By using the Pekeris approximation, we present solutions of the Dirac equation with the generalized Woods-Saxon potential with arbitrary spin-orbit coupling number κ 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 p-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, p-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 VS≈VV and pseudospin symmetry occurs when VS≈-VV, [11, 12], where S is scalar potential and VV is vector potential. Pseudospin symmetry is exact under the condition of d(V(r)+S(r))/dr=0, and spin symmetry is exact under the condition of d(V(r)-S(r))/dr=0 [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 (n,l,j=l+1/2) and (n-1,l+2,j=l+3/2), where n, l, and j are single nucleon radial, orbital, and total angular quantum numbers, respectively. The total angular momentum is j=l~+s~, where l~=l+1 is pseudo-angular momentum and s~ 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]:
(1)V(r)=-V01+e(r-R0)/a-Ce(r-R0)/a(1+e(r-R0)/a)2,
where V0 and C determine the potential depth, R0 is the width of the potential, a is the surface thickness, and 0<C<150 MeV [42]. In Figure 1, we illustrated the shape of the Woods-Saxon potential (C=0) and its generalized form in (1) (C≠0).
Shape of the generalized Woods-Saxon potential.
The purpose of this work is devoted to studying the spin symmetry solutions of the Dirac equation for arbitrary spin-orbit coupling quantum number κ 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 κ 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 S(r) and vector potential V(r) is
(2)[cα→·p→+β(Mc2+S(r))]ψ(r)=[E-V(r)]ψ(r),
where E is the relativistic energy of the system and p→=-i∇→ is the three-dimensional momentum operator. α→ and β are the 4×4 usual Dirac matrices given as
(3)α→=(0σ→σ→0),β=(I00-I),
where σ→ is Pauli matrix and I is 2×2 unitary matrix. The total angular momentum operator J→ and spin-orbit operator K=(σ→·L→+1), where L→ is orbital angular momentum operator and commutes with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are κ=(j+1/2)>0 and κ=-(j+1/2)<0 for unaligned spin j=l-1/2 and the aligned spin j=l+1/2, respectively. (H2,K,J2,Jz) can be taken as a complete set of the conservative quantities. Thus, the Dirac spinors can be written according to radial quantum number n and spin-orbit coupling number κ as follows:
(4)ψnκ(r→)=(fnκ(r→)gnκ(r→))=(Fnκ(r)rYjml(θ,φ)iGnκ(r)rYjml~(θ,φ)),
where fnκ(r→) is the upper (large) component and gnκ(r→) is the lower (small) component of the Dirac spinors. Yjml(θ,φ) and Yjml~(θ,φ) are the spherical harmonic functions coupled to the total angular momentum j and its projection m on the z 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 Fnκ(r) and Gnκ(r) as(5a)(ddr+κr)Fnκ(r)=1ℏc[Mc2+Enκ-Δ(r)]Gnκ(r),(5b)(ddr-κr)Gnκ(r)=1ℏc[Mc2-Enκ+Σ(r)]Fnκ(r),
where(6a)Δ(r)=V(r)-S(r),(6b)Σ(r)=V(r)+S(r).
Eliminating Fnκ(r) and Gnκ(r) 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:(7a)[d2dr2-κ(κ-1)r2]Gnκ(r)+[-1ℏ2c2(Mc2+Enκ-Δ(r))(M-Enκ+Σ(r))-1ℏ2c2(dΣ(r)/dr)(d/dr-κ/r)Mc2-Enκ+Σ(r)]Gnκ(r)=0,(7b)[d2dr2-κ(κ+1)r2]Fnκ(r)+[-1ℏ2c2(Mc2+Enκ-Δ(r))(Mc2-Enκ+Σ(r))+1ℏ2c2(dΔ(r)/dr)(d/dr+κ/r)Mc2+Enκ-Δ(r)]Fnκ(r)=0,
where κ(κ-1)=l~(l~+1) and κ(κ+1)=l(l+1).
