Z. Phys. A 359, 19–22 (1997) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1997

Microscopic aspects of identical bands in78Sr and 78Rb∗

A. Petrovici1,2, K.W. Schmid2, Amand Faessler2

1 Institute for Physics and Nuclear Engineering, R-76900 Bucharest, Romania2 Institut fur Theoretische Physik, Universit¨at Tubingen, D-72076 T¨ubingen, Germany

Received: 26 June 1996 / Revised version: 14 March 1997Communicated by D. Schwalm

Abstract. Recent investigations of the shape transition andshape coexistence phenomena dominating the structure of theeven–evenN ' Z nuclei in theA ' 80 mass region are ex-tended to the odd-odd nucleus78Rb. Special attention is paidto the structure of some “identical” bands which have been re-cently identified in78Sr and78Rb. The ground band of78Sr andthe yrast as well as excited negative parity bands in78Rb arestudied within the EXCITED VAMPIR approximation usingcomplex Hartree-Fock-Bogoliubov transformations in a rel-atively large model space. The results are compared with theavailable experimental data. The emerging picture reveals therole played by the strong quadrupole deformation on the ap-pearance of identical bands as well as the influence of the shapecoexistence on their evolution. Predictions for the electromag-netic and alignment properties of the bands are presented.

PACS: 21.10.-h; 27.50.+e

1 Introduction

The properties of theN ' Z nuclei in theA ∼ 80 regionhave attracted considerable attention in the last decade, boththeoretically as well as experimentally. These nuclei have veryelongated ground-state shapes with the Sr and Zr isotopes in-terpreted as approaching deformations as large asβ2 ∼ 0.4[1,2]. Large deformed shell gaps in the single-particle spec-trum stabilize these shapes much like the subshell closuresassociated with the superdeformed orbitals. Because of thesegaps rapid changes in shape are seen when the particle numberchanges by only two units. Changes in spin manifest them-selves in quite dramatic changes in shape sometimes, too, andmany nuclei in this region exibit shape coexistence phenom-ena. In this context the recently identified so called identicalbands in78Sr and78Rb [3] become a challenge for theoreticalmodels. Besides that, the occurence of identical bands itselfis still an open question [4].

? Work supported by the Institute for Physics and Nuclear Engineering,Bucharest, Romania and the DFG, Germany

Recently we obtained a consistent microscopic picture fora couple of doubly even N'Z nuclei from Z=36 to Z=42 in-cluding the78Sr isotope [1]. This was achieved by a completelymicroscopic variational approach in which all the essentialdegrees of freedom are dynamically determined by the cho-sen Hamiltonian. In each investigated nucleus the lowest fewstates of a given spin and positive parity were approximatedwithin thecomplexEXCITED VAMPIR approach [5], whichlike all the complex versions of the various models of theVAMPIR family [6] includes already in the mean field theproton-neutron interaction and unnatural-parity correlations.In the present study now the negative parity bands in the odd-odd nucleus78Rb are investigated. The yrast sequence (fromspin 5− to 15−) is considered as a candidate for a band be-ing identical to the ground-state band up to spin 12+ in theeven-even nucleus78Sr.

The theoretical spectra are compared with the experimen-tal data presently available. Furthermore predictions for thealignment and electromagnetic properties of the bands are pre-sented. In Sect. 2 we give a rough outline of the theoreticalapproach, define the model space and give some details of theeffective Hamiltonian being used. In Sect. 3 we present thenthe results obtained for the investigated bands. Some conclu-sions are given in Sect. 4.

2 The theoretical framework

As in the earlier calculations for nuclei out of theA ∼ 70 massregion [1,7-9] also in the present work a40Ca core is used andas single particle basis states the 1p1/2, 1p3/2, 0f5/2, 0f7/2,1d5/2 and 0g9/2 oscillator orbits for both protons and neutronsare taken. The corresponding single-particle energies are (inunits of the oscillator energy ¯hω) 0.040, -0.270, 0.300, -0.560,0.157 and 0.029 for the proton, and -0.070, -0.332, 0.130,-0.690, 0.079 and -0.043 for the neutron levels, respectiv-ely.

