- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

ISRN High Energy Physics

Volume 2013 (2013), Article ID 987632, 7 pages

http://dx.doi.org/10.1155/2013/987632

## Generalized Nuclear Woods-Saxon Potential under Relativistic Spin Symmetry Limit

Physics Department, Shahrood University of Technology, Shahrood 3619995161, Iran

Received 28 January 2013; Accepted 13 February 2013

Academic Editors: C. Ahn and C. A. d. S. Pires

Copyright © 2013 M. Hamzavi and A. A. Rajabi. 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.

#### Abstract

By using the Pekeris approximation, we present solutions of the Dirac equation with the generalized Woods-Saxon potential with arbitrary spin-orbit coupling number under spin symmetry limit. We obtain energy eigenvalues and corresponding eigenfunctions in closed forms. Some numerical results are given too.

#### 1. Introduction

The Dirac equation, which describes the motion of a spin 1/2 particle, has been used in solving many problems of nuclear and high-energy physics. One of the most important concepts for understanding the traditional magic numbers in stable nuclei is the spin symmetry breaking [1–4]. On the other hand, the -spin symmetry observed originally almost 40 years ago as a mechanism to explain different aspects of the nuclear structure is one of the most interesting phenomena in the relativistic quantum mechanics. Within the framework of Dirac equation, -spin symmetry used to feature deformed nuclei, superdeformation, and to establish an effective shell model [5–8], whereas spin symmetry is relevant for mesons [9, 10]. Spin symmetry occurs when and pseudospin symmetry occurs when , [11, 12], where is scalar potential and is vector potential. Pseudospin symmetry is exact under the condition of , and spin symmetry is exact under the condition of [13]. For the first time, the spin symmetry tested in the realistic nuclei [14], and then this symmetry is investigated by examining the radial wave functions [15–17]. The pseudospin symmetry refers to a quasi-degeneracy of single nucleon doublets with nonrelativistic quantum numbers and , where , , and are single nucleon radial, orbital, and total angular quantum numbers, respectively. The total angular momentum is , where is pseudo-angular momentum and is pseudospin angular momentum [12, 18–20].

On the other hand, some typical physical models have been studied like harmonic oscillator [19, 20], Woods-Saxon potential [21, 22], Morse potential [23–25], Eckart potential [26–28], Pöschl-Teller potential [29], Manning-Rosen potential [30], and so forth [31–36].

The interactions between nuclei are commonly described by using a potential that consists of the Coulomb and the nuclear potentials. These potentials are usually taken to be of the Woods-Saxon form. The Coulomb plus Woods-Saxon potentials are well known as modified Woods-Saxon potential that plays a great role in nuclear physics. Fusion barriers for a large number of fusion reactions from light to heavy systems can be described well with this potential [37–40]. But the modified Woods-Saxon potential has no exact or approximate solutions. The generalized Woods-Saxon is near the modified Woods-Saxon potential, and therefore we can solve the generalized Woods-Saxon instead of the modified Woods-Saxon potential. The form of the generalized Woods-Saxon potential is as follows [41, 42]: where and determine the potential depth, is the width of the potential, is the surface thickness, and MeV [42]. In Figure 1, we illustrated the shape of the Woods-Saxon potential () and its generalized form in (1) ().

The purpose of this work is devoted to studying the spin symmetry solutions of the Dirac equation for arbitrary spin-orbit coupling quantum number with the generalized Woods-Saxon potential, which was not considered before.

This paper is organized as follows. In Section 2, we briefly introduced the Dirac equation with scalar and vector potential with arbitrary spin-orbit coupling number under spin symmetry limit. The generalized parametric Nikiforov-Uvarov method is presented in Section 3. The energy eigenvalue equations and corresponding eigenfunctions are obtained in Section 4. In this section, some remarks and numerical results are given too. Finally, conclusion is given in Section 5.

