2 shell

Eur. Phys. J. Special Topics 150, 97–98 (2007) c EDP Sciences, Springer-Verlag 2007
DOI: 10.1140/epjst/e2007-00276-6

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Microscopic calculations for upper-f p, g9/2 shell nuclei: Ge & Se J.P. Draayer and K.P. Drumev Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA Received: January 31, 2007 Abstract. Shell-model calculations for isotopes of Ge and Se are reported where valence nucleons beyond the N = 28 = Z core occupy levels of the normal parity upper-f p shell (f5/2 , p3/2 , p1/2 ) and the unique parity g9/2 intruder configuration. Results are given for realistic interactions of the Kuo-Brown-3 type with various model space truncations that key in on the number of nucleon pairs allowed to occupy the intruder level. Electromagnetic (E2 & M1) rates as well as decay probabilities are calculated, some of which are key in determining the structure of “waiting point” nuclei that regulate certain nucleo-astrosynthesis processes. The role of the intruder level, which is treated on an equal footing with the normal parity levels, is shown to be important for reproducing structural details. The levels of the upper-f p shell are handled within the framework of a normal ls-coupled basis as well as its pseudo-SU(3) counterpart, and respectively, the g9/2 as a single level and as a member for the complete gds shell. The second of these two approaches, namely, the SU(3) picture, allows one to better probe the effect of deformation. PACS. 21.60.Fw Models based on group theory – 21.10.Re Collective levels

1 Towards an extended SU(3) shell model Until recently, SU(3) shell-model calculations – real SU(3) [1] for light nuclei and pseudo-SU(3) [2] for heavy nuclei – have been performed in either only one space (protons and neutrons filling the same shell, e.g. the ds shell) or two spaces (protons and neutrons filling different shells, e.g. for rare earth and actinide nuclei). Various results for low-energy features, like energy spectra and electromagnetic transition strengths, have been published over the years [3]. These applications confirm that the SU(3) model works well for light nuclei and the pseudoSU(3) scheme, under an appropriate set of assumptions, for rare earth and actinide species, but it has not been applied to nuclei with mass numbers A = 56 to A = 100, which is an intermediate region where conventional wisdom suggests the assumptions that underpin the use of SU(3)-based methodologies in other regions break down. In particular, in this intermediate region the g9/2 intruder level that penetrates down from the shell above due to the strong spin-orbit splitting appears to be as spectroscopically relevant to the overall dynamics as the normal parity f5/2 , p3/2 , p1/2 levels. Specifically, in this region the effect of the intruder level cannot be ignored or mimicked through a “renormalization” of the normal-parity dynamics which is how it has been handled to date in other regions. The upper-f p, g9/2 shell is the lightest region of nuclei where the intruder level must be taken into account. Its presence poses a significant challenge. For example, if the pseudo-SU(3) symmetry proves to be a good scheme for

characterizing upper-f p shell configurations, should one integrate the g9/2 intruder into this picture by treating it as a single j-shell that is independent of couplings to the other members (g7/2 , d5/2 , d3/2 , s1/2 ) of the gds shell, or should one take the complete gds shell into account? To benchmark the benefit of using the SU(3) scheme in this region (pseudo-SU(3) for the upper-f p shell and normal SU(3) for the g9/2 configurations, extended to the full gsd shell), we first generated results using the standard m-scheme representation for two nuclei, 64 Ge and 68 Se, with the 8 and 12 valence nucleons, respectively, distributed across the p1/2 , p3/2 , f5/2 , g9/2 model space with the f7/2 level considered to be full and frozen [4]. The Hamiltonian we used was a G-matrix with a phenomenologically adjusted monopole part [5] which describes the experimental energies well. Different cuts of the full model space were made to estimate the single-particle occupancy. The fractional occupancy for the low-lying states was found to be below about 0.3 particles for the intruder level. Although this number is small, the presence of the intruder among the occupied levels was found to be an important element of the theory. Most importantly, the results of this m-scheme study showed that the calculated distribution of single-particle occupancies for 64 Ge and 68 Se can be achieved with two or less nucleons in the intruder level. The pseudo-SU(3) symmetry in the states of different bands in 64 Ge was then studied using a renormalized version of the realistic interaction in the pf5/2 space. The distribution of the second order Casimir operator C2 of SU(3) in the ground state (g.s.) and gamma bands yielded

The European Physical Journal Special Topics 3. 5

64Ge

3 ]

contributions of 50–60% from the leading pseudo-SU(3) irreducible representation (irrep) which suggests that the pseudo-SU(3) symmetry is quite good. This, along with the observation from above that only a relatively few intruder level configurations are needed in realistic calculations for 64 Ge and 68 Se, led us to conclude that a symmetry-adapted, truncated set of basis states might suffice for realizing the structure of the nuclei in this region.

