Pseudospin and spin symmetry in the relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential
aa r X i v : . [ nu c l - t h ] J a n Pseudospin and spin symmetry in the relativisticgeneralized Woods-Saxon potential includingCoulomb-like tensor potential
J. Akbar, A. Suparmi ∗ , C. Cari Department of Physics, Sebelas Maret University, Surakarta, Indonesia
Abstract
The Dirac Equation is solved approximately for relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential in exact pseudospin andspin symmetry limits. The bound states energy eigenvalues are found by usingwavefunction boundary conditions, and corresponding radial wavefunctions areobtained in terms of hypergeometric function. Some numerical examples aregiven for the dependence of bound states energy eigenvalues on quantum num-bers and potential parameters.
Keywords:
Pseudospin symmetry; Spin symmetry; Dirac Equation;Woods-Saxon potential; Tensor potential
1. Introduction
The pseudospin symmetry concept was introduced about 40 years ago throughthe observation of quasi-degeneracy in some nuclei between single nucleon statewith quantum numbers ( n, l, j = l + 1 /
2) and ( n − , l + 2 , j = l + 3 / n , l and j are the radial, the orbital, and the total angular momentum quan-tum numbers, respectively. These two states are known as pseudospin dou-blets and have the same ”pseudo” orbital angular momentum quantum num-ber, ˜ l = l + 1, and pseudospin quantum number ˜ s = 1 /
2. The exact pseudospin ∗ Corresponding author
Email address: [email protected] (A. Suparmi)
Preprint submitted to Elsevier January 5, 2021 ymmetry occurs when doublets with j = ˜ l ± ˜ s are degenerate [1, 2]. Thisdoublet structure has been successfully explained the nuclear physics phenom-ena, including deformation, superdeformation, identical bands, and magneticmoments [3–8]. Meanwhile, the spin symmetry concept is used to explain theabsence of spin-orbital splitting between a single nucleon state with oppositespin ( n, l, j = l ± / Σ ( r ) = S ( r ) + V ( r ) = constant, and spin symmetryoccurs when ∆ ( r ) = S ( r ) − V ( r ) = constant, where S ( r ) is attractive scalarpotential and V ( r ) is repulsive vector potential. The pseudospin and spin sym-metry are exact under conditions d Σ ( r ) /dr = 0 and d ∆ ( r ) /dr = 0, respectively[11–13]. Since that time, the Dirac equation’s analytical solutions have beeninvestigated by many papers for solvable potentials, such as Woods-Saxon po-tential [14–18], Pseudoharmonic potential [19], Mie-Type potential [20], Morsepotential [21], Eckart potential [22], and Asymmetrical Hartmann potential [23],by different methods.Tensor coupling under the conditions pseudospin and spin symmetry hasbeen studied in references [24–27]. They have found that tensor interaction canremove the degeneracy between two states in pseudospin and spin doublets. Onthe other hand, the tensor coupling has been applied to the nucleon in effec-tive chiral lagrangians for nuclei [28] and describing the bound states of bothnucleons and antinucleons in the relativistic Hartree approach [29]. Thus, itis very convenient to use nuclear potentials like the generalized Woods-Saxonpotential and tensor potential for investigating the energy eigenvalues of nucleiand their corresponding eigenfunction. The generalized Woods-Saxon poten-tial consists of Woods-Saxon plus Woods-Saxon surface potential. This surfaceterm might be a useful model for describing the surface interactions betweennucleons. The solution of s -wave Dirac equation for Woods-Saxon potential has2een solved by Guo et al. [30] and is discussed in [31, 32], they found that thecondition of existing bound state energy only exist for V + S < C ps < r = R [14]. In the last three references, they carefullyexamine the asymtotic behavior of the wave function in vicinity r = R usingsome relations of the hypergeometric function. Our purpose here is to go fur-ther and investigate the solution of Dirac equation for generalized Woods-Saxonpotential including Coulomb-like tensor potential by using this wave functionboundary conditions.The present work is organized as follows: In Section 2, we give theoreticalformalism for the Dirac equation. In Section 3 and 4, we solved and analyzedthe Dirac equation’s solution for generalized Woods-Saxon potential includingCoulomb-like tensor potential in the limit of exact spin and pseudospin symme-try. The conclusions are summarized in Section 5.
