Cite this Article
Onate C. A., Onyeaju M. C., Ikot A. N., Ebomwonyi O., Idiodi J. O. A.. Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries. Communications in Theoretical Physics, 2018, 70(3): 294  
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Dirac Equation with a New Tensor Interaction under Spin and Pseudospin Symmetries
Onate C. A.1, *, Onyeaju M. C.2, Ikot A. N.2, Ebomwonyi O.3, Idiodi J. O. A.3
Physics Programme, Department of Physical Sciences, Landmark University, PMB 1001, Omu-Aran, Nigeria
Theoretical Physics Group, Physics Department, University of Port Harcourt, Nigeria
Department of Physics, University of Benin, Benin City, Nigeria

 

† Corresponding author. E-mail: oaclems14@physicist.net

Abstract
Abstract

The approximate analytical solutions of the Dirac equation under spin and pseudospin symmetries are examined using a suitable approximation scheme in the framework of parametric Nikiforov-Uvarov method. Because a tensor interaction in the Dirac equation removes the energy degeneracy in the spin and pseudospin doublets that leads to atomic stability, we study the Dirac equation with a Hellmann-like tensor potential newly proposed in this study. The newly proposed tensor potential removes the degeneracy from both the spin symmetry and pseudospin symmetry completely. The proposed tensor potential seems better than the Coulomb and Yukawa-like tensor potentials.

Keyword:wave equation;eigensolutions;Dirac equation;Nikiforov-Uvarov method;tensor interaction
1 Introduction

The study of Dirac equation with spin and pseudospin symmetries have been a line of great interest in the recent years.[1–15] Dirac equation as a relativistic wave equation, has been used to explain the behaviour of particles in the atomic domain. The symmetry limits (spin and pseudospin) of the Dirac equation introduced in nuclear theory[16–18] are used to explain the real characteristics of deformed nuclei,[19] superdeformation[20] and established effective shell model coupling scheme.[21] Owning to these importance, Dirac equation has been given serious attention using different methodologies such as Nikiforov-Uvarov method,[22–25] supersymmetry quantum mechanics,[26–28] asymptotic iteration method,[29–32] and others. However, one of the fundamental challenging problems facing the study of spin and speudospin symmetries of the Dirac equation is the production of the energy degenerate states which brings about instability of atoms. To solve this problem, a tensor interaction was introduced. The most popularly used tensor term is the Coulomb-like potential. It became clear that the energy degeneracies reduced due to the inclusion of the Coulomb-like tensor term in the Dirac equation. Recently, Hassanabadi et al.,[33] reported the use of Yukawa-like tensor potential as the tensor term. The Yukawa-like tensor potential also removed some energy degeneracy doublets in the spin and pseudospin symmetries. Motivated by this, we propose a new tensor potential as Hellmann-like tensor interaction and investigate the spin and pseudospin symmetries of the Dirac equation with the newly proposed tensor term. This is to ensure a more removal or complete removal of the energy degenerate states in both the spin and pseudospin symmetries.

2 Parametric Nikiforov-Uvarov Method

To use this powerful and elegant method, Tezcan and Sever[34] derived the following from the conventional Nikiforov-Uvarov method

The parameters c1, c2, and c3 are the basic parametric constants in which ξ1, ξ2, and ξ3 depend upon. From Eq. (1), they obtain the bound-state energy condition as

and the corresponding wave function as

the parametric constants in Eqs. (1), (2), and (3) are defined as[29] follows:

2.1 Dirac Equation in the Presence of Tensor Term

The Dirac equation with both vector potential V(r), scalar potential S(r) and tensor potential U(r) has the form:[35–36]

where E is the relativistic energy of the system, M is the mass of the particle p = −i∇ is the three dimensional momentum operator. The total angular momentum Jand the spin coupling , where L· is the orbital angular momentum of the spherical nucleus commute with the Dirac Hamiltonian. The eigenvalues of the spin-orbit coupling operator are k = (j + 1/2) > 0 and k = −(j + 1/2) < 0 for aligned spin j = ℓ + 1/2 and unaligned j = ℓ − 1/2. The set (H,K,J2,Jz) forms a complete set of conserved quantities. Hence, the spinors are given as[35–37]

where and are upper and lower components of the Dirac spinors respectively. and represent spin and pseudospin spherical harmonics respectively, and m is the projection of the angular momentum on the z-axis. Simplifying Eq. (6) further gives the two coupled differential equations whose solutions are the upper and lower radial wave functions Fnk(r) and Gnk(r) as

