Study of medium-spin states of neutron-rich 87, 89, 91Rb isotopes

. Excited states of the rubidium isotopes 87 , 89 , 9137 Rb have been studied at the INFN National Laboratory. Measurements of the γ -ray decay of fragments produced in binary grazing reactions resulting from the interaction of a beam of 530-MeV 96 Zr ions with a 124 Sn target have been complemented by studies of the γ -ray decay of ﬁssion fragments produced in the interaction of a beam of 230-MeV 36 S ions with a thick 176 Yb target. The structure of the yrast states of 87 , 89 , 9137 Rb has been discussed within the context of spherical shell-model and cranked Nilsson-Strutinsky calculations.


Introduction
The study of neutron-rich nuclei in the neighbourhood of mass number A ∼ 100 has been the subject of growing interest in recent years, motivated by the onset of collectivity in the region and the goal of understanding the underlying microscopic structure. Since these nuclei lie on the pathway of the rapid neutron capture process (r-process) [1], there is also a nuclear astrophysical interest in the structure of such nuclei. The experimental study of medium to high-spin states in this region of the chart of nuclides has been advanced through γ-ray spectroscopy studies associated with the population of the nuclei of interest using binary grazing reactions [2][3][4], fusion-fission, and spontaneous fission reactions [5,6]. Recent studies of even-even nuclei with A ∼ 100 have explored the influence of the neutron (h 11/2 , g 7/2 ) and proton (g 9/2 ) orbitals on the evolution of deformation [7,6]; however there is currently a lack of experimental information for odd-even nuclei with N ≥ 50 .
The region of the nuclear chart centred around mass number A = 100, between the 28 < Z < 50 and 50 < N < 82 major shell closures, is predicted to have large quadrupole deformation [8,9]. Within this area of the nuclear landscape, there is a well established region of large prolate deformation in the Sr and Zr isotopes, first observed experimentally in 1970 [10]. There is a sudden onset of quadrupole deformation at N = 60 in strontium (Z = 38) and zirconium (Z = 40); the energy of the first 2 + state drops from 815 keV in 96 Sr 58 to 144 keV in 98 Sr 60 and from 1233 keV in 98 Zr 58 to 212 keV in 100 Zr 60 [11]. The sudden onset of deformation between N = 58 and N = 60 is quenched with increasing Z and this can be seen in the behaviour of the measured two-neutron separation energies as a function of neutron number for Z in the range from 36 to 45 [12]. For the odd-A isotopes of Rb, the calculations of Möller and Nix [8] predict a gradual increase in ground-state deformation from 92 Rb 55 to 96 Rb 59 , with the deformation remaining essentially constant and large (ground-state electric quadrupole moment ≈ 2 -3 b) from 97 Rb 60 to 102 Rb 65 . Mean-square charge radii [13][14][15] and two-neutron separation energies [12,16] confirm a sudden structural change at N = 60. A recent Coulomb-excitation measurement [17] on neutron-rich 97 Rb 60 and 99 Rb 62 has provided the first strong experimental evidence for enhanced quadrupole collectivity through the observation of rotational sequences. In addition, an excited rotational band was observed in 96 Rb 59 [18], providing evidence for shape coexistence. As will be seen later, theoretical calculations performed as part of the present work suggest the existence of shape coexistence in the lighter Rb isotopes. In contrast to the sudden onset of deformation at N = 60, for the lighter isotopes of Rb it is expected that, as neutrons are added to the 2d 5/2 shell, there will be a tendency for the nucleus in its ground state to acquire a quadrupole deformation in order to remove the degeneracy associated with the occupancy of the 2d 5/2 orbit; this is the nuclear analog of the Jahn-Teller effect [19,20]. There is indeed evidence from the systematic behaviour of the mean square charge radii of the ground states of the isotopes of 38 Sr, 37 Rb, and 36 Kr that there are small departures from a spherical shape as the neutron number increases from N = 50 to N = 58. This is manifest in the slow increase of mean-square charge radii with increasing neutron number, measured in laser-spectroscopy experiments [13][14][15].
In the neighbouring isotones of 87 Rb 50 with Z = 36 and 38, namely 86 Kr and 88 Sr, the energies of the first 2 + excited states are large, 1564.6 keV and 1836.1 keV, respectively, as is expected for a spherical shell closure. For the neighbouring isotones of 89,91,93 Rb 52,54,56 with Z = 36 and Z = 38, the first 2 + energies are reasonably constant, about 1700 keV for the Kr isotopes and about 830 keV for the isotopes of Sr. For these isotopes of Kr and Sr, the ratio of the excitation energy of the first 4 + state to that of the first 2 + state lies in the range from 1.99 to 2.56. The increasing ratio with increasing neutron number may indicate a tendency towards a deformed shape as the 1d 5/2 neutron shell is being filled. Indeed, in the isotopes of Kr, the B(E2; 0 + → 2 + ) values [21] increase from 7.7(8) W.u. for 88 Kr 52 to 16.8(6) W.u. for 92 Kr 56 , indicating an increasing quadrupole collectivity with increasing neutron number. On the other hand, for the isotopes of Sr with N = 52, 54, and 56, the B(E2; 0 + → 2 + ) values [21] are subject to large experimental errors, namely 8.5 +3.7 −2.0 W.u., 7.7 +5.5 −2.3 W.u., and 7.8 +5.2 −2.2 W.u., respectively, which makes it difficult to draw definitive conclusions about how collectivity changes with increasing occupancy of the 2d 5/2 neutron orbit in this case.
Here, we present the results of experimental and theoretical studies of the medium-spin structure of the 87,89,91 Rb isotopes. The main aim of this work is the study of the evolution of the level structure with increasing neutron number and a comparison with shell-model (SM) calculations. In sect. 2, a description of the experiments is given. Experimental results are presented in sect. 3 and discussed within the context of the results of shell-model and Cranked Nilsson-Strutinsky (CNS) calculations in sect. 4 and 5, respectively. Finally, a summary appears in sect. 6.

Experimental procedure
Two experimental setups using beams delivered by the combination of the XTU-Tandem and ALPI accelerators at the INFN Legnaro National Laboratory have been used here for the study of neutron-rich nuclei in the A ∼ 100 region. In the first experiment, a beam of 96 Zr ions at 530 MeV (5.52 MeV/nucleon) incident on a thin 124 Sn target was utilized to initiate binary grazing reactions. The target, which had a thickness of 300 µg cm −2 , was isotopically enriched to 94.6% and had a carbon backing of thickness 40 µg cm −2 . Projectile-like binary reaction fragments were detected by the Prisma magnetic spectrometer [23,24], at a laboratory grazing angle of 38 • with respect to the beam direction, covering a solid angle of ∼ 80 msr (with angular acceptances of ∆θ ∼ 12 • and ∆φ ∼ 22 • ) and with a momentum acceptance of ∆p/p = ±10%. At the entrance to Prisma, projectile-like ions are registered by a two-dimensional position-sensitive micro-channel plate (MCP) detector [25] providing a fast time signal for timeof-flight (ToF) and (x i , y i ) initial position measurements. The ions then pass through quadrupole singlet and magnetic dipole elements, and are finally detected and identified at the focal plane detector system. This consists of a ten-section multiwire parallel-plate avalanche counter (MWPPAC) [26] providing a timing signal, in order to perform time-of-flight measurements in coincidence with the MCP, and (x f , y f ) signals, followed by a 10 × 4 array of transverse-field multi-parametric Ionization Chambers (IC), which give the energy loss (∆E) and the total energy (E) of the ions [25,26]. The mass resolution of Prisma was ∆A/A = 1/190, allowing a clean identification of the detected reaction products from Z = 24 (Cr) to Z = 42 (Mo). The associated γ rays from target-and projectile-like products were detected in coincidence using the Clara [27] array of 25 escape-suppressed Ge Clover (Euroball-type) detectors. The detectors of the Clara array form a hemisphere covering a large azimuthal angle range, between θ Clara = 98.8 • and 174.1 • , with respect to the entrance aperture of the Prisma magnetic spectrometer. The Clara array has a photopeak efficiency of about 3% and a peak-to-total ratio of 0.45 for 60 Co 1332-keV γ rays. Ion tracking through the Prisma spectrometer together with time-of-flight measurements were used to determine the velocity vector of the projectile-like fragments and this allows the Doppler correction of γ-ray energies to be performed on an event-by-event basis [28]. The statistics in the experiment were not adequate to perform a γ-γ coincidence analysis. In the discussion below, we will refer to this setup as the Prisma/Clara experiment. Figure 1 shows an example of the quality of the data from the experiment; the mass spectrum of rubidium isotopes (Z = 37) is shown in the inner figure. It is possible to identify masses from A = 83 to A = 94. The main figure shows the associated γ-ray spectrum obtained by gating on mass A = 87 ( 87 Rb).