2. Dirac Equation
The general form of Dirac equation for fermionic massive spin − / V ( ~r ) and an attractive scalar potential S ( ~r ) can be written as ( c = ~ = 1) (cid:0) ~α · ~p + β ( M + S ) + V − iβ~α · ˆ rU ( r ) (cid:1) ψ = E ψ. (1)For spherical nuclei, the total angular momentum operator ~J and the spin-orbitcoupling operator ~ K commute with the Dirac Hamiltonian, where β, ~σ and ~L are3he Dirac matrix, Pauli matrix, and orbital angular momentum, respectively.The eigenvalues of the spin-orbit coupling operator are κ = l > κ = − ( l + 1) < j = l − /
2) and aligned spin ( j = l + 1 / j and κ , the wavefunctions canbe written in the following form: ψ nκ ( ~r ) = f nκ g nκ = 1 r F nκ ( r ) Y ljm ( θ, φ ) iG nκ ( r ) Y ˜ ljm ( θ, φ ) , (2)where n is the radial quantum number, and m is the projection of angularmomentum on the third axis. The angular part can be splitting off and leavingthe upper and lower radial wavefunctions F nκ ( r ) and G nκ ( r ) as (cid:18) ddr + κr − U ( r ) (cid:19) F nκ ( r ) = (cid:0) M + E − ∆ ( r ) (cid:1) G nκ ( r ) , (3) (cid:18) ddr − κr + U ( r ) (cid:19) G nκ ( r ) = (cid:0) M − E + Σ ( r ) (cid:1) F nκ ( r ) , (4)where ∆ ( r ) = V ( r ) − S ( r ) and Σ ( r ) = V ( r )+ S ( r ). Then, by eliminating G nκ ( r )in Eq.(3) and F nκ ( r ) in Eq.(4), one can obtain Schr¨odinger-like equations forthe upper and the lower components of the radial wavefunctions " d dr − κ ( κ + 1) r + (cid:18) κr − U ( r ) − ddr (cid:19) U ( r )+ d ∆ ( r ) dr (cid:0) ddr + κr − U ( r ) (cid:1) M + E − ∆ ( r ) F nκ ( r )+ (cid:0) E + M − ∆ ( r ) (cid:1)(cid:0) E − M − Σ ( r ) (cid:1) F nκ ( r ) = 0 , (5) " d dr − κ ( κ − r + (cid:18) κr − U ( r ) + ddr (cid:19) U ( r ) − d Σ ( r ) dr (cid:0) ddr − κr + U ( r ) (cid:1) M − E + Σ ( r ) G nκ ( r )+ (cid:0) E + M − ∆ ( r ) (cid:1)(cid:0) E − M − Σ ( r ) (cid:1) G nκ ( r ) = 0 , (6)The solutions of Eq.(5) can be obtained with κ ( κ + 1) = l ( l + 1) and its eigenval-ues E nκ = E ( n, l ( l + 1)) depend only on n and l . In the exact of spin symmetry,the states with j = l ± / l = 0. Meanwhile, the solutionsof Eq.(6) can also be obtained with κ ( κ −
1) = ˜ l (˜ l + 1) and its eigenvalues4 nκ = E ( n, ˜ l (˜ l + 1)) depend only on n and ˜ l . By using condition in the exact ofpseudospin symmetry, the states with j = ˜ l ± / l = 0.
