ARTICLE

Relativistic symmetries of fermions in the background of the inverselyquadratic Yukawa potential with Yukawa potential as a tensorMajid Hamzavi and Sameer M. Ikhdair

Abstract: In the presence of spin and pseudo-spin symmetries, we obtain approximate analytical bound state solutions to theDirac equation with scalar–vector inverse quadratic Yukawa potential including a Yukawa tensor interaction for any arbitraryspin–orbit quantum number, �. The energy eigenvalues and their corresponding two-component spinor wave functions areobtained in closed form using the parametric Nikiforov–Uvarov method. It is noticed that the tensor interaction removes thedegeneracy in the spin and p-spin doublets. Some numerical results are obtained for the lowest energy states within spin andpseudo-spin symmetries.

PACS Nos.: 03.65.Ge, 03.65.Fd, 03.65.Pm, 02.30.Gp.

Résumé : En présence de symétries de spin et de pseudo-spin, nous obtenons une solution analytique approximative pour lesétats liés d’une équation de Dirac avec un potentiel de Yukawa quadratique inverse scalaire–vecteur, incluant une interactiontenseur de Yukawa pour toute valeur du nombre quantique de spin–orbite �. La méthode de Nikiforov–Uvarov nous permetd’obtenir les valeurs propres d’énergie et, sous forme analytique, leurs spineurs a deux composantes correspondants. On noteque l’interaction tensorielle lève la dégénérescence dans les doublets spin et p-spin. Nous complétons avec quelques résultatsnumériques pour les plus bas états d’énergie a symétries de spin et de pseudo-spin. [Traduit par la Rédaction]

1. IntroductionRecently, tensor potentials were introduced into the Dirac

equation with the substitution p¡ p � im��U�r�r and a spin–orbitcoupling is added to the Dirac Hamiltonian [1, 2]. Lisboa et al. [3]have studied a generalized relativistic harmonic oscillator forspin-1/2 particles by considering a Dirac Hamiltonian that con-tains quadratic vector and scalar potentials together with a lineartensor potential under the conditions of pseudospin symmetry(PSS) and spin symmetry (SS). Alberto et al. [4] studied the contri-bution of the isoscalar tensor coupling to the realization of thePSS in nuclei. Later on Akçay showed that the Dirac equation withscalar and vector quadratic potentials and a Coulomb-like tensorpotential can be solved exactly [5]. Moreover, the Dirac equationwas solved exactly with tensor potential containing a linear andCoulomb-like terms [6].

Since the discovery of Ginocchio, that PSS in nuclei can beunderstood easily in terms of the Dirac equation with large scalarand vector potentials. An increasing number of investigations ofthe SS and PSS in several physical potential models have beenreported [7–14]. Recently, we solved the Dirac equation with theKillingbeck (harmonic oscillator plus Cornell) potential and ob-tained the quasi-exact bound state solutions for relativistic energyeigenvalues and wave functions under SS and PSS in the frame ofthe series expansion method [15]. Very recently, Liang et al [16]studied the origin of PSS and its breaking mechanism by combin-ing supersymmetry (SUSY), quantum mechanics, perturbationtheory, and the similarity renormalization group (SRG) method.The Schrödinger equation is taken as an example, correspondingto the lower-order approximation in transforming a Dirac equa-tion into diagonal form by using SRG. It is shown that while the SSconserving term appears in the single-particle Hamiltonian, H,the PSS-conserving term appears naturally in the SUSY partner

Hamiltonian, H. The eigenstates of Hamiltonians H and H areexactly one-to-one identical except for the so-called intruderstates. The origin of PSS deeply hidden in H can be traced in itsSUSY partner Hamiltonian, H.

The PSS and SS are hot topics in nuclear structure physics.Analytical solutions of the Dirac Hamiltonian with potentials rel-evant to physics are very helpful for the study of these symme-tries. In the present work, we obtain approximate analyticalsolutions of the Dirac equation with the inverse quadratic Yukawa(IQY) potential including the Yukawa-like tensor interaction un-der the previously mentioned SS and PSS conditions. The IQYpotential has been recently introduced in ref. 17 with the simpleform

v(r) � �v0

e�2�r

r2v0 � 0 (1)

where � is the screening parameter and v0 is the depth of thepotential. This potential has a stronger singularity of order two atr = 0 than the well-known Yukawa potential of soft singularity[18, 19]. Furthermore, when potential (1) is inserted into the Diracequation, it gives the bound states for SS and PSS cases because itis finite

v(r) � �v0� 1

r2�

2�

r � (2)

in the low screening regime. The Dirac equation was solved forthe inversely quadratic Yukawa potential in the presence ofCoulomb-like tensor interaction. Later on Ikhdair and Hamzavi

Received 12 April 2013. Accepted 14 August 2013.

M. Hamzavi. Department of Physics, University of Zanjan, P. O. Box 45371-38791, Zanjan, Iran.S.M. Ikhdair. Department of Physics, Faculty of Science, an-Najah National University, Nablus, West Bank, Palestine; Department of Physics, Near East University, Nicosia, Northern Cyprus,Turkey.

Corresponding author: Majid Hamzavi (e-mail: majid.hamzavi@gmail.com).