Data from the Prisma/Clara experiment permit the unambiguous identification of γ rays associated with the decay of excited states of rubidium isotopes of a specific mass number. In addition, measurements of γ-ray multipolarity are possible based on γ-ray asymmetry ratios, defined as R asym = I γ (θ 1 )/I γ (θ 2 ). Here I γ (θ 1 ) and I γ (θ 2 ) are the intensities of the same γ-ray transition measured in any of the detectors at θ 1 = 98.8 • , while 98.8 • ≤ θ 2 ≤ 174.1 • ; θ i corresponds to the angle between the Clover detector and the centre of the entrance aperture of the Prisma spectrometer. Typical R asym values of 1.3 and 0.7 are expected for stretched quadrupole and dipole transitions, respectively [30].
While the Prisma/Clara experiment enables the unambiguous assignment of γ-ray transitions to each rubidium isotope studied here, the reaction yield was insufficient to allow the production of γ-ray coincidence matrices from which reliable level schemes can be constructed. However, γ-γ and γ-γ-γ coincidence data were available from a thick target experiment carried out using the Ge multi-detector array Gasp in configuration I. The array consists of 40 escape-suppressed coaxial hyperpure germanium detectors and an 80 element inner BGO ball for γ-ray multiplicity and total-energy measurements. The total photopeak efficiency of the Ge array is about 3 % at a γ-ray energy of 1332 keV ( 60 Co), and the mean peak-tototal ratio is about 60-65% at 1332 keV for the Comptonsuppressed γ-ray spectra [31]. In the second experiment, a beam of 36 S ions at an energy of 230 MeV (6.39 MeV/u) was delivered onto a thick target of 176 Yb. The target was isotopically enriched to 97.8% and was of thickness 14 mg cm −2 with an isotopically enriched 208 Pb (98.7%) backing of thickness 35 mg cm −2 ; the backing was sufficiently thick to stop all forward-moving reaction products. The fusion-fission reaction with the beam and target combination populates nuclei in the region of interest [32]. Data from the Gasp experiment were used to establish coincidence relationships between γ rays through the analysis of γ-γ and γ-γ-γ coincidence matrices and cubes. In the discussion below, we will refer to this setup as the Gasp experiment.
The Gasp data were also used in the measurement of γ-ray angular correlations for 87 Rb and 89 Rb; however, the statistics were not sufficiently good to measure angular correlations for 91 Rb. When two consecutive γ rays, namely γ 0 and γ 1 , are detected by any pair of Gasp detectors separated by an angle θ, the angular distribution of the radiation is described by the function W(t 0 , t 1 , θ) = λmax λ q λ A λλ (t 0 , t 1 )P λ (cos θ) [33], where t 0 (t 1 ) denotes the properties of the γ-ray transition γ 0 (γ 1 ). The coefficients A λλ are tabulated in ref. [34], and the attenuation coefficients q λ were calculated following the procedure of ref. [33]. For the Gasp array in configuration I, these values are q 0 = 1.0, q 2 = 0.909 and q 4 = 0.602. The coefficients take into account the effect of the finite size of the Ge detectors, and the effects of choosing ± 10 • as the allowed range for the separation angle between pairs of detectors. Two-fold γ-ray coincidences were sorted into three two-dimensional matrices. For the first matrix, the first and second γ rays correspond to their detection in two Ge detectors with a separation angle of θ s = 90 • ± 10 • . The  second matrix corresponds to γ rays observed in a pair of detectors with θ s = 120 • ±10 • and 60 • ±10 • , while, for the third matrix, θ s = 150 • ± 10 • and 30 • ± 10 • . The matrices contained approximately 4× 10 9 , 9× 10 9 , and 5× 10 9 events, respectively. For a specific cascade I 0 − → I 2 , where I 0 , I 1 , and I 2 are the spins of the initial, intermediate, and final states, a gate on γ 1 was made on the first axis of the three matrices and, for each projected one-dimensional γ-ray spectrum, the number of γ 0 counts was obtained. This gives the quantities N (90 • ), N (120 • ), and N (150 • ), which were then normalized. The normalization factors were obtained through an angular correlation analysis of known stretched E2-E2 and E1-E2 cascades in adjacent even-even nuclei ( 88 Sr, 104 Ru, and 100 Mo); the results of ref. [34] were used in the normalization procedure. The normalized yields, N (θ) (normalized to unity at 90 • ), were fitted to W(t 0 , t 1 , θ) to obtain the coefficients A λλ /A 00 , which can be compared to the predicted values given in ref. [34]. Where possible, the assigned γ-ray multi-polarities were determined by combining the results of the angular correlation measurements from the Gasp experiment with the asymmetry ratios from the Prisma/Clara experiment. Previously determined multipolarities, where available, are also presented.

Experimental results
The partial level schemes of 87,89,91 Rb nuclei from the present work, shown in figs. 2, 4, and 8, respectively, are informed by published work, and based on γ-ray coincidence relations, intensity balance considerations, transition energy relations, and by the observation that binary grazing reactions [35][36][37][38] and fusion-fission reactions [5] preferentially populate yrast and near-yrast states. The level energies, the corresponding depopulating γ-ray energies, relative γ-ray intensities, asymmetry ratios, proposed multipolarity and spin and parity assignments are summarised in tables 1, 3, and 5 for 87 Rb, 89 Rb, and 91 Rb, respectively. Transition multipolarities from recent nuclear data evaluations and from other published works are presented in the final column of each table. The angular correlation coefficients for 87 Rb and 89 Rb are presented in tables 2 and 4, respectively. Not all known levels of 87,89,91 Rb are shown in figs. 2, 4, and 8. Low-spin non-yrast states are, in general, not populated in the present work and are not included in the level schemes presented here. Thus, for example, the low lying excited states of 87 Rb at excitation energies of 845 keV (J π = 1/2 − ), 1349 keV (J π unknown), 1389 keV (J π = 3/2 − ), and 1463 keV (J π = 1/2 − ) [39] are not included in the level scheme of fig. 2.