3. The pseudospin symmetry
In the case of exact pseudospin symmetry ( d Σ ( r ) /dr = 0 , i.e., Σ ( r ) = C ps =const), we can assume ∆ ( r ) and U ( r ) are potential of the generalized Woods-Saxon type and potential of the Coulomb-like tensor, respectively, defined asfollow Σ ( r ) = C ps , ∆ ( r ) = − ∆ e r − Ra − ∆ e r − Ra (cid:16) e r − Ra (cid:17) , U ( r ) = − Hr , (7)where ∆ and ∆ represent the depth of the potential well. The parameters a , R , and H are the diffusivity of nuclear surface, the width of the potential well,and constants, respectively. On the other hand, Eq.(6) becomes d dr − ( κ + H )( κ + H − r + ( E − M − C ps ) E + M + ∆ e r − Ra + ∆ e r − Ra (cid:16) e r − Ra (cid:17) G nκ ( r ) = 0 . (8)There is no analytical solution of Eq.(8) with this potential model for ( κ + H )( κ + H − = 0 because the pseudospin-orbit coupling term ( κ + H )( κ + H − /r . Therefore, we shall use the Pekeris approximation [34] by introducingthe notations x = r − RR , ν = Ra , (9)and expanding the centrifugal potential in a series around x = 0 ( r ≈ R ) as V so ( x ) = ( κ + H )( κ + H − R (1 + x ) − = ˜ γ (cid:0) − x + 3 x + · · · (cid:1) . (10)where ˜ γ = ( κ + H )( κ + H − /R . According to the Pekeris approximation, weshall replace the potential V so ( x ) with expression V ∗ so = ˜ γ D + D e νx + D (1 + e νx ) ! , (11)5here D , D , and D are constants. We also expand this new form of centrifu-gal potential V ∗ ( x ) in a series around the point x = 0 ( r ≈ R ) V ∗ so = ˜ γ (cid:20)(cid:18) D + D D (cid:19) − ν D + D ) x + D ν x + · · · (cid:21) . (12)Comparing the equal powers of Eqs.(10) and (12), we get D = 1 − ν + 12 ν , D = 8 ν − ν , D = 48 ν , (13)Now, we introduce a new variable z = (cid:16) e r − Ra (cid:17) − to derive the solution ofEq.(8). Then Eq.(8) becomes (cid:20) d dz + (1 − z ) z (1 − z ) ddz + 1 z (1 − z ) (cid:0) − (˜ γD + ˜ µ ) a z + ( ˜ ̟ + ˜ µ − ˜ γD ) a z − (cid:16) ˜ γD − ˜Λ (cid:17) a (cid:1)(cid:21) G nκ ( z ) = 0 . (14)where˜ ̟ = ( E − M − C ps ) ∆ , (15) ˜Λ = ( E + M ) ( E − M − C ps ) , (16)˜ µ = ( E − M − C ps ) ∆ , (17)Eq.(14) can be reduced to a hypergeometric equation by setting the followingfactorization G nκ ( z ) = z ˜ σ (1 − z ) ˜ τ w ( z ) (18)with˜ σ = r(cid:16) ˜ γD − ˜Λ (cid:17) a , (19) ˜ χ = q(cid:0) ˜ ̟ − ˜ γ ( D + D ) (cid:1) a , (20)˜ τ = p ˜ σ − ˜ χ , (21)Then Eq.(14) becomes z (1 − z ) w ′′ ( z ) + h ˜ c − (cid:16) ˜ a + ˜ b + 1 (cid:17) z i w ′ ( z ) − ˜ a ˜ bw ( z ) = 0 , (22)where˜ η = 12 p γD + ˜ µ ) a , (23)6 a = ˜ σ + ˜ τ + 12 − ˜ η, (24)˜ b = ˜ σ + ˜ τ + 12 + ˜ η, (25)˜ c = 2˜ σ + 1 , (26)The complete solution to Eq.(22) is w ( z ) = C F (cid:18) ˜ σ + ˜ τ − ˜ η + 12 , ˜ σ + ˜ τ + ˜ η + 12 , σ + 1; z (cid:19) + C z − µ F (cid:18) − ˜ σ + ˜ τ − ˜ η + 12 , − ˜ σ + ˜ τ + ˜ η + 12 , − σ + 1; z (cid:19) . (27)For bound states E < M , and by using boundary condition z → r →∞ ) , G nκ ( z ) →
0, the allowed solution is G nκ ( z ) = z ˜ σ (1 − z ) ˜ τ F (cid:18) ˜ σ + ˜ τ − ˜ η + 12 , ˜ σ + ˜ τ + ˜ η + 12 , σ + 1; z (cid:19) . (28)Considering another boundary condition near the origin z → r →
