Indian J Phys DOI 10.1007/s12648-014-0592-5

ORIGINAL PAPER

Scattering phase shifts of Dirac equation with Manning-Rosen potential and Yukawa tensor interaction S Ortakaya1*, H Hassanabadi2 and E Maghsoodi2 1

Department of Physics, Faculty of Science, Erciyes University, 38039 Kayseri, Turkiye

2

Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran Received: 29 May 2014 / Accepted: 20 August 2014

Abstract: Bound-state solutions of the Dirac equation with Yukawa tensor interaction and Manning-Rosen potential are obtained for any arbitrary state. The energy eigenvalues and the corresponding eigenfunctions are obtained using the parametric Nikiforov–Uvarov method. Thereby, the radial wavefunctions of scattering states are obtained in terms of hypergeometric functions. Next, using the basic properties of the hypergeometric function, the phase-shifts are reported. In addition, some numerical results are included in the case of pseudospin and spin symmetry limits. Keywords:

Dirac equation; Manning-Rosen potential; Pseudospin and spin symmetry; Tensor interaction

PACS Nos.: 03.65.Ge; 03.65.Pm; 02.30.Gp

1. Introduction In the relativistic quantum mechanics, one of the most significant wave equations, is the Dirac equation, which describes the dynamics of spin-1/2 particles. Solutions of the Dirac equation for the pseudospin and spin symmetries play an important role in describing the nuclear shell structure [1, 2]. Pseudospin doublets are based on the small energy difference between nuclear energy levels with quantum numbers n; l; j ¼ l þ 12 and n  1; l þ 2; j ¼ l þ 32, where n, l and j are single nucleon radial, orbital and total angular quantum numbers, respectively [3]. It has been obtained that exact pseudospin symmetry occurs in the Dirac equation, when V (r) ? S(r) = constant [4, 5]. On the other hand, the spin symmetry occurs, when V (r) - S(r) = constant [6–10]. Many authors have studied the analytically solvable wave equations such as Schro¨dinger, Klein- Gordon and Dirac equations for various potentials [11–31]. Some authors have investigated the pseudospin and spin symmetries with several potentials including the Po¨schl-Teller [19], Yukawa [20, 21], Woods-Saxon [22], harmonic oscillator [23, 24], Hulthe´n [25, 26], Eckart [27], Po¨schl-Teller double-ring shaped Coulomb [28] and Cornell [29]. In recent years, various

methods have been used to solve the Dirac equation for solvable potentials under pseudospin and spin symmetries. These methods include the Nikiforov–Uvarov (NU) approach, supersymmetric quantum mechanics (SUSYQM), asymptotic iteration (AIM), Laplace transform, etc. [32–36]. In the present paper, our aim is to study the solutions of the Dirac equation with the Manning-Rosen potential (MRP) for any spin–orbit quantum number j. Under the conditions of the spin symmetry and pseudospin symmetry, we have investigated the bound-state energy eigenvalues and corresponding upper and lower spinor wavefunctions using the NU method. The solution of Dirac equation with MRP under pseudospin and spin symmetries, has been obtained in recent years. Wei et al. [16] have investigated Dirac equation with the potential in the framework of pseudospin symmetry. Likewise, Hassanabadi et al. [17] have studied the Dirac equation with actual and general MRP in the presence of Coulomb-like tensor interaction by applying SUSYQM. This paper is organized as follows: In the Sect. 2, we give a brief introduction of the Dirac equation under tensor potential. In the Sect. 3, we obtain approximate solutions to the Dirac equation for pseudospin and spin symmetries, when a Yukawa tensor interaction is present. The numerical values of relativistic energies are given in Sect. 4. Furthermore, the phase shifts are obtained in Sect. 5. The last section is devoted to conclusions.

*Corresponding author, E-mail: [email protected]

Ó 2014 IACS

S Ortakaya et al. ~

2. Dirac equation under tensor interaction

‘ ‘ ~:^ ðh; uÞ ¼ Yjm ðh; uÞ; ðr r Þ Yjm

The Dirac equation with scalar potential S(r), vector potential V (r) and a tensor potential U(r) is given by [37]

we arrive at the following coupled differential equations for the upper and lower radial wavefunctions   d j þ  UðrÞ Fnk ðrÞ ¼ ðM þ Enk  DðrÞÞGnk ðrÞ; dr r