#### 2. Dirac Equation under Spin and Pseudospin Symmetry Limit

The Dirac equation with scalar potential and vector potential is where is the relativistic energy of the system and is the three-dimensional momentum operator. and are the usual Dirac matrices given as where is Pauli matrix and is unitary matrix. The total angular momentum operator and spin-orbit operator , where is orbital angular momentum operator and commutes with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are and for unaligned spin and the aligned spin , respectively. can be taken as a complete set of the conservative quantities. Thus, the Dirac spinors can be written according to radial quantum number and spin-orbit coupling number as follows: where is the upper (large) component and is the lower (small) component of the Dirac spinors. and are the spherical harmonic functions coupled to the total angular momentum and its projection on the axis. Substituting (4) into (2) with the usual Dirac matrices, one obtains two coupled differential equations for the upper and the lower radial wave functions and as

where

Eliminating and from (5a) and (5b), we obtain the following second-order Schrödinger-like differential equations for the upper and lower components of the Dirac wave functions, respectively:

where and .

##### 2.1. Spin Symmetry Limit

In the spin symmetry limit or constant [14], then (7b) becomes where and for and , respectively. In (8), can be taken as generalized Woods-Saxon potential, and it is reduced to where

For the solution of (9), we will use the Nikiforov-Uvarov method which is briefly introduced in the following section.

##### 2.2. Pekeris-Type Approximation to the Spin-Orbit Coupling Term

Because of the spin-orbit coupling term, that is, , (9) cannot be solved analytically, except for . Therefore, we shall use the Pekeris approximation [43] in order to deal with the spin-orbit coupling terms, and we may express the spin-orbit term as follows: In addition, we may also approximately express it in the following way: where , , and is constant () [44–48]. If we expand the expression of (12) around up to the second-order term and next compare it with (11), we can obtain expansion coefficients , , and as follows: Now, we can take the potential (12) instead of the spin-orbit coupling potential (11).

#### 3. The Parametric Generalization Nikiforov-Uvarov Method

This powerful mathematical tool solves second-order differential equations. Let us consider the following differential equation [49–51]: According to the Nikiforov-Uvarov method, the eigenfunctions and eigenenergies, respectively, are where In the rather more special case of [50, 51],

and, from (15), we find for the wave function

#### 4. Bound States of the Generalized Woods-Saxon Potential with Arbitrary

Substituting (12) into (9) and using transformation , we find the following second order deferential equation for the upper component of the Dirac spinor as Comparing the previous equation with (14), one can find the following parameters: and also From (16), (21), and (22), we obtained the closed form of the energy eigenvalues for the generalized Woods-Saxon potential in spin symmetry as Recalling and from (10a) and (10b) the previous equation becomes a quadratic algebraic equation in . Thus, the solution of this algebraic equation with respect to can be obtained in terms of particular values of and . In Table 1, we have calculated some energy levels of the generalized Woods-Saxon potential under the spin symmetry limit for different values of . The empirical values that can be found in [52], as fm, fm, u, and (MeV), are used. Here, is the mass number of target nucleus, and . From Table 1, we can see that pairs ,, , , and so forth are degenerate. Thus, each pair is considered as spin doublet and has negative energy, and also when increases, the energy decreases. The reason is that when increases, the depth of the potential well becomes deeper and then the energy levels decrease. In Figure 2, the results are shown as a function of . As we saw in Table 1, when the depth of the potential well increases, then the energy levels decrease. In Figure 3, the results are presented as a function of width of the potential . Here, when the width of the potential increases, the energy levels decrease. Finally we showed the numerical results as a function of surface thickness in Figure 4, and one can observe that when the surface thickness increases, the energy levels decrease too.

To find corresponding wave functions, referring to (15), (21), and (22), we find the upper component of the Dirac spinor as or equivalently Finally, the lower component of the Dirac spinor can be calculated as where [20].

#### 5. Conclusion

In this paper we have studied the spin symmetry of a Dirac nucleon subjected to scalar and vector generalized Woods-Saxon potentials. The quadratic energy equation and spinor wave functions for bound states have been obtained by parametric form of the Nikiforov-Uvarov method. It is shown that there exist negative-energy bound states in the case of exact spin symmetry (). We gave some numerical results of the energy eigenvalues too.

#### Acknowledgment

The authors thank the kind referee for positive and invaluable suggestions, which improved this paper greatly.