Levels and transitions in 87 Rb
The nucleus 87 Rb, with neutron number 50, is a semimagic nucleus. The neutron shell gap here is of the order of 4.5 MeV [40] and, consequently, the low-energy region of the level scheme is expected to be described in terms of the occupation of the 1f 5/2 , 2p 3/2 , 2p 1/2 , and 1g 9/2 proton shell-model orbits by the nine extra-core (Z = 28) protons and an inert neutron closed-shell core. The ground state, as expected from the shell-model orbits available above Z = 28, has a J π value of 3/2 − , corresponding to the occupation of the 2p 3/2 orbital by three protons. On the same basis, we would expect the first excited 5/2 − state at an excitation energy of 402 keV to correspond to the four-particle one-hole configuration π(1f 5/2 ) −1 (2p 3/2 ) 4 , the non-yrast 1/2 − state at E x = 845 keV, unobserved in the present work, to correspond to the configuration π(2p 1/2 ) 1 (2p 3/2 ) 2 , and the first 9/2 + state at E x = 1134 keV to correspond to the occupation of the 1g 9/2 orbital by the odd proton. Proton pickup and stripping reactions [41][42][43][44] are consistent with the above description of the structure of the states. The spectroscopic factors, C 2 S, for proton stripping to the 3/2 − , 1/2 − , and 9/2 + states are 0.34, 1.02, and 0.99, respectively. The J π = 5/2 − state at 402 keV is also populated in proton stripping, contrary to the simple shell-model description of the 86 36 Kr ground state configuration. This indicates that 86 36 Kr has, in its ground state, components of the form π(2p 3/2 ) 4 (1f 5/2 ) −2 . Higher-lying excited states, and especially those of high spin, are expected to have components in their wave functions corresponding to promotion of one or more neutrons across the N = 50 shell gap. The shell-model calculations which will be presented here, in sect. 4 of the paper, do not allow neutrons (protons) to cross the N = 50 (Z = 50) gap; consequently, one should expect to observe a significant disagreement between experiment and shell model with increasing excitation energy and increasing spin, which might reasonably be interpreted as corresponding to the onset of np-nh neutron correlations (where np stands for number of particles and nh the number of holes). Such correlations correspond to the breaking of the N = 50 neutron core.
The low-spin states of 87 Rb have been studied in β decay [45,46], in Coulomb excitation [47], in inelastic hadron  scattering [48][49][50], in single-proton transfer reactions [41][42][43][44], and in a (γ, γ ) experiment [51]. The experimental study of medium-high spin yrast and near-yrast states above the previously known J π = 9/2 + isomeric state at E x = 1578 keV has, more recently, been carried out using fusion-fission and multinucleon transfer reactions. Three fusion-fission experiments involving the projectiletarget combinations of 24 Mg + 176 Yb at 134.5 MeV, 23 Na + 176 Yb at 129 MeV, and 18 O + 208 Pb at 91 MeV [29] were used to populate a range of nuclei, including 87 Rb; the γray decay of the final nuclei was studied using the Gammasphere Ge multi-detector array of escape-suppressed Ge detectors. States of 87 Rb were populated up to an excitation energy of 7241 keV. Multinucleon transfer reactions initiated by the interaction of a beam of 460-MeV 82 Se ions with a thick target of 192 Os have also been used to populate states of 87 Rb up to an excitation energy of 6822 keV [22]; in this experiment, the γ-ray decay of the binary reaction fragments was measured using the Gasp array. A recent evaluation of experimental data for the known nuclides of mass number 87 has also recently been published [39]. In relation to the structures above the isomeric state, the results from the present work agree, with few exceptions, with previous works, and a few new transitions have been added to the level scheme. As discussed earlier, fusion-fission and multinucleon transfer reactions selectively populate yrast and near-yrast states; consequently the states populated directly do not normally correspond to those populated in other processes, such as β decay and single nucleon transfer reactions. The proposed level scheme based on the present work is presented in fig. 2, together with the level scheme based on refs. [22,29]. The energy levels in fig. 2, corresponding to previous published work, are those of Fotiades et al. [29]. The work of Zhang et al. [22] extended the previously known level scheme above the 9/2 + isomer to Table 1. The energies of excited states of 87 Rb (Ei) together with transition energies (Eγ) and relative intensities (Iγ) of γ rays placed in the level scheme, asymmetry ratios (Rasym), and the proposed multipolarities, spins and parities of the initial and final states. Multipolarities from the recent compilation by Johnson et al. [39] are given in the final column. See text for details. Relative intensities are based on the results of the Gasp experiment. Quantities in italics correspond to new level energies or to hitherto unobserved γ-ray transitions.
255.6(2  Table 2. γ-ray angular correlations for two consecutive γ rays in a cascade in 87 Rb. See text for details. the 6822-keV level. The subsequent study by Fotiades et al. [29] added further to the level scheme. Multipolarity assignments made by Zhang et al. [22] were based on the measurement of angular distribution ratios from oriented states (ADO) in addition to the observed decay branches. On the other hand, Fotiades et al. [29] suggested spin and parity assignments for four levels above the isomer (at excitation energies of 3002, 3098, 3409, and 3644 keV) which were based on a comparison with the excited states of 85 Kr and 89 Y above the 9/2 + isomer and on a comparison with shell-model calculations for 87 Rb. The assignments here agree with those of Zhang et al. In fig. 2, the spin and parity assignments above the 9/2 + isomer and corresponding to the previous work are those of Zhang et al., where avail-able; for the E x = 3098-keV state the J π value is that of Fotiades et al. [29]. In table 1, the measured γ-ray energies and relative intensities are based on the results of the Gasp experiment. Here, the focus of discussion will be related mainly to the level structure above the isomeric state.
As mentioned above, fig. 1 shows a γ-ray spectrum from the Prisma/Clara experiment corresponding to the population of 87 Rb. An example of a double-gated γ-ray spectrum from the Gasp experiment is presented in fig. 3; the gates in this case were set on the photopeaks at energies of 506 keV and 235 keV. The coincidence relationships between 87 Rb γ rays were established from such high-quality double-gated spectra.