2.1. Spin Symmetry Limit
In the spin symmetry limit dΔ(r)/dr=0 or Δ(r)=Cs= constant [14], then (7b) becomes
(8)[d2dr2-κ(κ+1)r2-1ℏ2c2(Mc2+Enκ-Cs)××(Mc2-Enκ+Σ(r))d2dr2]Fnκ(r)=0,
where κ=l and κ=-l-1 for κ<0 and κ>0, respectively. In (8), Σ(r) can be taken as generalized Woods-Saxon potential, and it is reduced to
(9)[d2dr2-κ(κ+1)r2+η(V01+e(r-R0)/a+Ce(r-R0)/a(1+e(r-R0)/a)2)-ξ2]Fnκ(r)=0,
where(10a)η=1ℏ2c2(Enκ+Mc2-Cs),(10b)ξ2=1ℏ2c2(Mc2-Enκ)(Mc2+Enκ-Cs).
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, κ(κ+1)/r2, (9) cannot be solved analytically, except for κ=0,-1. 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:
(11)Vso(r)=κ(κ+1)r2=κ(κ+1)(1+x/R0)2≅κ(κ+1)R02(1-2xR0+3(xR0)2+⋯).
In addition, we may also approximately express it in the following way:
(12)V~so(r)=κ(κ+1)r2≅κ(κ+1)R02(D0+D11+eνx+D2(1+eνx)2),
where x=r-R0, ν=1/a, and Di is 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), we can obtain expansion coefficients D0, D1, and D2 as follows:
(13)D0=1-4νR0-12ν2R02,D1=8νR0-48ν2R02,D2=48ν2R02.
Now, we can take the potential V~so (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]:
(14)(d2dz2+α1-α2zz(1-α3z)ddz+1z2(1-α3z)2×[-p2z2+p1z-p0]d2dz2)ψ(z)=0.
According to the Nikiforov-Uvarov method, the eigenfunctions and eigenenergies, respectively, are
(15)ψ(z)=zα12(1-α3z)-α12-α13/α3Pn(α10-1,α11/α3-α10-1)(1-2α3z),(16)α2n-(2n+1)α5+(2n+1)(α9-α3α8)+n(n-1)α3+α7+2α3α8-2α8α9=0,
where
(17)α4=12(1-α1),α5=12(α2-2α3),α6=α52+p2,α7=2α4α5-p1,α9=α3α7+α32α8+α6,α8=α42+p0,α10=α1+2α4+2α8,α11=α2-2α5+2(α9+α3α8),α12=α4+α8,α13=α5-(α9+α3α8).
In the rather more special case of α3=0 [50, 51], (18a)limα3→0Pn(α10-1,α11/α3-α10-1)(1-2α3z)=Lnα10-1(α11z),(18b)limα3→0(1-α3z)-α12-α13/α3=eα13z,
and, from (15), we find for the wave function
(19)ψ=zα12eα13zLnα10-1(α11z).
4. Bound States of the Generalized Woods-Saxon Potential with Arbitrary <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M97"><mml:mrow><mml:mi>κ</mml:mi></mml:mrow></mml:math></inline-formula>
Substituting (12) into (9) and using transformation z=eνx, we find the following second order deferential equation for the upper component of the Dirac spinor as
(20){d2dz2+1+zz(1+z)ddz-κ(κ+1)ν2z2R02(D0+D1(1+z)+D2(1+z)2)+1ν2z2(ηV0(1+z)+ηCz(1+z)2-ξ2)}F(z)=0.
Comparing the previous equation with (14), one can find the following parameters:
(21)α1=1,α2=-1,α3=-1,p2=(κ(κ+1)ν2R02D0+ξ2ν2),p1=-κ(κ+1)ν2R02(2D0+D1)+1ν2(ηV0+ηC-2β2),p0=κ(κ+1)ν2R02(D0+D1+D2)-1ν2(ηV0-ξ2),
and also
(22)α4=0,α5=12,α6=14+p2,α7=-p1,α8=p0,α9=p2-p1+p0+14,α10=1+2p0,α11=-2+2(p2-p1+p0+1/4-p0),α12=p0,α13=12-(p2-p1+p0+1/4-p0).