#### References

- O. Haxel, J. H. D. Jensen, and H. E. Suess, “On the “magic numbers” in nuclear structure,”
*Physical Review*, vol. 75, no. 11, p. 1766, 1949. View at Publisher · View at Google Scholar · View at Scopus - M. Goepper, “On closed shells in nuclei. II,”
*Physical Review*, vol. 75, no. 12, pp. 1969–1970, 1949. View at Publisher · View at Google Scholar - W. Zhang, J. Meng, S. Q. Zhang, L. S. Geng, and H. Toki, “Magic numbers for superheavy nuclei in relativistic continuum Hartree-Bogoliubov theory,”
*Nuclear Physics A*, vol. 753, no. 1-2, pp. 106–135, 2005. View at Publisher · View at Google Scholar - J. Meng, H. Toki, S. G. Zhou, S. Q. Zheng, W. H. Long, and L. S. Geng, “Relativistic continuum Hartree Bogoliubov theory for ground-state properties of exotic nuclei,”
*Progress in Particle and Nuclear Physics*, vol. 57, no. 2, pp. 470–563, 2006. View at Publisher · View at Google Scholar - J. N. Ginocchio, “Relativistic symmetries in nuclei and hadrons,”
*Physics Reports*, vol. 414, no. 4-5, pp. 165–261, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Bohr, I. Hamamoto, and B. R. Mottelson, “Pseudospin in rotating nuclear potentials,”
*Physica Scripta*, vol. 26, no. 4, p. 267, 1982. View at Publisher · View at Google Scholar - J. Dudek, W. Nazarewicz, Z. Szymanski, and G. A. Leander, “Abundance and systematics of nuclear superdeformed states; relation to the pseudospin and pseudo-SU
_{3}symmetries,”*Physical Review Letters*, vol. 59, no. 13, pp. 1405–1408, 1987. View at Publisher · View at Google Scholar · View at Scopus - D. Troltenier, C. Bahri, and J. P. Draayer, “Generalized pseudo-SU
_{3}model and pairing,”*Nuclear Physics A*, vol. 586, no. 1, pp. 53–72, 1995. View at Scopus - P. R. Page, T. Goldman, and J. N. Ginocchio, “Relativistic symmetry suppresses quark spin-orbit splitting,”
*Physical Review Letters*, vol. 86, no. 2, pp. 204–207, 2001. View at Publisher · View at Google Scholar - K. T. Hect and A. Adler, “Generalized seniority for favored $J\ne 0$ pairs in mixed configurations,”
*Nuclear Physics A*, vol. 137, no. 1, pp. 129–143, 1969. View at Publisher · View at Google Scholar - A. Arima, M. Harvey, and K. Shimizu, “Pseudo
*LS*coupling and pseudo SU_{3}coupling schemes,”*Physics Letters B*, vol. 30, no. 8, pp. 517–522, 1969. View at Publisher · View at Google Scholar - J. N. Ginocchio, “Pseudospin as a relativistic symmetry,”
*Physical Review Letters*, vol. 78, no. 3, Article ID 439, 436 pages, 1997. View at Publisher · View at Google Scholar - J. N. Ginocchio, “Relativistic symmetries in nuclei and hadrons,”
*Physics Reports*, vol. 414, no. 4-5, pp. 165–261, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. G. Zhou, J. Meng, and P. Ring, “Spin symmetry in the antinucleon spectrum,”
*Physical Review Letters*, vol. 91, no. 26, Article ID 262501, 4 pages, 2003. View at Publisher · View at Google Scholar - X. T. He, S. G. Zhou, J. Meng, E. G. Zhao, and W. Scheid, “Test of spin symmetry in anti-nucleon spectra,”
*European Physical Journal A*, vol. 28, no. 3, pp. 265–269, 2006. View at Publisher · View at Google Scholar - C. Y. Song, J. M. Yao, and J. Meng, “Spin symmetry for anti-lambda spectrum in atomic nucleus,”
*Chinese Physics Letters*, vol. 26, no. 12, Article ID 122102, 2009. View at Publisher · View at Google Scholar - C. Y. Song and J. M. Yao, “Polarization effect on the spin symmetry for anti-Lambda spectrum in ${}^{16}\text{O}+\overline{\lambda}$ system,”
*Chinese Physics C*, vol. 34, no. 9, pp. 1425–1427, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, and A. Arima, “Pseudospin symmetry in relativistic mean field theory,”
*Physical Review C*, vol. 58, no. 2, pp. R628–R631, 1998. View at Publisher · View at Google Scholar - R. Lisboa, M. Malheiro, A. S. de Castro, P. Alberto, and M. Fiolhais, “Pseudospin symmetry and the relativistic harmonic oscillator,”