51

Can. J. Phys. 92: 51–58 (2014) dx.doi.org/10.1139/cjp-2013-0176 Published at www.nrcresearchpress.com/cjp on 22 August 2013.

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introduced and investigated a more general form of the Yukawapotential within the Coulomb-like tensor interaction [20]. The aimof the present work is to obtain approximate relativistic energyeigenvalues and their two-components wave functions for a spin-1/2 fermion in presence of scalar–vector IQY potential fields in-cluding Yukawa tensor interaction in presence of SS and PSS.Because the Dirac–IQY problem within Yukawa tensor potentialcannot be solved exactly, one has to resort to the Greene–Alrichusual approximation scheme [21] by which we need to take smallrange for �r. Hence, the physical motivation for using the tensorinteraction as Yukawa potential instead of the Coulomb potentialis that it can be easily reduced into a Coulomb-like tensor inter-action under this approximation scheme and generate smallnumerical corrections as we shall see later in our numerical cal-culations. Very recently, Aydogdu et al. [22] have also used thestandard Yukawa potential [18, 19] as a tensor interaction in theirwork instead of the usually used Coulomb potential.

The structure of this paper is organized as follows. In Sect. 2, webriefly introduce the Dirac equation with scalar, vector, and ten-sor potentials for arbitrary spin–orbit quantum number � includ-ing tensor interaction in the presence of SS and PSS limits. Theoutlines of the parametric Nikiforov–Uvarov (pNU) method arepresented in Sect. 3. The energy eigenvalue equations and corre-sponding wave functions expressed in terms of the Jacobi polyno-mial are obtained in Sect. 4. Some numerical values for the SS andPSS (fermionic and antifermionic) energy states are obtained anda discussion to the degeneracy of doublet states is also given inpresence and absence of the tensor interactions. We end up withsome concluding remarks in Sect. 5.

2. Dirac equation including tensor couplingThe Dirac equation for fermionic massive spin-1/2 fermions

moving in an attractive scalar S(r) repulsive vector V(r) and tensorU(r) potentials is (� = c = 1) [23]

[� ·p �(M S(r)) � i�� · rU(r)](r) � [E � V(r)](r) (3)

where E is the relativistic energy of the system, p = –i� is thethree-dimensional momentum operator and M is the mass of thefermionic particle. Further, � and � are the 4 × 4 Dirac matricesgiven by

� � �0 �

� 0 � � � �I 00 �I � (4)

where I is 2 × 2 unitary matrix and � are three-vector spin matri-ces

�1 � �0 11 0 � �2 � �0 �i

i 0 � �3 � �1 00 �1 � (5)

where � is the 2 × 2 matrix with the Pauli matrices in the diagonal.The total angular momentum operator J and spin–orbit operatorK = −�(� · L + 1), where L is orbital angular momentum of thespherical nucleons commute with the Dirac Hamiltonian and � isthe 4 × 4 matrix with the Pauli matrices in the diagonal. Theeigenvalues of spin–orbit coupling operator are � = (j + 1/2) > 0 and� = –(j + 1/2) < 0 for unaligned spin j = l – 1/2 and the aligned spinj = l + 1/2, respectively. (H, K, J2, Jz) can be taken as the complete setof the conservative quantities. Thus, the spinor wave functionscan be classified according to their angular momentum j, spin–orbit quantum number � and the radial quantum number n canbe written as follows:

n�(r) � � fn�(r)gn�(r)

� �1

r� Fn�(r)Yjml (�, �)

iGn�(r)Yjml (�, �) � (6)

where fn�(r) is the upper (large) component and gn�(r) is the lower

(small) component of the Dirac spinors; Yjml ��, �� and Yjm

l ��, �� arespin and p-spin spherical harmonics, respectively, and m is theprojection of the angular momentum on the z-axis. Substituting(6) into (3) and using the following relations:

(� ·A)(� ·B) � A ·B i� · (A × B) (7a)

(� ·P) � � · r�r ·P i� ·L

r� (7b)

together with the following properties [5]:

� ·LYjml (r�) � (� � 1)Yjm

l (r�) � ·LYjml (r�) � �(� 1)Yjm

l (r�)

� · rYjml (r�) � �Yjm

l (r�) � · rYjml (r�) � �Yjm

l (r�)(8)

one obtains two coupled differential equations for upper andlower radial wave functions Fn�(r) and Gn�(r) as

� d

dr

�

r� U(r)�Fn�(r) � (M En� � (r))Gn�(r) (9a)

� d

dr�

�

r U(r)�Gn�(r) � (M � En� �(r))Fn�(r) (9b)

where

(r) � V(r) � S(r) (10a)

�(r) � V(r) S(r) (10b)

stand for the difference and the sum potentials, respectively.Eliminating Fn�(r) and Gn�(r) from (9), we finally obtain the follow-ing two Schrödinger-like differential equations for the upper andlower radial spinor components:

� d2

dr2�

�(� 1)

r2

2�

rU(r) �

dU(r)

dr� U2(r)�Fn�(r)

d (r)/dr

M En� � (r)� d

dr

�

r� U(r)�Fn�(r) � [(M En� � (r))

