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Hindawi Publishing CorporationISRN High Energy PhysicsVolume 2013, Article ID 987632, 7 pageshttp://dx.doi.org/10.1155/2013/987632

Research ArticleGeneralized Nuclear Woods-Saxon Potential under RelativisticSpin Symmetry Limit

M. Hamzavi and A. A. Rajabi

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

Correspondence should be addressed to M. Hamzavi; [email protected]

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 CommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

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

1. Introduction

The Dirac equation, which describes the motion of a spin 1/2particle, has been used in solving many problems of nuclearand high-energy physics. One of themost important conceptsfor understanding the traditional magic numbers in stablenuclei is the spin symmetry breaking [1–4]. On the otherhand, the 𝑝-spin symmetry observed originally almost 40years ago as a mechanism to explain different aspects of thenuclear structure is one of themost interesting phenomena inthe relativistic quantummechanics.Within the framework ofDirac equation, 𝑝-spin symmetry used to feature deformednuclei, superdeformation, and to establish an effective shellmodel [5–8], whereas spin symmetry is relevant for mesons[9, 10]. Spin symmetry occurs when𝑉

𝑆≈ 𝑉𝑉and pseudospin

symmetry occurs when 𝑉𝑆≈ −𝑉𝑉, [11, 12], where 𝑆 is scalar

potential and 𝑉𝑉is vector potential. Pseudospin symmetry is

exact under the condition of 𝑑(𝑉(𝑟) + 𝑆(𝑟))/𝑑𝑟 = 0, and spinsymmetry is exact under the condition of 𝑑(𝑉(𝑟)−𝑆(𝑟))/𝑑𝑟 =0 [13]. For the first time, the spin symmetry tested in therealistic nuclei [14], and then this symmetry is investigated byexamining the radial wave functions [15–17].The pseudospinsymmetry refers to a quasi-degeneracy of single nucleondoublets with nonrelativistic quantum numbers (𝑛, 𝑙, 𝑗 = 𝑙 +

1/2) and (𝑛 − 1, 𝑙 + 2, 𝑗 = 𝑙 + 3/2), where 𝑛, 𝑙, and 𝑗 are singlenucleon radial, orbital, and total angular quantum numbers,respectively. The total angular momentum is 𝑗 = 𝑙 + ��, where𝑙 = 𝑙 + 1 is pseudo-angular momentum and �� is pseudospinangular momentum [12, 18–20].

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

The interactions between nuclei are commonly describedby using a potential that consists of the Coulomb and thenuclear potentials. These potentials are usually taken tobe of the Woods-Saxon form. The Coulomb plus Woods-Saxon potentials are well known as modified Woods-Saxonpotential that plays a great role in nuclear physics. Fusionbarriers for a large number of fusion reactions from light toheavy systems can be described well with this potential [37–40]. But the modifiedWoods-Saxon potential has no exact orapproximate solutions.The generalizedWoods-Saxon is nearthe modified Woods-Saxon potential, and therefore we cansolve the generalized Woods-Saxon instead of the modifiedWoods-Saxon potential. The form of the generalizedWoods-Saxon potential is as follows [41, 42]:

𝑉(𝑟) = −

𝑉0

1 + 𝑒(𝑟−𝑅0)/𝑎

−

𝐶𝑒(𝑟−𝑅0)/𝑎

(1 + 𝑒(𝑟−𝑅0)/𝑎)

2, (1)

where𝑉0and𝐶 determine the potential depth,𝑅

0is the width

of the potential, 𝑎 is the surface thickness, and 0 < 𝐶 <

150MeV [42]. In Figure 1, we illustrated the shape of theWoods-Saxon potential (𝐶 = 0) and its generalized form in(1) (𝐶 = 0).

2 ISRN High Energy Physics

1 2 3 4 5 6 7 8 9 10𝑟 (fm)

𝑉(𝑟)

−10

−20

−30

−40

−50

𝐶 = 0

𝐶 = 50

𝐶 = 100

Figure 1: Shape of the generalized Woods-Saxon potential.