~:^ ~Þ ¼ ½E  VðrÞwðr ~Þ; ½~:p a ~ þ bðM þ SðrÞÞ  iba r UðrÞwðr ð1Þ ~ and M denote the relativistic energy of where E, ~ p ¼ ir the system, three-dimensional momentum operator and mass of the particle, respectively. ~ a and b are 4 9 4 Dirac matrices,     0 ~ r I 0 ~ a¼ ; b¼ ; ð2Þ ~ r 0 0 I where I is 2 9 2unitary matrix and spin matrices are       0 1 0 i 1 0 r1 ¼ ; r2 ¼ ; r3 ¼ ; 1 0 i 0 0 1 ð3Þ for a spherical nuclei, the total angular momentum ~ J and ~ ~ ~ spin–orbit operator K ¼ bðR: L þ IÞ, where R are the 4 9 4 matrices, which have as block diagonal matrices ~ r Pauli matrices and ~ L is the orbital angular momentum operator, which commute with Dirac Hamiltonian. The eigenvalues of spin–orbit coupling operator are j ¼ ðj þ 12Þ\0 and j ¼ ðj þ 12Þ [ 0 for unaligned spin j ¼ ‘  12 and aligned spin j ¼ ‘ þ 12, respectively. The set ðH; K; J 2 ; Jz Þ is taken as the complete set of conservative quantities. Thus, the Dirac-spinors can be written as 0 1 ~Þ ‘ Fnk ðr ! Yjm ðh; uÞ C B ~Þ fnk ðr r C; ~Þ ¼ wnk ðr ð4Þ ¼B @ A ~Þ ‘~ Gnk ðr ~Þ gnk ðr Yjm ðh; uÞ i r ~Þ is the upper (alternatively called large in the where fnk ðr ~Þ is the lower (small) jargon) component and gnk ðr ‘~ component of the Dirac spinors. Y‘jm(h, u) and Yjm ðh; uÞ are spin and pseudospin spherical harmonics, respectively, and m is the projection of the angular momentum along the z-axis. Using ~Þðr ~~ ~Þ ¼ ~ ~ þ ir ~:A ~:B ~:ðA ðr A:B BÞ; ! ~ r: ~ L ~:p ~Þ ¼ ~ ~þ i ðr r:^ r r^:p ; r

~

~

 d j  þ UðrÞ Gnk ðrÞ ¼ ðM  Enk þ RðrÞÞFnk ðrÞ; dr r ð12Þ

where D(r) = V(r) - S(r) and R(r) = V(r) ? S(r) ~Þ þ Ginocchio et al. [3] has investigated that Vðr ~Þ  0 leads to pseudospin symmetry in nuclei. Meng Sðr et al. [4, 5] have presented that exact pseudospin symmetry ~Þ occurs in the Dirac equation, when d½Vðr~ÞþSðr ¼ 0 or dr ~Þ þ Sðr ~Þ ¼ constant after these pioneering studies, numerVðr ous works have been made to study the pseudospin sym~Þ ~Þ  Sðr ~Þ ¼ metry and spin symmetry d½Vðr~ÞSðr ¼ 0 or Vðr dr constant in nuclei and in the Dirac phenomenology. We ~Þ  Sðr ~Þ ¼ Cs solve Dirac equation for two conditions: Vðr ~Þ þ Sðr ~Þ ¼ Cps , where Csand Cpsare constants. and Vðr These constants can be chosen zero for bound systems whose potentials go to zero at infinity [41]. In addition, these limits have been used to investigate the spectrum of a nucleon or an antinucleon in the mean field of nucleons. The pseudospin symmetry occurs in the Dirac equation, ~Þ ¼ Cps ¼ Constant and pseudo-orbital angular when Rðr momentum is normal orbital angular momentum of the lower component of the Dirac spinor. The spin symmetry ~Þ ¼ Cs ¼ Constant occurs in the Dirac equation, when Dðr and orbital angular momentum is normal orbital angular momentum of the upper component of the Dirac spinor. Thus, we deal with the lower and upper components of the Dirac spinor to investigate the pseudospin and spin symmetries in the Dirac phenomenology. From Eqs. (11) and (12), we have 

ð6Þ 

~Þ Y ‘ ðh; uÞ ¼ ðj  1ÞY ‘ ðh; uÞ; ~:L ðr jm jm

ð7Þ

~Þ Y ‘ ðh; uÞ ¼ ðj þ 1ÞY ‘ ðh; uÞ; ~:L ðr jm jm

ð8Þ

‘~ ~:^ ðh; uÞ ðr r Þ Yjm

ð9Þ

‘ ¼ Yjm ðh; uÞ;

ð11Þ 

ð5Þ

as well as

ð10Þ

d 2 jðj þ 1Þ 2j dUðrÞ  U 2 ðrÞ  þ UðrÞ  r2 r dr dr 2   dDðrÞ d j dr þ þ  UðrÞ gFnj ðrÞ M þ Enj  DðrÞ dr r ¼ ðM þ Enj  DðrÞÞðM  Enj þ RðrÞÞFnj ðrÞ;

ð13Þ

d 2 jðj  1Þ 2j dUðrÞ  U 2 ðrÞ  þ UðrÞ þ r2 r dr dr 2   dRðrÞ d j dr þ  þ UðrÞ gGnj ðrÞ M  Enj þ RðrÞ dr r ¼ ðM þ Enj  DðrÞÞðM  Enj þ RðrÞÞGnj ðrÞ;