0) andapplied to Eq.(28), yields G nκ ( z ) ∼ (1 − z ) ˜ τ Γ(2˜ σ + 1)Γ( − τ )Γ (cid:0) ˜ σ − ˜ τ + ˜ η + (cid:1) Γ (cid:0) ˜ σ − ˜ τ − ˜ η + (cid:1) + (1 − z ) − ˜ τ Γ(2˜ σ + 1)Γ(2˜ τ )Γ (cid:0) ˜ σ + ˜ τ − ˜ η + (cid:1) Γ (cid:0) ˜ σ + ˜ τ + ˜ η + (cid:1) . (29)By applying the boundary condition G nκ (1) = 0 in the neighborhood of r = 0and using an approximation 1 − z = e − R/a for realistic nuclei, we getΓ(2˜ τ )Γ (˜ σ + ˜ η ′ + 1 − ˜ τ ) Γ (˜ σ − ˜ η ′ − ˜ τ )Γ( − τ )Γ (˜ σ + ˜ η ′ + 1 + ˜ τ ) Γ (˜ σ − ˜ η ′ + ˜ τ ) e τRa = − , (30)where ˜ η ′ = ˜ η − /
2. We note that ˜ σ − ˜ χ <
0, the parameter ˜ τ becomesimaginary˜ τ = iλ, λ = p ˜ χ − ˜ σ , (31)Therefore, from Eq.(30), we can get the energy eigenvalues equation in the caseof exact pseudospin symmetry as follow˜ ξ − ˜ ζ − ˜ ω + λRa = (cid:18) n + 12 (cid:19) π ; n = 0 , ± , ± , ± , . . . (32)where˜ ξ = arg Γ(2 iλ ) , ˜ ζ = arg Γ (˜ σ + ˜ η ′ + 1 + iλ ) , ˜ ω = arg Γ (˜ σ − ˜ η ′ + iλ ) , (33)7nd corresponding the lower spinor component G nκ ( r ) = ˜ N (cid:16) e r − Ra (cid:17) − (˜ σ + iλ ) e iλ ( r − R ) a F (cid:18) ˜ a, ˜ b, ˜ c ; 11 + e r − Ra (cid:19) . (34)We also obtain the upper spinor component by using Eq.(4) as follow F nκ ( r ) = ˜ NM − E + C ps (cid:16) e r − Ra (cid:17) − (˜ σ + iλ ) e iλ ( r − R ) a × − κ + Hr + iλa − ˜ σ + iλa (cid:16) e − ( r − R ) a (cid:17) F (cid:18) ˜ a, ˜ b, ˜ c ; 11 + e r − Ra (cid:19) + ˜ a ˜ b ˜ c F (cid:18) ˜ a + 1 , ˜ b + 1 , ˜ c + 1; 11 + e r − Ra (cid:19) . (35)where F nκ is admissible for E 6 = M + C ps and only for bound negative energystates solutions [35]. Moreover, we can reduce the energy eigenvalues in Eq.(32)into Schr¨odinger equation by using these transfomations E + M − C s → µ ′ , ( κ + H )( κ + H + 1) → l ′ ( l ′ + 1) , E − M → E NR and if we take ∆ = 0, then Eq.(32)becomesarg Γ (cid:16) ia p µ ′ ( E NR + ∆ ) − l ′ ( l ′ + 1)( D + D + D ) /R (cid:17) − arg Γ (cid:16) a p l ′ ( l ′ + 1) D /R − µ ′ E NR + p l ′ ( l ′ + 1) D a /R + 1 /
4+ 1 / ia p µ ′ ( E NR + ∆ ) − l ′ ( l ′ + 1)( D + D + D ) /R (cid:17) − arg Γ (cid:16) a p l ′ ( l ′ + 1) D /R − µ ′ E NR − p l ′ ( l ′ + 1) D a /R + 1 /
4+ 1 / ia p µ ′ ( E NR + ∆ ) − l ′ ( l ′ + 1)( D + D + D ) /R (cid:17) + p µ ′ ( E NR + ∆ ) R − l ′ ( l ′ + 1)( D + D + D ) = (2 n ′ − π . (36)where E NR , µ ′ , l ′ , and n ′ are the non-relativistic energy, reduced mass, theazimuthal quantum number, and the principal quantum number, respectively.The result in Eq.(36) consistent with expression (38) of ref. [33] if we take thespin orbit parameter V (0) LS = 0. 8able 1. The bound states energy eigenvalues in the unit of MeV in the case of exact pseudospin symmetry˜ l n, κ < l, j ) ˜ E n,κ< H =0) ˜ E n,κ< H = − . n − , κ > l + 2 , j + 1) ˜ E n − ,κ> H =0) ˜ E n − ,κ> H = − . − − − − , p / − . − . − − − − , p / − . − . − − − − , p / − . − . , − s / − . − . , d / − . − . , − s / − . − . , d / − . − . , − s / − . − . , d / − . − . , − p / − . − . , f / − . − . , − p / − . − . , f / − . − . , − p / − . − . , f / − . − . , − d / − . − . , g / − . − . , − d / − . − . , g / − . − . , − d / − . − . , g / − . − . M = 939 . c , a = 0 .
