Deformation and superdeformation: The shell model way

ELSEVIER

Nuclear

Physics

A73 1 (2004)

339-346 www.elsevier.coin/locate/npe

Deformation Alfred0

and Superdeformation:

The Shell Model

Way

Paves”* de Fisica Tebrica, Aut6noma de Madrid,

“Departamento

Universidad

28049, Madrid,

Spain

Present large scale shell model calculations give a very accurate and comprehensive description of light and medium light nuclei. Full pf-shell calculations have made it possible to describe many collective features in an spherical shell model context. Calculations including two major oscillator shells can cope with superdeformed bands. The underlying symmetries are seen to be variants of Elliott’s SU3. 1. INTRODUCTION The spherical shell model approach to the nuclear dynamics is one of the basic tenets of nuclear structure theory. It has three main ingredients; the effective interaction, the valence space and the computational tools that make it possible to solve the huge secular problems involved. The effective interaction used in shell model calculations can be split in two parts[l]; the monopole hamiltonian that determines the evolution of the spherical single particle orbits and the relative location of the different configurations and the multipole Hamiltonian “the correlators” (pairing, quadrupole, etc.). The monopole field fixes the geometry of the single particle orbits around the the Fermi level, while the multipole shapes the spectra (rotors, vibrators etc.) depending on the particular mean field geometry. A new generation of shell model codes, that can treat problems involving basis dimensions as large a few billions, has opened the possibility to access new regions of the chart of nuclides [2]. Full OtZw calculations in the pf-shell have demonstrated the ability of the spherical shell model to treat in an unified framework all the variety of nuclear excitations; single particle modes, neutron-neutron and proton-neutron pairing correlations, rotational bands based upon well deformed intrinsic states, etc. Other collective manifestations such as backbending, alignment, superfluid moments of inertia, etc, until now confined to the realm of heavy nuclei and treated by deformed mean field models, have been experimentally found in medium-light nuclei using the large y-detectors Gammasphere and Euroball [4]. Th e s h e 11model calculations have predicted or explained this full panoply of effects. 2. THE

YRAST

BAND

OF *‘Cr

The paradigm of nuclear rotor amenable to an spherical shell model description is probably *%r. In the pf shell space it has four valence neutrons and four valence neutrons. ‘Work

supported

in part

by a grant

0375-9474/$ - see front matter doi:10.1016/j.nuclphysa.2OO3.11.045

number

Q 2004 Published

BFM2000-30, by Elsevier

of MCyT B V

Spain

340

A. Paves/Nuclear

Physics A731 (2004) 339-346

In figure 1 we present the y-decay energies along the yrast band in a backbending plot. The experimental data are from [4]. The shell model calculation with the interaction KB3 reproduces the experiments to the tiniest details. We have plotted also the results of the Cranked-Hartree Fock Bogoliuvov calculation with the Gogny force of ref. [5], that is relevant because it badly misses the energies (the spectrum is very much compressed) while providing a good reproduction of the quadrupole properties. We realized that the CHFB calculation did not include the neutron proton pairing properly, thus making the moment of inertia far too big. To verify that the default is related to the method and not to the interaction, we have computed the two-body matrix elements of the Gogny force with suitable single particle wave functions, and used them in the shell model calculation. As can be seen in the figure, the resulting energies are excellent and the problem is settled in favor of the force.

0 Exp q SM-K03 A CHFB v SM-GOGNY 21 0.0

a’

, $5

, 1 .o

I 2.0 E, WV

I 3.0

4.0

Figure 1. The yrast band of 48Cr; experiment vs. the shell model calculations and the Gogny force and the CHFB results also with the Gogny force.

with

KB3

In table 1 we compare the calculated B(E2) values with the experimental ones from [6]. We use standard effective charges Sq,=Sq,=O.5. We extract the intrinsic quadrupole moments (static (s) and dynamic (t)) f rom the laboratory frame quantities using Qo(s) = (J + 1) (2J + 3) 3K2 _ J(J + 1) B(E2,

J +

J - 2) = $

e21(JK201J

Q-vdJL

- 2, K)12 Q,,(t)2

K + 1

K # l/2,

1;

(1) (2)

Notice the perfect agreement between theory and experiment and the existence of a very robust intrinsic structure that persist until the backbending region, as it is shown by the constancy and almost equality of the intrinsic static and dynamic quadrupole moments.