*Physical Review C*, vol. 69, no. 2, Article ID 024319, 15 pages, 2004. View at Publisher · View at Google Scholar · View at Scopus - A. S. de Castro, P. Alberto, R. Lisboa, and M. Malheiro, “Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: the case of the relativistic harmonic oscillator,”
*Physical Review C*, vol. 73, no. 5, Article ID 054309, 13 pages, 2006. View at Publisher · View at Google Scholar - R. D. Woods and D. S. Saxon, “Diffuse surface optical model for nucleon-nuclei scattering,”
*Physical Review*, vol. 95, no. 2, pp. 577–578, 1954. View at Publisher · View at Google Scholar - J. Y. Guo and Z. Q. Sheng, “Solution of the Dirac equation for the Woods-Saxon potential with spin and pseudospin symmetry,”
*Physics Letters A*, vol. 338, no. 2, pp. 90–96, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Agboola, “Spin symmetry in the relativistic
*q*-deformed morse potential,”*Few-Body Systems*, vol. 52, no. 1-2, pp. 31–38, 2012. View at Publisher · View at Google Scholar - W. C. Qiang, R. S. Zhou, and Y. Gao, “Application of the exact quantization rule to the relativistic solution of the rotational Morse potential with pseudospin symmetry,”
*Journal of Physics A*, vol. 40, no. 7, pp. 1677–1685, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Bayrak and I. Boztosun, “The pseudospin symmetric solution of the Morse potential for any $\kappa $ state,”
*Journal of Physics A*, vol. 40, no. 36, pp. 11119–11127, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Soylu, O. Bayrak, and I. Boztosun, “$\kappa $ state solutions of the Dirac equation for the Eckart potential with pseudospin and spin symmetry,”
*Journal of Physics A*, vol. 41, no. 6, Article ID 065308, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Jia, P. Guo, and X. L. Peng, “Exact solution of the Dirac-Eckart problem with spin and pseudospin symmetry,”
*Journal of Physics A*, vol. 39, no. 24, pp. 7737–7744, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. H. Zhang, X. P. Li, and C. S. Jia, “Analytical approximation to the solution of the Dirac equation with the Eckart potential including the spin-orbit coupling term,”
*Physics Letters A*, vol. 372, no. 13, pp. 2201–2207, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Xu, S. He, and C. S. Jia, “Approximate analytical solutions of the Dirac equation with the Pöschl-Teller potential including the spin-orbit coupling term,”
*Journal of Physics A*, vol. 41, no. 25, Article ID 255302, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. F. Wei and S. H. Dong, “Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry,”
*Physics Letters A*, vol. 373, no. 1, pp. 49–53, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. F. Wei and S. H. Dong, “Pseudospin symmetry in the relativistic Manning-Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal term,”
*Physics Letters B*, vol. 686, no. 4-5, pp. 288–292, 2010. View at Publisher · View at Google Scholar · View at Scopus - O. Aydodu and R. Sever, “Pseudospin and spin symmetry in the Dirac equation with Woods-Saxon potential and tensor potential,”
*European Physical Journal A*, vol. 43, no. 1, pp. 73–81, 2010. View at Publisher · View at Google Scholar - C. Berkdemir, “Pseudospin symmetry in the relativistic Morse potential including the spin-orbit coupling term,”
*Nuclear Physics A*, vol. 770, no. 1-2, pp. 32–39, 2006. View at Publisher · View at Google Scholar - M. Hamzavi, A. A. Rajabi, and H. Hassanabadi, “Exact spin and pseudospin symmetry solutions of the Dirac equation for Mie-type potential including a coulomb-like tensor potential,”
*Few-Body Systems*, vol. 48, no. 2, pp. 171–182, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Hamzavi, A. A. Rajabi, and H. Hassanabadi, “Exact pseudospin symmetry solution of the Dirac equation for spatially-dependent mass Coulomb potential including a Coulomb-like tensor interaction via asymptotic iteration method,”
*Physics Letters A*, vol. 374, no. 42, pp. 4303–4307, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Hamzavi, A. A. Rajabi, and H. Hassanabadi, “Exactly complete solutions of the dirac equation with pseudoharmonic potential including linear plus coulomb-like tensor potential,”