65 fm, R = 7 fm, ∆ = 750 MeV, ∆ = 50 MeV, C ps = − ~ c = 197 . E = E − M .From Table 1, one can observe that the energies of all nucleon states increasewith increases in quantum numbers n or ˜ l . Furthermore, pseudospin doubletshave energy degeneracy in the absence of the tensor potential ( H = 0), but in thecase H = 0, the energy degeneracy between pseudospin doublets is removed.Therefore, we can say that the tensor interactions remove energy degeneracyfor pseudospin doublets, and this present result agrees with the previous one[16, 24–27]. On the other hand, we can see from Table 1. that the tensor in-teraction ( H = 0) make energies of aligned and unaligned pseudospin move inopposite direction. This can be explained by considering energy eigenvalue inEq.(32); the energy eigenvalues depend on pseudospin dependent term 2 κH in9 γ . The factor κ takes negative and positive values depending on the pseudospinalignment. Then, we also observe that the parameter of tensor potential for H < (cid:16) ∆ ˜ E = ˜ E ˜ lj =˜ l − / − ˜ E ˜ lj =˜ l +1 / (cid:17) . Ẽ ( M e V ) p f p f H Figure 1. Pseudospin energy eigenvalues as a function of H for the pseudospindoublets (2 p / , f / ) and (3 p / , f / )We investigate the behavior of pseudospin energy eigenvalues in the pres-ence of tensor potential in Figure 1. for pseudospin doublets (2 p / , f /
2) and(3 p / , f / ). From Figure 1, the energy splittings of pseudospin doublets for(3 p / , f / ) have a greater value than (2 p / , f / ) in the case of H = 0.Furthermore, this energy splitting of pseudospin doublets increases while H de-creases. It can be understood again by the dependence of energy eigenvaluesin Eq.(32) with pseudospin dependent term 2 κH in ˜ γ . So even though the en-ergies of pseudospin aligned states decrease as H increases, the energies of thepseudospin unaligned states can still increase with increasing H .10 (MeV) Ẽ ( M e V ) p f p f Figure 2. Pseudospin energy eigenvalues as a function of ∆ for the pseudospindoublets (2 p / , f / ) and (3 p / , f / ) Δ (MeV) Ẽ ( M e V ) p f p f Figure 3. Pseudospin energy eigenvalues as a function of ∆ for the pseudospindoublets (2 p / , f / ) and (3 p / , f / )11he sensitivity of the pseudospin energy eigenvalues to the width of potentialwell ∆ and ∆ for H = − . ∆ or ∆ without thepresence of tensor potential, there is no energy splitting that occurs betweenpseudospin doublets. From Figure 2, we observe that both of the pseudospinenergy splitting and eigenvalues for all of the pseudospin doublets have a slightdecrease when ∆ increases. Meanwhile, in Figure 3, the energy eigenvalues for1 f / and 2 f / states have a slight decrease to the vary of ∆ than 2 p / and3 p / states, but the energy splitting for all of the pseudospin doublets have aslight increase as ∆ increases. Therefore, from Figures 2 and 3, both of thepseudospin energy splitting and eigenvalues in the presence of tensor potentialare insensitive to the changes of ∆ or ∆ .