For the effective two-body interaction, too, the renormal-ized G-matrix out of [1,8,9] is taken. It consists out of anuclear matter G-matrix derived from the Bonn One-Boson-Exchange potential [10] modified by two short range (0.707fm) Gaussians for the isospinT=1 proton-proton and neutron-

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Fig. 1.The theoretical spectrum of78Srfor even spin positive parity states andthe lowest odd- and even-spin neg-ative parity bands in78Rb are com-pared with the recent experimental re-sults [3,12]. The calculated states areobtained within the EXCITED VAM-PIR approximation based oncomplexHFB transformations. The labels (oifor odd- andei for even-spin bands)are used to indicate the correspondencebetween the experimental and the the-oretical bands. Strong M1 transitions(larger than one W.u.) are indicated bya label m

neutron matrix elements with strengths of -40 MeV and-30 MeV, respectively, and the enhancement of the isoscalarspin 0 and 1 particle-particle matrix elements by an addi-tional Gaussian with the same range and a strength of -180 MeV. Furthermore, the interaction contains an isospin-independent Gaussian-shape two-body spin-orbit force (range0.5 fm, strength -1500 MeV), and monopole shifts of -300keV for all the diagonal isospin T=0 matrix elements of theform 〈0g9/20f ; IT = 0|G|0g9/20f ; IT = 0〉 with 0f denot-ing either the 0f5/2 or the 0f7/2 orbit, and -500 keV in the〈1p1d5/2; IT = 0|G|1p1d5/2; IT = 0〉matrix elements, where1p denotes either the 1p1/2 or the 1p3/2 orbit, which have beenintroduced in the earlier calculations to influence the onset ofdeformation.

This Hamiltonian was adjusted in various variational cal-culations usingcomplexmean fields [1,7-9] and gave a ratherconsistent picture for the ground and excited bands in even-evenN ' Z isotopes from Z=36 to Z=42 [1].

In order to investigate the microscopic structure of thelowest even spin positive parity states in78Sr and the lowestnegative parity bands in78Rb first the VAMPIR solutions start-ing from intrinsically prolate and oblate deformed trial con-figurations are constructed. From these the most bound onesare selected. These solutions are the optimal representation ofthe yrast states by single symmetry projected Hartree-Fock-Bogoliubov quasi-particle determinants.

For excited states of a given symmetry then the EXCITEDVAMPIR approach is used which constructs each additionalexcited state by an independent variational calculation im-posing orthogonality with respect to the solutions already ob-tained. Finally the residual interaction between the various so-lutions with the same symmetry is diagonalized. In the present

work up to three such EXCITED VAMPIR states for eachsymmetry are used.

3 Results and discussion

Along these lines the lowest even spin positive parity statesup to angular momentum 12+ in 78Sr and the lowest negativeparity bands up to spin 17− in 78Rb were calculated. The maingoal was to get some microscopic insight into the structure ofthe so called identical bands in78Sr and78Rb, which have beenrecently identified experimentally [3].

For all the investigated states the oblate minima are muchhigher in energy than the prolate deformed ones. We found [1]that the ground-state band in78Sr is strongly prolate deformedand, in agreement with the experimental observation [3], theside bands are very high in energy with respect to the yrastone. So, e.g., the second 8+ state being prolate deformed, too,occurs 2.2 MeV above the first one. Consequently, no shapemixing effects were found in78Sr. The resulting even spinground band is compared with the experimental data [3,11] inFig. 1.

