Solutions of the Schrodinger Equation for the Hulthen-Type Potential Plus Modified Kratzer Potential: Application to Ionic Crystals
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The approximate analytical solution of the radial Schrodinger equation has been extended to ionic crystal using the Hulthen-type plus modified Kratzer potential model, within the framework of Nikiforov-Uvarov method using Greene-Aldrich approximation. The aim of combining these potentials is to have a wide application. The energy eigenvalues for NaCl, NaF, NaBr and NaI ionic crystals were computed for various vibrational and rotational quantum numbers. Special cases were considered when the potential parameters were altered, resulting into Hulthen-Type Potential and Modified Kratzer Potential. Their energy eigenvalues expressions and numerical computations agreed with the already existing literatures. Also, spectroscopic parameter for ionic crystals were used in plotting graphical variation of the bound state energy eigenvalues for the ionic crystals with different potential parameters and quantum numbers were discussed. Our results are in agreement with the reports of other researchers.
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Abramowitz, M. & Stegun, I. A., (1964) Handbook of Mathematical Functions with formulas, Grapgs and Mathematical tables. (New York: Dover).
Ahmadov, A. Aslanova, S. Orujova, M. S. Badalov, S. Dong, S. H. (2019) approximation bound state solution of Klein-Gordon equation with linear combination of Hulthen and Yukawa potentials, Phys. Lett. A 383 (24) 3010-3017.
Akpan, I. O., Antia,A. D., Ikot, A. N.( 2012) Bound state solutions of Klein-Gordon equation with deformed equal scalar and vector eckart potential using newly improved approximation scheme, ISRN High Energy phys. 2012
Anita, A. D. Essien, I. Umoren, E. Eze, C. (2015) Approximate of the non-relativistic Schrödinger equation with inversely quadratic Yukawa potential plus Mobius square potential via parametric Nikiforov-Uvarov method. Adv. Phys. Theor. Appl. 44 (1) p.
Arda, A. Sever, R. (2009) Approximate i-state solution to the Klein-Gordon equation for modified woods-saxon potential with position dependent mass, Int. J. Mod. Phys. A 24 (20n21) 3985-3994.
Badalov, V. Ahmadov, H. Badalov, S. (2010) Any i-state solution to the Klein-Gordon equation for modified Woods-Saxon potential, Int. J. Mod. Phys. E 19 (07) 1463-1475.
Bayrak, O., kocak, G., Boztosun, I., (2006) J. Phys, A Math. Gen 39 6955.
Bayrak, O., Boztosun, I., Ciftci, H., (2007), Exact analytical solutions to the Kratzer potential by the asymptotic iteration method, international Journal of Quantum chemistry 107 (3) 540-544. https:doi.org/10.1002/qua.21141.
Berezin, A. A. (1986) Phys.Rev. B, 33 pp. 2122
Berkdemir, C.,Berkdemir, A., Han,J., (2006) Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential, Chem. Phys. Lett. 417 (4-60) 326-328
Chen, C Y Lu, F L and Sun, D S., (2008) Cent Eur. J. Phys. 6 884
Ciftei, H., Hall, R. L., Saad, N.,(2003), Asymtotic iteration method for eigenvalue problems, Journal of Physics A: Mathematical and General 36 (47) 11807-11816 https://doi.org//10.1088/03050083.
Chun-Feng, H., Zhong-Xiang, Z. and Yan L. (1991) Bound state of the Klein-Gordon with vector and scalar Woods-Saxon potentials, Acta Physica Sinica 8 (8) 561-565. https://doi.org/10.1088/1004-423X/8/8/001.
Chun-Feng, H.., Zhong-Xiang, Z., and Yan L.I (1991)Acta Physica Sinica 8, 561
Das, T. & Arda, A, (2017), Klein-Gordon equation for a charged particle in space varying electromagnrtic fields- a systematic study via Laplace transform, Chin. J. Phys.55 (2) 310-317
Dong,S.,.Lozada-Cassou, M (2006) exact solutions of Klein-Gordon equation with scalar and vector ring-shaped potentials ,Phys. Scr. 74 (2) 285.