As noted earlier, the state at E x = 1578 keV with J π = 9/2 + is isomeric, with a half-life of 6(1) ns [50]. In the Prisma/Clara experiment, 87 Rb ions (v/c ∼ 9.7%) travel a distance of about 17 cm from the target position in 6 ns. Consequently, due to the isomerism of the decaying state, the measured intensity of the 1175-keV transition (and of the subsequent 402-keV transition) is considerably reduced in relation to the number of emitted γ rays. The spectroscopic factor for population of the state in the direct single-proton transfer reaction, 86 Kr( 3 He, d) 87 Rb, is 0.99 [43], as noted earlier, and this indicates that the state has a pure proton 1g 9/2 configuration. The π(1g 9/2 ) state is also observed in the other N = 50 isotones, 89 Y [52] and 85 Br [53]. In 89 Y, the 9/2 + state appears at an excitation energy of 909 keV and the measured half life is 15.663(5) s [54]. The spectroscopic factor, measured in the 88 Sr( 3 He, d) 89 Y proton stripping reaction, is C 2 S = 0.74, again indicating that the state is a fairly pure single proton 1g 9/2 excitation. 84 Se is not a stable nucleus and the 84 Se( 3 He, d) 85 Br proton stripping reaction has not been reported in the literature to date. However, the 86 Kr(d, 3 He) 85 Br proton pickup reaction has been studied by May and Lewis [55] and by Pfeiffer et al. [56]. Within the context of the simple shell model, the 1g 9/2 proton orbital is empty in the 86 Kr ground state and, consequently, no = 4 proton pickup strength should be observed. Contrary to this expectation, = 4 strength was observed by May and Lewis [55] for an excited state of 85 Br at E x = 2.310(31)MeV with a spectroscopic strength of C 2 S = 1.12, corresponding to a proton 1g 9/2 occupation probability of V j 2 = 0.11 in the ground state of 86 Kr. However, a subsequent measurement by Pfeiffer et al. [56], which used a beam of vector-polarised deuterons, assigned a J π of 3/2 − to a state at 2.3 MeV. In the 2014 evaluation of A = 85 nuclides by Singh and Chen [54], which post-dates the two above mentioned 86 Kr(d, 3 He) studies, a J π of 3/2 − has been assigned to a state at E x = 2.310(31) MeV on the basis of the work of Pfeiffer et al. [56]. An excited state at E x = 1859 keV is a possible candidate for the π(1g 9/2 ) state of 85 Br [54] and the state has tentatively been included in a study of the systematics of the π(1g 9/2 ) orbital in the 78 Ni region by Bączyk et al. [57]. For the isotopes of Rb, Br, and As, the excitation energies of the lowest 9/2 + state relative to the first 5/2 − state show a minimum near neutron number 42. The isomeric state in 87 Rb is connected to the 5/2 − state by an M2 transition of energy 1175 keV, and to the ground state by a weak 1577-keV E3 transition. The branch to the ground state was first identified by Fotiades et al. [29]. The B(M2; 9/2 + → 5/2 − ) value of 0.12(2) W.u. is not particularly hindered, but nevertheless is in agreement with the range of hindrance factors for M2 transitions in this mass region [58,59]. The measured ratio I γ (1578 keV→ 0 keV)/I γ (1578 keV→ 402 keV) of 0.12 ± 0.01 [39] has been combined with the known lifetime of the state at E x = 1578 keV in a determination of B(E3;1578 keV→0 keV). The resulting value, 0.89(0.16) e 2 fm 6 (1.98(0.36) W.u.), falls within the distribution of E3 strength for nuclei in the mass region [58,59]. There is a close-lying state at E x = 1578.0 keV (J π = 1/2 − , 3/2 − ) [39], with energy separation from the 9/2 + state of only 0.1 keV, which has been populated in β decay from 87 Kr [45,46,39], in the 87 Rb(γ, γ ) reaction [51], and in the 88 Sr(d, 3 He) proton pickup reaction [41,42,44]; the state also decays both to the ground state and to the 402-keV first excited state. Since the state is non-yrast, its population in the present work is not expected.
We now comment on those transitions observed here for the first time and on transitions for which we are able to propose multipolarity assignments. No definitive parity assignments are possible from the present results; suggested electric or magnetic multipolarity assignments are therefore tentative. Multipolarity assignments have previously been proposed [39], see column 7 of keV) are of mixed dipole/quadrupole multipolarity. The measured γ-ray asymmetry ratios from the present work (column 4 of table 1), although subject to large statistical uncertainty, are consistent with these earlier assignments. Based on these multipolarity assignments, the proposed J π values of the states with excitation energies of 3409, 3644, 4150, 4854, 5026, and 5480 keV are 13/2 + , 15/2 + , 17/2 + , 19/2 + , 21/2 + , and 23/2 + respectively, in agreement with the proposed assignments of Zhang et al. [22]. The multipolarity of the 1340.1-keV transition is unknown; consequently the J π value of the 6821-keV level cannot be assigned. The most probable spin values, based on the population characteristics of the reactions used here, are (25/2) or (27/2). The 420.4-keV transition has a measured asymmetry consistent with a dipole multipolarity and it can therefore be speculated that the spin of the 7241-keV state is (27/2) or (29/2). It is noted again that the proposed assignments have been influenced by the observation that fusion-fission reactions favour the population of yrast and near-yrast states. The 1831.0-keV transition which connects the 13/2 + state at 3409 keV to the 9/2 + isomeric state has a measured asymmetry which is consistent with a quadrupole transition, in agreement with the previously proposed spin values of the states. Finally, the transitions of energy 1423.8 keV and 1520.6 keV which feed the 9/2 + state have measured asymmetries consistent with mixed dipole/quadrupole and quadrupole, respectively, consistent with the J π values of fig. 2. The angular correlation results presented in table 2 are consistent with these suggested multipolarities. The M2 multipolarity assignment is based on the lifetime of the 9/2 + isomeric state, discussed earlier. The proposed E3 multipolarity of the 1577 keV transition, also discussed above, is based on the J π values of the two states connected by the transition.
The transition of energy 306 keV which populates the 13/2 + state at E x = 3409 keV is reported here for the first time. The E x = 3707(8) keV state was populated in an L = 4 transition in a 87 Rb(p,p') experiment [48], suggesting a J π value of 11/2 − ; we presume that this is 37 52 Rb 89 previous work present work the same state observed here at E x = 3715 keV, which has tentatively, in fig. 2, been assigned a J π value of (11/2 − ). The γ-ray transitions at 671 keV, 451 keV, and 1608 keV are reported here for the first time. It was not possible to identify transitions above the level at E x = 7241 keV. In ref. [29] 865-keV and 935-keV transitions from the levels at E x = 6345 keV and 5789 keV, respectively, were reported; these transitions are not observed in the present work.

89 Rb experimental results
On the basis of the simple shell model, we expect the ground state of 89 Rb to be described in terms of the configuration Low-lying excited states correspond to the occupation of the f pg-shell by the nine extra-core protons and rearrangement of the two extra-core neutrons within the sdg shell. Within the context of the shell-model calculations presented here, and based on these configurations, it is expected that high-spin states of 89 Rb will be described more accurately than was the case for 87 Rb, where promotion of neutrons across the N = 50 shell gap is not allowed. As we shall see later, this is indeed the case.
A study of the excited states of 89 Rb was recently carried out by Astier et al. [60], by Bucurescu et al. [3], and by Pawlat et al. [61]. An evaluation of A = 89 nuclides has also recently been undertaken by Singh and Chen [54]. The work of Astier et al. [60] post-dates the evaluation. In the work of Bucurescu et al., 90 Zr projectiles incident on a 208 Pb target and 82 Se projectiles incident on a target of 238 U were used to populate the final nuclei in binary grazing reactions; projectile-like species were detected and identified using the Prisma magnetic spectrometer at the INFN Legnaro National Laboratory. The γ-ray decay of the binary reaction fragments was measured using the Clara multi-detector Ge γ-ray array. Double and triple γ-ray coincidences were obtained using an additional thick target experiment which involved a beam of 82 Se ions incident on a target of 192 Os; the production of the 89 Rb isotope was again via deep inelastic processes but, in addition, fusion-evaporation reactions with the 16 O target contaminant leads to the population of 89 Rb. In the experiment of Pawlat et al. [61], neutronrich 89,91 Rb nuclei were populated as fission products in heavy-ion reactions and decay γ rays were detected using the Gammasphere Ge multi-detector array. In the work of Bucurescu et al. [3], states were observed up to an excitation energy of 4033 keV and the highest spin assignment was (19/2, 21/2) whereas, in the work of Pawlat et al., the level structure was extended up to an excita- Table 3. The energies of excited states of 89 Rb (Ei) together with transition energies (Eγ) and relative intensities of γ rays (Iγ) placed in the level scheme, based on the results of the Gasp experiment, and asymmetry ratios (Rasym). Proposed multipolarities, based on the present work, appear in column five of the , followed by the proposed spins and parities of the initial and final states. The final column presents multipolarity assignments from the recent evaluation by Singh [54] and by Astier et al. [60] (in bold font).