*International Journal of Modern Physics A*, vol. 26, no. 7-8, pp. 1363–1374, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. L. Tian, N. Wang, and Z. X. Li, “Spread and quote-update frequency of the limit-order driven Sergei Maslov model,”
*Chinese Physics Letters*, vol. 24, no. 8, p. 905, 2007. View at Publisher · View at Google Scholar - N. Wang, K. Zhao, W. Scheid, and X. Wu, “Fusion-fission reactions with a modified Woods-Saxon potential,”
*Physical Review C*, vol. 77, no. 1, Article ID 014603, 11 pages, 2008. View at Publisher · View at Google Scholar - N. Wang and W. Scheid, “Quasi-elastic scattering and fusion with a modified Woods-Saxon potential,”
*Physical Review C*, vol. 78, no. 1, Article ID 014607, 7 pages, 2008. View at Publisher · View at Google Scholar - I. Boztosun, “New results in the analysis of ${}^{16}\text{O}+{}^{28}\text{S}\text{i}$ elastic scattering by modifying the optical potential,”
*Physical Review C*, vol. 66, no. 2, Article ID 024610, 6 pages, 2002. View at Publisher · View at Google Scholar - H. Fakhri and J. Sadeghi, “Supersymmetry approaches to the bound states of the generalized Woods-Saxon potential,”
*Modern Physics Letters A*, vol. 19, no. 8, pp. 615–625, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Berkdemir, A. Berkdemir, and R. Sever, “Editorial Note: polynomial solutions of the Schrödinger equation for the generalized Woods-Saxon potential [Phys. Rev. C 72, 027001 (2005)],”
*Physical Review C*, vol. 74, no. 3, Article ID 039902, 2006. View at Publisher · View at Google Scholar · View at Scopus - C. L. Pekeris, “The rotation-vibration coupling in diatomic molecules,”
*Physical Review*, vol. 45, no. 2, pp. 98–103, 1934. View at Publisher · View at Google Scholar · View at Scopus - V. H. Badalov, H. I. Ahmadov, and A. I. Ahmadov, “Analytical solutions of the schrödinger equation with the Woods-Saxon potential for arbitrary l state,”
*International Journal of Modern Physics E*, vol. 18, no. 3, pp. 631–641, 2009. View at Publisher · View at Google Scholar · View at Scopus - V. H. Badalov, H. I. Ahmadov, and S. V. Badalov, “Any l-state analytical solutions of the Klein-Gordon equation for the Woods-Saxon potential,”
*International Journal of Modern Physics E*, vol. 19, no. 7, pp. 1463–1475, 2010. View at Publisher · View at Google Scholar · View at Scopus - O. Aydodu and R. Sever, “Pseudospin and spin symmetry in the Dirac equation with Woods-Saxon potential and tensor potential,”
*European Physical Journal A*, vol. 43, no. 1, pp. 73–81, 2010. View at Publisher · View at Google Scholar - M. R. Pahlavani and S. A. Alavi, “Solutions of Woods-Saxon potential with spin-orbit and centrifugal terms through Nikiforov-Uvarov method,”
*Communications in Theoretical Physics*, vol. 58, no. 5, p. 739, 2012. View at Publisher · View at Google Scholar - M. R. Pahlavani and S. A. Alavi, “Study of nuclear bound states using mean-field Woods-Saxon and spin-orbit potentials,”
*Modern Physics Letters A*, vol. 27, no. 29, Article ID 125016, 13 pages, 2012. View at Publisher · View at Google Scholar - A. F. Nikiforov and V. B. Uvarov,
*Special Functions of Mathematical Physics*, Birkhäuser, Berlin, Germany, 1988. View at MathSciNet - C. Tezcan and R. Sever, “A general approach for the exact solution of the Schrödinger equation,”
*International Journal of Theoretical Physics*, vol. 48, no. 2, pp. 337–350, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. M. Ikhdair, “Rotational and vibrational diatomic molecule in the Klein-Gordon equation with hyperbolic scalar and vector potentials,”
*International Journal of Modern Physics C*, vol. 20, no. 10, p. 1563, 2009. View at Publisher · View at Google Scholar - C. M. Perey, F. G. Perey, J. K. Dickens, and R. J. Silva, “11-MeV proton optical-model analysis,”
*Physical Review*, vol. 175, no. 4, pp. 1460–1475, 1968. View at Publisher · View at Google Scholar