× (M � En� �(r))]Fn�(r) �(� 1) � l(l 1) (11)

and

� d2

dr2�

�(� � 1)

r2

2�

rU(r)

dU(r)

dr� U2(r)�Gn�(r)

d�(r)/dr

M � En� �(r)� d

dr�

�

r U(r)�Gn�(r) � [(M En� � (r))

× (M � En� �(r))]Gn�(r) �(� � 1) � l(l 1) (12)

respectively. The quantum number � is related to the SS quantumnumbers l by

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� � �(l 1) � ��j 1

2�(s1/2, p3/2, etc.) j � l

1

2aligned spin (� � 0)

l � �j 1

2�(p1/2, d3/2, etc.) j � l �

1

2unaligned spin (� � 0)

and to the PSS quantum number l by

� � � l � ��j 1

2�(s1/2, p3/2, etc.) j � l �

1

2aligned p-spin (� � 0)

(l 1) � �j 1

2�(d3/2, f5/2, etc.) j � l

1

2unaligned p-spin (� � 0)

where � = ±1, ±2, … For example, (1s1/2, 0d3/2) and (1p3/2, 0f5/2) can beconsidered as p-spin doublets.

2.1. PSS LimitThe origin of relativistic nature of PSS was started by Ginocchio

with the 1997 (see ref. 8, for example). Based on the relativisticmean field (RMF) approach the existence of the broken PSS wasinvestigated. Both spherical RMF and constrained deformed RMFcarried out for spherical and deformed sample nuclei. The quasi-degenerate PSS doublets are confirmed to exist near the Fermisurface for both spherical and deformed nuclei [24]. The mecha-nism behind the PSS was also studied; the PSS was shown to beconnected with the competition between the centrifugal barrierand the p-spin orbital potential, which is mainly decided by thederivative of the difference between the scalar and vector poten-tials. With the scalar and vector potentials derived from the self-consistent relativistic Hartree–Bogoliubov calculations, the PSSand its energy dependence in real nuclei were discussed in ref. 24.

It is worthy to mention that Ginocchio showed that there was aconnection between PSS and the time component of a vector po-tential and the scalar potential are nearly equal (i.e., S(r) ≈ –V(r))[8, 25, 26]. After that, Meng et al. derived that if dV�r� S�r��/dr �d��r�/dr � 0or�(r) =Cps =constant, thePSSisexact intheDiracequation(i.e., Cps = 0) [24, 27].

For more discussion on PSS, one can refer to refs. 28, 29. In thissubsection, we take attractive scalar S(r) repulsive vector V(r) andtensor U(r) potentials as [17, 21, 22]

V(r) � V0

e�2�r

r2S(r) � �S0

e�2�r

r2(13)

U(r) � �U0

e��r

rU(r) � �U0�1

r� �� (14)

giving bound state solutions, and hence the difference potentialbecomes

(r) � � 0

e�2�r

r2 (r) � � 0lim

r¡∞

e�2�r

r2¡ 0 (15)

where 0 = −(V0 + S0). The nodal structure for the exact PSS is thesame as the one for the central IQY potential which go to zero atinfinity. Thus, under this symmetry, (12) can be rewritten as

� d2

dr2�

�(� � 1)

r2� 2�U0

e��r

r2 U0��e��r

r

e��r

r2 �� U0

2e�2�r

r2� � 0

e�2�r

r2� �2�Gn�(r) � 0 (16)

where we defined � � En� � M � Cps, �2 � �M En���M � En�

Cps�. Further, � � � l for � < 0 and � � l 1 for � > 0.

2.2. SS limitIn this type of symmetry, taking d (r)/dr = 0 or (r) = Cs =

constant (for a more discussion on SS, one can refer to refs. 30–33)and inserting the sum potential as

�(r) � ��0

e�2�r

r2�(r) � ��0lim

r¡∞

e�2�r

r2¡ 0 (17)

where �0 = S0 – V0. The nodal structure for the exact SS (i.e., Cs = 0)is the same as the one for the central IQY potential, which goes tozero at infinity. Thus, we can rewrite (11) as

� d2

dr2�

�(� 1)

r2� 2�U0

e��r

r2� U0��e��r

r

e��r

r2 �� U0

2e�2�r

r2 ��0

e�2�r

r2� �2�Fn�(r) � 0 (18)

where � = M + En� – Cs and �2 = (M – En�)(M + En� – Cs). Also, � = l and� = –l – 1 for � < 0 and � > 0, respectively.

Note that (16) and (18) cannot be solved analytically because ofthe centrifugal and the pseudo-centrifugal terms, respectively. Sowe shall use the following approximations [17, 21, 22, 34–36] todeal with the centrifugal term:

1

r2≈ 4�2 e�2�r

(1 � e�2�r)2(19a)

and

1

r2≈ 4�2 e��r

(1 � e�2�r)2(19b)

which are found to be nearly the same in contribution. Here, theusual approximation, (19a), was first introduced by Greene andAldrich in ref. 21. Because our approximation is only valid forsmall values of �r, then we can expand the exponential term inthe numerator so that we can approximate IQY potential and theYukawa tensor interaction as

V(r) � �V0

r2

C

rU(r) � �

A

r B A � U0

B � �U0 C � 2�V0

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Obviously, the tensor interaction reduces into Coulomb-like tensorinteraction when �r becomes small. Thus, the motivation behindchoosing the Yukawa tensor potential is that at small values of �r,the Yukawa tensor potential reduces to the Coulomb-like tensor inter-action. To show the accuracy of the present approximation at small �r,we plot the centrifugal term and its approximations (19) in Fig. 1.

