Collective bands in 84Sr

Nuclear Physics A501 (1989) 311-318 North-Holland, Amsterdam

C O L L E C T I V E B A N D S IN 84Sr*

R. SAHU

Physics Department, Berhampur University, Berhampur-760007, Orissa, India Received 27 February 1989 (Revised 25 April 1989) Abstract: The structure of the collective bands in S4Sr is investigated within the framework of the

deformed configuration-mixing shell model based on Hartree-Fock states. The configuration space consists of the single-particle orbits lp3/2, 01"5/2, lp~/2 and 0g9/2. An effective interaction, given by Kuo and modified by Bhatt, has been used. The calculated levels, having a similar structure, are grouped into collective bands. The predicted lowest K = 0+ band, the quasi-gamma band and two negative-parity bands agree reasonably well with experiment. The structure of the 8+ triplet state, predicted from our theoretical analysis, is in agreement with the results obtained from the g-factor measurement of these states. The calculated B(E2) values also agree well with experiment.

I. I n t r o d u c t i o n

The nuclei in the mass region A = 6 0 - 9 0 have b e e n the subject of extensive e x p e r i m e n t a l a n d theoretical t r e a t m e n t in the last few years. These nuclei exhibit a variety of n u c l e a r p h e n o m e n a like coexistence of shapes, large g r o u n d - s t a t e deformation, b a n d crossing, rapid variations of structure with c h a n g i n g n u c l e a r n u m b e r , etc. The n e u t r o n - d e f i c i e n t s t r o n t i u m isotopes such as 8°Sr have b e e n e x p e r i m e n t a l l y f o u n d to be a m o n g the m o s t - d e f o r m e d nuclei in this region. O n the other h a n d , SSSr with a closed n e u t r o n shell at N = 50 a n d a fairly good p r o t o n - s u b s h e l l closure at Z = 38, is f o u n d to be d o u b l y magic and, hence, spherical. The n u c l e u s S4Sr is of p a r t i c u l a r interest since it is located at the b o u n d a r y b e t w e e n the spherical a n d d e f o r m e d s t r o n t i u m isotopes. H i g h - s p i n states in 848r have b e e n e x p e r i m e n t a l l y investigated by D e w a l d et al. l) by m e a n s of c o n v e n t i o n a l i n - b e a m g a m m a - r a y spectroscopy with (HI, x n ) reactions. They have f o u n d a close-lying 8 + triplet state above the yrast 6 + state. They have interpreted the b a n d b u i l t on the first 8 + state to be a n e u t r o n - a l i g n e d b a n d a n d that built o n the s e c o n d 8 + state to be a p r o t o n - a l i g n e d b a n d . The third 8 + state is c o n s i d e r e d to b e l o n g to the g r o u n d b a n d . In a d d i t i o n , they have observed a quasig a m m a b a n d a n d two negative-parity b a n d s . Low-spin states of this n u c l e u s were also studied by Rester et al. 2) using the (a, c~') experiments, a n d by Ball et aL 3) * Supported by the Department of Science and Technology, Government of India. 0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

