Influence of Tensor Interaction on Evolution of Nuclear Shells

doi:10.4236/jmp.2013.45B007

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Journal of Modern Physics, 2013, 4, 33-36

doi:10.4236/jmp.2013.45B007 Published Online May 2013 (http://www.scirp.org/journal/jmp)

Influence of Tensor Interaction on Evolution of

Nuclear Shells

Rupayan Bhattacharya

Department of Physics, University of Calcutta, Kolkata, India

Email: [email protected]

Received 2013

ABSTRACT

Using Skyrme’s density dependent interaction the evolution of nuclear shells has been studied in Hartree-Fock formal-

ism. Optimization of the strength of tensor interaction has been done in reproducing the observed splitting of shell

model states of 40,48Ca, 56Ni and 208Pb. Spin-orbit splitting in Ca-isotopes, 56Ni, 90Zr, 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.

Keywords: Tensor Interaction; Hartree-Fock Theory; Nuclear Shells

1. Introduction

The important role of the tensor component of the nu-

cleon-nucleon interaction information of shells beyond

the conventional ones was pointed out in the seventies

[1-6]. 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 self-

consistent mean field calculations generated lot of inter-

est in the nuclear structure study circle.

In order to study the evolution of shells with the inclu-

sion 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.

2. Theoretical Formulation

The tensor interaction in HF theory is given by [5]

VT =

(1)

The expressions for proton (- neutron) spin-orbit po-

tential is

L. S (2)

(3)

The final spin-orbit potential component is given by

where Jq(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 1f7/2 – 1f5/2 , 2p3/2 – 2p1/2 and 1d5/2 –

1d3/2 levels were performed for isoscalar spin-saturated

N = Z nucleus 40Ca. It has been observed that the separa-

tion energies of the spin-orbit doublets decrease with

reduction in values of W0 (- 4/3 ) and a reduction is

required for agreement with the experimental results.

After fixing the first coupling constant , other tensor

R. BHATTACHARYA

coupling constants were fixed by optimizing the 1f7/2-

1f5/2 difference in 56Ni and 48Ca nuclei. Our final adjust-

ment 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 208Pb, a very well studied nucleus experimentally.

3. Results

The difference in energies for some spin-orbit partner

shell model states near the Fermi-surface of 208Pb are

shown in Table 1 for a few standard Skyrme force pa-

rameters 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 cal-

culated splitting of the spin-orbit partners of proton and

neutron single particle states 16O, 40,48Ca, 56Ni, 90Zr, 132Sn

along with the empirical values [12,13]. Neutron sin-

gle-particle energy differences between 1i13/2 and 1h9/2

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.

Table 1. Splitting of spin-orbit doublets in 208Pb.

Energy in MeV

P Sly5 Sly4 KDE KDEt SKP SKPtEXP

Δ1h 5.90 6.22 5.77 4.87 5.24 5.30 5.56

Δ2d 1.98 1.89 1.99 1.57 1.55 1.41 1.33

Δ2f 2.68 2.61 2.70 2.17 2.15 1.98 1.93

Δ3p 1.05 1.02 1.04 0.80 0.76 0.69 0.84

Δ1h 5.80 5.59 5.75 5.48 5.18 5.09 5.10

Δ1i 7.64 7.25 7.67 7.10 6.62 6.39 6.46

Δ2f 3.10 1.96 3.16 2.47 2.30 2.12 2.03

Δ2g 3.71 3.57 3.77 2.96 2.93 2.73 2.51

Δ3p 1.22 1.13 1.25 0.94 0.83 0.74 0.90

Δ3d 1.75 1.67 1.60 1.31 1.27 1.15 0.97

Figure 1. Energy differences of 1i13/2 and 1h9/2 states in N =

82 isotones.

Table 2. Calculated values of spin-orbit splitting along with

empirical values of doubly shell closed nucle i O, Ca, Sn, Zr.

Energy difference(T)

NucleusOrbitals

SKP-T KDE-T

Energy

difference(E)

d 4.89 4.77 5.08

16O

d 4.65 4.54 4.97

p 1.87 1.63 2.00

f 6.45 6.24 5.64

2p 1.65 1.46 1.72

40Ca

f 6.13 6.0 6.05

2p 2.33 2.03 1.77

f 7.59 7.73 8.01

2p 1.24 1.29 2.14

48Ca

f 5.03 5.27 4.92

p 1.83 1.69 1.88

f 6.42 6.97 6.82

p 1.58 1.52 1.83

56Ni

f 6.09 6.55 7.01

d 2.65 2.53 2.43

1g 7.34 7.55 7.07

d 1.69 1.72 2.03

1g 5.28 5.36 5.56

90Zr

f 3.84 3.94 4.56

f 2.68 2.45 2.00

3p 0.93 0.75 0.81

 h　 7.37 7.44 6.68

d 2.08 2.09 1.93

132Sn

d 1.72 1.86 1.75

g 4.62 5.15 5.33

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 ex-

perimental 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 no-

menclature 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 1d5/2 and the last occupied state 1p1/2 con-

stitute 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 clo-

sure at 24O as observed in the recent experimental results.

R. BHATTACHARYA 35

The gap for Z = 20 isotopes and N = 20 isotonic chain

is the energy difference between the proton (neutron)

single particle states 1f7/2 and 1d3/2. 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 52–56Ti isotopes with intermediate-energy Coulomb

excitation and absolute B (E2; 0+ → 21

+ ) transition rates

have been obtained. These data confirm the presence of a

sub-shell closure at neutron number N = 32 in neu-

tron-rich nuclei above the doubly magic nucleus 48Ca.