Eliminate Gnk(r) and Fnk(r) from Eqs. (7) and (8) respectively, we have[30–31]

2.2 Spin Symmetry Limit

The spin symmetry limit occurs when dΔ(r)/dr = 0 and Σ(r) = V(r). Hence, Eq. (9) becomes

where we have defined the following for mathematical simplicity:

2.3 Pseudospin Symmetry Limit

The pseudospin symmetry limit occurs when dΣ(r)/dr = 0 and Δ(r) = V(r). Thus, Eq. (10) turns to

where λ1 = M − Enk + Cp, λ3 = (k + H)(k + H − 1).

3 Approximate Solutions of the Dirac Equation

In this section, we obtain the energy equation and the corresponding wave spinors for the two symmetry limits of the Dirac equation in a close and compact form. It is noted that Eqs. (11) and (15) for spin and pseudospin symmetry limits respectively cannot be solved for k ≠ 0 without the use of approximation scheme. Here, we resort to employ the following approximation scheme:[38–39]

which is valid for δ ≪ 1. In this study, the interacting potential called the modified Eckart potential is of the form

3.1 Solutions of the Spin Symmetry

To obtain the solutions of the spin symmetry, we substitute Eqs. (16) and (17) into Eq. (11) and by defining a variable of the form y = e−δr, we have

where

Comparing Eq. (18) with Eq. (1), and by using Eq. (4), we obtain the following

Substituting Eq. (22) for ci (i = 1,2, . . ., 9) into Eq. (2), we have energy equation for the spin symmetry as

Substituting Eq. (22) for ci (i = 10, . . ., 13) into Eq. (3), we have the upper component of the wave function as

The lower component of the wave function is given as

3.2 Solutions of the Pseudospin Symmetry

To obtain the solutions of the pseudospin symmetry, we substitute Eqs. (16) and (17) into Eq. (15) and by defining a variable of the form y = e−δr, we have

where

Comparing Eq. (26) with Eq. (1), and by using Eq. (4), we obtain the following

Substituting Eq. (30) for ci (i = 1,2, . . ., 9) into Eq. (2), we have energy equation for the pseudospin symmetry as

Substituting Eq. (30) for ci (i = 10, . . ., 13) into Eq. (3), we have the upper component of the wave function as

The lower component of the wave function is given as

4 Conclusion

The energies for the spin symmetry limit and the pseudospin symmetry limit are presented in Tables 1 and 2 respectively for H = 0, H = 0.5, and H = 1. The energies for H = 0 are obtained in the absence of tensor interaction. The usual energy degenerate states 0p3/2 = 0p1/2, 1p3/2 = 1p1/2, 2p3/2 = 2p1/2, 3p3/2 = 3p1/2, 0d5/2 = 0d3/2, 1d5/2 = 1d3/2, 2d5/2 = 2d3/2, 3d5/2 = 3d3/2, 0f5/2 = 0f7/2, 1f5/2 = 1f7/2, 2f5/2 = 2f7/2, 3f5/2 = 3f7/2 are obtained for the spin symmetry limit. For H = 0.5, it is observed that the energy degeneracy are removed. Similarly, when H is set to be 1, it is equally observed that the energy degeneracy are completely removed. In the pseudospin symmetry limit, when H = 0, which means absence of tensor interaction, we have the following energy degenerate states1s1/2 = 0d3/2, 1p3/2 = 0f5/2, 1d5/2 = 0g7/2, 1f7/2 = 0h9/2, 2s1/2 = 1d3/2, 2p1/2 = 1f5/2, 2d5/2 = 1g7/2, 2f7/2 = 1h9/2. However, when we set H to 0.5 and 1 respectively i.e. when Hellmann-like tensor interaction is included, the whole energy degenerate states are removed. It seen that of all the tensor potential used such as Coulomb and Yukawa, the Hellmann-like tensor potential is more effective.

Table 1

Energy in the spin symmetry limit for M = 1, Cs = 5, δ = 0.01, and V = 5δ fm−1.

