Journal of Modern Physics, 2013, 4, 33-36
doi:10.4236/jmp.2013.45B007 Published Online May 2013 (
Influence of Tensor Interaction on Evolution of
Nuclear Shells
Rupayan Bhattacharya
Department of Physics, University of Calcutta, Kolkata, India
Received 2013
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 =
The expressions for proton (- neutron) spin-orbit po-
tential is
L. S (2)
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
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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
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)
d 4.89 4.77 5.08
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
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
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
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
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
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.
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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.
GAP (N = 8) (MeV)
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)
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
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Fellowship [No. F.6-34/2011(SA – II)].
[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.
[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.
[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,
[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.
[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.
[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.
[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.
[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. 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
[13] N. Schwierz, I. Wiedenhover and A. Volya,
[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