312

R. Sahu / Collective bands

using the (p, t) reactions. The latter experiment identifies three excited 0 + states and four 2 + states below 3.5 MeV excitation energy. The systematics of the quasi-gamma band for 74Se, 78'S°Kr and Sasr were analyzed by Yoshikawa et al. 4) using the (p, 2ny) reaction. Very few theoretical calculations have been attempted for this nucleus. Shell-model calculations have been carried out for 8o 86Sr by Ogawa 5) and Kitching et al. 6) In Ogawa's calculation 9°Zr is treated as the core. The two proton holes are assumed to have the configuration (lpl/2, lp3/z, 0f5/2) 2 while the neutron holes are restricted only to the 0g9/2 hole orbit. On the other hand, Kitching and coworkers assumed a closed proton shell at Z =38 and only the 0g9/2 and lp~/2 neutron shells were considered. Both the calculations gave good agreement with experiment for the low-spin states of 86Sr. However, the description of the isotopes s4's2'8°Sr was less successful, even for the low-spin states. An IBM calculation has been performed by Dewald et aL J) for S4Sr, which has satisfactorily described the ground band and gamma band. However, none of the theoretical models described above give any insight into the structure of the bands built on the 8* states in S4Sr. Hence, it would be quite interesting to perform a microscopic calculation which can provide information regarding the structure of these collective bands. We recall that in ref. 7) a similar situation was observed in 68Ge and the experimental group had assigned one of the 84 states to proton alignment. Later theoretical and experimental work showed this to be incorrect. The deformed configuration-mixing shell model, based on Hartree-Fock states, has been found to be quite successful in explaining many features of the nuclei in the region A = 60-90 [refs. 8 ,~)]. Hence, we feel that it would be worthwhile to apply this model to understand the 8+ triplet state and other characteristics of Sasr. Sect. 2 contains the results of the present calculation. Sect. 3 gives a brief summary and a discussion of the results.

2. Deformed shell model calculations

As in our earlier calculations, we consider 5*Ni as the inert core with active nucleons distributed in lp3/2 of 5/2, lp~/2 and 0gw2 single-particle orbitals. The single-particle energies for the first three orbitals are taken, from 57Ni data ~), to be 0.0, 0.78 and 1.08 MeV. Experimentally the energy of the gg/~ orbital is not well defined. Fortier and Gales ~) found a weakly excited 2+ level at about 3 MeV. For lighter nuclei, like Ge and Se isotopes, we found it in refs. 7~9) appropriate to take the single-particle energy of this level to be around 4.5 MeV. However, our recent studies of ~°Zr [ref. "~)] suggest that this energy is lower for heavier nuclei. In the present calculation, we take this value to be 3.5 MeV. We assume the single-particle energies for both protons and neutrons to be the same. We use a modified Kuo interaction in our calculation. The same effective interaction was also used by us in our earlier works 9,.~,12).

R, Sahu / Collective bands

313

In our calculation, the lowest-energy prolate intrinsic state is obtained by performing an axially symmetric Hartree-Fock calculation. The Hartree-Fock single-particle spectrum for 84Sr is shown in fig. 1. The protons are distributed entirely in the pf single-particle orbits while the neutrons occupy both the pf and g9/2 orbits, the k = ~7+ and 9+ orbits being empty. For calculating positive-partity states, we consider four excited intrinsic Hartree-Fock states in addition to the lowest one. We obtain two excited intrinsic states with K = 0 + by placing two or four protons in the g9/2 orbitals and then performing a self-consistent HF calculation for the rest of the particles (tagged HF). A similar tagged HF calculation gives a K = 8+ intrinsic state in which a neutron is placed in each of k =~7 + and 9~+ orbits. In addition a K = 2 + intrinsic state is considered by placing a proton in k = I - and ~1 orbits. Good angular-momentum states are projected from each of these intrinsic states. However, these angular momentum states projected from different intrinsic states are, in general, not orthogonal to each other. Hence, a standard band mixing calculation

84Sr

9+ 0.0

7+

E 9-I-

>-

: " L. .. . . ....

L9

n~ bJ Z W

7+

Z o

5.0

x w

5

....

5+ 3+ 1+ 1

<

5+ 3+ 1+

:,; :-"

5 3

0 0

0 0

0

0

-"~-

-~'~

3 1

;'--

~-:

1

1,3 -

I0.C

1

E

-----64.77,

Q .= 1 5 . 5 7 , K =

0+

Fig. 1. The spectra of single-particle HF orbits for the lowest-energy prolate intrinsic state. The numbers next to the levels denote 2k values. The protons are represented by circles and the neutrons by crosses. The positive-parity levels are marked by a plus sign. All other levels have negative parity. The HF energies in MeV and mass quadrupole moments (b 2) are also shown.