3. Parametric NU methodThis powerful mathematical tool is usually used to solve

second-order differential equations. At first we consider the fol-lowing differential equation [37]:

n′′(s)

�(s)

�(s)n

′ (s) �(s)

�2(s)n(s) � 0 (20)

where �(s) and ��s� are polynomials with at most of second degree,and ��s� is a first-degree polynomial. To make the application ofthe NU method rather simpler and direct without need to checkthe validity of our solution. We present a recipe for the method.Hence, we start by writing the general form of the Schrödinger-like equation (20) in a more general form as [37–40]

n′′(s) � c1 � c2s

s(1 � c3s)�n

′ (s) ��p2s2 p1s � p0

s2(1 � c3s)2 �n(s) � 0 (21)

satisfying the wave functions

n(s) � �(s)yn(s) (22)

Furthermore, we compare (21) with its counterpart (20) to obtainthe following parameter values:

�(s) � c1 � c2s �(s) � s(1 � c3s) �(s) � �p2s2 p1s � p0

(23)

Now, following the NU method [37], we obtain the energy equa-tion [38–40]

c2n � (2n 1)c5 (2n 1)� c9 � c3 c8� n(n � 1)c3 c7

2c3c8 � 2 c8c9 � 0 (24)

and the corresponding wave functions

�(s) � sc10(1 � c3s)c11 �(s) � sc12(1 � c3s)

c13 c12 � 0

c13 � 0 yn(s) �1

�(s)

dn

dsn�n(s)�(s)� � Pn

(c10,c11)(1 � 2c3s)

c10 � �1 c11 � �1 n�(s) � Nn�sc12(1 � c3s)c13Pn

(c10,c11)(1 � 2c3s)(25)

where Pn��,���x�, (� > –1, � > –1, and x � [–1, 1]) is the Jacobi polyno-

mial, �(s) is the weight function and �(s) is defined in (23). Theparametric constants are given as follows:

c4 �1

2(1 � c1) c5 �

1

2(c2 � 2c3) c6 � c5

2 p2

c7 � 2c4c5 � p1 c8 � c42 p0 c9 � c3(c7 c3c8) c6

c10 � c1 2c4 � 2 c8 � 1 � �1

c11 � 1 � c1 � 2c4 2

c3 c9 � �1 c3 ≠ 0

c12 � c4 � c8 � 0 c13 � �c4 1

c3� c9 � c5� � 0

c3 ≠ 0

(26)

where c12 > 0, c13 > 0, and s � [0, 1/c3], c3 ≠ 0.In the special case where c3 = 0, the wave function (22) becomes

limc3¡0

Pn(c10,c11)(1 � 2c3s) � Ln

c10(c11s) limc3¡0

(1 � c3s)c13 � ec13s

(s) � Nsc12ec13sLnc10(c11s)

(27)

where Ln�����x� is the Laguerre polynomial.

4. Dirac–IQY potential including Yukawa tensorproblem

4.1. PSS bound statesIn this subsection we will obtain the energy eigenvalues and the

corresponding wave functions for the PSS limit of the IQY poten-tial in the presence of the Yukawa tensor interaction potential(16). Hence, substituting (19) into (16) gives

� 1

4�2

d2

dr2� �(� � 1)

e�2�r

(1 � e�2�r)2� 2�U0

e�2�r

(1 � e�2�r)2

U0

2

e�2�r

(1 � e�2�r) U0

e�2�r

(1 � e�2�r)2� U0

2 e�4�r

(1 � e�2�r)2

� � 0

e�4�r

(1 � e�2�r)2�

�2

4�2�Gn�(r) � 0 (28)

and followed by making the transformation s = e–2�r, where s �(0, 1), one obtains

d2Gn�(s)

ds2

1 � s

s(1 � s)

dGn�(s)

ds

1

s2(1 � s)2�p2s

2 p1s � p0�× Gn�(s) � 0 s � (0, 1) (29)

with

Fig. 1. The centrifugal term r–2 (red solid curve) and itsapproximations in (19) with � = 0.1 fm–1.