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 interac-

tion is taken into consideration

Figure 2. Comparison of calculated gap evolution of Z = 8

isotones with and without tensor interaction and experi-

mental data.

68101

GAP (N = 8) (MeV)

SKP

SKP-T

EXPT

Figure 3. Gap evolution of N = 8 isotones with and without

tensor interaction along with experimental data.

Figure 4. Comparison of calculated gap evolution of Z = 20

isotones with and without tensor interaction and experi-

mental data.

12 14 16 18 20 22 24 26 28 30

GAP (N = 20) (M eV)

SKP

SKP-T

EXP

Figure 5. Gap evolution of N = 20 isotones with and without

tensor interaction and exper i mental data.

4. Conclusions

It has been shown in this work that inclusion of tensor

interaction in the Skyrme-Hartree-Fock theory induces a

change in the spin-orbit potential of nucleus and hence

improves the single particle structure of shell closed

nucleus considerably and the optimized coefficients of

tensor interaction reproduces the evolution of shell gaps

for Z, N = 8, 20, two most important low mass shell

closed nuclei and generates the spin-orbit splitting of

some shell model states near the Fermi surface of some

shell closed nuclei. Furthermore, it also reproduces the

trend in change of energy differences in 1h11/2 and 1g9/2

states of Sn isotopes.

5. Acknowledgements

The author thanks the UGC for support by the Emeritus

R. BHATTACHARYA

Fellowship [No. F.6-34/2011(SA – II)].

REFERENCES

[1] D. Vautherin and D. M. Brink, “Hartree-Fock Calcula-

tions with Skyrme’s Interaction,” Physics Letters B, Vol.

32, No. 3, 1970, pp. 149-153.

doi:10.1016/0370-2693(70)90458-2

[2] D. Vautherin and D. M. Brink, “Hartree-Fock Calcula-

tions with Skyrme’s Interaction.Ⅰ. Spherical Nuclei,”

Physical Review C, Vol. 5, No. 3, 1972, pp. 626-647.

doi:10.1103/PhysRevC.5.626

[3] T. Otsuka, et al., “Magic Numbers in Exotic Nuclei and

Spin-Isospin Properties of the NN Interaction,” Physical

Review Letters

, Vol. 87, No. 8, 2001,

082502.

doi:10.1103/PhysRevLett.87.082502

[4] G. G. Colo, H. Sagawa, S. Fracasso and P. F. Bortignon,

“Spin-Orbit Splitting and the The Tensor Component of

the Skyrme Interaction,” Physics Letters B, Vol. 646, No.

5-6, 2007, pp. 227-231.

doi.org/10.1016/j.physletb.2007.01.033

[5] D. M. Brink and F. Stancu, “Evolution of Nuclear Shells

with the Skyrme Density Dependent Interaction,” Physi-

cal Review C, Vol. 75, 2007, 064311.

doi:10.1103/PhysRevC.75.064311

[6] T. Lesinski, M. Bender, K. Bennaceur, T. Duguet and J.

Meyer, “Tensor Part of The Skyrme Energy Density

Functional: Spherical Nuclei,” Physical Review C, Vol.

76, 2007, p. 014312. doi:10.1103/PhysRevC.76.014312

[7] M. Zalewski, J. Dobaczewski, W. Satula and T. R.

Werner, “Spin-Orbit and Tensor Mean-Field Effects on

Spin-Orbit Splitting Including Self-Consistent Core

Polarization,” Physical Review C, Vol. 77, 2008, p.

024316.doi:10.1103/PhysRevC.77.024316

[8] Y. Z. Wang, J. Z. Gu, J. M. Dong and X. Z. Zhang, “Sys-

tematic Study of Tensor Effects in Shell Evolution,”

Physical Review C, Vol. 83, 2011,054305.

doi:10.1103/PhysRevC.83.054305

[9] J. Dobaczewski, H. Flocard and J. Treiner, “Har-

tree-Fock-Bogolyubov Description of Nuclei Near The

Neutron-Drip Line,” Nuclear Physics A, Vol. 422, No. 1,

1984, pp. 103-139. doi:10.1016/0375-9474(84)90433-0

[10] E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R.

Schaeffer, “A Skyrme Parametrization from Subnuclear

to Neutron Star Densities Part Ⅱ. Nuclei Far from Sta-

bilities,” Nuclear Physics A, Vol. 635, No. 1-2, 1998, pp.

231-256. doi:10.1016/S0375-9474(98)00180-8

[11] B. A. Brown, “New Skyrme Interaction for Normal and

Exotic Nuclei,” Physical Review C, Vol. 58, 1998, pp.

220-231. doi:10.1103/PhysRevC.58.220

[12] B. K. Agrawal, S. Shlomo and V. Kim Au, “Determina-

tion of the Parameters of A Skyrme Type Effective Inter-

action Using the Simulated Annealing Approach,” Physi-

cal Review C, Vol. 72, 2005, 014310

doi:10.1103/PhysRevC.72.014310

[13] N. Schwierz, I. Wiedenhover and A. Volya,

arXiv:0709.3525

[14] A. Oros, Ph.D thesis, University of Koln, 1996.

[15] D. C. Dinca, et al., “Reduced Transition Probabilities to

the First 2 State in 52,54,56Ti and Development of Shell

Closures at N = 32, 34,” Physical Review C, Vol. 71,

2005, 041302.doi:10.1103/PhysRevC.71.041302