A. Poves/Nucleav Table 1 48Cr; quadrupole

properties

J

B(E%,

2 4 6 8 10 12 14 16

321(41) 330( 100) 300(80) 220(60) 185(40) 170(25) lOO(16) 37(6) ’

Physics A731 (2004) 339-346

341

of the yrast band

B(E2)th 228 312 311 285 201 146 115 60

Q&l

Qds)

107 105 100 93 77 65 55 40

103 108 99 93 52 12 13 15

104 104 103 102 98 80 50 40

Therefore we can conclude that 48Cr is a good rotor and that the spherical shell model can describe all its properties. In the last column of the table we can see that the quadrupole properties are as well described in an smaller space, comprising only the Aj=2 aligned orbits. We have followed through this hint, proposing that the deformation in this region is due to a variant of Elliott’s SU3 that we have dubbed Quasi-SU3 [7].

3. SUPERDEFORMED

BANDS

IN 36Ar AND

40Ca

The existence of excited deformed bands in spherical nuclei is a well documented fact, dating back to the 60’s. A classical example is provided by the four particle four holes and eight particles eight holes states in 160 , starting at 6.05 MeV and 16.75 MeV of excitation energy [8,9]. H owever, it is only recently that similar bands, of deformed and even superdeformed character, have been discovered in other medium-light nuclei such as 56Ni [lo], 36Ar [II] and 40Ca [12] and explored up to high spin. One characteristic feature of these bands is that they belong to rather well defined spherical shell model configurations; for instance, the deformed excited band in 56Ni can be associated with the configuration (1f7,z)12 (2ps,z, ifs/z, 2~r,z)~ while the (super)deformed band in 36Ar has the structure (sd)16 (~f)~. The states we aim to are dominantly core excitations from the s&shell to the pf-shell. The natural valence space would thus comprise both major oscillator shells. However, the inclusion of the lds,z orbit in the valence space produces a huge increase in the size of the basis and massive center of mass effects, thus we are forced to exclude it from the valence space; this is equivalent to take a closed core of 28Si. We are aware that this truncation will reduce slightly the quadrupole coherence of the solutions. Our valence space will consist of the orbits 2sr/z, ld3/z, lf7,2, 2~312, lf512, 2~r/~. The effective interaction is the same used in ref. [ll] denoted sdpf.sm. In the 36Ar case we proceed to solve the secular problem in the 4p-4h space. In figure 2 the calculated energy levels are compared with the experimental results in a backbending plot. The agreement is excellent, except at J=12 where the data show a clear backbending while the calculation produces a much smoother upbending pattern. In the experimental data there is a close-by second lO+ state, therefore, the discrepancy may be due to the lack of mixing in the calculation.