.
Table 2

Energy in the pseudospin symmetry limit for M = 1, Cs = −5, δ = 0.01, and V = 2 fm−1.

.
Reference
[1] Ikot A. N. Maghsoodi E. Zarrinkamar S. Hassanabadi H. Indian J. Phys. 88 2014 283
[2] Ikot A. N. Hassanabadi H. Etuk S. E. et al. Indian J. Phys. 89 2015 289
[3] Shurifl S. Tajic F. Hamzavi M. Ikhdair S. M. Zeit. Naturf. A 70 2015 499
[4] Eshghi M. Mehbran H. Ikhdair S. M. Acta Mathematicae Applicatae Sinica, English Series 31 2015 1131
[5] Ikot A. N. Zarrinkamar S. Zare S. Hassanabadi H. Chin. J. Phys. 54 2016 968
[6] Onyeaju M. C. Ikot A. N. Onate C. A. et al. Eur. Phys. J. Plus. 132 2017 302
[7] Onate C. A. Ojonubah J. O. Int. J. Mod. Phys. E 24 2015 1550020
[8] Onate C. A. Onyeaju M. C. Ikot A. N. Ann. Phys. 375 2016 239
[9] Hamzavi M. Ikhdair S. M. Ita B. I. Phys. Scr. 85 2012 045009
[10] Aydoğdu O. Sever R. Few-Body Syst. 47 2010 193
[11] Wei G. F. Dong S. H. Phys. Lett. A 373 2008 49
[12] Wei G. F. Dong S. H. Phys. Lett. A 373 2009 2428
[13] Wei G. F. Dong S. H. Phys. Scr. 81 2010 035009
[14] Wei G. F. Dong S. H. Phys. EPJA 46 2010 207
[15] Wei G. F. Dong S. H. EPL 87 2009 40004
[16] Ikhdair S. M. Falaye B. J. Chin. Phys. 421 2013 84
[17] Wei G. F. Dong S. H. Eur. J. Phys. A 43 2010 185
[18] Arima A. Harvey M. Shimizu K. Phys. Lett. B 30 1969 517
[19] Dudek J. Nazarewicz W. Szymanski Z. Leander G. A. Phys. Rev. Lett. 59 1987 1405
[20] Bohr A. Hamamoto I. Mottelson B. R. Phys. Scr. 26 1982 267
[21] Troltenier D. Bahri C. Draayer J. P. Nucl. Phys. A. 586 1995 53
[22] Nikiforov A. F. Uvarov V. B. Special Functions in Mathematical Physics Birkhauser Basel 1988
[23] Ikhdair S. M. Phys. Scr. 83 2011 015010
[24] Miranda M. G. Sun G. H. Dong S. H. Int. J. Mod. Phys. E 19 2010 123
[25] Hamzavi M. Rajabi A. A. Hassanabadi H. Int. J. Mod. Phys. A 26 2011 1363
[26] Onate C. A. Oyewumi K. J. Falaye B. J. Few-Body Syst. 55 2014 61
[27] Wei G. F. Dong S. H. Phys. Lett. B 686 2010 288
[28] Jia C. S. Chen T. He S. Phys. Lett. A 377 2013 682
[29] Soylu A. Bayrak O. Boztosun I. J. Math. Phys. 48 2007 082302
[30] Falaye B. J. Oyewumi K. J. Ibrahim T. T. et al. Can. J. Phys. 91 2013 98
[31] Falaye B. J. Oyewumi K. J. Ikhdair S. M. Hamzavi M. Phys. Scr. 89 2014 115204
[32] Bayrak O. Soylu A. Boztosun I. J. Math. Phys. 51 2010 112301
[33] Hassanabadi H. Maghsoodi E. Zarrinkamar S. Commun. Theor. Phys. 58 2012 807
[34] Tezcan C. Sever R. Int. J. Theor. Phys. 48 2009 337
[35] Ginocchio J. N. Phys. Rep. 414 2005 165
[36] Ginocchio J. N. Leviatan A. Phys. Lett. B 425 1998 1
[37] Hassanabadi H. Maghsoodi E. Zarrinkamar S. Rahimov H. Mod. Phys. Lett. A 26 2011 2703
[38] Qiang W. C. Dong S. H. Phys. Lett. A 363 2007 169
[39] Qiang W. C. Dong S. H. Phys. Lett. A 368 2007 13