ð14Þ

~ ‘~þ 1Þ and j(j - 1) = ‘(‘ ? 1). where jðj þ 1Þ ¼ ‘ð

Scattering phase shifts of Dirac equation

3. Solutions of Dirac equation 3.1. Pseudospin symmetry The pseudospin symmetry occurs, when dRðrÞ dr ¼ 0, i.e., RðrÞ ¼ Cps ¼ constant the difference potential D(r) is taken as MRP [16] " # h2  aða  1Þ A DðrÞ ¼  ; ð15Þ 2Mb2 ðer=b  1Þ2 er=b  1 where a and A are dimensionless parameters and b is related to the range of the potential. The Yukawa-type tensor potential is given as UðrÞ ¼ V0

edr ; r

ð16Þ

putting the above potentials into Eq. (14), we find 

2

Fig. 1 Behavior of f(r) and its approximations for d = 0.05 fm-1

 dr

3

d2 jðj  1Þ edr V 2 e2dr dedr r þ e  2jV0 2  0 2 þ V0 7 6 dr 2  2 r r r r2 7 6 7 6 ! 7 6 4dr 2dr e e 5 4 ps ps þ ðEnj  M  Cps Þ Enj þ M  V1 þ V 2 2 2dr 1e ð1  e2dr Þ

Dps 2 ¼

Gps nj ðrÞ ¼ 0;

ð17Þ 1 is a parameter having dimensions of where d ¼ 2b A inverse length, V1 ¼ aða1Þ 2Mb2 and V2 ¼ 2Mb2 (in the atomic units ⁄ = 1). Now, we consider the two proper approximations

1 4d2 e2dr  ; r 2 ð1  e2dr Þ2

ð18Þ

Dps 3 ¼

ps  ðM  Enj þ Cps Þ  ps 2M þ 2Enj  V2  jðj  1Þ 2  4d  3  2j  V0 ; ð22Þ 2 ps  ðM  Enj þ Cps Þ  ps : M þ Enj 2 4d

ð23Þ

In the parametric generalization of NU method, we consider a an equation of the form [25, 33, 38]. d2 a1  a2 s d n s2 þ n2 s  n3 wn ðsÞ þ 1 wn ðsÞ þ wn ðsÞ 2 sð1  a3 sÞ ds ds ½sð1  a3 sÞ2 ¼ 0:

ð24Þ

2 dr

1 4d e  ; r 2 ð1  e2dr Þ2

ð19Þ

the above approximations for d = 0.05 fm-1are plotted in Fig. 1. Using above approximations and defining the new variable x = e-2dr, we have "

 ps 2 d2 ð1  x Þ d 1 ps  þ þ D1 x þ Dps 2 x  D3 2 2 2 xð1  xÞ dx x ð1  xÞ dx Gps nj ðrÞ

#

1 a4 ¼ ð1  a1 Þ; 2 1 a5 ¼ ða2  2a3 Þ; 2 a6 ¼ a25 þ n1 ; a7 ¼ 2a4 a5  n2 ; a8 ¼ a24 þ n3 :

a9 ¼ a3 a7 þ a23 a8 þ a6 ; pffiffiffiffiffi a10 ¼ a1 þ 2a4 þ 2 a8 ; pffiffiffiffiffi pffiffiffiffiffi a11 ¼ a2  2a5 þ 2ð a9 þ a3 a8 Þ; pffiffiffiffiffi a12 ¼ a4 þ a8 ; pffiffiffiffiffi pffiffiffiffiffi a13 ¼ a5  ð a9 þ a3 a8 Þ:

¼ 0;

ð25Þ ð20Þ

where Dps 1 ¼

The parameters relevant to Eq. (24) are given by.

ps  ðM  Enj þ Cps Þ  ps M þ Enj  V1  V2 þ V0 ðV0 2 4d 1 þ Þ; 2 ð21Þ

By using NU method [38], we obtain energy eigenvalue equation as follows: pffiffiffiffiffi pffiffiffiffiffi a2 n  ð2n þ 1Þa5 þ ð2n þ 1Þð a9 þ a3 a8 Þ þ nðn  1Þa3 pffiffiffiffiffiffiffiffiffi þ a7 þ 2a3 a8 þ 2 a8 a9 ¼ 0; and the general solution reads as

ð26Þ

S Ortakaya et al.