342

A. Poves/Nucleav

Physics A731 (2004) 339-346 Ar-36,

SD-band

1

18

14 12

42Oi

Figure

I 0

2. The superdeformed

1000

2000

3000 E-gamma (keV)

4000

5000

band in 36Ar; exp. vs. 4p-4h calculation

In table 2 we have collected the quadrupole properties of the superdeformed band. We have computed the B(E2) ‘s and the spectroscopic quadrupole moments QY using standard effective charges 6q,=Sq,=O.5. The intrinsic (transition) quadrupole moments Qo(t) are extracted as in the 48Cr case. Both Qo(s) and &e(t) are nearly equal and reasonably constant up to the backbending region where they diverge indicating the onset of the alignment regime. The comparison with the experimental B(E2)‘s from ref. [13] is remarkably good. The value of the intrinsic quadrupole moment corresponds roughly to a deformation /7=0.5, so we can speak of a nearly superdeformed band up to J=lO-12. We move now to the study of the superdeformed band of 4oCa, recently measured at Gammasphere [12]. When we analise the maximum quadrupole content of the different np-nh spaces in 4oCa either in the SU(3) limit or reasoning in terms of Nilsson diagrams, we conclude that thk only possible candidate to produce a superdeformed band is the 8p-8h space. This surmise is borne out by the calculations and therefore we shall refer from now on to the 8p-8h results. In 40Ca the valence space adopted for 36Ar leads to very large basis dimensions (2 x 10” for the complete calculation of 12 particles). The dimension of the 8p-8h space is still very large (4.5 x 10’) therefore we have to draw in our experience in the pf-shell, to reduce still a bit the dimensions (to 2.2 x 10’) by limiting the maximum number of particles in the lfsi2 and 2pi,z orbits to two, a truncation that produces results that are already very close to the complete ones. The gamma-ray energies along the yrast sequence are compared to the experimental results (Band 1 in ref. [la]) in fig. 3 in the form of a backbending plot. The calculation reproduces very satisfactorily the experimental results. The only difference is the change of slope in the experimental curve at J=lO, not reproduced by the calculation; this is

A. PoueslNucleav Table 2 Quadrupole efm2) J 2 4 6 8 10 12 14 16

properties

of the 4p-4h

B(E2)(J+J-2) EXP

Physics A731 (2004) 339-346

configuration’s

TH 315 435 453 429 366 315 235 131

372(59) 454(67) 440(70) 316(72) 275(72) 232(53) >84

yrast-band

Qapec

343

in 36Ar (in e2fm4 and

Qo(s)

-36.0 -45.9 -50.7 -52.8 -52.7 -53.0 -54.3 -56.0

126 126 127 125 121 119 120 122

Qo(t) 126 124 120 114 104 96 82 61

L”

24 c 22 201616-

J

14 12 IO8642OO

Figure

3. The superdeformed

I 1000

I 2000

I 3000

E-gamma

(in keV)

I 4000

5

10

band in 40Ca; exp. vs. 8p-8h calculation

probably due to local mixing. Notice that the experimental band is very regular, showing no backbending up to the highest measured spin, J=16, contrary to the situation in 48Cr [4] where the backb en d’mg takes place at J=12. The calculated band is also very regular and backbends only at J=20 which is the band termination for the configuration (1f7j2)* (Id312 2s1$‘. The yrast state with J=22 corresponds to configurations of the type and beyond. The delay in the alignment is surely due to the (1f7/2)7 2P3/2 (l&/z h/2)-' extra collectivity induced by the presence of sd particles in Pseudo-SU(3) orbitals. In table 3 we have collected the quadrupole properties of the superdeformed band. We have computed the B(E2) ‘s and the spectroscopic quadrupole moments Qs as in the 3GAr case. As expected, both &O(S) and Qo(t) are very large, nearly equal and reasonably constant up to the band termination, a fact that supports the existence of a robust intrinsic

344

A. Paves/Nuclear

Table 3 Quadrupole properties sdpf valence space J 2 4 6 8 10 12 14 16 18 20 22 24

B(E2)(J

Physics A731 (2004) 339-346

of the 8p-8h configuration’s --t J-2) 589 819 869 860 823 760 677 572 432 72 8 7

Qspec -49.3 -62.4 -68.2 -70.9 -71.6 -71.3 -71.1 -72.2 -75.0 -85.1 -79.1 -81.5

yrast-band

in 4oCa, calculated

in the

Q,(t)

Qob)