Ei (keV)
Eγ (  tion energy of 7391 keV and the highest spin assignment was (25/2 + ) (E x =5606 keV). In the most recent work of Astier et al. [60], high-spin states of 88 Kr and of 89 Rb were studied using the 18 O + 208 Pb fusion-fission reaction. Gamma rays emitted from the fission fragments were detected with the Euroball array of escape-suppressed Ge γ-ray detectors at the IReS Strasbourg Laboratory. States of 89 Rb were studied to an excitation energy of 7389 keV (J π = (29/2 + )), with the order of the 686-and 1098-keV transitions at the top of the previously established high-spin decay sequence inverted. In addition, a few new transitions were observed, both in the high-spin and lowspin parts of the level scheme. Gamma-ray angular correlation measurements for the most intense γ-ray transitions were used to determine the dipole or quadrupole nature of transitions. The level structure of 89 Rb was earlier studied using β decay [62] and in the 86 Kr(α,p) reaction [54]. In the latter work, J assignments up to 11/2 were proposed. The level scheme of fig. 4, corresponding to previously published work, is based on the publications of Astier et al. [60], Bucurescu et al. [3], and Pawlat et al. [61] and the J π values of Astier et al. [60] are those that are given. There is a disagreement between the work of Astier et al. [60] and those of Bucurescu et al. [3] and Pawlat et al. [61] in relation to the J π assignment for the state at E x = 586 keV; the latter publications have assigned a J π value of (7/2), whereas Astier et al. [60] have assigned 5/2 − . It is noted that the evaluation of Singh [54], which does not include the publication of Astier, has a spin assignment of 7/2( − ) for the state. Figure 5 presents γ-ray spectra corresponding to two different reaction Total Kinetic Energy Loss (TKEL) values [63], in coincidence with residual nuclei of 89 Rb detected and identified at the focal plane of Prisma. In the γ-ray spectrum of fig. 5a, the 809-, 836-, 1193-, and 1572-keV transitions show an enhancement in yield relative to the other transitions compared with the same transitions in the spectrum of fig. 5b, indicating that they are emitted from higher-lying states (with higher TKEL values). From fig. 5 it is possible, for example, to deduce that the 836-keV γ ray is emitted from a state lying higher in energy than is the case for the 809-keV γ ray. Figure 6 presents two examples of double-gated γ-ray spectra from the Gasp experiment. The upper spectrum, fig. 6a, corresponds to the first gate set on the 809-keV photopeak, in coincidence with gates set on the 198-keV and 264-keV photopeaks. The lower spectrum, fig. 6b, corresponds to the first gate set on the 836-keV transition in coincidence with the 1193-keV transition. The analysis of such spectra forms the basis for establishing coincidence relationships between transitions of 89 Rb. The identification of 89 Rb γ rays in the Prisma/Clara experiment, in combination with the coincidence data from the Gasp experiment, allows us to construct the level scheme presented in fig. 4. Measured asymmetry ratios (table 3) and γ-γ angular correlation measurements (table 4) have resulted in proposed multipolarity assignments (see column 5 of table 3).
The E x = 1195-keV isomeric state was assigned a J π value of 9/2 + by Astier et al. [60] and a lifetime of a few nanoseconds by Bucurescu et al. The lifetime is consistent with an M2 multipolarity for the 9/2 + → 5/2 − 974-keV transition. In the more recent work of Pawlat et al. [61], the half-life was measured to be 8(2) ns. In the present  work, see fig. 5, the 975-keV transition is not seen in the Prisma/Clara experiment, which indicates that most of the γ-ray decay occurs outside the Clara reaction chamber; consequently, it can be concluded that the half-life is longer than circa 6 ns. A more precise half-life measurement has been made here for the state using data from the Gasp experiment (A preliminary report on the measurement was given in ref. [64]). Briefly, in fig. 7 a Getime signal versus E γ matrix is shown. In the creation of the matrix, only γ rays in coincidence with the 221-keV, 5/2 − → 3/2 − , transition and any of the γ rays of energy 809 keV (13/2 + → 9/2 + ), 836 keV (17/2 + → 13/2 + ), and 1193 keV (21/2 + → 17/2 + ) were included. From fig. 7 it is clear that the 975-keV transition is isomeric. The presence of known isomers in the data (T 1/2 = 103(3) ns in 97 Zr and T 1/2 = 6(1) ns in 87 Rb) allowed an internal calibration [64,65]  In the present work, we have identified for the first time a very weak γ-ray branch (E γ = 1195 keV) from the isomeric state to the ground state, presumably of E3 multipolarity. This branch was also observed, it will be recalled, in 87 Rb.
The transitions above the 9/2 + state, with E γ = 809, 836, 1193, and 1572 keV, have a quadrupole character confirmed by the asymmetry ratios (table 3) and the γ-γ angular correlation measurements (table 4). On the basis that the sequence of four transitions is of E2, rather than M2 character, we assign J π values of 13/2 + , 17/2 + , 21/2 + , and 25/2 + to the states of excitation energy 2005, 2841, 4033, and 5605 keV, respectively. For the 809-and 836-keV transitions, for example, the predicted half-lives (assuming a hindrance factor of 10) for M2 transitions are 45 ns and 38 ns, respectively, which excludes the possibility of the transitions being of M2 multipolarity. The same spin-parity assignments were made to the first three states of this sequence by Pawlat et al. [61], again based on the results of a γ-ray angular correlation analysis. Astier et al. [60], on the basis of a measurement of the angularcorrelation of successive γ rays, assigned a quadrupole multipole order to three of the transitions, but the 1572-keV transition remained unassigned. In the work of Bucurescu et al. [3], the experimental γ-ray angular distribution data were unable to yield a definitive multipolarity assignment for the 1193-keV transition, although the 809and 836-keV transitions were unambiguously assigned as quadrupole. The 686-keV transition has previously been observed by Pawlat et al. and, more recently, by Astier et al., but no multipolarity assignment was made. From its measured R asym value of 0.63 (17) (see table 3), a dipole character can be assigned to the transition, leading to a proposed (27/2 + ) spin assignment for the 6291-keV state. The highest state reported at E x = 7390 keV deexcites through the emission of a 1099-keV γ ray. Here, the placement of the 686-and 1099-keV γ-ray transitions in the level scheme agrees with the work of Astier et al. [60], but not with that of Pawlat et al. [61]. However, the evidence for this placement, based on measured γ-ray intensities, is much more convincing in the present work than in that of Astier et al., where the two intensities are the same, within the quoted experimental errors. The multipolarity of the 1099-keV transition is currently unknown; Astier et al. [60] have proposed a J π value of (29/2 + ) for the 7390-keV state; this does not appear to be supported by experimental measurement. From the Prisma/Clara setup it is possible to observe γ rays at energies of 111, 286, and 433 keV; these three transitions belong to the 89 Rb nucleus, but they could not be placed in the level scheme. In the recent evaluation of Singh [54], they are not listed in the previously observed 89 Rb γ-ray transitions.
For the 586-keV transition, the observed quadrupole nature (see tables 3 and 4), together with the predicted long half-life for an M2 transition (assuming a hindrance factor of 10), namely 230 ns, indicates an E2 multipolarity for this transition. On this basis, it is proposed that the J π value of the state at E x = 586 keV is 7/2 − . This is in disagreement with the 5/2 − spin assignment for the state made by Astier et al. [60].
In Section 4, the structure of 89 Rb will be discussed in the context of the shell model.