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p2 � U0�U0 1

2� � 0

�2

4�2

p1 � ��(� � 1) � U0�2� �3

2�

2�2

4�2p0 �

�2

4�2

(30)

where G(s = 0) = G(s = 1) = 0 Now, we apply the pNU method bycomparing (29) and (21), we can obtain the parametric coefficients;namely, ci (i = 1, 2, 3) as follows:

c1 � 1 c2 � 1 c3 � 1

The values of coefficients ci (i = 4, 5, …, 13) are found from (26) as

c4 � 0 c5 � �1

2c6 �

1

4 U0�U0

1

2� � 0

�2

4�2

c7 � �(� � 1) U0�2� �3

2� �

2�2

4�2c8 �

�2

4�2

c9 � p2 � p1 p0 1

4� �� �

1

2 U0�2

� 0

c10 � ��

�c11 � 2�� c12 � �

�

2�c13 � ��

1

2

(31)

where

�� � ��� �1

2 U0�2

� 0 (32)

Expression (24) gives the energy eigenvalues of the IQY poten-tial within the Yukawa tensor interaction in presence of PSSlimit as

�n 1

2 ��� �

1

2 U0�2

� 0 ��

2��2

� U0�U0 1

2� � 0

�2

4�2(33)

which is valid only for a few lowest energy states. Some numer-ical results are calculated from the preceding equation anddisplayed in Table 1. Notice that we obtain these numericalresults in the presence of PSS and in the absence of the Yukawatensor potential form. We take a set of parameter values:M = 5.0 fm–1, Cps = 0, V0 = 2.9 fm–1, S0 = 2.5 fm–1, and � = 0.1 fm–1.In the absence of the tensor potential (U0 = 0), the existence ofthe degeneracy is obviously seen in the following p-spin dou-blets (ns1/2, (n – 1)d3/2),(np3/2, (n – 1)f5/2), (nd5/2, (n – 1)g7/2), (nf7/2,(n – 1)h9/2) and so on where n = 1, 2, …

In Fig. 2, we study the effect of the tensor potential on the p-spindoublet splitting by considering the following pairs of orbitals:(1d5/2, 0g7/2), (2f7/2, 1h9/2). It is observed that in the absence of thetensor interaction (i.e., U0 = 0), members of p-spin doublets havesame energy. However, in the presence of the tensor interactionU0 ≠ 0, the degeneracies are completely removed.

Next we calculate the corresponding wave functions. By refer-ring to relation (25), one finds the necessary functions

�(s) � s�/�(1 � s)2�� �(s) � s�/2�(1 � s)��1/2

yn(s) � Pn(�/�,2��)(1 � 2s) (34)

Finally, using Gn�(s) = �(s)yn(s), we obtain the lower spinorcomponent of the Dirac wave function by means of relation (25)as

Gn�(s) � Nn�s�/2�(1 � s)��1/2Pn(�/�,2��)(1 � 2s) (35)

or equivalently

Gn�(r) � Nn�e�� r(1 � e�2�r)��1/2Pn

(�/�,2��)(1 � 2e�2�r) r � (0, ∞)(36)

where Nn� is the normalization constant. It is obvious that wavefunction (36) satisfies the boundary conditions at r � (0, ∞) so it iswell-behaved inside the interval and finite on the edges (i.e.,Gn�(0) = 0 when r ¡ 0 and Gn�(r) ¡ 0 when r ¡ ∞). On the otherhand, the upper spinor component of the Dirac wave function canbe calculated via (9b) as

Fn�(r) �1

M � En� Cps� d

dr�

�

r� U0

e��r

r�Gn�(r) (37)

where E ≠ M + Cps.

4.2. SS bound statesFollowing the same procedures as in the previous subsection by

substituting (19) into (18) and making the change of variablest = e–2�r, t � (0, 1), one obtains

� d2

dt2

(1 � t)

t(1 � t)

d

dt

1

t2(1 � t)2�2�2

4�2� �(� 1) � (2� 1)U0 �

U0

2 �t

�1

t2(1 � t)2�U0

2 �U0

2

�2

4�2� ��0�t2 �

�2

4�2�Gn�(r) � 0 (38)

giving

Table 1. Bound state energy eigenvalues (fm–1) of the exact pseudo-spin symmetry for several values of n and � with U0 = 0.1 and Cps = 0.

l n, � < 0 (l, j) En,�<0 U0 ≠ 0 En,�<0 U0 = 0 n–1, � > 0 (l + 2, j + 1) En–1,�>0 U0 ≠ 0 En–1,�>0 U0 = 0

1 1, −1 1s1/2 −4.991 224 698 −4.990 991 885 0, 2 0d1/2 −4.990 824 466 −4.990 991 8852 1, −2 1p3/2 −4.988 603 650 −4.988 207 060 0, 3 1f5/2 −4.987 882 048 −4.988 207 0603 1, −3 1d5/2 −4.984 258 919 −4.983 667 952 0, 4 0g7/2 −4.983 157 468 −4.983 667 9524 1, −4 1f7/2 −4.977 906 709 −4.977 094 596 0, 5 0h9/2 −4.976 373 865 −4.977 094 5961 2, −1 2s1/2 −4.979 127 920 −4.978 797 859 1, 2 1d3/2 −4.978 558 921 −4.978 797 8592 2, −2 1p3/2 −4.975 475 018 −4.974 937 186 1, 3 1f5/2 −4.974 495 552 −4.974 937 1863 2, −3 1d5/2 −4.969 703 290 −4.968 940 001 1, 4 1g7/2 −4.968 280 126 −4.968 940 0014 2, −4 1f7/2 −4.961 652 133 −4.960 648 157 1, 5 1h9/2 −4.959 756 362 −4.960 648 157

