Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 49 (2016)

Shell-model ﬁts for N = 82 isotones
M. Honma,∗1 T. Otsuka,∗2,∗3,∗4 T. Mizusaki,∗5 Y. Utsuno,∗6,∗3 N. Shimizu,∗3 and M. Hjorth-Jensen∗4,∗7,∗8
Owing to the well-established shell gap at Z = 50
and 82, the shell model has been successfully applied
for describing the structure of the N = 82 isotones
with 50 < Z < 82 by taking the valence single-particle
orbits 0g7/2 , 1d5/2 , 0h11/2 , 2s1/2 and 1d3/2 on top ofan inert 132 Sn core. The N = 82 isotones show various interesting features. For example, the excitationenergy of the ﬁrst 2+ state increases as a function ofZ and shows a prominent maximum at Z = 64, suggesting the development of a subshell gap. The lighter
isotones with Z < 64 are dominated by mixed conﬁgurations of the lower two orbits (g7/2 and d5/2 ) withexcitations to the upper orbits, while for the heavier
ones the unique-parity high-j orbit h11/2 plays a crucial role. The systematics of yrast spectra and theproperty of some isomers are successfully explained by
the seniority scheme. In addition, the development of
the octupole collectivity has been observed in the lowlying states connected by strong E3 transitions.
It is interesting to investigate to what extent suchvariety of nuclear structures can be described within
the shell-model framework. The eﬀective interaction
is a key ingredient to address this problem, which canbe either determined purely empirically1) or derived
microscopically from the realistic nucleon-nucleon potential. The latter attempt has been extensively developed,2,3) and the importance of the three-body force
has been suggested to account for the deviation of
the shell-model results from the experimental observations. Another possible approach for a better description of the experimental data is to modify such a microscopically determined eﬀective interaction4) by ﬁtting
the calculated energies to the experimental data. In
this report, we present such ﬁtting calculations for theN = 82 isotones. The similar analysis for Sn isotopes
was reported previously.5)
The single-particle energies of the d5/2 , d3/2 , andh11/2 orbits relative to the g7/2 are taken from the experimental energy spectra of 133 Sb,7) and only that of
the s1/2 orbit is determined by the ﬁt. Starting fromthe microscopically derived eﬀective interaction basedon the N3LO potential,6) we have carried out iterativeﬁtting calculations. In the latest ﬁt, 26 linear combina∗1
∗2
∗3
∗4
∗5
∗6
∗7
∗8

tions of the ﬁt parameters (i.e. 160 two-body matrix
elements with A−0.3 mass dependence and the singleparticle energy of the s1/2 orbit) have been modiﬁed,
and we have attained a rms error of 142 keV for 338
energy data of 22 isotones.
Figure 1 illustrates the quality of the ﬁt. It canbe seen that the overall agreement between the shellmodel results and the experimental data is quite goodincluding the “magic”-like feature at Z=64. The ﬁt is
successful also for odd-Z cases. The semi-empirical approach presented here would be useful not only for theanalysis of experimental data but also as a reference forclarifying various eﬀects such as the three-body forcesthat are not included in the present microscopic approach. It is also interesting to extend the scope of
this interaction beyond the semi-magic nuclei by combining it with a suitable proton-neutron interaction.
6512N=82++Ex (MeV)

432
1010+8+64++2

52 54 56 58 60 62 64 66 68 70 72
proton number

Fig. 1. Comparison of excitation energies of even-spin yrast
states between the experimental data (symbols) andthe shell-model results (lines). Experimental data aretaken from Ref.8) Open symbols indicate that the spinor parity assignment is uncertain. The shell-model results are obtained by using the eﬃcient shell-model code
MSHELL649) .

References
1) B. H. Wildenthal: Phys. Rev. Lett. 22, 1118 (1969).
2) A. Holt et al.: Nucl. Phys. A 618, 107 (1997).
3) L. Coraggio et al.: Pnys. Rev. C 80, 044320 (2009).
4) M. Hjorth-Jensen et al.: Phys. Rep. 261, 125 (1995).
5) M. Honma et al.: RIKEN Accel. Prog. Rep. 45, 35 (2012).
6) D. R. Entem et al.: Phys. Rev. C 68, 041001(R) (2003).
7) M. Sanczhez-Vega et al.: Phys. Lett. 80, 5504 (1998).
8) Data extracted using the NNDC WorldWideWeb sitefrom the ENSDF database.
9) T. Mizusaki et al.: MSHELL64 code (unpublished).

Center for Mathematical Sciences, University of Aizu
Department of Physics, University of Tokyo
Center for Nuclear Study, University of Tokyo
National Superconducting Cyclotron Laboratory, Michigan
State University
Institute of Natural Sciences, Senshu University
Advanced Science Research Center, Japan Atomic Energy
AgencyDepartment of Physics and Astronomy, Michigan State University
Department of Physics, University of Oslo- 77 -