The effect of tensor interaction in splitting the energy levels of relativistic systems

In this paper we solve analytically Dirac equation for Eckart plus Hulthen potentials with Coulomb-like and Yukawa-like tensor interaction in the presence of Spin and Pseudo-spin Symmetry for any k number. The Parametric Nikiforov-Uvarov method is used to obtain the energy Eigen-values and wave functions. We also discuss the energy Eigen-values and the Dirac spinors for the Eckart plus Hulthen potentials for the spin and pseudo-spin symmetry with PNU method. To show the accuracy of the present model, some numerical results are shown in both pseudo-spin and spin symmetry limits.

In this paper, we attempt to solve approximately Dirac wave equation for k≠0 for Eckart plus Hulthen potentials for the spin and pseudo-spin symmetry with a tensor potential by using the Formula method. The organization of this paper is as follows: in Section 2, the Formula method is reviewed [44]. In section 3 we review Basic Dirac Equations briefly. In section 4.1 and 4.2, solutions of Dirac wave equation for the spin and pseudo-spin symmetry of these potentials in the presence of Coulomb-like tensor interaction are presented, respectively. In section 5.1 and 5.2 solutions of Dirac wave equation for the spin and pseudo-spin symmetry of these potentials in the presence of Coulomb-like plus Yukawa-like tensor interaction are presented, respectively. In section 6 we provide results and discussion. The conclusion is given in Section 7.

Review of Formula Method
The Formula method has been used to solve the Schrodinger, Dirac and Klein-Gordon wave equations for a certain kind of potential. In this method the differential equations can be written as follows [44] The solution provides a valuable means for checking and improving models and numerical methods introduced for solving complicated quantum systems.

Basic Dirac Equations
In the relativistic description, the Dirac equation of a single-nucleon with the mass moving in an attractive scalar potential S(r) and a repulsive vector potential V (r) can be written as [45] Where E is the relativistic energy, M is the mass of a single particle and α and β are the 4 ×4 Dirac matrices. For a particle in a central field, the total angular momentum J and K ( . Where Δ(r)=V(r)-S(r) and ∑(r)=V(r)+S(r) are the difference and the sum of the potentials V(r) and S(r), respectively and U(r) is a tensor potential. We obtain the second-order Schrodingerlike equation as:

Solution Spin Symmetric with Coulomb-like tensor interaction
Under the condition of the spin symmetry, i. e.
The potential ∑(r) is taken as the Eckart [42,43] Where the parameters q1, q2, v0 and v1 are real parameters, these parameters describe the depth of the potential well, and the parameter α is related to the range of the potential.
For the tensor term, we consider the Coulomb-like potential [46], Where Rc is the coulomb radius, Zf and Zg stand for the charges of the projectile particle f and the target nucleus g, respectively.
Equation (15) is exactly solvable only for the case of k = 0. In order to obtain the analytical solutions of Eq. (15), we employ the improved approximation scheme suggested by Greene and Aldrich [47,48] and replace the spin-orbit coupling term with the expression that is valid The behavior of the improved approximation is plotted in Fig. 1. We can see the good agreement for small α values.
The behavior approximation for By applying the transformation s exp( 2 r)    Eq. (15) brings into the form: 2 n,k n,k n,k 22 Where the parameters A, B and C are considered as follows: Now by comparing Eq. (17) with Eq. (1), we can easily obtain the coefficients ki (i = 1, 2, 3) as follows: The values of the coefficients ki (i = 4, 5) are also found from Eq. (4) as below: Thus, by the use of energy equation Eq. (2) for energy Eigen-values, we find: In Tables 1-3, we give the numerical results for the spin symmetric energy Eigen-values (in units of fm −1 ).
We have obtained the energy Eigen-values and the wave function of the radial Dirac equation for Eckart plus Hulthen potentials with Coulomb-like tensor interaction in the presence of the spin symmetry for k≠0.
We have obtained the energy Eigen-values and the spinors of the radial Dirac equation for Eckart plus Hulthen potentials with Coulomb-like tensor interaction in the presence of the pseudo-spin symmetry for k≠0. We show in Fig (2) behavior energy for various H in the spin and pseudo-spin symmetry.

Solution Spin symmetry with Coulomb plus Yukawa-like tensor interaction
In this section for the spin symmetry, we consider Δ(r) = Cs , ∑(r) and U(r) as the following: Where H and A are the real parameters, substitution of Eq. (35) and (36) Where the parameters χ2, χ1 and χ0 are considered as the follows: By comparing Eq. (37) with Eq. (1), we can easily obtain the coefficients ki (i = 1, 2, 3) as follows: k1=k2= k3=1 (39) The values of the coefficients ki (i = 4, 5) are also found from Eq.(4) as below:   The effects of the Yukawa-like tensor interactions on the upper and lower components for the spin symmetry are shown in Figs. 3.
In Fig.5, 6 and 7 we showed that degeneracy is removed by tensor interaction. Furthermore, the amount of the energy difference between the two states in the doublets increases with increasing H and A.

Results and Discussion
We obtained the energy Eigen-values in the absence and the presence of the Coulomb-like tensor potential for various values of the quantum numbers n and k. In table 1 and 4 in the absence of the tensor interaction (H = 0), the degeneracy between spin doublets and pseudospin doublets are observed. For example, we observe the degeneracy in (1p1/2, 1p3/2), (1d3/2, 1d5/2)..., etc in the spin symmetry, and we observe the degeneracy in (1s1/2, 1d3/2), (1p3/2, 1f5/2)..., etc in the pseudo-spin symmetry. When we consider the tensor interaction for example by parameter H=0.65, the degeneracy is removed. In table 2 and 3 for the spin symmetry also in   table 5 and 6 for the pseudo-spin symmetry, we show that degeneracy exist between spin doublets for several of parameters α and M, and we show that degeneracy is removed in the present of tensor interaction. In Fig (2 1f5/2) and (1d5/2, 1g7/2) and spin doublets (1f5/2, 1f7/2) and (1d3/2, 1d5/2) for the effects of the Yukawa-like tensor interactions and for Yukawa plus Coulomb-like tensor interaction are given in Figs. 5 and 6. In Fig. 7 we show that the coulomb tensor interaction is stronger than Yukawa-like tensor interaction for remove degeneracy.

Conclusions
In this paper, we have discussed approximately the solutions of the Dirac equation for Eckart plus Hulthen potentials with Spin Symmetry and Pseudo-spin Symmetry for k≠0. We obtained the energy Eigen-values and the wave function in terms of the generalized polynomials functions via the Formula method. To show the accuracy of the present model, some numerical values of the energy levels are shown in figure 3, 4, 5 and 6. We have showed that the energy degeneracy in pseudo-spin and spin doublets is removed by the tensor interaction effect.