314

R. Sahu / Collective bands

is p e r f o r m e d to o r t h o n o r m a l i z e these states 13). If O ~ ( a ) is the o r t h o n o r m a l i z e d wave f u n c t i o n for a given J, a n d 6MK(~7) J is the wave function p r o j e c t e d from the intrinsic state )(k(~7), then the overlap, B ~ ( , , a)--- (~'~,, ( , ) 1 4 , ~ , ( a ) ) ,

(1)

gives the a m p l i t u d e o f the o r t h o g o n a l i z e d eigenstate ~bM(a) J c o n t a i n e d in the state J J ~MK(rl). This a m p l i t u d e Bk('q, a ) also is a m e a s u r e o f the p r o b a b i l i t y that a given o r t h o n o r m a l i z e d eigenstate b e l o n g s to a p a r t i c u l a r intrinsic state. The quantity ( l - I B m ] 2) gives the intensity in the state ~b~a(a) o f all the states o r t h o g o n a l to J ~mK(r/) a n d thus p r o v i d e s a m e a s u r e o f b a n d mixing. In o u r c a l c u l a t i o n all the states having a similar p a t t e r n o f B~(~7, a ) values are classified into one band. F o r the c a l c u l a t i o n o f the n e g a t i v e - p a r i t y b a n d s , we c o n s i d e r five intrinsic states. A m o n g t h e m one intrinsic state has K = 2 , two o f these have K = 3 , one has K = 4 a n d a n o t h e r intrinsic state has K = 5 . These intrinsic states are o b t a i n e d by placing one p r o t o n in the 89/2 orbital a n d an o d d n u m b e r o f p r o t o n s in the p f orbit. However, one can also o b t a i n n e g a t i v e - p a r i t y intrinsic states by p l a c i n g three p r o t o n s in the g9/2 orbital. H o w e v e r , such t h r e e - q u a s i p a r t i c l e states lie at an excitation energy o f m o r e t h a n 4 MeV as is e v i d e n t from the H F single-particle spectrum. Hence, we do not c o n s i d e r such t h r e e - q u a s i p a r t i c l e states. We project out g o o d a n g u l a r m o m e n t u m states from each o f these intrinsic states a n d then mix t h r o u g h the h a m i l t o n i a n using a s t a n d a r d b a n d - m i x i n g calculation. As has been d i s c u s s e d in our earlier paper'S), we do not c o n s i d e r the o b l a t e states since they are not likely to occur at low energy in this region.

2.1. ENERGY SPECTRUM The c a l c u l a t e d levels are classified into different collective b a n d s as d e s c r i b e d above. These collective b a n d s are shown in fig. 2a. We have identified, in a d d i t i o n to the g r o u n d b a n d a n d the b a n d s built on the 8 + states, a K = 2 + b a n d and two n e g a t i v e - p a r i t y bands. The e x p e r i m e n t a l l y o b s e r v e d collective b a n d s have been shown in fig. 2b. The overall a g r e e m e n t is quite satisfactory. The b a n d - m i x e d wave functions o b t a i n e d in o u r c a l c u l a t i o n give i n f o r m a t i o n r e g a r d i n g the structure o f these b a n d s . These wave functions o f the states o f the g r o u n d b a n d a n d the three 8+ states are given in table 1. An analysis o f the wave functions suggests that all the c a l c u l a t e d levels o f the g r o u n d b a n d up to J = 6 ' m a i n l y b e l o n g to the K = 0 intrinsic state r e p r e s e n t e d by the serial no. 1. This intrinsic state is the lowest-energy p r o l a t e H F state. H o w e v e r , we observe that the b a n d m i x i n g increases with the increase o f spin. F o r e x a m p l e , the J = 0 + state o f the g r o u n d b a n d has more than 81% p r o b a b i l i t y o f the intrinsic state 1. H o w e v e r , for J = 4 + or J - 6 +, this p r o b a b i l i t y decreases to a b o u t 55%. The first 8' state has a b o u t 49% p r o b a b i l i t y o f the intrinsic state 5 which is the n e u t r o n a l i g n e d K = 8 + state. However, there is a very strong mixing o f the intrinsic state 1. We identify this state to be m a i n l y a n e u t r o n - a l i g n e d

