Comparison with the results of the Nilsson-Strutinsky method

The Nilsson-Strutinsky(NS) method is a convenient and well-established method to treat nuclear deformations. Because NS computation is much simpler than self-consistent-field calculations, it seems worth doing another systematic survey of SD isomers using the NS method in order to compare the results with those of the Skyrme-HF method for the same nuclei. Our survey using NS method provides a useful overview of the situation outside the region investigated in section 4 of our paper [TTO98].

We have utilized a program for the standard Nilsson-Strutinsky calculation [NTS69] provided by Y.R. Shimizu [Shi97], which takes into account two axially symmetric deformations, i.e., the quadrupole deformation epsilon_2 and the hexadecapole deformation epsilon_4. For each value of epsilon_2, the value of epsilon_4 is optimized so as to minimize the total energy. The standard values given in Table 1 of Ref. [BR85] are used for the parameters kappa_N and mu_N of the Nilsson potential. The pairing correlation is active for single-particle levels within +/- 1.2 hbar omega from the Fermi level, while the strengths of the pairing force are determined such that the smoothed pairing gap becomes

bar{Delta} = 13 A^{-1/2} MeV.

The parameters of the macroscopic part [MS67] are

a_s=17.9439 MeV,
kappa_s=1.7826,
R_c=1.2249 A^{1/3} fm.

See Ref. [BRA91] for calculational details.

With this model one can calculate the entire region of the nuclear chart, i.e., from the proton drip line to the neutron drip line: The model does not suffer from the problem of neutron pairing in neutron-rich nuclei explained in section 4 of our paper [TTO98] because the Nilsson potential does not have a continuum spectrum. Concerning the expected enhancement of the pairing due to the coupling with the continuum states, however, the present model simply neglects its influences.

The calculation for 2000 even-even nuclei can be completed in a few hours with an ordinary personal computer owing to the simpleness of the NS method itself and also to the specialization of the code to non-rotating axially symmetric states.

Fig. 1 in formats of JPEG(143KB) or GIF(497KB) or PS(1.2MB) displays the resulting deformations epsilon_2, excitation energies E^*, and barrier heights E_B of the SD minima at epsilon_2 > 0.35. Let us discuss the results according to the grouping employed in section 4 of our paper [TTO98].

• 1. 38 <= Z <= 40, 36 <= N <= 38
  The area of this island of zero-spin SD states is much smaller than in the Skyrme-HF results.
• 2. 48 <= Z <= 50, 44 <= N <= 48
  The number of nuclei is reduced from 10 to 4.
• 3. 58 <= Z <= 60, 60 <= N <= 66
  None of the 6 nuclei were found to display SD isomers in the NS model.
• 4. 70 <= Z <= 80, 80 <= N <= 102
  The number of nuclei is almost unchanged (thirty), but the number of neutron deficient (rich) nuclei is decreased (increased). The appearance of SD around Pb-Hg-Pt isotopes is quite similar in the Skyrme-HF and the NS methods. However, the details are different. For example, the PES of the Skyrme-HF has a smaller E^* and a larger E_B than that of the NS model for the Pb isotopes.
• 5. Z+ N > ~200
  One can see that the group No. 5 found in the Skyrme-HF is a part of a huge area extending to Z>82 and/or N>126. In the actinide region there exist fission isomers for almost all the nuclei except in a rectangle-like area Z >= 102 and N <= 186, where large deformation minima do not exist and the nucleus goes into fission directly from the ground state.
• 6. 38 <= Z <= 40, 60 <= N <= 68
  None of 8 nuclei in this region show SD isomers.
• 7,8. 36 <= Z <= 62, 78 <= N <= 82
  This region extending vertically includes small islands of Nos. 7 and 8 listed in section 4 of our paper [TTO98].
• 9. Z=68, 86 <= N <= 88
  Compared with the island No. 9 of the Skyrme-HF results, SD isomers are found at Z and N values differing by +4 and -4, respectively.
• 10. 46 <= Z <= 54, 98 <= N <= 102
  The nine nuclei above the neutron-drip line have zero-spin SD states. The deformations are in the range 0.63 <= delta <= 0.64.
• 11. 54 <= Z <= 66, 118 <= N <= 130
  Each of twenty-seven nuclei above the neutron-drip line has a large-deformation (0.36 <= delta <= 0.41) solution mostly in the ground state.

References

[TTO98]

Satoshi Takahara, Naoki Tajima and Naoki Onishi, submitted to Nuclear Physics A.

[NTS69]

S.G. Nilsson, C.F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I. Lamm, P. Moeller and B. Nilsson, Nucl.Phys. A131 (1969) 1.

[Shi97]

Y.R. Shimizu, private communication.

[BR85]

T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14.

[MS67]

W.D. Myers and W.J. Swiatecki, Ark. Phys. 36 (1967) 343.

[BRA91]

T. Bengtsson, I. Ragnarsson and S. Aberg, in ``Computational Nuclear Physics 1'', ed. K. Langanke, J.A. Maruhn and S.E. Koonin, (Springer-Verlag, Berlin, 1991) 51.