Influence of Tensor Interaction on Evolution of Nuclear Shells

Using Skyrme’s density dependent interaction the evolution of nuclear shells has been studied in Hartree-Fock formalism. Optimization of the strength of tensor interaction has been done in reproducing the observed splitting of shell model states of Ca, Ni and Pb. Spin-orbit splitting in Ca-isotopes, Ni, Zr, N = 82 isotones, Sn-isotopes and evolution of gaps in Z, N = 8, 20 have been reanalyzed with the inclusion of tensor interaction. For doubly shell closed nuclei, it has been observed that tensor interaction is sensitive to spin saturation of nuclear shells.


Introduction
The important role of the tensor component of the nucleon-nucleon interaction information of shells beyond the conventional ones was pointed out in the seventies [1][2][3][4][5][6].Study of shell evolution and modification of magic numbers in exotic nuclei after inclusion of the tensor component of the nucleon-nucleon interaction in the selfconsistent mean field calculations generated lot of interest in the nuclear structure study circle.
In order to study the evolution of shells with the inclusion of tensor force in Skyrme-Hartree-Fock theory we have studied the spin-orbit splitting of some shell model states of doubly shell closed nuclei, shell gap evolutions of Z, N = 8, 20 nuclei, difference in single particle(-hole) energy levels near the Fermi surfaces of Z = 50 isotopes, Ca -isotopes and N = 82 isotones.

Theoretical Formulation
The tensor interaction in HF theory is given by [5] The expressions for proton (-neutron) spin-orbit potential is (3) The final spin-orbit potential component is given by where J q(q') (r) is the proton or neutron spin-orbit density defined as We have used different sets of parameters (SKP [8], SLY5 [9], SKX [10], KDEO [11]) for our calculations.A schematic pairing force of BCS type has been used to take care of pairing correlation.To get the optimized spin-orbit constant , variational studies of the spin-orbit splitting of 1f 7/2 -1f 5/2 , 2p 3/2 -2p 1/2 and 1d 5/2 -1d 3/2 levels were performed for isoscalar spin-saturated N = Z nucleus 40 Ca.It has been observed that the separation energies of the spin-orbit doublets decrease with reduction in values of W 0 (-4/3 ) and a reduction is required for agreement with the experimental results.After fixing the first coupling constant , other tensor coupling constants were fixed by optimizing the 1f 7/2 -1f 5/2 difference in 56 Ni and 48 Ca nuclei.Our final adjustment of the tensor coupling constants was done after calculating the spin-orbit splitting of several shell model states near the Fermi-surface of the stable shell closed nuclei 208 Pb, a very well studied nucleus experimentally.

Results
The difference in energies for some spin-orbit partner shell model states near the Fermi-surface of 208 Pb are shown in Table 1 for a few standard Skyrme force parameters from where we see that after inclusion of tensor interaction in SKP set, a good agreement with empirical values has been achieved.In Table 2 we present the calculated splitting of the spin-orbit partners of proton and neutron single particle states 16 O, 40,48 Ca, 56 Ni, 90 Zr, 132 Sn along with the empirical values [12,13].Neutron single-particle energy differences between 1i 13/2 and 1h 9/2 states in N = 82 isotones calculated with and without inclusion of tensor force along with experimental values [12] have been presented in Figure 1 In Figure 2 we present the gap evolution for Oxygen (Z = 8) isotopes.Empirical values are taken from ref. [12] and [13] In Figure 2 we have presented the gap evolution for Z = 8 with and without tensor forces.For comparison experimental data sets have also been shown.The gap here means the difference in single particle energies of the first particle state above and the last hole state below the Fermi surface of the nuclei under consideration.Since we are dealing with shell closed nuclei, the shell model nomenclature has been used for the particle (-hole) states.For the Z = 8 isotopes and N = 8 (Figrue 3) isotones the energy difference between the first unoccupied proton (neutron) level 1d 5/2 and the last occupied state 1p 1/2 constitute the gap.Inclusion of tensor interaction changes the curvature of the shell evolution of the graph correctly and also produces the kink at N = 16 indicating a shell closure at 24 O as observed in the recent experimental results.
The gap for Z = 20 isotopes and N = 20 isotonic chain is the energy difference between the proton (neutron) single particle states 1f 7/2 and 1d 3/2 .In the gap evolution of Z = 20 isotopes as shown in Figure 4, inclusion of tensor interaction increases the gap, as more and more neutrons are added in the system.We find the gap reaches a maximum at N = 32.Dinca et al. [14] have studied the even 52-56 Ti isotopes with intermediate-energy Coulomb excitation and absolute B (E2; 0 + → 2 1 + ) transition rates have been obtained.These data confirm the presence of a sub-shell closure at neutron number N = 32 in neutron-rich nuclei above the doubly magic nucleus 48 Ca.
In Figure 5 we present the gap evolution of N = 20 isotones for Si to Fe nuclei.Though in the experimental data one observes a peak at Z = 20, we have obtained a change in the gradient at that point when tensor interaction is taken into consideration

Acknowledgements
The author thanks the UGC for support by the Emeritus

Figure 2 .Figure 3 .
Figure 2. Comparison of calculated gap evolution of Z = 8 isotones with and without tensor interaction and experimental data.

Figure 4 .Figure 5 .
Figure 4. Comparison of calculated gap evolution of Z = 20 isotones with and without tensor interaction and experimental data.