Wn ðsÞ ¼ sa12 ð1  a3 sÞa12 ða13 =a3 Þ Pnða10 1;ða11 =a3 Þa10 1Þ ð1  2a3 sÞ;

Ds1 ¼

ð27Þ

s  ðM þ Enj  Cs Þ  s M  Enj þ V1 þ V2 þ V0 ðV0 2 4d 1  Þ; 2

ða 1;ða =a Þa 1Þ

11 3 10 where Pn 10 ð1  2a3 sÞ is Jacobi function. We can apply parametric NU method to Eq. (20). Thus, we obtain the energy eigenvalue equation as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi! 1 1 ps ps 2 n þ n þ þ ð2n þ 1Þ þ Dps Dps 1 þ D3  D2 þ 3 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ps ps ps 1 ps ps ps þ D1 þ D3  D2 ¼ 0; ð28Þ  D2 þ 2D3 þ 2 D3 4

and the lower and upper components of the wavefunction are ffi pffiffiffiffiffi  ps ps ps 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2d Dps r 2dr 2þ 4þD1 þD3 D2 3 ð r Þ ¼ N e 1  e Gps nj nj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 1 ps ps ps ð29Þ þ Dps  2 F1 ðn; n þ 2 D3 þ 2 1 þ D3  D2 4 qffiffiffiffiffiffiffi 2dr þ 1; 2 Dps Þ: 3 þ 1; e   1 d j edr ps    V Fnj ðrÞ ¼  Gps 0 ps nj ðrÞ: r M  Enj þ Cps dr r ð30Þ 3.2. Spin symmetry In the spin symmetry limit, the difference potential is taken as a constant, i.e. dDðrÞ dr ¼ 0 or DðrÞ ¼ Cs ¼ constant: Here we take MRP as. e4dr

e2dr RðrÞ ¼ V1  V2 ; 2 1  e2dr ð1  e2dr Þ

ð31Þ

ð34Þ Ds2 ¼

s  ðM þ Enj  Cs Þ  s 2M  2Enj þ V2  jðj þ 1Þ 2  4d  3  2j þ V0 ; 2

ð35Þ Ds3 ¼

ðM þ

s Enj 2

4d

 Cs Þ 

 s : M  Enj

If we use same procedure given in previous section, the energy equation for the spin symmetry limit is obtained as ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffisffi 1 1 2 s s s þ n þ n þ þ ð2n þ 1Þ þ D1 þ D3  D2 þ D3 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 þ Ds1 þ Ds3  Ds2 ¼ 0; ð37Þ  Ds2 þ 2Ds3 þ 2 Ds3 4 The upper and lower components of Dirac wavefunction are obtained as follows: ffiffiffiffis  1  pQ 1þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þQs þQs Qs s 3 Fnj 1  e2dr 2 4 1 3 2 ðrÞ ¼ Nnj e2dr rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffisffi 1 þ Ds1 þ Ds3  Ds2  2 F1 ðn; n þ 2 D3 þ 2 4 pffiffiffiffiffiffi þ 1; 2 Ds3 þ 1; e2dr Þ; ð38Þ

Gsnj ðrÞ

  dr  1 d j e s þ þ V0 ¼ ðrÞ: Fnj s M þ Enj  Cs dr r r ð39Þ

Substitution of Eqs. (31) and (16) into Eq. (13) yields.  dr  3 d 2 jðj þ 1Þ edr V02 e2dr de r þ edr   2jV   V 0 0 6 dr 2 7 r2 r2 r2 r2 6 7 6 7 ! 6 7 4dr 2dr e e 4 5 s s  V2  ðM þ Enj  Cs Þ M  Enj þ V1 2 2dr 2dr 1e ð1  e Þ

ð36Þ

2

s ðrÞ ¼ 0; Fnj

ð32Þ Now by considering the previous approximations and x = e-2dr into Eq. (32), we have # "  s 2  d2 ð1  x Þ d 1 s s þ þ D1 x þ D2 x  D3 dx2 xð1  xÞ dx x2 ð1  xÞ2 s Fnj ðrÞ ¼ 0;

ð33Þ where

4. Results and discussion We have obtained the energy eigenvalues in the absence (V0 = 0) and the presence (V0 = 0) of the Yukawa-like tensor potential for various values of the quantum numbers n and j. The results are reported in Tables 1–4 for MRP in the relativistic symmetries. One can clearly see that there is a degeneracy among the pseudospin and spin doublets and that these degeneracies are changed or removed in the presence of the tensor interaction. A few numerical values of energies for pseudospin and spin symmetry limits can be compared with those reported earlier [17]. Within the framework of earlier [17], we only consider the Yukawa interaction. The potential parameters are taken as M = 1 fm-1, b = 1 fm, d = 0.1 fm-1, a = 1.5, A = -5 (i.e. V1 = -0.0858 fm-1

Scattering phase shifts of Dirac equation Table 1 The energy eigenvalues (in fm-1) of pseudospin symmetry for several n and j values with parameters M ¼ 1 fm1 , d = 0.025, V1 ¼ 0:0009375 fm1 ; V2 ¼ 0:03815 fm1 , Cps ¼ 6 fm1 in the presence of Yukawa tensor interaction (V0 ¼ 0:75 fm1 ) el

n, j \ 0

(l, j)

Enj \ 0

n - 1, j [ 0

(l ? 2, j ? l)