The purpose of this work is devoted to studying thespin symmetry solutions of the Dirac equation for arbitraryspin-orbit coupling quantum number 𝜅 with the generalizedWoods-Saxon potential, which was not considered before.

This paper is organized as follows. In Section 2, webriefly introduced the Dirac equation with scalar and vec-tor potential with arbitrary spin-orbit coupling number 𝜅under spin symmetry limit. The generalized parametricNikiforov-Uvarov method is presented in Section 3. Theenergy eigenvalue equations and corresponding eigenfunc-tions are obtained in Section 4. In this section, some remarksand numerical results are given too. Finally, conclusion isgiven in Section 5.

2. Dirac Equation under Spin andPseudospin Symmetry Limit

The Dirac equation with scalar potential 𝑆(𝑟) and vectorpotential 𝑉(𝑟) is

[𝑐�� ⋅ �� + 𝛽 (𝑀𝑐2+ 𝑆(𝑟))] 𝜓(𝑟) = [𝐸 − 𝑉(𝑟)] 𝜓(𝑟) , (2)

where 𝐸 is the relativistic energy of the system and �� = −𝑖∇ isthe three-dimensional momentum operator. �� and 𝛽 are the4 × 4 usual Dirac matrices given as

�� = (

0 ��

�� 0) , 𝛽 = (

𝐼 0

0 −𝐼) , (3)

where �� is Pauli matrix and 𝐼 is 2 × 2 unitary matrix. Thetotal angular momentum operator �� and spin-orbit operator𝐾 = (��⋅��+1), where �� is orbital angularmomentumoperator

and commutes with Dirac Hamiltonian. The eigenvalues ofspin-orbit coupling operator are 𝜅 = (𝑗 + 1/2) > 0 and 𝜅 =−(𝑗 + 1/2) < 0 for unaligned spin 𝑗 = 𝑙 − 1/2 and the alignedspin 𝑗 = 𝑙 + 1/2, respectively. (𝐻2, 𝐾, 𝐽2, 𝐽

𝑧) can be taken as

a complete set of the conservative quantities. Thus, the Diracspinors can be written according to radial quantum number𝑛 and spin-orbit coupling number 𝜅 as follows:

𝜓𝑛𝜅(

𝑟) = (

𝑓𝑛𝜅 (

𝑟)

𝑔𝑛𝜅 (

𝑟)) = (

𝐹𝑛𝜅 (

𝑟)

𝑟

𝑌𝑙

𝑗𝑚(𝜃, 𝜑)

𝑖

𝐺𝑛𝜅 (

𝑟)

𝑟

𝑌��

𝑗𝑚(𝜃, 𝜑)

) , (4)

where 𝑓𝑛𝜅( 𝑟) is the upper (large) component and 𝑔𝑛𝜅( 𝑟) is thelower (small) component of the Dirac spinors. 𝑌𝑙

𝑗𝑚(𝜃, 𝜑) and

𝑌��

𝑗𝑚(𝜃, 𝜑) are the spherical harmonic functions coupled to the

total angular momentum 𝑗 and its projection𝑚 on the 𝑧 axis.Substituting (4) into (2) with the usual Dirac matrices, oneobtains two coupled differential equations for the upper andthe lower radial wave functions 𝐹

𝑛𝜅(𝑟) and 𝐺

𝑛𝜅(𝑟) as

(

𝑑

𝑑𝑟

+

𝜅

𝑟

)𝐹𝑛𝜅 (

𝑟) =

1

ℎ𝑐

[𝑀𝑐2+ 𝐸𝑛𝜅− Δ (𝑟)] 𝐺𝑛𝜅 (

𝑟) , (5a)