R. Sahu / Collective bands

315

band. The s e c o n d 8 + state has m o r e than 90% probability o f the intrinsic state 2 which is an excited K = 0 + state obtained by placing two protons into the g9/2 orbital. Hence, w e find the band built on this 8 + state to be mainly a proton-aligned band. The third 8 + state is a highly m i x e d state. It has only 36% probability o f the intrinsic state 5 and about 30% probability each o f intrinsic state nos. 1 and 4. D e w a l d etaL 1) have also given a similar interpretations for these 8 + states. Recently Kucharska et aL 14) have m e a s u r e d the g-factors of the first two 8 + states. They have found that the g-factor o f the first 8 + state is negative and that for the second 8 + state is positive. From this they have c o n c l u d e d that the first 8 + state belongs to a neutron-aligned band and the s e c o n d 8 + state to a proton-aligned band. This agrees with our predictions regarding the structure of these levels. Our calculated K = 2 + band mainly originates from the intrinsic state with K = 2 +. The calculated band head o f this band is about 0.4 M e V higher than the experimental value. H o w e v e r , the relative spacings have been well reproduced. D e w a l d et al. 1) have observed two negative-parity bands, one starting with J = 3 level and another with J = 5- level. The band built on the J = 5- level is well reproduced in our calculation. The 5 , 7-, 9 and 11- levels of the band built on the J 3 level =

(a)

=

84Sr

14+ io+

12+

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10.2 5.0

+ 10+

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]E >r~

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3.2

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z

11.7

8+

~10.5

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/ / /

/ ~ 2/ .8

0

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{"g9/=)

3+

P<

22.9 4+

,...;,

12

9+ .._.~ '

3-.4:~14. 6 8- i ~2 3~ ~ - -1_ _ lO.8

2.9 4.3 4.3 5.6 7.3

o ~'~13.3

7-

2 419.3

uJ

2+ 12.1 0.0

0+ D.C.M.

Fig. 2a.

The different collective bands obtained from our microscopic calculation. The quantities near the arrows represent the B(E2) values in W.u.

R. Sahu / Collective bands

316 (b)

84Sr

14+

( 1 2 ) - 12 + 11 14 5.0

10 10~"

¢1

9

10+--------[--

~E v

34

g+

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8

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(7) +

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5

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26 5.0

0+ EXpT.

Fig. 2b. The experimental collective bands observed by Dewald et aL ~). The quantities near the arrows represent B(E2) values in W.u. are also well r e p r o d u c e d . H o w e v e r , in o u r c a l c u l a t i o n the J = 3 level lies higher in energy than the 5 a n d 7 levels. In o u r earlier c a l c u l a t i o n o f ~'6Zn [ref. ,5)], we h a d a similar feature also. This 3 b a n d has been i n t e r p r e t e d by D e w a l d et al. to be o f o c t u p o l e nature. We feel that the full collectivity o f the 3 state c a n n o t be r e p r o d u c e d in o u r limited configuration space. If the higher ds orbits were i n c l u d e d , the 3 w o u l d d e v e l o p the full collectivity a n d w o u l d be at a much lower energy. P r a h a r a j ,6) has recently s h o w n that nuclei in this region are soft with respect to o c t u p o l e d e f o r m a t i o n , but he has a much larger configuration space. D e w a l d el al. have also p e r f o r m e d an IBM c a l c u l a t i o n for the yrast b a n d a n d K - 2 + b a n d . The a g r e e m e n t with e x p e r i m e n t is quite satisfactory. Ball et al. ~) have m e a s u r e d in a ( p , t ) r e a c t i o n three excited 0 + states a n d three excited 2 T states b e t w e e n 1.5 a n d 3.5 MeV. The a b o v e IBM c a l c u l a t i o n correctly predicts these levels. In o u r c a l c u l a t i o n we find that, in the g r o u n d state o f S4Sr, all the p r o t o n s are d i s t r i b u t e d in the p f orbits (fig. 1). Since S6Sr is e x p e c t e d to be m o r e spherical than S4Sr, all the p r o t o n s in the g r o u n d state o f S~'Sr s h o u l d also be d i s t r i b u t e d in the p f orbits. Actual c a l c u l a t i o n also confirms this. The excited intrinsic states in which there are two or m o r e p r o t o n s in the g9/2 orbital w o u l d be o r t h o g o n a l to the a b o v e