A completely different picture is obtained for the nega-tive parity bands in78Rb. The lowest three solutions for theodd spin states from spin 5− to spin 17− are very close inenergy. The lowest three 9− solutions are separated by only250 keV. For the 11−, 7−, and 13− states 400, 500 and 600keV, respectively, are obtained. The maximal separation ofthe lowest three solutions is 900 keV and occurs for spin 17−.Even after the diagonalization of the residual interaction theresulting states are still bunched in a small excitation energyinterval for each spin though strongly mixed for particular val-ues of angular momentum. The results for the energeticallylowest two odd and even spin sequences are again displayed

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Table 1.Spectroscopic quadrupole momentsQsp2 (in efm2) of ground-band(gb) states for the nucleus78Sr and the negative parity bands of78Rb presentedin Fig. 1. As effective chargesep = 1.5 anden = 0.5 have been used

78Sr c78RbI[ h] gb o1 e1 o2 e2

2+(5−) −85.2 −120.5 −106.06− −125.0 −109.0

4+(7−) −100.8 −128.0 −111.08− −129.0 −111.0

6+(9−) −117.8 −126.0 −112.410− −128.0 −107.0

8+(11−) −122.3 −127.4 −105.612− −122.0 −108.0

10+(13−) −123.7 −124.4 −104.214− −116.0 −113.0

12+(15−) −118.2 −121.0 −103.316− −120.0 −107.0

Table 2.B(E2;I → I − 2) values (in W.u.) for some states of the nuclei78Sr and78Rb. For78Rb the strengths for the secondary branches indicatedin Fig. 1 are given in parenteses. As effective charges againep = 1.5 anden = 0.5 have been used

78Sr c78RbI[ h] gb o1 e1 o2 e2

4+(7−) 126 151 1198− 154 115

6+(9−) 137 136 79(15)10− 141 67

8+(11−) 140 105(26) 48(28)12− 125 79(16)

10+(13−) 137 143 10114− 94(24) 110

12+(15−) 115 138 9916− 114 99

in Fig. 1. The experimental data for Rb have been taken from[3,12]. It is worthwhile to mention here that a higher precisionfor the energy intervals in78Rb presenting strong mixing forsome spins could be achieved increasing the dimension of themany-nucleon basis, but this procedure increases very muchthe computing time without changing the qualitative conclu-sions.

It should be mentioned that one of the three calculatedbands (the odd spin sequence denoted byo1 and its even spinanalogue labellede1 in the following) is considerably moredeformed than the other two and displays similar quadrupole,octupole and hexadecapole deformations in theintrinsic sys-tem as obtained for the ground-state band in78Sr. Furthermorethe proton contribution to the pairing gap is rapidly decreasingwith increasing spin in this band while for the other bands itremains almost constant. Up toIπ ≤ 9− this band is yrast. Inthe final wave functions for these states the dominant symme-try projected determinant exhausts 81% for spin 9−, 90% for11− and 13−, 85% and 67% for 15− and 17−, respectively.The corresponding even spin band is yrast up to spin 12−.Here the dominant component exhausts 90% for spin 8− and10−, 74%, 55%, and 79% for 12−, 14− and 16−, respectively.

As already mentioned, the second pair of negative paritybands displays other properties. The quadrupole deformationis smaller, and the pair correlations are not reduced with in-

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Fig. 2. The alignment plot representing the angular momentum contributionof the nucleons (neutrons-ν, protons-π) filling the f5/2 and/org9/2 sphericalbasis states in the direction of the total angular momentumI for the ground-state band in78Sr is compared with those for the odd spin sequenceo1 (opensymbols) and the corresponding even spin sequencee1 (full symbols) in 78Rb

creasing spin. On the other hand the odd (o2) and even (e2)spin sequences show a rather similar configuration mixing asobtained in theo1 ande1 bands discussed above.