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p2 � U0�U0 �1

2� � ��0

�2

4�2

p1 � ��(� 1) � �2� 3

2�U0

2�2

4�2p0 �

�2

4�2

(39)

To avoid repetition in our solution, we calculate the parametricconstants as

c1 � 1 c2 � 1 c3 � 1 c4 � 0 c5 � �1

2

c6 �1

4 U0�U0 �

1

2� � ��0

�2

4�2

c7 � �(� 1) U0�2� 3

2� �

2�2

4�2c8 �

�2

4�2

c9 � p2 � p1 p0 1

4� ��

1

2 U0�2

� ��0

c10 � ��

�c11 � 2�� c12 � �

�

2�c13 � ��

1

2

�� � ��� 1

2 U0�2

� ��0

(40)

Therefore, the energy states under the SS case can be easily ob-tained as

�n 1

2 ���

1

2 U0�2

� ��0 ��

2��2

� U0�U0 �1

2� � ��0

�2

4�2(41)

The upper spinor component of the Dirac wave function becomes

Fn�(r) � Nn�e��r(1 � e�2�r)��1/2Pn(�/�,2��)(1 � 2e�2�r) r � (0, ∞)

(42)

and Nn� is the normalization constant. Here we have used thetransformations [41, 42]

En� ¡ �En� V(r) ¡ �V(r) (i.e., � ¡ �� 0 ¡ �0)

Cps ¡ �Cs � ¡ � � ¡ � 1 Gn�(r) ¡ Fn�(r)(43)

The upper spinor component of the wave function is finite at r = 0and r ¡ ∞ (i.e., Fn�(r = 0) = 0 and Fn�(r ¡ ∞) ¡ 0). Furthermore, thelower spinor component of the Dirac wave function is obtainedvia (9a) as

Gn�(r) �1

M En� � Cs� d

dr

�

r U0

e��r

r�Fn�(r) (44)

where E ≠ –M + Cs.In Table 2, we use the same parameters as in the previous sub-

section. We can observe that every doublet members: (np1/2, np3/2),(nd3/2, nd5/2), and (nf5/2, nf7/2) where n = 0, 1, 2, …, have same energywithout the tensor potential (U0 = 0). Thus, they are to be viewed asspin doublets, (i.e., the state 1p1/2 with n = 1 and � = 1 forms a spindoublet with the 1p3/2 state with n = 1 and � = –2). On the otherhand, in the presence of the tensor interaction (U0 ≠ 0), one canobserve degeneracy between every pair of spin doublets is re-moved.

In Fig. 3, we have investigated the effect of the tensor potentialon the spin doublet splitting by considering the following orbitalpairs: (1p3/2, 1p1/2) and (1f7/2, 1f5/2), and one can see that the resultsobtained in the spin symmetric limit resemble the ones observedin the p-spin symmetric limit.

For these Yukawa-type scalar S(r) and vector V(r) potentials,which go to zero at large distances, the conditions for exact PSSand SS symmetry are that �(r) = V(r) + S(r) and (r) = V(r) – S(r)respectively (i.e., the constants Cps = 0 and Cs = 0). This is true forevery potential that goes to zero at large distances. So constantsdifferent from zero do not correspond to exact PSS or SS symme-tries for such potentials because in that case they would not go tozero to finite value.

Accordingly, it is well-known that the energy eigenvalues areusually negative for antifermionic states in the PSS while positivefor fermionic states in the SS case. This is clearly demonstratedwhen Cs = 0, or Cps = 0 as shown in Tables 1 and 2. Our energyeigenvalues in Tables 1 and 2, show that we have negative spec-trum for the PSS energy states and positive one for the SS case. Ourresults are in agreement with normal case mentioned before.Moreover, the numerical results show that, in the exact p-spinsymmetry limit, the degenerate states are as follows: whenU0 = 0f (ns1/2, (n –1)d3/2), (np3/2, (n – 1)f5/2), (nd5/2, (n – 1)g7/2), (nf7/2,(n – 1)h9/2), and so on where n = 1, 2, 3, … The degenerate states inthe spin symmetry limit are U0 = 0 f (np3/2, np1/2), (nd5/2, nd3/2),(nf7/2, ng7/2), (nh11/2, nh9/2), and so on where n = 1, 2, 3, …

Fig. 2. Contribution of the tensor potential parameter to the energy levels in the case of pseudo-spin symmetry.

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Note that Alberto et al. [43–45] in their recent works remarkedthat it is possible to have positive energy bound solutions in exactPSS condition for confining potentials of any shape includingnaturally those used in hadrons physics from nuclear to quarkmodels. Because this does not happen for potentials going to zeroat large distances, nevertheless they showed that these results arestill valid for negative energy bound state solutions for antifermi-ons.

5. ConclusionIn this work, we have studied the energy eigenvalues and wave

functions of the Dirac particle in the scalar–vector inversely qua-dratic Yukawa fields plus a standard Yukawa tensor interaction inview of spin and p-spin symmetries with arbitrary spin–orbitquantum number, �. To the best of our knowledge, all works doneon the Dirac equation are almost relevant to linear or Coulombtensor interactions. However, in the present study, the tensorinteraction is taken as the Yukawa potential. This work is animproved work of ref. 17 where authors used a different kind oftensor potential as Coulomb-like form. We used a powerful math-ematical tool (i.e., the parametric NU method) in solving secondorder differential equations. Also, from our numerical results andfigures, it is found that the tensor interaction removes degenera-cies between each p-spin or spin doublet members.