317

R. Sahu / Collective bands

TABLE 1 The band-mixed wave function B~(r/, a) of the levels of the ground band and three 8+ levels. The first two rows represents the serial number of the five intrinsic HF states used and the K quantum number of the corresponding intrinsic states. The band-mixed wave functions are given from the third to the seventh column. The quantities in parentheses represent the experimental energies of the dilterent levels. j

0+ 2+ 4+ 6+ 8~ 8~ 83

B~(~,a)

Energy (MeV)

0.0 (0.0) 1.028 (0.793) 2.055 (1.768) 2.917

(2.808) 3.430 (3.271 ) 3.969 (3.332) 4.814 (4.029)

I:K=0 +

2:K=0 +

3:K=0 +

4:K=2 +

5:K=0 +

0.929

-0.368

-0.027

0.0

0.0

-0.817

0.525

0.044

0.252

0.0

-0.739

0.654

0.054

0.209

0.0

-0.758

0.645

0.042

0.219

0.0

-0.724

0.281

0.0 l 0

0.387

-0.722

0.112

0.948

0.040

-0.104

0.300

-0.550

-0.124

-0.007

0.515

0.619

lowest-energy intrinsic states in which protons occupy only the pf orbits. Hence, the g o o d angular m o m e n t u m states which originate from such excited intrinsic states with two or more protons in the g9/2 orbital would not be excited in the 86Sr(p, t)S4Sr reactions. Thus, the 0 + and 2 + states in 84Sr, excited in these reactions and lying between 1.5 and 3.5 MeV excitation energies, must originate from the intrinsic states in which all the protons are in the pf orbits. From fig. 1, it is clear that we can obtain such intrinsic states by promoting two protons from the 2- or 3 occupied orbits to the ½- unoccupied orbit. However, we have not considered such intrinsic states in our band mixing calculation since we are mainly interested in understanding the structure o f 8 + states and other basic features such as band structures. A more elaborate calculation with more intrinsic states will be carried out later.

2.2. B(E2) VALUES T h e c a l c u l a t e d a n d e x p e r i m e n t a l B ( E 2 ) v a l u e s a r e g i v e n in fig. 2a a n d 2b. In o u r c a l c u l a t i o n , w e h a v e t a k e n t h e e f f e c t i v e c h a r g e o f p r o t o n to b e 1.6e a n d o f t h e n e u t r o n t o b e 1.0e. T h e s e e f f e c t i v e c h a r g e s w e r e a l s o u s e d in o u r e a r l i e r c a l c u l a t i o n s . As has b e e n d i s c u s s e d earlier, we have classified the different levels into collective b a n d s o n t h e b a s i s o f t h e b a n d m i x e d w a v e f u n c t i o n s . S i n c e all t h e levels o f a c o l l e c t i v e b a n d h a v e s i m i l a r b a n d - m i x e d w a v e f u n c t i o n s a n d h e n c e s i m i l a r structure,

318

R. Sahu / Collective bands

they s h o u l d be c o n n e c t e d by a strong E2 transition. A c t u a l c a l c u l a t i o n shows that the i n t r a b a n d E2 t r a n s i t i o n p r o b a b i l i t i e s are always larger than the i n t e r b a n d transition p r o b a b i l i t i e s by an o r d e r o f m a g n i t u d e . The c a l c u l a t e d B(E2, 2 ~ ~ 0 +) is s m a l l e r t h a n e x p e r i m e n t by a b o u t a factor o f two. H o w e v e r , all o t h e r E2 transition p r o b a b i l i t i e s o f the g r o u n d b a n d a n d K = 8 + b a n d s are r e a s o n a b l y well r e p r o d u c e d . The B ( E 2 ) values for the K = 2 + b a n d a n d K = 5- b a n d have not been e x p e r i m e n t a l l y m e a s u r e d . We have m a d e p r e d i c t i o n s for these two bands. The B(E2) values for the K = 3 b a n d agrees in t r e n d with e x p e r i m e n t .