(

𝑑

𝑑𝑟

−

𝜅

𝑟

)𝐺𝑛𝜅 (𝑟) =

1

ℎ𝑐

[𝑀𝑐2− 𝐸𝑛𝜅 + Σ (𝑟)] 𝐹𝑛𝜅 (

𝑟) , (5b)

where

Δ (𝑟) = 𝑉 (𝑟) − 𝑆 (𝑟) , (6a)

Σ (𝑟) = 𝑉 (𝑟) + 𝑆 (𝑟) . (6b)

Eliminating 𝐹𝑛𝜅(𝑟) and 𝐺

𝑛𝜅(𝑟) from (5a) and (5b), we obtain

the following second-order Schrodinger-like differentialequations for the upper and lower components of the Diracwave functions, respectively:

[

𝑑2

𝑑𝑟2−

𝜅 (𝜅 − 1)

𝑟2

]𝐺𝑛𝜅 (

𝑟)

+ [−

1

ℎ2𝑐2(𝑀𝑐2+ 𝐸𝑛𝜅− Δ (𝑟)) (𝑀 − 𝐸

𝑛𝜅+ Σ (𝑟))

−

1

ℎ2𝑐2

(𝑑Σ (𝑟) /𝑑𝑟) (𝑑/𝑑𝑟 − 𝜅/𝑟)

𝑀𝑐2− 𝐸𝑛𝜅+ Σ (𝑟)

]𝐺𝑛𝜅 (

𝑟) = 0,

(7a)

[

𝑑2

𝑑𝑟2−

𝜅 (𝜅 + 1)

𝑟2

]𝐹𝑛𝜅 (

𝑟)

+ [−

1

ℎ2𝑐2(𝑀𝑐2+ 𝐸𝑛𝜅− Δ (𝑟)) (𝑀𝑐

2− 𝐸𝑛𝜅+ Σ (𝑟))

+

1

ℎ2𝑐2

(𝑑Δ (𝑟) /𝑑𝑟) (𝑑/𝑑𝑟 + 𝜅/𝑟)

𝑀𝑐2+ 𝐸𝑛𝜅− Δ (𝑟)

] 𝐹𝑛𝜅 (

𝑟) = 0,

(7b)

where 𝜅(𝜅 − 1) = 𝑙(𝑙 + 1) and 𝜅(𝜅 + 1) = 𝑙(𝑙 + 1).

ISRN High Energy Physics 3

2.1. Spin Symmetry Limit. In the spin symmetry limit 𝑑Δ(𝑟)/𝑑𝑟 = 0 or Δ(𝑟) = 𝐶

𝑠= constant [14], then (7b) becomes

[𝑑2

𝑑𝑟2−

𝜅 (𝜅 + 1)

𝑟2

−

1

ℎ2𝑐2(𝑀𝑐2+ 𝐸𝑛𝜅− 𝐶𝑠)

× (𝑀𝑐2− 𝐸𝑛𝜅+ Σ (𝑟)) ] 𝐹𝑛𝜅 (

𝑟) = 0,

(8)

where 𝜅 = 𝑙 and 𝜅 = −𝑙−1 for 𝜅 < 0 and 𝜅 > 0, respectively. In(8), Σ(𝑟) can be taken as generalizedWoods-Saxon potential,and it is reduced to

[

𝑑2

𝑑𝑟2−

𝜅 (𝜅 + 1)

𝑟2

+𝜂(

𝑉0

1 + 𝑒(𝑟−𝑅0)/𝑎

+

𝐶𝑒(𝑟−𝑅0)/𝑎

(1 + 𝑒(𝑟−𝑅0)/𝑎)

2) − 𝜉2]𝐹𝑛𝜅 (

𝑟) = 0,

(9)

where

𝜂 =

1

ℎ2𝑐2(𝐸𝑛𝜅 +𝑀𝑐

2− 𝐶𝑠) ,

(10a)

𝜉2=

1

ℎ2𝑐2(𝑀𝑐2− 𝐸𝑛𝜅) (𝑀𝑐

2+ 𝐸𝑛𝜅 − 𝐶𝑠) .