Energy [MeV

98

( 3+ ) 3+

(4+)

(2+)

2+

1. 5 1

3. 5

Energy [MeV

4+

3+

2

4

]

χ − Q.Q + aJ 2 + bKJ2 2

gamma band 4+

(2+)

2+

3 2. 5

2+

pf 5/ 2 g 9/ 2 -> e ef f =0.5 SU(3)->e ef f =0.2 0+ 0+

0+

pf5/2

Exp (6+) (5-) (6+)

0+ (6+) ( 4+ )

1

SU(3)

4+ 2+ 3+ 3+ 1+ 2+ 4+

2+ 4+

2 1. 5

2+

4+

(4+) 2+

(2+) (2+) (2+)

0+

0+

2+

2+

0+

0+

0. 5 0

0+

Fig. 1. Excitation spectra and B(E2) strengths for 64 Ge and 68 Se using the extended SU(3) theory compared with experiment and m-scheme shell-model results using G-matrix input.

(1)

The model space consists of two parts for each particle type, a normal (N) parity pseudo-shell (f5/2 , p3/2 , p1/2 → d˜5/2 , d˜3/2 , s˜1/2 ) and a unique (U) parity shell composed of levels of opposite parity from the complete gds shell above. The proton and neutron quantum numbers are indicated by aσ = {{aσN , aσU }ρσ (λσ , µσ ), Sσ }, where aστ = {Nστ [fστ ]αστ (λστ , µστ )Sστ } are the basis-state labels for the four spaces in the model (σ stands for π or ν, and τ stands for N or U). The Hamiltonian


στ (Hsp − GS στ † S στ ) − GS στ † S σ τ H= σ,τ

(4+)

0

2 Basics of the model and some results

|aπ ; aν , ρ(λ, µ)kL, S; JM
.

pf 5 / 2 g 9 / 2 SU(3 ) Ex p 4 + 4+

0. 5

68Se

The usual pseudo-SU(3) shell model was extended to take the intruder level into account explicitly. Many-particle basis states are built as SU(3) proton (π) and neutron (ν) coupled states with well-defined particle number and good total angular momentum.

2. 5

g.s. band

σ
=σ
,τ

(2)

includes spherical Nilsson single-particle energies as well as the quadrupole-quadrupole and pairing interactions plus two rotor-like terms that are diagonal in the SU(3) basis. The single-particle terms together with the proton and neutron pairing interactions mix the SU(3) basis states allowing for a realistic description of the energy spectra of the nuclei. The parameters in the Hamiltonian were fixed by systematics [6] and a fit was performed in order to find the best parameter values for the two rotorlike terms. Calculations were carried out using a set of basis states with (pseudo-) spin zero and one proton and neutron configurations. Since the most important configurations are those with highest spatial symmetry, we ignored configurations that involve an odd number of particles in any of the four spaces and only considered the interplay among those having zero or two protons and/or neutrons in the unique space. Then, from all the possible couplings we chose those irreps with the highest value of the second order Casimir operator of pseudo-SU(3) in the normal parity space and SU(3) in the unique parity space. Configurations with no particles in the unique space were found to lie lowest and determine, by-and-large, the structure of the lowest energy eigenstates. For 64 Ge, those with two and four particles in the unique space begin to

play a role at higher energies. Since the spin-orbit interaction does not mix states from these groups it does not have an effect on the low-lying states. The small amount of mixing between configurations with different number of particles is due to the pair scattering between the spaces. Excitation spectra as well as electromagnetic transition strengths were calculated and some of the results are presented in figure 1. The agreement of the calculated results with experiment is reasonable for the lowest states from the g.s. and gamma bands of both 64 Ge and 68 Se. An expected effect is that the effective charge is now smaller since more of the collectivity from the unique parity space has been properly taken into account. Some B(M1) transition strengths were also successfully described. In conclusion, the extended version of the SU(3) model gives a better description of the collectivity properties of the systems considered, and lead to a reduction in the value for the effective charge. This work was supported by the US National Science Foundation, Grant Numbers 0140300 & 0500291 and the Southeastern Universities Research Association (SURA).

References 1. J.P. Elliott, Proc. R. Soc. Lond. Ser. A 245, 128 (1958); ibid. 245, 562 (1958) 2. R.D. Ratna Raju, J.P. Draayer, K.T. Hecht, Nucl. Phys. A 202, 433 (1973) 3. C. Vargas, J.G. Hirsch, J.P. Draayer, Nucl. Phys. A 690, 409 (2001); ibid. 697, 655 (2002); G. Popa, J.G. Hirsh, J.P. Draayer, Phys. Rev. C 62, 064313 (2000); C. Vargas, J.G. Hirsch, T. Beuschel, J.P. Draayer, Phys. Rev. C 61, 031301 (2000) 4. M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 77, 3315 (1996) 5. E. Caurier, F. Novacki, A. Poves, J. Retamosa, Phys. Rev. Lett. 77, 1954 (1996) 6. P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1979)