From (16), (21), and (22), we obtained the closed form of the energy eigenvalues for the generalized Woods-Saxon potential in spin symmetry as
(23)(n+12+(κ(κ+1)/ν2R02)D2-ηC/ν2+1/4+(κ(κ+1)/ν2R02)(D0+D1+D2)-(1/ν2)(ηV0-ξ2))2=(κ(κ+1)/ν2R02)D0+ξ2/ν2.
Recalling η and ξ from (10a) and (10b) the previous equation becomes a quadratic algebraic equation in Enκ. Thus, the solution of this algebraic equation with respect to Enκ can be obtained in terms of particular values of n and κ. In Table 1, we have calculated some energy levels of the generalized Woods-Saxon potential under the spin symmetry limit for different values of C. The empirical values that can be found in [52], as r0=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 target nucleus, and R0=r0A1/3. From Table 1, we can see that pairs (np1/2,np3/2),(nd3/2,nd5/2), (nf5/2,nf7/2), (ng7/2,ng9/2), and so forth are degenerate. Thus, each pair is considered as spin doublet 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 the energy levels decrease. In Figure 2, the results are shown as a function of V0. 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 R0. Here, when the width of the potential increases, the energy levels decrease. Finally we showed the numerical results as a function of surface thickness a in Figure 4, and one can observe that when the surface thickness increases, the energy levels decrease too.
The approximate bound state energy eigenvalues (in unit of MeV) of the generalized Woods-Saxon potential within the spin symmetry limit for several values of n and κ.
l
n,κ<0,κ>0
State
EnκC=0
EnκC=50 MeV
EnκC=100 MeV
EnκC=150 MeV
1
0, −2, 1
0p3/2,0p1/2
−939.0045195
−939.0074037
−939.0102535
−939.0130695
2
0, −3, 2
0d5/2,0d3/2
−938.5761054
−938.5822985
−938.5884052
−938.5944276
3
0, −4, 3
0f7/2,0f5/2
−938.4033046
−938.4111487
−938.4188786
−938.4264972
4
0, −5, 4
0g9/2,0g7/2
−938.532527
−938.5397300
−938.5468274
−938.5538221
1
1, −2, 1
1p3/2,1p1/2
−906.9952581
−907.4067290
−907.8063989
−908.1948041
2
1, −3, 2
1d5/2,1d3/2
−905.7621398
−906.1743738
−906.5749872
−906.9645019
3
1, −4, 3
1f7/2,1f5/2
−904.2904841
−904.7048150
−905.1077390
−905.4997600
4
1, −5, 4
1g9/2,1g7/2
−902.7908284
−903.2078994
−903.6137936
−904.0089945
1
2, −2, 1
2p3/2,2p1/2
−835.7682302
−837.2498452
−838.6826152
−840.0691370
2
2, −3, 2
2d5/2,2d3/2
−834.4172283
−835.8982294
−837.3306378
−838.7170294
3
2, −4, 3
2f7/2,2f5/2
−832.5628129
−834.0439814
−835.4769039
−836.8641261
4
2, −5, 4
2g9/2,2g7/2
−830.3570797
−831.8396528
−833.2743934
−834.6638100
Computed energies as a function of V0.
Computed energies as a function of R0.
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 Dirac spinor as
(24)Fnκ(z)=zα12(1-α3z)-α12-α13/α3Pn(α10-1,α11/α3-α10-1)×(1-2α3z)=zξ3(1+z)1/2+(κ(κ+1)/ν2R02)D2-(ηC/ν2)+1/4×Pn(2p0,-2(κ(κ+1)/ν2R02)D2-ηC/ν2+1/4)(1+2z)
or equivalently
(25)Fnκ(r)=ep0((r-R0)/a)(1+e(r-R0)/a)1/2+(κ(κ+1)/ν2R02)D2-ηC/ν2+1/4×Pn(2p0,-2(κ(κ+1)/ν2R02)D2-ηC/ν2+1/4)(1+2e(r-R0)/a).
Finally, the lower component of the Dirac spinor can be calculated as
(26)Gnκ(r)=ℏcMc2+Enκ-Cs(ddr+κr)Fnκ(r),
where Enκ≠-Mc2+Cs [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 (Cs=0). 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.
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