(10b)

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

2.2. Pekeris-Type Approximation to the Spin-Orbit CouplingTerm. Because of the spin-orbit coupling term, that is, 𝜅(𝜅 +1)/𝑟2, (9) cannot be solved analytically, except for 𝜅 = 0, −1.

Therefore, we shall use the Pekeris approximation [43] inorder to deal with the spin-orbit coupling terms, and we mayexpress the spin-orbit term as follows:

𝑉so (𝑟) =𝜅 (𝜅 + 1)

𝑟2

=

𝜅 (𝜅 + 1)

(1 + 𝑥/𝑅0)2

≅

𝜅 (𝜅 + 1)

𝑅2

0

(1 − 2

𝑥

𝑅0

+ 3(

𝑥

𝑅0

)

2

+ ⋅ ⋅ ⋅) .

(11)

In addition, we may also approximately express it in the fol-lowing way:

��so (𝑟) =𝜅 (𝜅 + 1)

𝑟2

≅

𝜅 (𝜅 + 1)

𝑅2

0

(𝐷0+

𝐷1

1 + 𝑒𝜈𝑥+

𝐷2

(1 + 𝑒𝜈𝑥)2) ,

(12)

where 𝑥 = 𝑟 − 𝑅0, 𝜈 = 1/𝑎, and 𝐷

𝑖is constant (𝑖 = 0, 1, 2)

[44–48]. If we expand the expression of (12) around 𝑥 = 0 up

to the second-order term and next compare it with (11), wecan obtain expansion coefficients𝐷

0,𝐷1, and𝐷

2as follows:

𝐷0= 1 −

4

𝜈𝑅0

−

12

𝜈2𝑅2

0

,

𝐷1=

8

𝜈𝑅0

−

48

𝜈2𝑅2

0

,

𝐷2 =

48

𝜈2𝑅2

0

.

(13)

Now, we can take the potential ��so (12) instead of the spin-orbit coupling potential (11).

3. The Parametric GeneralizationNikiforov-Uvarov Method

This powerful mathematical tool solves second-order differ-ential equations. Let us consider the following differentialequation [49–51]:

(

𝑑2

𝑑𝑧2+

𝛼1− 𝛼2𝑧

𝑧 (1 − 𝛼3𝑧)

𝑑

𝑑𝑧

+

1

𝑧2(1 − 𝛼3𝑧)

2

× [−𝑝2𝑧2+ 𝑝1𝑧 − 𝑝0] )𝜓 (𝑧) = 0.

(14)

According to the Nikiforov-Uvarov method, the eigenfunc-tions and eigenenergies, respectively, are

𝜓 (𝑧) = 𝑧𝛼12(1 − 𝛼

3𝑧)−𝛼12−𝛼13/𝛼3

𝑃(𝛼10−1,𝛼11/𝛼3−𝛼10−1)

𝑛(1 − 2𝛼

3𝑧) ,

(15)

𝛼2𝑛 − (2𝑛 + 1) 𝛼5

+ (2𝑛 + 1) (√𝛼9− 𝛼3√𝛼8)

+ 𝑛 (𝑛 − 1) 𝛼3+ 𝛼7+ 2𝛼3𝛼8− 2√𝛼8

𝛼9= 0,

(16)

where

𝛼4=

1

2

(1 − 𝛼1) , 𝛼

5=

1

2

(𝛼2− 2𝛼3) ,

𝛼6= 𝛼2

5+ 𝑝2, 𝛼

7= 2𝛼4𝛼5− 𝑝1,

𝛼9 = 𝛼3𝛼7 + 𝛼

2

3𝛼8 + 𝛼6,

𝛼8= 𝛼2

4+ 𝑝0,

𝛼10= 𝛼1+ 2𝛼4+ 2√𝛼8

,

𝛼11 = 𝛼2 − 2𝛼5 + 2 (√𝛼9 + 𝛼3√𝛼8) ,

𝛼12= 𝛼4+ √𝛼8

,

𝛼13 = 𝛼5 − (√𝛼9 + 𝛼3√

𝛼8) .