91 Rb experimental results
States of 91 Rb with spin values up to 7/2 were previously studied [69,70] using β − -γ coincidences from the decay of 91 Kr. More recently, fusion-fission and multinucleon transfer reactions were used to populate medium to high-spin states. In the work of Pawlat et al. [61], referred to above in the discussion of 89 Rb, states of 91 Rb were populated up to an excitation energy of 6239 keV and J π assignments were proposed, based on γ-γ angular correlation measurements, for excited states up to 4097 keV (J π = 21/2 + ). The spontaneous fission of 252 Cf was used by Hwang et al. [66] in a study of 91,92,93 Rb; in this work, states of 91 Rb were observed up to an excitation energy of 4097 keV with the γ-ray decay studied using the Gammasphere array. The spins of positive-parity states of 91 Rb were assigned from a comparison with those of 89 Rb. The most recent study of medium-spin states of 91,93,95 Rb was made by previous work present work 91 37 Rb 54 Simpson et al. [67]; states of 91 Rb were populated using the spontaneous fission of 248 Cm and 252 Cf sources and levels with excitation energy up to 5298 keV were studied. In the work of Simpson et al. [67], no γ-ray angular correlation data could be collected for the observed transitions of 91 Rb because of the low population of 91 Rb and contaminating transitions from other nuclei. The most recent evaluation of experimental nuclear structure and decay data for all known A = 91 nuclides was published by Baglin [68] in 2013. The J π values of states from previous studies presented in fig. 8 were taken from the published level scheme of Pawlat et al. [61], with the exception of the 722-keV state, where the J π value is that of Hwang et al. [66]. For the 9/2 + isomeric state and the decay sequence built on it, the J π assignments made by Hwang et al. agree with those of Pawlat et al., but the former work places the J π assignments in parentheses. The excitation energies of states, corresponding to previous work in fig. 8, were taken from the work of Simpson et al. [67]. The excitation energy of the state at 6239 keV, unobserved by Simpson, was taken from Pawlat et al. [61]. In fig. 9, the γ-ray energy spectrum in coincidence with mass A = 91 and Z = 37 from the Prisma/Clara experiment is shown. The peak to background ratio is inferior to   Table 5. The energies (Ei) of excited states in 91 Rb together with transition energies (Eγ), relative γ-ray intensities (Eγ) (based on the Gasp experiment), and γ-ray asymmetry ratios (Rasym). Proposed multipolarities, based on the present work, appear in column five of the table, followed by the proposed spins and parities of the initial and final states. The final two columns present the J π and multipolarity assignments from the 2013 Nuclear Data Sheets evaluation [68]. Quantities in italics correspond to new level energies or previously unobserved γ-ray transitions.
multipolarity [68] 108.91 (19) 108.90 (20) 40 (5) 0.50 (12) (20) 501.80 (20) 18 (3) 1.65(48) E2 141.5(10) < 1 9/2 + → 7/2 − (9/2 + ) 412.20 (20) 5(2) 706.50 (20) 100 (12)   that of the γ-ray spectra presented earlier for the 87,89 Rb isotopes. The level scheme of fig. 8, corresponding to the present work, is informed by earlier work, and is based on the relative intensities of transitions in the γ-ray energy spectra corresponding to different values of TKEL, on γ-ray intensity balance, and, importantly, on the results of an analysis of 91 Rb triples coincidences from the Gasp experiment. Table 5 presents the energies (E i ) of excited states in 91 Rb together with transition energies (E γ ), relative γ-ray intensities (I γ ) (based on the Gasp experiment), and γ-ray asymmetry ratios (R asym ). Proposed multipolarities, informed by previous work and based on the present work, appear in column five of table 5, followed by the proposed spins and parities of the initial and final states. The final two columns present the J π and multipolarity assignments from the 2013 Nuclear Data Sheets evaluation [68]. Quantities in italics correspond to new level energies or previously unobserved γ-ray transitions. Figure 10 presents an example of a double-gated γ-ray spectrum from the Gasp experiment in which the first gate is set on the 706-, 1138-, and 474-keV photopeaks and the second gate is set on the 412-keV photopeak. The intensities of the γ-ray transitions at energies of 474, 1119, and 728 keV are, within the experimental errors, the same. Consequently, in the present work, we are unable to establish the order of the transitions within the 91 Rb level scheme. Otherwise, the present level scheme agrees with the results of the earlier work of Pawlatet al. [61], Hwang et al. [66], and Simpsonet al. [67]. The 502-keV level, previously observed in the beta decay of 91 Kr [69,70], was also populated here and, in addition, new levels are tentatively proposed at E x = 993 keV and at 2167 keV. It is noted that the 502-keV level was not observed in the published work of Pawlatet al. [61], Hwang et al. [66], and Simpson et al. [67].
The transitions of energy 502 and 993 keV, which directly populate the ground state, have asymmetry ratios (see table 5) which are consistent with quadrupole multipolarity. An M2 assignment is excluded as a consequence of the long half lives for M2 transitions (480 and 16 ns, respectively, with an assumed hindrance factor of 10). Consequently, a J π value of (7/2 − ) is assigned to each decaying state, at excitation energies of 501.8 and 993.1 keV. A  [68] 505.3(2)-keV transition, with an asymmetry ratio (see table 5) consistent with quadrupole multipolarity remains unplaced in the level scheme. The 505-keV photopeak appears in the γ-ray spectrum measured in coincidence with 91 Rb ions detected at the focal plane of Prisma, fig. 9, and this observation excludes a multipolarity of M2 for the transition (470 ns lifetime, assuming a hindrance factor of 10). The 993-keV state has not previously been observed; the 271-and 993-keV γ rays which deexcite the state have not previously been attributed to 91 Rb [68]. The γ-ray photopeak at an energy of 1025 keV, corresponding to the depopulation of the J π = 9/2 + state at 1134 keV, can be seen in the thick target Gasp experiment, but not in the Prisma/Clara experiment; this observation is consistent with the known isomerism of the state [69,67]. Previous measurements of half-lives have yielded values of 17.0(8) ns [69] and 16(1) ns [67]. For transitions above the 9/2 + isomeric state, the asymmetry ratios allow an assignment of multipolarity for the γ rays at energies of 706, 1138, 1119, 474, and 728 keV.
With the exception of the 474-keV, (23/2 + ) → (21/2 + ) transition, for which a mixed M1/E2 character is assigned, the asymmetry ratios are consistent with quadrupole multipolarity. M2 transitions, with an assumed hindrance factor of 10, for these quadrupole transitions, have half lives in the range from 8 to 90 ns and, on this basis, we propose here E2 assignments. The tentative J π values for the 4097-, 4571-, and 5299-keV states, shown in fig. 8, are based on these multipolarity assignments.