AcknowledgementsWe would like to thank the kind referees for their positive

suggestions and critical comments which have improved the pres-

ent manuscript. SMI acknowledges the partial support providedby the Scientific and Technological Research Council of Turkey.

References1. M. Moshinsky and A. Szczepaniak. J. Phys. A: Math. Gen. 22, L817 (1989).

doi:10.1088/0305-4470/22/17/002.2. V.I. Kukulin, G. Loyola, and M. Moshinsky. Phys. Lett. A, 158, 19 (1991). doi:

10.1016/0375-9601(91)90333-4.3. R. Lisboa, M. Malheiro, A.S. de Castro, P. Alberto, and M. Fiolhais. Phys. Rev.

C, 69, 024319 (2004). doi:10.1103/PhysRevC.69.024319.4. P. Alberto, R. Lisboa, M. Malheiro, and A.S.de Castro. Phys. Rev. C, 71, 034313

(2005). doi:10.1103/PhysRevC.71.034313.5. J.D. Bjorken and S.D. Drell. Relativistic Quantum Mechanics. McGraw-Hill,

New York. 1964; H. Akçay. Phys. Lett. A, 373, 616 (2009). doi:10.1016/j.physleta.2008.12.029; H. Akçay. J. Phys. A: Math. Theor. 40, 6427 (2007). doi:10.1088/1751-8113/40/24/010.

6. O. Aydogdu and R. Sever. Few-Body Syst. 47, 193 (2010). doi:10.1007/s00601-010-0085-9; Eur. Phys. J. A, 43, 73 (2010). doi:10.1140/epja/i2009-10890-6.

7. J.N. Ginocchio, A. Leviatan, J. Meng, and S.-G. Zhou. Phys. Rev. C, 69, 034303(2004). doi:10.1103/PhysRevC.69.034303.

8. J.N. Ginocchio. Phys. Rev. Lett. 78, 436 (1997). doi:10.1103/PhysRevLett.78.436;Phys. Rep. 414, 165 (2005). PMID:10.1016/j.physrep.2005.04.003.

9. C.-S. Jia, X.L. Zeng, and L.T. Sun. Phys. Lett. A, 294, 185 (2002). doi:10.1016/S0375-9601(01)00840-4.

10. C.-S. Jia, P. Guo, and X.L. Peng. J. Phys. A: Math. Theor. 39, 7737 (2006).doi:10.1088/0305-4470/39/24/010.

11. C. Berkdemir. Nucl. Phys. A, 770, 32 (2006). doi:10.1016/j.nuclphysa.2006.03.001.

12. G.-F. Wei and S.H. Dong. Europhys. Lett. 87, 40004 (2009). doi:10.1209/0295-5075/87/40004.

13. G.-F. Wei and S.H. Dong. Phys. Lett. B, 686, 288 (2010). doi:10.1016/j.physletb.2010.02.070.

14. G.-F. Wei and S.H. Dong. Eur. Phys. J. A, 46, 207 (2010). doi:10.1140/epja/i2010-11031-0.

Table 2. Bound state energy eigenvalues (fm–1) of the exact spin symmetry for several values of n and � with U0 = 0.1 and Cs = 0.

l n, � < 0 (l, j = l + 1/2) En,�<0 U0 ≠ 0 En,�<0 U0 = 0 n, � > 0 (l, j = l–1/2) En,�>0 U0 ≠ 0 En,�>0 U0 = 0

2 0, −3 0d5/2 4.983 698 087 4.983 999 944 0, 2 0d3/2 4.983 750 343 4.983 999 9443 0, −4 0f7/2 4.979 788 117 4.979 191 416 0, 3 0f5/2 4.978 346 361 4.979 191 4164 0, −5 0g9/2 4.971 488 983 4.970 618 606 0, 4 0g7/2 4.969 533 173 4.970 618 6065 0, −6 0h11/2 4.960 822 667 4.959 727 267 0, 5 0h0/2 4.958 428 908 4.959 727 2672 1, −3 1d5/2 4.976 103 954 4.981 196 505 1, 2 1d3/2 4.980 263 518 4.981 196 5053 1, −4 1f7/2 4.963 644 002 4.971 978 460 1, 3 1f5/2 4.970 847 812 4.971 978 4604 1, −5 1g9/2 4.950 080 237 4.960 743 844 1, 4 1g7/2 4.959 420 103 4.960 743 8445 1, -6 1h11/2 4.934 732 228 4.947 508 217 1, 5 1h11/2 4.945 985 395 4.947 508 217

Fig. 3. Contribution of the tensor potential parameter to the energy levels in the case of spin symmetry.

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com

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MC

GIL

L U

NIV

ER

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/15/

14Fo

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15. M. Hamzavi, S.M. Ikhdair, and K.-E. Thylwe. Z. Naturforsch. 67a, 567 (2012).doi:10.5560/ZNA.2012-0046..

