations for heavy-quark bound states”, Physical Review D, 33, 3338 (1986). https://doi.org/10.1103/physrevd.33.3338
B. Grinstein, “A modern introduction to quarkonium theory”, International Journal of Modern Physics, 15, 461 (2000). https://doi.org/10.1142/S0217751X00000227
W. Lucha, F. Schoberl, and D. Gromes, “Bound states of quarks”, Physics Reports, 200, 127 (1991). https://doi.org/10.1016/0370-1573(91)90001-3
E. P. Inyang, A.N.Ikot, I.O.Akpan, J.E.Ntibi, E.Omugbe, and E. S. William, “Analytic study of thermal properties and masses of heavy mesons with quarkonium potential”, Result in Physics. (2022) 105754. https://doi.org/10.1016/j.rinp.2022.105754
E.P. Inyang, E.P. Inyang, I.O. Akpan, J.E. Ntibi, and E.S. William, “Masses and thermodynamic properties of a Quarkonium system”, Canadian Journal Physics, 99, 990 (2021). https://doi.org/10.1139/cjp-2020-0578
E.P. Inyang, E.O. Obisung, P.C. Iwuji, J.E. Ntibi, J. Amajama, and E.S. William, “Masses and thermal properties of a charmonium and Bottomonium mesons”, J. Nig. Soc. Phys. Sci. 4 (2022). https://doi.org/884.10.46481/jnsps.2022.884
C. Eckart, “The Penetration of a potential Barrier by Electrons”, Phys. Rev. 35, 1303 (1930). https://doi.org/10.1103/PhysRev.35.1303
H. Hellmann, “A New Approximation Method in the Problem of Many Electrons”, J. Chem. Phys. 3, 61 (1935). https://doi.org/10.1063/1.1749559
C.A. Onate, J.O. Ojonubah, A. Adeoti, E.J. Eweh, and M. Ugboja, “Approximate Eigen Solutions of D.K.P. and Klein-Gordon Equations with Hellmann Potential”, Afr. Rev. Phys. 9, 497 (2014). http://eprints.lmu.edu.ng/1594/1/ONATE%2032.pdf
E.P. Inyang, E.S. William, J.O. Obu, B.I. Ita, E.P. Inyang, and I.O. Akpan, “Energy spectra and expectation values of selected diatomic molecules through the solutions of Klein-Gordon equation with Eckart-Hellmann potential model”, Molecular Physics, 119(23), e1956615 (2021). https://doi.org/10.1080/00268976.2021.1956615
S.K. Nikiforov, and V.B. Uvarov. Special functions of Mathematical Physics, (Birkhauser, Basel, 1988)
E.P. Inyang, E.S. William, and J.A. Obu, “Eigensolutions of the N-dimensional Schrödinger equation` interacting with Varshni-Hulthen potential model”, Rev. Mexi. Fisi. 67, 193 (2021). https://doi.org/10.31349/RevMexFis.67.193
M. Abu-Shady, “N-dimensional Schrödinger equation at finite temperature using the Nikiforov-Uvarov method”, J. Egypt. Math. Soc. 23, 4 (2016). https://doi.org/10.1016/j.joems.2016.06.006
I.S. Gradshteyn, and I.M. Ryzhik, Tables of integrals, series and product, 7th ed. (Academic Press, Boston, 2007).
E.P. Inyang, E.P. Inyang, J. Karniliyus, J.E. Ntibi, and E.S. William, “Diatomic molecules and mass spectrum of heavy quarkonium system with Kratzer-screened Coulomb potential (KSCP) through the solutions of the Schrödinger equation”, European Journal of Applied Physics, 3, 55 (2021). https://doi.org/10.24018/ejphysics.2021.3.2.61
E.P. Inyang, P.C. Iwuji, J.E. Ntibi, E. Omugbe, E.A. Ibanga, and E.S. William “Quark-antiquark study with inversely quadratic Yukawa potential using Nikiforov-Uvarov-Functional analysis method”, East Eur. J. Phys. 2, 43 (2022), https://periodicals.karazin.ua/eejp/article/view/18535/16839
R. Olive, D.E. Groom, and T.G. Trippe, “Particle Data Group”, Chin. Phys. C, 38, 60 (2014).
M. Tanabashi, C.D. Carone, T.G. Trippe, and C.G. Wohl, Particle Data Group, Phys. Rev. D, 98, 546 (2018).
Copyright (c) 2022 Etido P. Inyang, Effiong O. Obisung, Eddy S. William, Ituen B. Okon
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