En

1

1, -1

1s1/2

-1.05394

0, 2

0d3/2

-1.02075

2

1, -2

1p3/2

-1.02959

0, 3

0f5/2

-1.00944

3

1, -3

1d5/2

-1.01420

0, 4

0g7/2

-1.00358

4

1, - 4

1f7/2

-1.00601

0, 5

0h9/2

-1.00084

1

2, -1

2s1/2

-1.02306

1, 2

ld3/2

-1.00870

2

2, -2

2p3/2

-1.01275

1, 3

1f5/2

-1.00338

3 4

2, -3 2, -4

2d5/2 2f7/2

-1.00562 -1.00181

1, 4 1, 5

1g7/2 1h9/2

-1.00079 -1.00000

- 1, j [ 0

Table 2 The energy eigenvalues (in fm-1) for pseudospin symmetry limit with parameters M ¼ 1 fm1 , V1 ¼ 0:0009375 fm1 ; V2 ¼ 0:03815 fm1 , Cps ¼ 6 fm1 d (fm-1)

Eps nj (V0 = 0)

Eps nj (V0 = 0.75)

1d5/2

1f5/2

1d5/2

1f5/2

0.0025

-1.27844

-1.24891

-1.28685

-1.24273

0.0050

-1.18598

-1.16202

-1.20386

-1.15013

0.0075 0.0100

-1.12225 -1.08101

-1.10728 -1.07219

-1.14327 -1.10109

-1.09400 -1.05992

0.0125

-1.05439

-1.04918

-1.07201

-1.03872

0.0150

-1.03687

-1.03372

-1.05180

-1.02512

0.0175

-1.02505

-1.02311

-1.03752

-1.01616

0.0200

-1.01691

-1.01568

-1.02725

-1.01015

0.0225

-1.01121

-1.01042

-1.01975

-1.00609

0.0250

-1.00719

-1.00668

-1.01420

-1.00338

Table 3 Energies in the spin Symmetry Limit for with parameters M ¼ 1 fm1 , d = 0.025, V1 ¼ 0:0009375 fm1 ; V2 = - 0.03815, Cs ¼ 6 fm1 : el

n, j \ 0

(l, j)

s En, j\0 fm-1(V0 = 0)

s En, j\0 fm-1(V0 = 0.75)

n, j [ 0

(l, j)

-1 s En, j [ 0 fm (V0 = 0)

Essn, j [ 0 fm-1 (V0 = 0.75)

1

0, -2

0p3/2

1.082878136

1.142299357

0, 1

0p1/2

1.082878136

1.046842121

2

0, -3

0d5/2

1.038370705

1.068862897

0, 2

0d3/2

1.038370705

1.021636230

3 4

0, -4 0, -5

0f7/2 0g9/2

1.017608322 1.007509494

1.031887740 1.014509795

0, 3 0, 4

0f5/2 0g7/2

1.017608322 1.007509494

1.009504849 1.003525691

1

1, -2

1p3/2

1.033724449

1.052471135

1, 1

1p1/2

1.033724449

1.020093016

2

1, -3

1d5/2

1.016515092

1.028711054

1, 2

1d3/2

1.016515092

1.009077483

3

1, -4

1f7/2

1.007194916

1.013715109

1, 3

1f5/2

1.007194916

1.003394589

4

1, -5

1g9/2

1.002472639

1.005745862

1, 4

1g7/2

1.002472639

1.000763443

1

2, -2

2p3/2

1.014497556

1.022348486

2, 1

2p1/2

1.014497556

1.00836008

2

2, -3

2d5/2

1.006681554

1.012295808

2, 2

2d3/2

1.006681554

1.003195716

3

2, -4

2f7/2

1.002329756

1.005370696

2, 3

2f5/2

1.002329756

1.000713224

4

2, -5

2g9/2

1.000383045

1.001685830

2, 4

2g7/2

1.000383045

1.000000531

and V2 = -2.5 fm-1), Cps = -1 fm-1 and Cs = 1 fm-1 1 (according to Ref [17], d 6¼ 2b ). In the absence of tensor interaction (V0 = 0), the pseudospin and spin energies

ps s are obtained as E1s ¼ 0:415015 fm1 and E0d ¼ 5=2 1=2 1 0:359555 fm , respectively. For the tensor parameter V0 = ps s 0.5, we obtain E1s ¼ 0:233954 fm1 and E0d ¼ 1=2 5=2

S Ortakaya et al. Table 4 The energy eigenvalues (in fm-1) for spin symmetry limit with parameters M ¼ 1 fm1 ,V1 ¼ 0:0009375 fm1 ; V2 = - 0.03815, Cs ¼ 6 fm1 : d (fm-1)

Esnj (V0 = 0)

Esnj (V0 = 0.75)