16. H. Liang, S. Shen, P. Zhao, and J. Meng. Phys. Rev. C, 87, 014334 (2013).doi:10.1103/PhysRevC.87.014334.

17. M. Hamzavi, S.M. Ikhdair, and B.I. Ita. Phys. Scr. 85, 045009 (2012). doi:10.1088/0031-8949/85/04/045009.

18. H. Yukawa. Proc. Phys. Math. Soc. Jpn. 17, 48 (1935). doi:10.1143/PTPS.1.1.19. H. Yukawa and S. Sakata. Proc. Phys. Math. Soc. Jpn. 19, 1084 (1935).20. S.M. Ikhdair and M. Hamzavi. Few-Body Syst. 53, 487 (2012). doi:10.1007/

s00601-012-0475-2.21. R.L. Greene and C. Aldrich. Phys. Rev. A, 14, 2363 (1976). doi:10.1103/

PhysRevA.14.2363.22. O. Aydogdu, E. Maghsoodi, and H. Hassanabadi. Chin. Phys. B, 22, 010302

(2013). doi:10.1088/1674-1056/22/1/010302.23. W. Greiner. Relativistic Quantum Mechanics. Springer, Berlin. 2000.24. G.A. Lalazissis, Y.K. Gambhir, J.P. Maharana, C.S. Warke, and P. Ring

Phys. Rev. C, 58, R45 (1998). doi:10.1103/PhysRevC.58.R45; J. Meng,K. Sugawara-Tanabe, S. Yamaji, P. Ring, and A. Arima. Phys. Rev. C, 58, R628(1998). doi:10.1103/PhysRevC.58.R628.

25. J.N. Ginocchio. Phys. Rep. 315, 231 (1999). doi:10.1016/S0370-1573(99)00021-6.26. J.N. Ginocchio. Nucl. Phys. A, 654, 663c (1999). doi:10.1016/S0375-9474(00)

88522-X..27. J. Meng, K. Sugawara-Tanabe, S. Yamaji, and A. Arima. Phys. Rev. C, 59, 154

(1999). doi:10.1103/PhysRevC.59.154.28. B.N. Lu, E.G. Zhao, and S.G. Zhou. Phys. Rev. Lett. 109, 072501 (2012). doi:10.

1103/PhysRevLett.109.072501. PMID:23006363.29. H. Liang, S. Shen, P. Zhao, and J. Meng. Phys. Rev. C, 87, 014334 (2013).

doi:10.1103/PhysRevC.87.014334. arXiv:1207.6211.30. S.G. Zhou, J. Meng, and P. Ring. Phys. Rev. Lett. 91, 262501 (2003). doi:10.1103/

PhysRevLett.91.262501. PMID:14754045.

31. X.T. He, S.G. Zhou, J. Meng, E.G. Zhao, and W. Scheid. Eur. Phys. J. A, 28, 265(2006). doi:10.1140/epja/i2006-10066-0.

32. C.Y. Song, J.M. Yao, and J. Meng. Chin. Phys. Lett. 26, 122102 (2009). doi:10.1088/0256-307X/26/12/122102.

33. C.Y. Song and J.M. Yao. Chin. Phys. C, 34, 1425 (2010). doi:10.1088/1674-1137/34/9/061.

34. M. Hamzavi, M. Movahedi, K.-E. Thylwe, and A.A. Rajabi. Chin. Phys. Lett. 29,080302 (2012). doi:10.1088/0256-307X/29/8/080302.

35. M. Karakoç and I. Boztosun. Int. J. Mod. Phys. E, 15, 1253 (2006). doi:10.1142/S0218301306004806.

36. S.M. Ikhdair. Cent. Eur. J. Phys. 10(2), 361 (2012). doi:10.2478/s11534-011-0121-5.37. A.F. Nikiforov and V.B. Uvarov. Special Functions of Mathematical Physics.

Birkhausr, Berlin. 1988.38. S.M. Ikhdair. Int. J. Mod. Phys. C, 20(10), 1563 (2009). doi:10.1142/

S0129183109014606.39. S.M. Ikhdair. Chem. Phys. 361, 9 (2009). doi:10.1016/j.chemphys.2009.04.023.40. S.M. Ikhdair. J. Math. Phys. 51, 023525 (2010). doi:10.1063/1.3293759.41. S.M. Ikhdair and R. Sever. J. Phys. A: Math. Theor. 44, 355301 (2011). doi:10.

1088/1751-8113/44/35/355301.42. S.M. Ikhdair and R. Sever. Appl. Math. Comput. 218, 10082 (2012). doi:10.1016/

j.amc.2012.03.073.43. P. Alberto, A.S. de Castro, and M. Malheiro. Phys. Rev. C, 87, 031301 (2013).

doi:10.1103/PhysRevC.87.031301.44. A.S. de Castro and P. Alberto. Phys. Rev. A, 86, 032122 (2012). doi:10.1103/

PhysRevA.86.032122.45. R. Lisboa, M. Malheiro, P. Alberto, M. Fiolhais, and A.S. de Castro. Phys. Rev.

C, 81, 064324 (2010). doi:10.1103/PhysRevC.81.064324.

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