(17)

In the rather more special case of 𝛼3= 0 [50, 51],

lim𝛼3→0

𝑃(𝛼10−1,𝛼11/𝛼3−𝛼10−1)

𝑛(1 − 2𝛼

3𝑧) = 𝐿

𝛼10−1

𝑛(𝛼11𝑧) , (18a)

lim𝛼3→0

(1 − 𝛼3𝑧)−𝛼12−𝛼13/𝛼3

= 𝑒𝛼13𝑧

, (18b)


Table 1: The approximate bound state energy eigenvalues (in unit of MeV) of the generalized Woods-Saxon potential within the spinsymmetry limit for several values of 𝑛 and 𝜅.

𝑙 𝑛, 𝜅 < 0, 𝜅 > 0 State 𝐸𝑛𝜅

𝐶 = 0

𝐸𝑛𝜅

𝐶 = 50MeV𝐸𝑛𝜅

𝐶 = 100MeV𝐸𝑛𝜅

𝐶 = 150MeV1 0, −2, 1 0𝑝

3/2, 0𝑝1/2

−939.0045195 −939.0074037 −939.0102535 −939.01306952 0, −3, 2 0𝑑

5/2, 0𝑑3/2

−938.5761054 −938.5822985 −938.5884052 −938.59442763 0, −4, 3 0𝑓

7/2, 0𝑓5/2

−938.4033046 −938.4111487 −938.4188786 −938.42649724 0, −5, 4 0𝑔

9/2, 0𝑔7/2

−938.532527 −938.5397300 −938.5468274 −938.55382211 1, −2, 1 1𝑝

3/2, 1𝑝1/2

−906.9952581 −907.4067290 −907.8063989 −908.19480412 1, −3, 2 1𝑑

5/2, 1𝑑3/2

−905.7621398 −906.1743738 −906.5749872 −906.96450193 1, −4, 3 1𝑓

7/2, 1𝑓5/2

−904.2904841 −904.7048150 −905.1077390 −905.49976004 1, −5, 4 1𝑔

9/2, 1𝑔7/2

−902.7908284 −903.2078994 −903.6137936 −904.00899451 2, −2, 1 2𝑝

3/2, 2𝑝1/2

−835.7682302 −837.2498452 −838.6826152 −840.06913702 2, −3, 2 2𝑑

5/2, 2𝑑3/2

−834.4172283 −835.8982294 −837.3306378 −838.71702943 2, −4, 3 2𝑓

7/2, 2𝑓5/2

−832.5628129 −834.0439814 −835.4769039 −836.86412614 2, −5, 4 2𝑔

9/2, 2𝑔7/2

−830.3570797 −831.8396528 −833.2743934 −834.6638100

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

𝜓 = 𝑧𝛼12𝑒𝛼13𝑧

𝐿𝛼10−1

𝑛(𝛼11𝑧) . (19)

4. Bound States of the GeneralizedWoods-Saxon Potential with Arbitrary 𝜅

Substituting (12) into (9) and using transformation 𝑧 = 𝑒𝜈𝑥,

we find the following second order deferential equation forthe upper component of the Dirac spinor as

{

𝑑2

𝑑𝑧2+

1 + 𝑧

𝑧 (1 + 𝑧)

𝑑

𝑑𝑧

−

𝜅 (𝜅 + 1)

𝜈2𝑧2𝑅2

0

(𝐷0 +

𝐷1

(1 + 𝑧)

+

𝐷2

(1 + 𝑧)2)

+

1

𝜈2𝑧2(

𝜂𝑉0

(1 + 𝑧)

+

𝜂𝐶𝑧

(1 + 𝑧)2− 𝜉2)}𝐹 (𝑧) = 0.