4. The spin symmetry
Now, we consider in the case of exact spin symmetry ( d ∆ ( r ) /dr = 0 , i.e., ∆ ( r ) = C s = const). Assuming Σ ( r ) and U ( r ) are potential of the general-ized Woods-Saxon type and potential of the Coulomb-like tensor, respectively,defined as follow ∆ ( r ) = C s , Σ ( r ) = − Σ e r − Ra − Σ e r − Ra (cid:16) e r − Ra (cid:17) , U ( r ) = − Hr , (37)where Σ and Σ represent the depth of the potential well. The parameters a , R , and H are the diffusivity of nuclear surface, the width of the potential well,and constants, respectively. Then, Eq.(5) is turned into the following form d dr − ( κ + H )( κ + H + 1) r + ( E + M − C s ) E − M + Σ e r − Ra + Σ e r − Ra (cid:16) e r − Ra (cid:17) F nκ ( r ) = 0 . (38)This Equation has the same form as the Eq.(8), so we applied the Pekerisapproximation using the same notations in Eq.(9) and introduce similar variable12 = (cid:16) e r − Ra (cid:17) − like in the previous sections, we obtain (cid:20) d dz + (1 − z ) z (1 − z ) ddz + 1 z (1 − z ) (cid:0) − ( γD + µ ) a z + ( ̟ + µ − γD ) a z − ( γD − Λ ) a (cid:1)(cid:21) F nκ ( z ) = 0 . (39)where γ = ( κ + H )( κ + H + 1) R , (40) ̟ = ( E + M − C s ) Σ , (41) Λ = ( E − M )( E + M − C s ) , (42) µ = ( E + M − C s ) Σ , (43)Eq.(39) have the same form as Eq.(14). Thus, we applied the same procedure asthat used in the previous section. So, the corresponding upper spinor componentis F nκ = N (cid:16) e r − Ra (cid:17) − ( σ + iδ ) e iδ ( r − R ) a F (cid:18) a ′ , b ′ , c ′ ; 11 + e r − Ra (cid:19) , (44)with σ = p ( γD − Λ ) a , (45) χ = q(cid:0) ̟ − γ ( D + D ) (cid:1) a , (46) τ = iδ, δ = p χ − σ , (47)and η = 12 p γD + µ ) a , (48) a ′ = σ + τ + 12 − η, (49) b ′ = σ + τ + 12 + η, (50) c ′ = 2 σ + 1 , (51)13ith Eq.(3), the lower spinor component is obtained as G nκ ( r ) = NM + E − C s (cid:16) e r − Ra (cid:17) − ( σ + iδ ) e iδ ( r − R ) a × κ + Hr + iδa − σ + iδa (cid:16) e − ( r − R ) a (cid:17) F (cid:18) a ′ , b ′ , c ′ ; 11 + e r − Ra (cid:19) + a ′ b ′ c ′ F (cid:18) a ′ + 1 , b ′ + 1 , c ′ + 1; 11 + e r − Ra (cid:19) . (52)where G nκ is admissible for E 6 = − M + C s and only for bound positive energystates solutions [35].The energy eigenvalues equation in the case of exact spin symmetry can alsobe obtained as follow ξ − ζ − ω + δRa = (cid:18) n + 12 (cid:19) π ; n = 0 , ± , ± , ± , . . . (53)where ξ = arg Γ(2 iδ ) , ζ = arg Γ ( σ + η ′ + 1 + iδ ) , ω = arg Γ ( σ − η ′ + iδ ) , (54)if we take l = H = Σ = 0 in Eq.(53), then Eq.(53) becomesarg Γ (cid:18) ia q ( M + E n, − − C s )( E n, − − M + Σ ) (cid:19) − (cid:18) a q ( M − E n, − )( M + E n, − − C s )+ ia q ( M + E n, − − C s )( E n, − − M + Σ ) (cid:19) − tan − q ( E n, − − M + Σ ) / ( M − E n, − )+ R q ( M + E n, − − C s )( E n, − − M + Σ ) = (cid:18) n + 12 (cid:19) π. (55)The result in Eq.(55) consistent with expression (19) of ref. [30]. Moreover, wecan reduce Eq.(53) into Schr¨odinger equation by using the same procedure asthat used to get Eq.(36), then the transformation of Eq.(53) will be consistentwith expression (38) of ref. [33] if we take the spin orbit parameter V (0) LS = 0.14able 2. The bound states energy eigenvalues in the unit of MeV in the case of exact spin symmetry l n, κ < l, j ) E n,κ< H =0) E n,κ< H = − . n, κ > l, j ) E n,κ> H =0) E n,κ> H = − . , − s / . . − − − − , − s / . . − − − − , − s / . . − − − − , − p / . . , p / . . , − p / . . , p / . . , − p / . . , p / . . , − d / . . , d / . . , − d / . . , d / . . , − d / . . , d / . . , − f / . . , f / . . , − f / . . , f / . . , − f / . . , f / . . M = 939 . c , a = 0 .
