Excited States and Collectivity in 88Se

1Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble Alpes, CNRS/IN2P3, Grenoble INP, F-38026 Grenoble, France 2Institut Laue-Langevin, 71 avenue des Martyrs, 38042 Grenoble Cedex 9, France 3Université de Strasbourg, IPHC, 23 rue du Loess 67037 Strasbourg, CNRS, UMR7178, 67037 Strasbourg, France 4Grand Accélérateur National d’Ions Lourds, CEA/DRF–CNRS/IN2P3, Boulevard Henri Becquerel, F-14076 Caen, France 5INFN, via Marzolo 8, 35131 Padova, Italy 6ELI-NP/IFIN-HH, 30 Reactorului Str., 077125 Magurele, Romania 7Faculty of Physics University of Warsaw, ulica Pasteura 5, PL-02-093 Warsaw, Poland


Introduction
A key question in nuclear-structure studies is how to microscopically describe the onset of collective behavior when moving away from shell closures in terms of the interactions between individual nucleons occupying single-particle states. Little is known about the neutron-rich A ∼ 90 nuclei North-East of 78 Ni which can possess collective states when just 4 valence protons and 2 neutrons are present [1]. The fragile nature of the 78 Ni core has been suggested as a likely reason for the presence of (5, 6 + ) states at an energy of ∼3 MeV in 82 Ge [2] and a tendency towards γ softness in 84 Ge [1]. Experimental data suggest the (2 + 1 ) state of 86 Ge may be deformed [3]. Recently evidence has been found for collective behavior in the neutron-rich Se isotopes, namely the beginning of a γ-band in 86 Se [4,5], and a j−1 ground-state spin of (3/2 + ) for 87 Se [6]. An 886.2-keV transition was assigned as the decay   of the (2 + 1 ) state in 88 Se [7] however this was recently attributed to 87 Se [6], leaving the level scheme of 88 Se unknown. Comprehensive shell-model (SM) and symmetry conserved configuration mixing-Gogny (SCCM) calculations have been published for 88 Se [8]. These predict that a significant amount of both quadrupole deformation and triaxiality should be present in 88 Se [8]. In this context, it is interesting to study the structure of 88 Se.
The neutron-rich A∼90 Se nuclei are populated with a higher fission yield in the 235 U(n th , f ) reaction than following the spontaneous fission of 248 Cm or 252 Cf, used in previous prompt-γ studies of this isotopic chain [4,6,7]. The 235 U(n th , f ) reaction has therefore been used, in conjunction with the EXILL Ge array [9], to study γ-rays emitted by 88 Se.

Experimental Technique
The experiment was carried out at the PF1B cold-neutron beam of the Institute Laue-Langevin in Grenoble, using the EXILL array of Ge detectors [9]. Thermal neutron-induced fission of a 235 U target was used to create 88 Se nuclei and prompt γ − γ − γ spectroscopy allowed decays from their excited states to be observed. More details on the experimental technique can be found in Ref. [10].

Experimental Results
Gates were set on known transitions from each of the isotopes 84−87 Se and the distribution of the average masses of the Ce partner isotopes obtained. From this it was found that 146 Ce is the most likely fission partner of 88 Se in the 235 U(n th , f ) reaction. As no transitions are known in 88 Se then gates were set on the intense 2 + →0 + (258.2 keV) and 6 + →4 + (502. the spectrum shown in Fig. 1. Here one observes lines previously attributed to 86 Se (4n channel) [4] and 87 Se (3n channel) [6], along with other transitions from 146 Ce [11]. Four unidentified lines are present at 589.4, 653.5, 961.9, and 1242.5 keV.
A double gate was then set on the new 589.4-and 961.9-keV lines and strong γ-ray transitions from the complementary fragments 145,146,147 Ce were present. From the measured average mass of the coincident Ce isotopes [145. 8(2)] this cascade was assigned to 88 Se. The above coincidence relationships, and further gating, has allowed the decay scheme of 88 Se shown in Fig. 1(b) to be constructed. This analysis is explained in more detail in Ref. [10] along with justifications for the proposed spin assignments.