1d5/2

0pl/2

1d5/2

0pl/2

0.0025

1.289174782

1.349612226

1.295257179

1.342675450

0.0050

1.209140080

1.307310824

1.224050401

1.286266765

0.0075 0.0100

1.149880343 1.107695769

1.265331026 1.226388468

1.170310607 1.130049974

1.231159784 1.183583324

0.0125

1.077966438

1.191829614

1.099899357

1.145018470

0.0150

1.056896126

1.162015813

1.077214665

1.114662512

0.0175

1.041770682

1.136732428

1.060012606

1.091019199

0.0200

1.030749962

1.115493131

1.046843492

1.072609672

0.0225

1.022605532

1.097727067

1.036660709

1.058202661

0.0250

1.016515092

1.082878136

1.028711054

1.046842121

0:1756738 fm1 . These results are in agreement with those reported earlier [17]. In the calculations, we have considered the potential parameters as M ¼ 1fm1 ; b = 20 fm, a = 1.5 fm-1, A = 30.52 (i.e. V1 = 0.0009375 fm-1, and V2 = 0.03815 fm-1), Cps ¼ 6 fm1 according to Ref. [16], we consider 1 d ¼ 2b . Also it is observed that the difference of energy eigenvalues between the degenerate states increases as V0 increases. We show the effect of tensor term on the boundstates for various values of the parameters d in Table 2 in pseudospin symmetry limit and Table 4 in spin symmetry limit. In Fig. 2(a), the wavefunctions are plotted for Pseudospin Symmetry Limit for 1p3/2with and without a tensor interaction and in Fig. 2(b), the wave functions are

zð1  zÞ

d

2

Gps nj ðzÞ dz2

z

dGps nj ðzÞ dz

þ

without a tensor interaction. It can be seen that tensor interaction affects only the shape of the wavefunctions and does not change the node structure of the radial upper and lower components.

5. Scattering state solutions 5.1. Pseudospin symmetry In this section, we have obtained the scattering states of MRP in the presence of Yukawa tensor interaction. Using a new variable of the form z = 1 - e-2dr, Eq. (17) becomes Eq. (40).

8 ps ps < ðEnj MCps2 ÞðEnj þMÞ

jðj1Þ2jV0 þV0 z ð1zÞ þ z 4d ps 2 ps ps ÞV2 :  ðEnj MCps2ÞV1 þ4d V02 ð1zÞ þ V0 þ ðEnj MC 2 z 4d 4d2

plotted for Pseudospin Symmetry Limit for 0g7/2 with and without a tensor interaction. The wavefunctions for spin Symmetry Limit for 1f7/2 are plotted in Fig. 3(a). Also in Fig. 3(b), we have portrayed the behavior of the wavefunctions in spin Symmetry Limit for 2p1/2 with and

9 = ;

Gps nj ðzÞ ¼ 0;

ð40Þ

To obtain hypergeometric differential equation from Eq. (40), we use a new transformation ps

ps

b ps c Gps nj ðzÞ ¼ z ð1  zÞ gnj ðzÞ;

ð41Þ

which gives hypergeometric-type equation for gnj(z) as

Scattering phase shifts of Dirac equation

Fig. 2 (a) Wavefunction for Pseudospin Symmetry Limit for 1p3/2 (b) Wavefunction for Pseudospin Symmetry Limit for 0g7/2 for M ¼ 1 fm1 , d = 0.025, V0 ¼ 0:75 fm1 ; V1 ¼ 0:0009375 fm1 ; V2 = 0.03815, Cps ¼ 6 fm1 :

Fig. 3 (a) Wavefunction for spin Symmetry Limit for 1f7/2 (b) Wavefunction for spin Symmetry Limit for 2p1/2 for M ¼ 1 fm1 , d = 0.025, V0 ¼ 0:75 fm1 ; V1 ¼ 0:0009375 fm1 ; V2 = -0.03815, Cs ¼ 6 fm1 :

9 d2 d cps ðcps  1Þ ðbps Þ2 > ps > ps ps > þ zð1  zÞ 2 þ ð2c  ð1 þ 2c þ 2b ÞzÞ þ = dz z dz ð1  zÞ gps nj ðzÞ ¼ 0; ps ps > > k k > > ps > ps ps ps ps ps ps ps ps 1 2 > þ þ k3 ; : c ðc  1Þ  b ðb  1Þ  2c b  c  b þ ð1  zÞ z 8 > > > <

by considering kps 1 ¼

ps ps ðEnj  M  Cps ÞðEnj þ MÞ ; 2 4d

ð43Þ

kps 2 ¼ jðj  1Þ  2jV0 þ V0 ðEps  M  Cps ÞV1 þ 4d2 V02  nj ; 4d2

ð42Þ

ð44Þ

S Ortakaya et al. ps ps ðEnj  M  Cps ÞðEnj þ MÞ 2 4d ps ðEnj  M  Cps ÞV1 þ 4d2 V02 V0 þ þ 2 4d2 ps ðEnj  M  Cps ÞV2 : þ 4d2

ps ps gps 3  g2 ¼ g1 ;

kps 3 ¼

ð45Þ

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1 þ 4kps 2 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi ik bps ¼  ; k ¼ 4d2 kps 1 ; 2d cps ¼