172 170 167 162 157 160 149 128 111

172 172 171 168 164 160 157 158 162

state. The calculated Qo value of 172 efm2 is in very good agreement with the experimental value Qo(t)=1.80?~:$, obtained from the fractional Doppler shifts. This experimental value corresponds to a deformation p-0.6, i.e. to a superdeformed shape. The calculation predicts a slight decrease of the deformation with increasing J, while experimentally it seems to remain constant until the highest measured spin state (J=16). This departure may be due to the blocking of the lds,s orbital. Despite that, the calculation at fix particle-hole number contains most of the relevant physics of the superdeformed band in 40Ca. The deformation that our calculation produces is probably the highest ever obtained in a shell model calculation describing a “bona fide” rotational band. As a matter of fact it almost saturates the SU(3) limit of the intrinsic quadrupole moment in these two major shells; Qo=226 efm2, or the -more realistic- quasi-SU(3)+pseudo-SU3 one (in this case we assume that the particles obey quasi-SU3 and the holes psudo-SU3, see [7]); Qs=180 efm2. At the band termination, the B(E2)‘s drop to zero while the spectroscopic quadrupole moments keep constant, reflecting the transition from the collective to the aligned regime. In 36Ar the SU(3) and q uasi-SU(3)+pseudo-SU3 limits are Qs=173 efm2 and Qs=136 efm2 respectively. The calculated value, Qs=126 efm2 is also very close to the quasi-SU(3)+pseudu-SU3 prediction. The final step consists in finding the mixing among the different np-nh configurations. This is a formidable task because the np-nh series is not easy to truncate. Besides the non yrast states lie in regions of high level densities making the computation very demanding. Thus, we have bee forced to reduce the valence space, eliminating the two upper pf-shell orbits The calculation of the three lower O+ is straightforward, however, the calculation of the excited states belonging to the 4p-4h or the 8p-8h bands is not, because these are most often drowned in a sea of other uninteresting states. To overcome this difficulty we select as starting vectors in the Lanczos procedure the eigenstates of the band obtained in the np-nh space. This choice accelerates the convergence of the states we are seeking

A. Paves/Nuclear

Physics A731 (2004) 339-346

345

&”

1816-

H H

EXP SM-FULL

1412-

J lo8642I OO

I 1000

I 2000

E-gamma

Figure space

4. The superdeformed

I 3000

4

IO

(in keV)

band in 40Ca. , exp. vs. full calculation

in the zbm2 valence

and makes it possible to keep track of them in case of fragmentation. We have used this method to obtain the mixed superdeformed band, starting with the 8p-8h states, The structure of the ground state is the expected one; about 60% closed shell, with mainly 2p2h mixing. The first excited Of is dominantly (60%) 4p-4h with mainly 6p-6h and 8p-8h mixing. The third one is 60% 8p-8h, and the mixing is dominantly 4p-4h. The excitation energies of the superdeformed band change little, but enough to improve the quality of the agreement with the experimental data of the 8p-8h calculation. (see figure 4). However, the quadrupole moments and transition probabilities get eroded -30% to SO%-. We think this is a spurious effect due to the space limitations (if not to some hidden defect of the interaction). Work is in progress to overcome these limitations and to understand in detail the mixing mechanism of the different bands in 40Ca

4. CONCLUSION We have addressed in this talk some issues related to the description of deformation by the spherical shell model. 48Cr provides the ref erence example of a rotor that can be described in a 07~~ calculation. It also suggest that quasiSU3 is the underlying symmetry for nuclear rotors. We have studied the many particle many hole configurations in 36Ar and 40Ca, to understand its recently discovered superdeformed excited bands. We have shown that the yrast bands of the 4p-4h and 8p-8h configurations in the sd-pf valence space reproduce very well the experimental results. In the two shell case, the symmetry at play is the combination of quasi-SU3 for the particles and pseudo-SU3 for the holes. Acknowledgements Madrid collaboration

The results presented in this talk stem from the Strasbourg(E. Caurier, F. Nowacki , A. P. Zuker and A. P.).

A. Poves/Nucleav

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Physics A731 (2004) 339-346

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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