Dong S.H., Gu, X. Y. (2008) J. Phys. Conf. Ser. 96, 012109
Egrifes, H., Demirhan, D, and Buyukkilic, F., (2000) Phys. Lett. A.275 229
Gu, X- Y. Dong, S- H. Ma, Z- Q. (2008) Energy spectra for modified Rosen-Morse potential solved by the exact quantization rule, J. phys. A Math. Theor. 42 (3) 035303.
Hamzavi, M., Thlwe, K. E and Rajabi, A., (2013), Aproximate Bound states solution of the Hellmann potential Communication in Theoretical Physics, 60 1
Hassannabadi,H., Rahimov,H., Zarrinkamar, S., (2011) approximate solution of Klein-Gordon equation with Kratzer potential, Adv. High Energy phys. 2011 p.
Hulthen, L. Sugawara, M. Flugge(ed), S. (1957) Handbuckder physic, Springer
Huseyin,R. & Sever. J. j.math. chem. 50 1938 (2012)
Ikhdair, S. M. (2009) Bound state solution of the Klein-Gordon equation for vector and scalar general Hulthen-type potentials in D-dimension, Int. J. Mod. Phys. C 20 (01) 25-45.
Ikhdair, S. M. Sever, R. (2007) Exact solution of the Klein-Gordon equation for the PT-symmetry Woods-saxon potential by the Nikiforov-Uvarov Method, Ann. Phys. 16 (3) 218-232.
Ikhdair, S. M. (2009) An improved approximation scheme for the centrifugal term and the Hulthen potential, Eur. Phys.J. A 39 (3) 307-314
Ikhdair, S.M. and Falaye, B. J., (2013) chem. Phys 421 84
Ikot, A. N. Maghsoodi E., Zarrinkamar, S. Hassanabadi, H. (2013) Relativistic spin and pseudospin symmetries of inverse quadratic Yukawa-plus Mobius square potentials including a Coulomb-like tensor interaction, Few-Body syst. 54 (11) 2027-2040.
Ikot, A. N., Akpabio, L.E. and Umoren, E. B., (2011), Exact solution of Schrödinger equation with inverted Woods-Saxon potential and Manning-Rosen potential, Journal Scientific. Research 3(1) 25-33
Jia, C. S., & Jia, Y. (2017), Eur. J. D. 71 3
Karayer, H., Demirha, D., & Buyukkilic, F. (2015). Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation. Journal of Mathematical Physics, 56, 063504, 1-14. https://dx.doi.org/10.1063/1.4922601
Kratzer, A, (1920), Die ultraroten rotationsspektren der halogenwasserstoffe, Z. Phys 3 289 -307
Lai, C.S. Lin, W. C. (1980)phys. Let. A, 78 pp. 335
Liu, H-B. Yi, L-Z. Jia, C-S. (2018) solutions of the Klein-Gordon equation with the Tietz potential energy, J. Math. Chem. 56 (10) 2982-2994
Ma, Z. Q. Dong, S. H. Gu, X. Y..,Yu, J., Lozada-Cassou,M., (2014)The Klein-Gordon equation with a Coulomb plus scalar potential in D-dimensions, Int. J. Mod. Phys. E 13 (03) 597-610.