ð46Þ

ð48Þ Recalling the form of the hypergometric equation [39], we rewrite Eq. (8) as   d2  ps  ps ps   d ps ps zð1  zÞ 2 þ g3  1 þ g1 þ g2 z  g1 g2 gps nj ðzÞ dz dz ¼ 0; ð49Þ

ð50Þ

with qffiffiffiffiffiffi kps 3 ; qffiffiffiffiffiffi ps ps gps kps 2 ¼c þb  3 ; ps ps gps 1 ¼c þb þ

ps gps 3 ¼ 2c ;

ð53Þ

hence, the radial wavefunction of the scattering states is  cps   ps ps ikr 2dr : 1  e2dr 2 F1 gps Gps nj ðrÞ ¼ Nnj e 1 ; g2 ; g3 ; 1  e ð54Þ we now study the asymptotic form of the above expression for large r and calculate the normalization constant of radial wavefunctions Nnj as well as the phase shifts. From Eqs. (51)–(53), we have  ps ps ps ps ps  ps gps ; ð55Þ 3  g2  g1 ¼ 2b ¼ g1 þ g2  g3 ps ps gps 3  g1 ¼ g2 ;

ð56Þ

ð58Þ

ps

ps

ps ps ps Cðgps 3 ÞCðg1 þ g2  g3 Þ ps ps Cðg1 ÞCðg2 Þ

ps ps ps ps ps ps  2 F1 ðgps 3  g1 ; g3  g2 ; g3  g1  g2 þ 1; 1  zÞ;

ð59Þ we can rewrite Eq. (54) as [39]  ps ps ðr!1Þ Cðgps ps ikr 3  g1  g2 Þ Gps ðrÞ  ! N e Cðg Þ nj ps ps ps : nj 3 Cðgps 3  g1 ÞCðg3  g2 Þ    ps ps Cðgps 2ikr 3  g1  g2 Þ þe ; ps ps ps Cðgps 3  g1 ÞCðg3  g2 Þ by taking

Cðgps gps gps Þ 3 1 2 gps ÞCðgps gps Þ Cðgps 3 1 3 2

Cðgps gps1 gps2 Þ id ¼ Cðgps g3 ps ÞCðg ps ps e g Þ 3

1

3

ð60Þ and

2

inserting in Eq. (54) we arrive at ps ps Cðgps ðr!1Þ ps ps 3  g1  g 2 Þ Gnj ðrÞ ! Nnj Cðg3 Þ ps ps ps ps Cðg3  g1 ÞCðg3  g2 Þ n o  eiðkrþdÞ þ eiðkrþdÞ

ps ps Cðgps p ps 3  g1  g2 Þ ¼ 2Nnj Cðg3 Þ ; ps ps ps ps sin kr þ d þ Cðg3  g1 ÞCðg3  g2 Þ 2 ð61Þ

ð51Þ ð52Þ

of

ps ps ps ps  2 F1 ðgps 1 ; g2 ; g1 þ g2  g3 þ 1; 1  zÞ ps

ð47Þ

properties

ps ps ps  ps ps ps  Cðgps 3 ÞCðg3  g1  g2 Þ g1 ; g2 ; g3 ; z ¼ ps ps ps Cðg3  g1 ÞCðg3  gps 2 Þ

þ ð1  yÞg3 g1 g2

Therefore, Eq. (42) is written as  d2 d zð1  zÞ 2 þ ð2cps  ð1 þ 2cps þ 2bps ÞzÞ  ððcps Þ2 : dz dz  Þ gps þ ðbps Þ2 þ 2cps bps  kps nj ðzÞ ¼ 0; 3

which possesses the solution  ps ps ps  2dr ; gps nj ¼ 2 F1 g1 ; g2 ; g3 ; 1  e

and by applying the following hypergeometric function [39, 40]  ps ps ps  2 F1 g1 ; g2 ; g3 ; 0 ¼ 1; 2 F1

where

ð57Þ

By comparing Eq. (61) with the boundary condition [39, jp 40] r ! 1 ) Gps nj ð1Þ ! 2sinðkr  2 þ dj Þ phase shifts and the normalization constant for pseudospin symmetry limit are given by. p dj ¼ ð j þ 1Þ þ d 2    qffiffiffiffiffiffi p ik ik ps ¼ ðj þ 1Þ þ arg C  arg C c þ  kps 3 2 d 2d  qffiffiffiffiffiffi ik  arg C cps þ þ kps ; 3 2d