3. Conclusion We have tried to u n d e r s t a n d the structure o f the collective b a n d s in S4Sr using o u r m i c r o s c o p i c model. We have p r e d i c t e d the b a n d built on first 8 + state to be m a i n l y a n e u t r o n - a l i g n e d b a n d . The b a n d built on the s e c o n d 8 + state is p r e d i c t e d to be a p r o t o n - a l i g n e d b a n d . The p r e d i c t i o n s r e g a r d i n g the structure o f the first two 8 + states are in a g r e e m e n t with the e x p e r i m e n t a l m e a s u r e m e n t s o f the g - f a c t o r o f the 8 + states by K u c h a r s k a et al. 14). We find the third 8 ~ state to be a highly m i x e d state. The c a l c u l a t e d K = 2 ~ b a n d and n e g a t i v e - p a r i t y b a n d s agree r e a s o n a b l y well with e x p e r i m e n t . The J = 3 level o f the K - 3 b a n d lies much higher in energy c o m p a r e d to e x p e r i m e n t . I n c l u s i o n o f d- a n d s-orbits in the basis m a y r e p r o d u c e the full collectivity o f this level a n d m a y bring it down. M o s t o f the B(E2) transition p r o b a b i l i t i e s are also in g o o d a g r e e m e n t with e x p e r i m e n t . The a u t h o r is t h a n k f u l to Prof. S. P. P a n d y a for critically r e a d i n g the m a n u s c r i p t a n d for giving v a l u a b l e suggestions.

References 1) A. Dewald, U. Kaup, W. Gast, A. Gelberg, H. W. Schuh, K. O. Zell and P. yon Brentano, Phys. Rev. C25 (1982) 226 2) A.C. Rester, B. Van Nooijen, P. Spilling, J. Konijn and J.J. Pinajian, Phys. Rev. C7 (1973) 210 3) J.B. Ball, J.J. Pinajian, J.S. Larsen and A.C. Rester, Phys. Rev. C8 (1973) 1438 4) N. Yoshikawa, Y. Shida, O. Hashimoto, M. Sakai and T. Numao, Nucl. Phys. A327 (1979) 477 5) K. Ogawa, Phys. lett. I]45 (1973) 214 6) J.E. Kitching et aL, Nucl. Phys. A177 (1971) 433 7) R. Sahu and S.P. Pandya, Nucl. Phys. A414 (1984) 240 8) D.P. Ahalpara and S. P. Pandya, J. of Phys. GI2 (1986) 15 9) R. Sahu, D.P. Ahalpara and S.P. Pandyna, J. of Phys. GI3 (1987) 603 10) R. Sahu and S.P. Pandya, J. of Phys. GI4 (1988) L165 11) S. Fortier and S. Gales, Nucl. Phys. A321 (1979) 137 12) D.P. Ahalpara, K.H. Bhatt and R. Sahu, J. of Phys. GI1 (1985) 735 13) A.K. Dhar, D.R. Kulkarni and K.H. Bhatt, Nucl. Phys. A238 (1975) 34(1 14) A.I. Kucharska, J. Billowes and C.J. Lister, Int. Workshop on nuclear structure of the zirconium region, Bad Honnef, 1988, (Springer, Berlin, 1989) 15) D.P. Ahalpara, K.H. Bhatt, S.P. Pandya and C.R. Praharaj, Nucl. Phys. A371 (1981) 210 16) C.R. Praharaj, private communication