Advances in High Energy Physics

Volume 2015 (2015), Article ID 915796, 17 pages

http://dx.doi.org/10.1155/2015/915796

## Spin and Pseudospin Symmetry in Generalized Manning-Rosen Potential

Department of Physics, Mersin University, 33143 Mersin, Turkey

Received 8 May 2015; Accepted 18 June 2015

Academic Editor: Shi-Hai Dong

Copyright © 2015 Hilmi Yanar and Ali Havare. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Spin and pseudospin symmetric Dirac spinors and energy relations are obtained by solving the Dirac equation with centrifugal term for a new suggested generalized Manning-Rosen potential which includes the potentials describing the nuclear and molecular structures. To solve the Dirac equation the Nikiforov-Uvarov method is used and also applied the Pekeris approximation to the centrifugal term. Energy eigenvalues for bound states are found numerically in the case of spin and pseudospin symmetry. Besides, the data attained in the present study are compared with the results obtained in the previous studies and it is seen that our data are consistent with the earlier ones.

#### 1. Introduction

Spin and pseudospin symmetries are symmetries of the Dirac Hamiltonian. Spin symmetry leads to degeneracy between two states with quantum numbers and . These two states are considered as a spin doublet with . , , , and are radial, orbital angular momentum, total angular momentum, and spin quantum numbers, respectively. The spin doublet or spin symmetry is used to explain the spectrum of antinucleon in a nucleus [1–4] and small spin-orbit splitting in hadrons [5]. Pseudospin symmetry causes degeneracy between two states with quantum numbers and . These two states are regarded as a pseudospin doublet with quantum numbers , where and are pseudoorbital angular momentum and pseudospin quantum numbers, respectively [6, 7]. The pseudospin doublet or pseudospin symmetry is used to explain deformation [8], identical bands [9–11], magnetic moment [12–14], and effective shell-model [15] in the nuclear structures.

Pseudospin and pseudoorbital angular momentum were introduced for the first time to explain experimental observation of quasi-degeneracy between and states of single-nucleon in nuclei [6, 7]. Pseudospin symmetry was discussed firstly in nonrelativistic framework [6]. However, in the 1990s, Blokhin et al. found a connection between pseudospin symmetry and relativistic mean field theory [16–18]. Then Ginocchio recognized that the pseudoorbital angular momentum is the orbital angular momentum of lower component of the Dirac spinor [19]. He showed that the pseudospin symmetry occurs in nuclei when the sum of the scalar potential and vector potential is approximately equal to zero [1, 2, 19–22]. Then, Meng et al. proved that the exact pseudospin symmetry exists when or in the Dirac equation [23, 24]. Also they introduced that the exact spin symmetry occurs when or in the Dirac equation [3]. For two symmetries, the Dirac Hamiltonian is invariant under the algebra [25]. After these studies, in recent years, spin and pseudospin symmetric solutions of the Dirac equation have been obtained and investigated by using different methods for various external potentials [26–66]. In [26], Wei and Dong have studied the Manning-Rosen potential under the spin symmetry limit. Also they have examined pseudospin symmetric solutions and energy eigenvalues for the Manning-Rosen potential [27]. Moreover the relativistic symmetries of the Manning-Rosen potential have been investigated by considering different approximation scheme for the centrifugal term [28, 29, 32, 33].

In this paper we have considered the new suggested generalized Manning-Rosen potential [67]. The new suggested generalized Manning-Rosen potential is defined in the following form:

Generalized Manning-Rosen potential includes some molecular potentials which are Manning-Rosen [68–73], Morse [74–77], hyperbolic Pöschl-Teller [78], -parameter hyperbolic Pöschl-Teller [79], and Kratzer-Fues [80–82] potentials. Also it can be reduced to Woods-Saxon [35–39, 83] and Hulthen [40–45, 84] potentials which are used in the explanation of the nuclear structures and to the Yukawa potential that represents the interactions between nucleons in nuclei [55–59, 85].