Nnj

ð62Þ  ffiffiffiffiffiffi ffiffiffiffiffiffi p p    ik ps ik  kps kps 1 C cps þ 2d 3 C c þ 2d þ 3   ¼ : Cð2cps Þ C ikd ð63Þ

Scattering phase shifts of Dirac equation

5.2. Spin symmetry Using a new variable of the formz = 1 - e-2dr, Eq. (32) becomes

ks2 ¼ jðj þ 1Þ  2jV0  V0 ðEs þ M  Cs ÞV1 þ 4d2 V02  nj ; 4d2

9 8 s s ðEnj þ M  Cs ÞðM  Enj Þ z jðj þ 1Þ  2jV0  V0 > > > >  þ = s s ð1  zÞ z d2 Fnj ðzÞ dFnj ðzÞ < 4d2 s Fnj þ zð1  zÞ z ðzÞ ¼ 0; 2 2 2 s s > > dz dz > ; :  ðEnj þ M  Cs ÞV1 þ 4d V0 ð1  zÞ  V0 þ ðEnj þ M  Cs ÞV2 > z 2 4d2 4d2

ks3 ¼

Introducing the trial wavefunction s

s

s s Fnj ðzÞ ¼ zc ð1  zÞb fnj ðzÞ;

ð65Þ

Eq. (56) takes the form d2 d fzð1  zÞ 2 þ ð2cs  ð1 þ 2cs þ 2bs ÞzÞ  ððcs Þ2 dz dz s þ ðbs Þ2 þ 2cs bs  ks3 Þgfnj ðzÞ ¼ 0;

s s ðEnj þ M  Cs ÞðM  Enj Þ 2 4d s ðEnj þ M  Cs ÞV1 þ 4d2 V02 V0 þ  2 4d2 s ðEnj þ M  Cs ÞV2 : þ 4d2

whose solution is the second-type hypergeometric function   s fnj ðzÞ ¼ 2 F1 gs1 ; gs2 ; gs3 ; z ; ð67Þ Eq. (65) has the form of a hypergeometric function and thus by comparison, we obtain qffiffiffiffiffi gs1 ¼ cs þ bs þ ks3 ; ð68Þ qffiffiffiffiffi ð69Þ gs2 ¼ cs þ bs  ks3 ; ð70Þ

and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

1 þ 1 þ 4ks2 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffi ik bs ¼  ; k ¼ 4d2 ks1 ; 2d cs ¼

ð71Þ ð72Þ

where ks1

s ðEs þ M  Cs ÞðM  Enj Þ ¼  nj ; 2 4d

ð64Þ

ð75Þ

The radial wavefunction is written as  cs   s Fnj ðrÞ ¼ Nnj eikr 1  e2dr 2 F1 gs1 ; gs2 ; gs3 ; 1  e2dr : ð76Þ

ð66Þ

gs3 ¼ 2cs ;

ð74Þ

In analogy with the previous section, we can write Cðgs3  gs1  gs2 Þ ðr!1Þ s Fnj ðrÞ ! Nnj Cðgs3 Þ Cðgs3  gs1 ÞCðgs3  gs2 Þ n o  eiðkrþdÞ þ eiðkrþdÞ  Cðgs3  gs1  gs2 Þ

s sin kr þ d þ p ; ¼ 2Nnj Cðg3 Þ Cðgs  gs ÞCðgs  gs Þ 2 3

1

3

2

ð77Þ From the boundary condition [39, 40] r ! 1 ) s Fnj ð1Þ ! 2sinðkr  jp phase-shifts and the 2 þ dj Þ, normalization constant for pseudospin symmetry limit are determined as   p p ik dj ¼ ðj þ 1Þ þ d ¼ ðj þ 1Þ þ arg C 2 2 d     ffiffiffiffi ffi q ik ik qffiffiffiffisffi s s s  arg C c þ  k3  arg C c þ þ k3 ; 2d 2d ð78Þ

ð73Þ

and

S Ortakaya et al.

Nnj

 pffiffiffiffiffi  pffiffiffiffiffi ik ik  ks3 C cs þ 2d þ ks3 1 C cs þ 2d   ¼ : Cð2cs Þ C ikd ð79Þ

In Eq. (30), one can see that Fps nj(r)is accetpable for Eps nj = M ? Cps, which is valid only for negative energy solutions. From Eq. (39), we see that Gsnj(r) is physically motivated for Es = Cs - M, which is only valid for positive energy solutions. Thus, the energy spectrum obtained in the pseudospin symmetry limit is negative and the energy spectrum obtained in the spin symmetry limit is positive [41, 42].

6. Conclusions In this paper, we have obtained the approximate solutions of the Dirac equation for the Manning-Rosen potential including a tensor Yukawa interaction within the framework of pseudospin and spin symmetry limits. We have obtained the energy eigenvalues and corresponding lower and upper wavefunctions in terms of the hypergeometric function. We have observed the Yukawa tensor interaction affects the degeneracy of the pseudospin and spin doublets.

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