Discussion
The systematic evolution of the experimental energies of the known (2 + 1 ) states in the 50 ≤ N ≤ 58 Ge, Se, Kr and Sr isotopes [12][13][14][15] are shown in Fig. 2. Here one can see that the E(2 + 1 ) values of each isotonic chain between N = 52 − 58 decrease when going from Sr down to Ge, implying an increase in quadrupole deformation. The E(4 + 1 )/E(2 + 1 ) energy ratios of the Se and Kr nuclei increase from 2.1-2.3 at N = 52, which are typical values for vibrational excitations, to 2.5-2.6 at N = 54, characteristic of γ-unstable or shape transitional nuclei. From their similar E(2 + 1 ) values, and E(4 + 1 )/E(2 + 1 ) ratios, it is likely that the low-energy states of 86,88 Se have comparable collective characteristics.
Raman's empirical formula, which relates β to E(2 + 1 ) and mass [16], allows β = 0.22(2) to be estimated for 88 Se. This is larger than β = 0.17 calculated by the FRDM model [17], possibly pointing to a faster onset of deformation than expected. It is also worth noting that the E(4 + 1 )/E(2 + 1 ) ratio for 88 Se is the highest one so far reported in this region for nuclei N < 60, pointing to increased rigidity.
The (2 + 2 ) level energy of 88 Se is one of the lowest known in the region. It lies below the (4 + 1 ) state, indicating triaxial softness. However the E(2 + 2 )/E(2 + 1 ) ratios presented in Fig. 2(c) show that several other nuclei in the region appear to be softer towards triaxial deformation.  To further investigate the properties of 88 Se the experimental level scheme has been compared to the results of a SM calculation displayed in Fig. 3. The valence space used is comprised of the π( f 5/2 , p 3/2 , p 1/2 , g 9/2 ) and ν(d 5/2 , d 3/2 , s 1/2 , g 7/2 , h 11/2 ) orbits. A recent effective interaction for this model space (Ni78-II) has been employed (see [10] and references therein).
The general features of the experimental level scheme of 88 Se are reproduced reasonably well by the calculation, including the E(4 + 1 )/E(2 + 1 ) and E(2 + 2 )/E(2 + 1 ) ratios, which are marked by crosses in Figs. 2(b) and (c), though some collectivity is missing. The agreement between the experimental and calculated E(2 + 1 ), E(2 + 2 ), and E(4 + 1 ) values is within the typical errors, though this worsens for higher energy states. Figure 3 also shows the predicted transition intensities as arrow widths and these include any contributions from both E2 and M1 multipolarities. The sum of the arrow widths out of each theoretical level is the same as that of its experimental counterpart and the agreement is mostly good. Effective charges of e π = 1.7e and e ν = 0.7e were used, the same as in previous works using both the Ni78-I and Ni78-II interactions [8,18]. These high effective charges indicate that orbits from outside the valence space also contribute to the collectivity present in 88 Se. Applying Kumar's formula [19] to the SM B(E2) values allows the intrinsic shape parameter β = 0.23 to be obtained for the ground state, consistent with empirically derived one. The calculated level sequence, with a doublet of 3 + , 4 + 2 states, is characteristic of a γ-unstable nucleus. This feature, however, is not confirmed in the present experimental scheme, due to fission favoring the population of yrast states.
When moving away from the Z = 28 and N = 50 shell closures the π f 5/2 , πp 3/2 and νd 5/2 , νs 1/2 orbits are the first to be filled. The proximity of the πp 1/2 and νd 3/2 , νg 7/2 orbits would lead to two pseudo-SU(3) blocks being formed, as considered in Ref. [8], and shown to be consistent with the SM diagonalization results for 88 Se [8,10]. An inspection of the SM wave functions shows that the πp f and νsd orbits are dominant and seem to be responsible for much of the collectivity present in 88 Se.  calculations have shown that the collectivity present is due to the νd 3 5/2 configuration [6,20], and in particular its coupling to π2 + states.

ACM Interpretation
The level scheme of 88 Se obtained in the present work has been interpreted within the framework of the Algebraic Collective Model (ACM). The ACM is the algebraic version of Bohr's collective model and its Hamiltonian, in terms of the deformation parameters β and γ, is where M is the effective mass and with Λ the Casimir operator [21]. The term cos 2 3γ allows the generation of a triaxial minimum in the potential energy surface, with κ determining the degree of deformation. This Hamiltonian is capable of describing a wide range of collective spectra and is also applicable in regions undergoing phase transitions from axial to triaxial deformation, or from spherical to deformed, where α is the control parameter.
The number of free parameters in the ACM means that multiple solutions are possible for 88 Se. The level energies of 88 Se were reproduced when this nucleus is described as either a γ-unstable or a transitional nucleus. Although the former description reproduces better the level energies the γ-branching ratios are far from the experimental ones. The opposite is true for a transitional picture.
For a transitional description of 88 Se parameters of M = 70, α = 0.55, χ = −0.8, and κ = 0 were used. The value of α ∼ 0.5 is typical for a transitional nucleus undergoing a phase change from a spherical shape to an axially deformed one. As α > 0.5 this implies non-zero static β deformation. A wide, shallow minimum is present in the potential energy surface of Fig. 4(b) at values of β = 0 to 0.3. Although a small minimum is present at γ = 0 • , practically this nucleus can be described as γ unstable.
The calculated relative reduced transition rates within the ground-state band are in fair agreement with those obtained using the SM, shown in Fig. 3. However, with the exception of the 2 + 2 → 0 + 1 transition, decays within the γ band, and out of it, are predicted to be much stronger. For example the predicted B(E2; 2 + 2 → 2 + 1 )/B(E2; 2 + 2 → 0 + 1 ) is much higher than the one obtained from the measured transition intensities. As this ratio changes rapidly as a function of α here, then a small increase in this parameter allows the experimental B(E2; 2 + 2 → 2 + 1 )/B(E2; 2 + 2 → 0 + 1 ) value to be reproduced. Further experimental measurements are necessary to definitively classify the nature of this nucleus.

Conclusion
Prompt γ − γ − γ coincidence measurements performed using the EXILL Ge array, following the coldneutron induced fission of a 235 U target, have allowed a first level scheme of 88 Se to be established. The energies of the (2 + 1 ) and (4 + 1 ) states are characteristic of a γ-unstable or transitional nucleus and hint at an increase in collectivity compared to 86 Se. The identification of a low-lying (2 + 2 ) level indicates the presence of γ vibrations in this nucleus. Shell-model predictions were found to reproduce many of the general properties of the 88 Se level scheme and show that the occupation of the π f 5/2 p and νsd orbits is mostly responsible for the collectivity present. Calculations performed using the ACM also reproduce the decay scheme of 88 Se and allow the structure of this nucleus to be classified as transitional, in line with the systematics. The ACM also predicts that 88 Se has non-zero static β deformation and is γ-unstable.