(20)

Comparing the previous equation with (14), one can find thefollowing parameters:

𝛼1= 1, 𝛼

2= −1, 𝛼

3= −1,

𝑝2= (

𝜅 (𝜅 + 1)

𝜈2𝑅2

0

𝐷0+

𝜉2

𝜈2) ,

𝑝1= −

𝜅 (𝜅 + 1)

𝜈2𝑅2

0

(2𝐷0+ 𝐷1) +

1

𝜈2(𝜂𝑉0+ 𝜂𝐶 − 2𝛽

2) ,

𝑝0 =

𝜅 (𝜅 + 1)

𝜈2𝑅2

0

(𝐷0 + 𝐷1 + 𝐷2) −

1

𝜈2(𝜂𝑉0 − 𝜉

2) ,

(21)

and also

𝛼4= 0, 𝛼

5=

1

2

,

𝛼6 =

1

4

+ 𝑝2, 𝛼7 = −𝑝1,

𝛼8= 𝑝0, 𝛼

9= 𝑝2− 𝑝1+ 𝑝0+

1

4

,

𝛼10 = 1 + 2√𝑝0,

𝛼11 = −2 + 2 (√𝑝2 − 𝑝1 + 𝑝0 + 1/4 − √𝑝0) ,

𝛼12 = √𝑝0,

𝛼13 =

1

2

− (√𝑝2 − 𝑝1 + 𝑝0 + 1/4 − √𝑝0) .

(22)

From (16), (21), and (22), we obtained the closed form ofthe energy eigenvalues for the generalized Woods-Saxonpotential in spin symmetry as

(𝑛 +

1

2

+ √(𝜅 (𝜅 + 1) /𝜈2𝑅2

0)𝐷2 − 𝜂𝐶/𝜈

2+ 1/4

+√(𝜅(𝜅 + 1)/𝜈2𝑅2

0)(𝐷0+ 𝐷1+ 𝐷2)−(1/𝜈

2)(𝜂𝑉0− 𝜉2))

2

= (𝜅 (𝜅 + 1) /𝜈2𝑅2

0)𝐷0+ 𝜉2/𝜈2.

(23)

Recalling 𝜂 and 𝜉 from (10a) and (10b) the previous equationbecomes a quadratic algebraic equation in 𝐸

𝑛𝜅. Thus, the

solution of this algebraic equation with respect to 𝐸𝑛𝜅

can beobtained in terms of particular values of𝑛 and 𝜅. InTable 1, wehave calculated some energy levels of the generalizedWoods-Saxon potential under the spin symmetry limit for different


45 50 55 60𝑉0 (MeV)

−903

−903.5

−904

−904.5

−905

−905.5

−906

−906.5

−907

−907.5

−908

𝐸𝑛𝜅(M

eV)

(1𝑝3/2, 1𝑝1/2)(1𝑑5/2, 1𝑑3/2)

(1𝑓7/2, 1𝑓5/2)(1𝑔9/2, 1𝑔7/2)

Figure 2: Computed energies as a function of 𝑉0.

5 5.5 6 6.5 7 7.5 8𝑅0 (fm)

−903

−903.5

−904

−904.5

−905

−905.5

−906

−906.5

−907

−907.5

−908

𝐸𝑛𝜅(M

eV)

(1𝑝3/2, 1𝑝1/2)(1𝑑5/2, 1𝑑3/2)

(1𝑓7/2, 1𝑓5/2)(1𝑔9/2, 1𝑔7/2)

Figure 3: Computed energies as a function of 𝑅0.

values of 𝐶. The empirical values that can be found in [52], as𝑟0 = 1.285 fm, 𝑎 = 0.65 fm,𝑀𝑛 = 1.00866 u, and 𝑉0 = 4.5 +