Therefore, the generalized Manning-Rosen potential is important for being able to explain both nuclear and molecular structures. If the generalized Manning-Rosen potential is considered as a nuclear potential, the spin and pseudospin symmetric solutions should be examined. Also spin and pseudospin symmetry can be investigated for diatomic molecular potentials by defining the reduced mass , where and are two nuclei masses of a diatomic molecule. In this context, it is aimed at investigating spin and pseudospin symmetric solutions for the generalized Manning-Rosen potential and at finding the bound state energy eigenvalues of the considered potential under the spin and pseudospin symmetry limits. Another aim is to examine the special cases of the generalized Manning-Rosen potential and to compare the obtained results for special cases with the results that are obtained in the previous studies and also to check up whether these results are compatible.

The scheme of paper is as follows. In Section 2, the Nikiforov-Uvarov method that we use to solve the Dirac equation is introduced briefly. In Section 3, the second-order differential equations are obtained from Dirac equation for the spin and pseudospin symmetry. In Sections 4 and 5, the spin and pseudospin symmetric solutions and bound state energy eigenvalues are found for the generalized Manning-Rosen potential, respectively. In Section 6, the effects of potential parameters on the energy eigenvalues are investigated. In Section 7, special cases of the generalized Manning-Rosen potential are introduced and energy eigenvalues are calculated for each special case in the spin and pseudospin symmetric limits. Also in the same section our energy eigenvalues are compared with the previous ones. Finally, an appraisal of the obtained results is made in Section 8.

#### 2. Nikiforov-Uvarov Method

This section has been prepared by using [86]. The Nikiforov-Uvarov method is used to solve the second-order differential equations which must be in the following form:where can be first-degree polynomial maximally. and can be polynomials no more than second degree. The function is defined asand the form of (2) is required to be covariant. In this case the following equation is obtained:where which can be first-degree polynomials maximally and that can be second-degree polynomials maximally are found as follows, respectively:by defining the polynomial asThe convenient must be found to obtain solutions of (4) or (2) and to study properties of solutions. Therefore the following relation is established between and :where is a constant. In this case (4) reduces to following equation:This is a hypergeometric type equation and solution of this equation is functions of hypergeometric type. By using (8), the polynomial is found:where is a first-degree polynomial. Therefore the expression in the square root in (10) must be written a square of a polynomial. Considering this property, the values are found for . Derivatives of the hypergeometric functions are also the hypergeometric functions. Accordingly, th derivative of satisfies the hypergeometric type equation. By using this property, a relation for is obtained:When the quantum mechanical problems are discussed, an energy relation is obtained by equating (11) to (12).

The hypergeometric type function is found by using the following Rodrigues relation for hypergeometric type equations:where is a weight function and satisfies the following equation:

#### 3. Dirac Equation

In the presence of an attractive scalar and a repulsive vector potentials, the Dirac equation for a nucleon with mass is written as follows () [27, 46, 87]:where and are the Dirac matrices, while is defined in terms of Pauli spin matrices, and is described in terms of unit matrices [87]:The spherically symmetric Dirac spinor wave function [1] is written aswhere and are the upper and lower radial wave functions, respectively. and are the spin and pseudospin spherical harmonics, respectively. is the eigenvalues of spin-orbit coupling operator . The values of are for unaligned spin and for aligned spin . Substituting (9) and (10) into (8) and by using the following relations [88]:together with the following properties [88]:the Dirac equation reduces to two coupled differential equations as follows:whereFrom these two coupled differential equations, two second-order differential equations have been obtained for the upper and lower radial wave functions, respectively:where and .

#### 4. Spin Symmetric Solution

In the spin symmetric case the sum of scalar potential with vector potential and the difference between these potentials are written as follows:where is a constant, is the nuclear radius, is the thickness of surface layer, and and are related with the potential dept for nuclei. Also and are dimensionless parameters and they are used to determine shape of the potential. Substituting (25) into (23), (23) reduces toTo solve this equation analytically, the following approximation which is called the Pekeris approximation [89] is applied to the centrifugal term:where , , and are constants and their values are found as follows by expanding the terms in (27):The terms in (27) are expanded in a series about because the nuclear distance does not fluctuate very far from the equilibrium position at [35]. Therefore the approximation in (27) gives good result only for . The equivalence of the Pekeris approximation to centrifugal term is shown in Figure 1. In this figure the blue curve shows the Pekeris approximation and the dotted curve indicates the centrifugal term.