65 fm, R = 7 fm, Σ = −
750 MeV, Σ = −
50 MeV, C s = 50 MeV, and ~ c = 197 . E = E + M .From Table 2, one can observe that the energies of all nucleon states decreasewith increases in quantum numbers n or l . Also, one can see that spin doubletshave energy degeneracy in the absence of the tensor potential ( H = 0), but thisdegeneracy is removed in the case of H = 0. Our results show that the energydegeneracy for spin doublets is removed by tensor interaction. This presentresult is in good agreement with those obtained previously [16, 24–27]. Like inthe case of exact pseudospin symmetry, we also found that the energies of thealigned and unaligned spin move in the opposite direction in the presence of ten-sor potential. This happens because the energy eigenvalues in Eq.(53) dependson the term 2 κH through γ . The spin-dependent term 2 κH takes negative andpositives values depending on the values of κ , respectively. Then, we can see15hat the choice for H < E = E lj = l − / − E lj = l +1 / ). H E ( M e V ) d d f f Figure 4. Spin energy eigenvalues as a function of H for the spin doublets(1 d / , d / ) and (0 f / , f / )We investigate how the spin energy eigenvalues dependent on the parameters H , Σ and Σ in Figures 4, 5, and 6, respectively. We consider the spin dou-blets (1 d / , d / ) and (0 f / , f / ) as an example. From Figure 4, the energysplitting of spin doublets (0 f / , f / ) have a greater value than (1 d / , d / )in the case of H = 0. Like in the case of exact pseudospin symmetry, the energysplitting of spin doublets increases while H decreases. This is because of thedependence of energy eigenvalues in Eq.(53) on the spin-dependent term 2 κH in γ . 16 (MeV) E ( M e V ) d d f f Figure 5. Spin energy eigenvalues as a function of Σ for the spin doublets(1 d / , d / ) and (0 f / , f / ) Σ (MeV) E ( M e V ) d d f f Figure 6. Spin energy eigenvalues as a function of Σ for the spin doublets(1 d / , d / ) and (0 f / , f / ) 17n Figures 5 and 6, we vary the width of potential well Σ and Σ in the pres-ence of tensor potential for H = − .
5, respectively, to investigate the sensitivityof spin energy splitting and eigenvalues. We observe that there is no energysplitting occurs between spin doublets when we vary the width of the potentialwell Σ and Σ without the presence of tensor potential. From Figure 5, one canobserve that the spin energy splitting have a slight increase with increasing Σ ,while the energy eigenvalues have a slight decrease with increasing Σ . Then,in Figure 6, the energy eigenvalues for 1 d / state have a slight decrease to thevary of Σ than the other states. Like in the case of exact pseudospin symme-try, from Figures 5 and 6, we found that both of the spin energy splitting andeigenvalues in the presence of tensor potential are insensitive to the changes of Σ or Σ .
5. Conclusions
In this work, we have obtained approximate analytical solutions of the Diracequation for the generalized Woods-Saxon potential including Coulomb-like ten-sor potential in the case of exact pseudospin and spin symmetry. By carefullyexamining the asymptotic behavior of the wave function in neighborhood r = R ,the bound states energy eigenvalues and corresponding radial wavefunctionshave been obtained in the case of exact pseudospin and spin symmetry limits.We have found that the tensor interactions ( H = 0) remove energy degeneracyfor pseudospin (spin) doublets. These present results are in good agreementwith those obtained previously [16, 24–27]. We have also found that the ener-gies of aligned and unaligned pseudospin (spin) move in the opposite directiondue to the pseudospin (spin) dependent term in the energy eigenvalues equation.We have investigated the dependence of pseudospin (spin) energy eigenvaluesand splitting on the parameters H, ∆ ( Σ ) and ∆ ( Σ ). In both of pseudospin(spin) doublets, the energy splitting increases while H decreases. Furthermore,in the presence of tensor potential, the pseudospin (spin) energy splitting andeigenvalues are insensitive to the changes of ∆ ( Σ ) or ∆ ( Σ ).18 cknowledgments This research was partly supported by the Mandatory Research Grant ofSebelas Maret University with contract number 452/UN27.21/PN/2020.
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