Shell-Model Calculations
Spherical shell-model calculations have been performed in order to understand the microscopic structure of the odd-A Rb isotopes studied here. The set of codes CENS (Computational Environment for Nuclear Structure) from Oslo University was utilized [72] in the calculations. An effective two-body interaction was derived microscopically from the Idaho N3LO nucleon-nucleon potential [73] using a V low-k renormalization procedure [74]. A model space outside a 78 28 Ni 50 core was used, with single-particle states π(1f 5/2 , 2p 3/2 , 2p 1/2 , 1g 9/2 ) and ν(2d 5/2 , 2d 3/2 , 3s 1/2 , 1g 7/2 ). Promotion of neutrons across the N = 50 shell gap is not allowed in these calculations. The available single-particle energies (SPE) used in the calculations were taken from ref. [71]. Those SPE not available in the literature were calculated in the Hartree-Fock approximation using the Skyrme SLy4 parameterization [75]   10% in the value of the SPE for the 1g 9/2 orbital, from -9.08 MeV [71] to -10.08 MeV, was introduced to reproduce the position of the 9/2 + isomeric level. The values of SPE are shown in table 6. Within the context of the sim- ple shell model, the isotopes of Rb with A = 87, 89, and 91 have, in their ground states, a proton configuration of (1f 5/2 ) 6 (2p 3/2 ) 3 (2p 1/2 ) 0 (1g 9/2 ) 0 and neutron configurations of (1g 9/2 ) 10 (2d 5/2 ) (N −50) (2d 3/2 ) 0 (3s 1/2 ) 0 (1g 7/2 ) 0 , where N = 50, 52, and 54 for 87 Rb, 89 Rb, and 91 Rb, respectively. The calculations which will now be discussed below show that this is a reasonably good approximation to the shell-model description of the ground-state configurations. To make the calculations possible, it was necessary to make a truncation of the occupation numbers. At most, three protons were allowed to occupy the 2p 1/2 and 1g 9/2 orbitals. The results of the shell-model calculations for 87 Rb and 89 Rb are presented in fig. 11. Figures 12 and 13 present the shell-model nucleon orbit occupations for the positive-parity states built on the J π = 9/2 + isomeric state for 87 Rb and 89 Rb, respectively. For these states, the figures also present the difference between the shell-model excitation energies and the experimental values. There are two 13/2 + states predicted and two ob-served in 87 Rb and both states are included in fig. 12. A horizontal bar has been drawn through the data points corresponding to the 13/2 + yrast state, which does not form part of the strong decay sequence built on the 9/2 + isomeric state. Similarly, there are two predicted and observed states of 89 Rb with J π = 25/2 + and, in fig. 13, the data points corresponding to the state of lower excitation energy are indicated by a horizontal bar. For 87 Rb and 89 Rb, the 3/2 − ground states are expected to correspond, to a proton hole in the 2p 3/2 shell, while the first excited 5/2 − states are expected to correspond to a vacancy in the 1f 5/2 proton shell, i.e. π(1f 5/2 ) 5 (2p 3/2 ) 4 . The results of the shell-model calculations performed here are in good agreement with these simple shellmodel orbit occupations; for the 3/2 − states of 87 Rb and 89 Rb, the shell-model occupations are 5.8 and 5.7, respectively for the proton 1f 5/2 orbital and 2.9 and 2.8, respectively, for the proton 2p 3/2 orbital. For the 5/2 − state, the corresponding proton occupancies are 5.0 and 5.9 for the 1f 7/2 orbital and 3.7 and 3.7 for the 2p 3/2 orbital. Within the uncertainties associated with the DWBA analysis of single-nucleon transfer reactions (principally associated with the description of the entrance and exit channels using optical-model potentials and the parameterisation of the Woods-Saxon bound-state potential), the conclusions from published proton stripping [43] and pickup [44] reactions are broadly in agreement with such descriptions of the ground state of 87 Rb and the excited state at E x = 402 keV with J π = 5/2 − . In addition, as noted earlier, the published proton stripping reaction 86 Kr( 3 He,d) 87 Rb populates the 1578-keV 9/2 + state with a spectroscopic strength of (2J + 1)S = 9.9 [43], which is consistent with the description of the state as a pure 1g 9/2 proton excitation. The 9/2 + state of 89 Rb at E x = 1195 keV was strongly populated in the 86 Kr(α,p) 89 Rb reaction in an L = 4 transfer with σ(exp)/σ(DWBA) = 4.3 [54]. Presumably the three-nucleon transfer process involves the transfer of a neutron pair into the 2d 5/2 orbital, coupled to spin zero, and the transfer of a proton into the 1g 9/2 orbital. For 87 Rb, 89 Rb, and 91 Rb, the low-lying negative parity states correspond to occupation of the 1f 5/2 , 2p 3/2 and 2p 1/2 proton orbitals, with the 1g 9/2 proton shell essentially empty.
The positive-parity decay sequences in 87 Rb, 89 Rb, and 91 Rb are built on the J π = 9/2 + isomeric state for which the 1g 9/2 orbit is occupied by one proton; within the shell-model configuration space, this is the only way in which positive-parity states can be formed. For 87 Rb, the positive-parity sequence up to the 17/2 + state is very well reproduced in the calculations (see figs. 11 and 12). This implies that these high-spin states are well described within the context of the shell model in terms of a single proton in the 1g 9/2 orbital coupled to excitations of the proton core, with neutrons occupying the lowest available shell-model orbitals. For the J π = 19/2 + and 21/2 + states, there is a significant disagreement between the shell-model and experimental excitation energies, and this is probably related to the restricted configuration space of the present shell-model calculations. Presumably, a more satisfactory shell-model description of these states involves promotion of a neutron into the available orbits beyond the N = 50 shell gap. This has been discussed in an earlier publication [22]. For 89 Rb, the overall agreement between experiment and shell model is reasonably good for positive parity states up to 25/2 + ; in this case neutrons are allowed to occupy the whole of the sdg shell. The excitation energy of the J π = 29/2 + state is not well reproduced; the difference between experimental and shell-model excitation energies is about 660 keV. 3 Figure 14 presents a comparison of the shell-model results for 91 Rb and experiment. It is again noted that the experimental J π values in parentheses are speculative. Since the main focus of interest here is related to the higher-spin states of 91 Rb, in fig. 14 the first two shellmodel states of a given J π value are included for states with J π ≥ 13/2 + . The results of two calculations, corresponding to neutron one-particle one-hole configurations and two-particle two-hole configurations are presented in the figure. In the neutron one-particle one-hole calculation, only one state with J π = 15/2 + was predicted. Unlike the situation in 87 Rb and in 89 Rb, the first excited J π = 5/2 − state of 91 Rb, observed experimentally at 109 keV, is not well reproduced in the shell-model calculation. Similarly, the excitation energy of the 9/2 + bandhead of the positive-parity decay sequence is not well reproduced, in contrast to 87 Rb and 89 Rb. It is possible, for each experimental state above the 9/2 + isomer, to identify an association of experimental and shell-model states but, given the number of relatively close lying high-spin shell-model states and the uncertainties in the experimental J π values, the confidence in doing so is considerably reduced compared to the association for 87 Rb and 89 Rb. For the experimental states at 1134 keV, 1841 keV, and 2978 keV, the J π values have been established as 9/2 + , 13/2 + , and 17/2 + , respectively. The corresponding shell-model states are those at excitation energies of 960, 1381, and 2547 keV, respectively, for the neutron one-particle one-hole calculation and 802, 1330, and 2574 keV for the neutron twoparticle two-hole calculation. The state observed in the present work at an excitation energy of 2167 keV has no assigned J π value; it is expected to be an yrast or near yrast state. On this basis, it is speculated that the associated shell-model state is either that at 2077 keV with J π = 13/2 + or that at 2243 keV with J π = 15/2 + .