Table 1 presents the calculated spectroscopic quadrupolemoments for the investigated states. It is clearly seen thatthe members of the ground state band of78Sr have similarquadrupole moments as the odd spin negative parity sequenceo1 in 78Rb as well as the corresponding even spin sequencee1. This becomes even more evident, if the corresponding “in-trinsic” quadrupole momentsQ0(I) = −(2I + 3)Qsp2 (I)/I arecompared. Thus, the quadrupole deformation for the 2+ statein 78Sr isβ = 0.36 and for the yrast 5− in 78Rb amounts toβ = 0.39. On the other hand the quadrupole moments for thesecond odd (o2) and even (e2) spin negative parity states in78Rb are considerably smaller than those of the78Sr groundband.

Table 2 displays the corresponding B(E2) values. Againsimilar values are obtained for the Sr-ground band and theo1 ande1 sequences in Rb. However, it becomes evedent, too,that for higher angular momenta this similarity is deterioratingbecause of the configuration mixing obtained for the Rb bands.

A particular way to investigate the microscopic structureof the discussed bands is offered by the alignment plot. Thealignment is defined as the angular momentum contributionof the nucleons filling a given spherical state in the directionof the total angular momentumI. The high j orbital whichcould be involved in a strong and sharp alignment in this massregion isg9/2 spherical basis state for both neutrons and pro-tons. In Fig. 2 we present the alignment plots for the nucleonsoccupying theg9/2 andf5/2 spherical orbitals for the states

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Fig. 3.Same as in Fig. 2, but for the second pair (o2 ande2) of negative paritybands in the nucleus78Rb

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Fig. 4.The alignment of all the nucleons occupying theg9/2 spherical orbitalrelative to the ground band in78Sr is plotted for the odd spin negative paritybands in78Rb. The same labels as in Fig. 1 are used

of the ground band in78Sr and the negative parity sequenceso1 ande1 in 78Rb. As can be seen in all these bands the maincontribution to the total angular momentum comes from theg9/2 protons. They display a rapid alignment with increasingangular momentum. On the other hand the contribution fromtheg9/2 neutrons shows only a moderate alignment while thecontributions from both thef5/2 protons and neutrons stayalmost constant in all the bands.

Figure 3 shows the alignment plots for the second pair ofnegative parity states in78Rb. As compared to Fig. 2 here asmaller alignment is observed for theg9/2 protons while thecontribution from the neutrons filling this orbital is increased.Again thef5/2 orbit stays almost inert for both protons andneutrons.

The angular momentum alignment relative to the groundband states in78Sr for the two odd spin negative parity se-

quences in78Rb is presented in Fig. 4 taking into accountonly the contributions of theg9/2 protons and neutrons. Asexpected the relative angular momentum alignment is largerfor the less deformed band. Note that the relative alignmentsfor both bands is less than the three units expected from the spindifference with respect to the Sr ground band. This indicatescontributions from other basis orbitals to the relative angularmomentum.

Finally we would like to mention that in Fig. 1 the strongestM1, ∆I = 1 transitions characterised by B(M1) values largerthan 1 W.u are indicated. The second pair of bands is connectedby M1 transitions, too, but these are a factor of 10 weakerthan the corresponding strengths connecting theo1 with thee1band. For the indicated M1 transitions the orbital contributionamounts to 40% - 50%.

4 Conclusions

In the present work thecomplexEXCITED VAMPIR approx-imation was used to investigate the structure of the identicalbands recently identified in the nuclei78Sr (ground state bandup to spin 12+) and78Rb (one of the two odd spin negativeparity bands populated in this nucleus, from spin 5− up tospin 15−). Indeed strong similarities between the calculatedground state band in78Sr and the most deformed negativeparity band in78Rb are found. However, at higher angularmomenta these similarities are deteriorating because of thestrong configuration mixing obtained for the negative paritysequences in Rb. More experimental information, especiallyabout the electromagnetic properties of the bands are neededfor a more detailled comparison. On the other hand the the-oretical calculations can be improved, too, by using a largernumber of symmetry-projected configurations for each partic-ular state under investigation.

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