Maghsoodi, E., Hassanabadi, H., Zarrinkamar, S., (2012), Spectrum of Dirac Equation under Deng-Fan scalar and vector potential and a coulomb Tensor interaction by SUSYQM, Few-Body system 53 (3-4) 525-538 https://doi.org/10.1007/s00601-0314-5
Nagiyev, S. M. Ahmadov, A. (2019) exact solution of the relativistic finite-difference equation for Coulomb plus a ring-shaped-like potential. Int. J. Mod. Phys. A 34 (17) 1950089
Nififorov,A. F, Uvarov, V. B, (1988), special functions of mathematical physics, 205 Springer
Nugraha, D. A. Suparmi, A. Cari, C. Pratiwi, B. N. (2017) Asymptotic iteration method for analytical solution of Klein-Gordon equation for trigonometric Poschl-Teller potentialin D-dimensions, journal of physics: conference series, IOP Publishing,
Oyewumi, k.J., Akinpelu, F.O., Agboola, A.D., (2008) Int.J. Theor. Phys 47, 1039
Onate, C., Onyeaju, M.,Ikot, A., Ebomwonyi, O, (2017), Eigen solution and entropy system for Hellmann potential in the presence of the Schrödinger equation, Eur. Phys. J. Plus 132 (11) 462
Onate, C., Ojonubah,J., Adeoti, A., Eweh, J.,Ugboja,M., (2014), Approximate eigen solutions of DKP and Klein-Gordon equation with Hellmann potential, Afr. Rev. Phys. J.9 (006) 497-504.
Payil, S. H (1984).J. phys. A, 17 pp. 575
Pekeris C.I. (1934) Phys. Rev. 45 98
Peng, X. L., Jiang, R.,Jia, C. S., Zhang, L. H., and Zhao, Y. L, (2018), Gibbs free energy of gaseous phosphorus dimer, Chem. Eng. Sci. 190 122
Popov, V.S. Wiengberg, V. M. (1985) phys. Let. A, 107 pp. 371
Pyykko,P., Jokiasaari, (1975) J. Chem. Phys, 10 pp. 293
Roy, B., Roychoudhury,R., (1987) J. phys. A, 20 pp. 3051
Qiang, W. C. and Dong, S. H., (2007), Analytical approximations to the solutions of the Manning-Morsen potential with Centrifugal term. Phys. Lett. A 368.13-17.
Qiang, W.C. Dong, S.H.Analytical , (2007)approximation to the solution of the Manning-Rosen potential with centrifugal term, Phys. Lett. A 368 (1-2) 13-17
Qiang, W.C. Dong, S.H.Analytical , (2007), Arbitrary i-state solution of the rotating Morse potential Rosen potential through the exact quantization rule method, Phys. Lett. A 363 (3) 169-176
Pearson, R. G., (1963), Hard and Soft Acids and Bases. Journal of the American Chemical Society. 85 (22) 3533-3539
Ramantswana, M. Rampho, G. J. Edet, C. O Ikot, A.N Okorie, U. S. Qadir, K.W. Abdullah, H.Y. (2023) Determination of thermodynamic properties of CrH,NiC, and CuLi Diatomic Molecules with the linear combination of Hulthen-type potential plus Yukawa potential, physics open, doi: https://doi.org/10.1016/j.physo.2022.100135.
Saad, N.,(2007) the Klein-Gordon equation with a generalized Hulthen potential in D-dimensions, Phys. Scr. 76 (6) 623.
Saad, N.,(2008), The Klein-Gordon equation with Kratzer potential in in D-dimensions, Central European Journal of Physics 6 (3) 717-729.
Sadeghi, J.,(2007), Factoeization method and solution of non-central modified Kratzer potential, Acta Physica Polonia A 112 (1) 23-28
Saleh, R., Kassem, M., Mabrouk, S., (2019) exact solution of nonlinear fractional order partial differential equations via singular manifold method, Chin. J. Phys. 61 290- 300
Szego, G. (1939), Orthogonal polynomials. American Mathematical society, New York.
Sherman, J. (1932), Crystal Energy of ionic compounds and thermochemical Applications, chemical Review. 11 (1) 93-170
Wei, G. F. Dong, S. H. (2010) Pseudospin symmetry in the relativistic Manning-Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal term, Phys. Lett. B 686 (4-5) 288-292
Yahya, W. A. & Oyewumi, K. J, (2015) Journal of the Association of Arab Univ. for basic and Appl. Sci. 1815