With neutron numbers in the range from 52 to 56, neutrons gradually fill the 2d 5/2 shell-model state. There will be a strong neutron-proton attractive tensor interaction [76] between the j > 2d 5/2 neutrons and the j < 1f 5/2 protons which will lead to an increased binding of the 5/2 − state. Figure 15(a) shows the evolution of the excitation energy difference between the 5/2 − state and the 3/2 − ground state for the Rb isotopes with N = 50, 52, 54, and 56. The decrease in E x (5/2 − ) -E x (3/2 − ) with increasing neutron number would appear to be consistent with the expected effects of the tensor interaction. It is also possible that the J π = 5/2 − states have components in their wave function corresponding to the coupling of a 2p 3/2 proton to the J π = 2 + 1 state of the core and this could potentially provide an explanation for the observed behaviour shown in fig. 15(a). However, a comparison of the observed energy trends in fig. 15(a) and fig. 15(b) leads to the conclusion that this explanation is incorrect. It is noted here that, while the shell-model results presented earlier are able to reproduce the energy difference between the 5/2 − and 3/2 − states of 87 Rb and 89 Rb, this is not the case for 91 Rb.

Cranked Nilsson-Strutinsky-Bogoliubov calculations
For the 87 37 Rb 50 nucleus, extensive Cranked Nilsson -Strutinsky -Bogoliubov (CNSB) calculations were carried out. The calculations are based on a modified version of the code used in ref. [77]. The new code uses a different strategy which is described in detail in ref. [78]. The calculations were performed in a grid of eight parameters, namely the pairing parameters for protons and neutrons, (∆ p , λ p ) and (∆ n , λ n ), and the deformation parameters (ε 2 , γ, ε 4 ), which are used as variational parameters, and the cranking frequency ω x . For each set of variational parameters, all low-lying many-quasiparticle configurations were created using blocking, and traced diabatically [77] (removing virtual crossings) as a function of the cranking frequency ω x . For each of these configurations, we performed particlenumber projection and interpolated the projected energy as a function of ω x to obtain values for fixed spins. In the final step of the calculation, the projected energy was minimized for each spin state (I ≈ I x ) with respect to all configurations of a given symmetry and the variational parameters. (In this section, we choose to use I, rather than J, to denote the total angular momentum quantum number of a state.)  Rb50 calculated in the CNSB formalism and labelled by the proton and neutron parity and signature, (π, α)p(π, α)n. Energies are given relative to an 0.01892 I (I + 1) reference. Experimental values assigned to π = + are drawn with squares. A normalization to the I π = 9/2 + state is used when comparing experimental data with calculations.
With the present method, information about individual configurations (occupancies of different orbitals) is lost but the advantage is that fully minimized yrast lines, corresponding to the 16 different parity and signature symmetry combinations, (π, α) p (π, α) n , can be obtained very efficiently in an automatic way. For selected (π, α) p (π, α) n configurations, the energies from the calculation in the CNSB formalism for 87 Rb are shown in fig. 16. In these calculations, we have considered a (ε 2 ,γ)-mesh of 99 points shown in fig. 17, with a minimization also in 4 ε 4 points at each (ε 2 ,γ) deformation. Furthermore, no interpolation has been carried through in the 10 ∆ p,n and 7 λ p,n points; instead the (λ p,n , ∆ p,n )-mesh point giving lowest energy has been chosen. This is a more stable method but, with a limited number of points in the mesh, it can lead to some irregularities in the energy curves of fig. 16. I π = 29/2 + Fig. 17. The calculated potential energy surface for 87 Rb at I = 29/2 for the (π, α)p(π, α)n = (+,1/2)(+,0) symmetry group. The contour line separation is 0.25 MeV.
In the CNSB calculations for 87 Rb, we have used the 'standard' [79] Nilsson single-particle parameters and for the rotating liquid drop energy, we have used the 1966 Myers-Swiatecki parameters [80] with the rigid-body moment of inertia calculated with a sharp surface with a radius parameter r 0 = 1.2 fm. For the pairing strength, we have used standard values specified in [77]. The CNSB calculations for 87 Rb, fig. 16, are in general agreement with observed yrast states, especially for states with I < 10. In both cases, one obtains irregular sequences of states typical for spherical or close-to-spherical deformations. Indeed, the yrast line up to I ≈ 20 is calculated to be composed of close-to-spherical states having ε 2 0.05. In fig. 16 we show only the yrast line for (π, α) p (π, α) n = (+, ±1/2)(+, 0) corresponding to the symmetry of the observed states, but the yrast line with opposite parity for the protons is also calculated to be close to spherical for I < 20 and to have about the same energy.
In fig. 16, the encircled points indicate states that are non-collective i.e calculated to have γ = 60 • and thus correspond to terminating states. The first such point is obtained for I = 9/2 corresponding to one proton in the 1g 9/2 orbital and four proton holes in the fp shell coupled to total angular momentum of zero. In order to get a better understanding of the terminating states, a comparison has been made with unpaired CNS calculations [79,81] where its is possible to fix configurations in terms of the number of particles in high-and low-j shells. In this way, the favoured terminations seen at I = 17/2 and I = 29/2 can be understood as corresponding to configurations with one and three 1g 9/2 protons, respectively. The I = 29/2 state and also the signature partner with I = 27/2 are calculated to be terminating states and are the highest spin states seen in experiment.
The energy surface drawn in fig. 17 for I = 29/2 shows both the terminating state and also a second more deformed minimum with ε 2 0.22 and γ = 0 • . The energy corresponding to the second minimum is also shown in fig. 16 where it is seen to become yrast around I = 20. A similar minimum is found with negative parity for protons and this energy is also displayed in fig. 16. However, these more deformed bands are calculated to be roughly 2 MeV higher in energy than the I = 29/2 state and are not seen, nor indeed expected to be seen, in the present experiment.
Calculations were also performed for the isotopes 89,90 Rb with similar results. In general, for the low-spin states we obtain weakly deformed yrast lines but where the additional neutrons added to the 2d 5/2 shell drives the shape to slightly larger deformations, as discussed within the context of the Jahn-Teller effect in the introduction, sect. 1.

Summary
This paper is concerned with a study of the medium-high spin states of the rubidium isotopes (Z = 37) with A = 87, 89, and 91. Excited states were populated in binary transfer reactions initiated through the interaction of 530-MeV 96 Zr ions with a thin target of 124 Sn. Use of the Prisma/Clara setup in the measurements allowed the unambiguous identification of the projectile-like nuclei together with the associated γ rays. Data from a second experiment, which used the Gasp array to measure γ rays from the decay of fission fragments produced in the reaction 36 S+ 176 Yb at 230 MeV, were used to build level schemes and to perform angular correlations measurements.
Experimental results have been compared with the results of spherical shell-model calculations, and cranked Nilsson-Strutinsky calculations. Shell-model calculations for 87 Rb and 89 Rb show a reasonably good agreement for the low-and most of the medium-spin states. Deviations between experiment and shell model for high-spin positive parity states of 87 Rb are possibly a consequence of the restricted shell-model space in the present work and may indicate the importance of crossing the neutron shell closure through np-nh neutron excitations for energies in excess of about 5 MeV. In 91 Rb it is possible, for each experimental state above the 9/2 + isomer, to identify an association of experimental and shell-model states but, given the number of relatively close lying high-spin shell-model states and the uncertainties in the experimental J π values, the confidence in doing so is considerably reduced compared to the association for 87 Rb and 89 Rb. In this respect, further experimental work on the level structure of 91 Rb would be helpful.
Cranked Nilsson-Strutinsky calculations, including pairing, were performed to describe medium-spin states in the nuclei. The results confirm the spherical and close-tospherical behaviour of 87,89,91 Rb. A very interesting second minimum is predicted for levels above the 29/2 + state; however, it was not possible to observe these levels in the present experiment and further experimental work is required to populate and study such non-yrast states.