0.13𝐴 (MeV), are used. Here, 𝐴 is the mass number of targetnucleus, and 𝑅

0= 𝑟0𝐴1/3. From Table 1, we can see that pairs

(𝑛𝑝1/2, 𝑛𝑝3/2),(𝑛𝑑3/2, 𝑛𝑑5/2), (𝑛𝑓5/2, 𝑛𝑓7/2), (𝑛𝑔

7/2, 𝑛𝑔9/2), and

so forth are degenerate. Thus, each pair is considered as spindoublet and has negative energy, and also when 𝐶 increases,the energy decreases. The reason is that when 𝐶 increases,the depth of the potential well becomes deeper and then theenergy levels decrease. In Figure 2, the results are shown as afunction of 𝑉0. As we saw in Table 1, when the depth of thepotential well increases, then the energy levels decrease. InFigure 3, the results are presented as a function of width of thepotential 𝑅

0. Here, when the width of the potential increases,

the energy levels decrease. Finally we showed the numericalresults as a function of surface thickness 𝑎 in Figure 4, andone can observe that when the surface thickness increases,the energy levels decrease too.

0.4 0.45 0.5 0.55 0.6 0.65𝑎 (fm)

−895

−900

−905

−910

−915

−920

−925

𝐸𝑛𝜅(M

eV)

(1𝑝3/2, 1𝑝1/2)(1𝑑5/2, 1𝑑3/2)

(1𝑓7/2, 1𝑓5/2)(1𝑔9/2, 1𝑔7/2)

Figure 4: Computed energies as a function of 𝑎.

To find corresponding wave functions, referring to (15),(21), and (22), we find the upper component of the Diracspinor as

𝐹𝑛𝜅 (

𝑧) = 𝑧𝛼12(1 − 𝛼

3𝑧)−𝛼12−𝛼13/𝛼3

𝑃(𝛼10−1,𝛼11/𝛼3−𝛼10−1)

𝑛

× (1 − 2𝛼3𝑧)

= 𝑧√𝜉3(1 + 𝑧)

1/2+√(𝜅(𝜅+1)/𝜈2𝑅2

0)𝐷2−(𝜂𝐶/𝜈

2)+1/4

× 𝑃

(2√𝑝0,−2√(𝜅(𝜅+1)/𝜈2𝑅2

0)𝐷2−𝜂𝐶/𝜈

2+1/4)

𝑛 (1 + 2𝑧)

(24)

or equivalently

𝐹𝑛𝜅 (

𝑟)

= 𝑒√𝑝0((𝑟−𝑅0)/𝑎)

(1 + 𝑒(𝑟−𝑅0)/𝑎

)

1/2+√(𝜅(𝜅+1)/𝜈2𝑅2


2+1/4

× 𝑃

(2√𝑝0,−2√(𝜅(𝜅+1)/𝜈2𝑅2


2+1/4)

𝑛 (1 + 2𝑒(𝑟−𝑅0)/𝑎

) .

(25)

Finally, the lower component of the Dirac spinor can becalculated as

𝐺𝑛𝜅 (

𝑟) =

ℎ𝑐

𝑀𝑐2+ 𝐸𝑛𝜅− 𝐶𝑠

(

𝑑

𝑑𝑟

+

𝜅

𝑟

)𝐹𝑛𝜅 (𝑟) , (26)

where 𝐸𝑛𝜅

= − 𝑀𝑐2+ 𝐶𝑠[20].

5. Conclusion

In this paper we have studied the spin symmetry of a Diracnucleon subjected to scalar and vector generalized Woods-Saxon potentials. The quadratic energy equation and spinorwave functions for bound states have been obtained byparametric formof theNikiforov-Uvarovmethod. It is shownthat there exist negative-energy bound states in the case ofexact spin symmetry (𝐶

𝑠= 0). We gave some numerical

results of the energy eigenvalues too.


Acknowledgment

Theauthors thank the kind referee for positive and invaluablesuggestions, which improved this paper greatly.

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Research Article Generalized Nuclear Woods-Saxon Potential ...downloads.hindawi.com/journals/isrn/2013/987632.pdf · nuclear potentials. ese potentials are usually taken to be of