Lifetime measurements of excited states in 169,171,173 Os Lifetime measurements of excited states in 169 , 171 , 173 Os: Persistence of anomalous B ( E 2 ) ratios in transitional rare earth nuclei in the presence of a decoupled i 13 / 2 valence neutron

Lifetimes of low-lying excited states in the ν i 13 / 2 + bands of the neutron-deﬁcient osmium isotopes 169 , 171 , 173 Os have been measured for the ﬁrst time using the recoil-distance Doppler shift and recoil-isomer tagging techniques. An unusually low value is observed for the ratio B ( E 2 ; 21 / 2 + → 17 / 2 + )/ B ( E 2 ; 17 / 2 + → 13 / 2 + ) in 169 Os, similar to the “anomalously” low values of the ratio B ( E 2 ; 4 + 1 → 2 + 1 )/ B ( E 2 ; 2 + 1 → 0 + gs ) previously observed in several transitional rare-earth nuclides with even numbers of neutrons and protons,

Lifetimes of low-lying excited states in the νi 13/2 + bands of the neutron-deficient osmium isotopes 169,171,173 Os have been measured for the first time using the recoil-distance Doppler shift and recoil-isomer tagging techniques. An unusually low value is observed for the ratio B(E2; 21/2 + → 17/2 + )/B(E2; 17/2 + → 13/2 + ) in 169 Os, similar to the "anomalously" low values of the ratio B(E2; 4 + 1 → 2 + 1 )/B(E2; 2 + 1 → 0 + gs ) previously observed in several transitional rare-earth nuclides with even numbers of neutrons and protons, including the neighbouring 168,170 Os. Furthermore, the evolution of B(E2; 21/2 + → 17/2 + )/B(E2; 17/2 + → 13/2 + ) with increasing neutron number in the odd-mass isotopic chain 169,171,173 Os is observed to follow the same trend as observed previously in the eveneven Os isotopes. These findings indicate that the possible quantum phase transition from a seniority conserving structure to a collective regime as a function of neutron number suggested for the even-even systems is maintained in these odd-mass osmium nuclei, with the odd valence neutron merely acting as a "spectator". As for the even-even nuclei, the phenomenon is highly unexpected for nuclei that are not situated near closed shells.

Introduction
The emergence of collective behaviour and deformation in atomic nuclei due to the residual interactions between valence particles outside closed-shell configurations represents one of the most important challenges for the description of finite many-body quantum systems [1,2]. Such effects imply that the wave function has spread out over multiple, coherent particle-hole components and are commonly associated with experimental observables such as a lowering of the first excited 2 + 1 state energy accompanied by an increase in the 2 + 1 → 0 + gs reduced electric quadrupole transition probability, B(E2), in atomic nuclei with even numbers of neutrons and protons. The E2 strength between quantum states in the nucleus is a particularly sensitive probe of the corresponding wave functions and it is consequently crucial for understanding collective phenomena. In general, a roughly inverse parabolic evolution of electric quadrupole strength is expected along isotopic chains as the nucleon numbers are varied between closed shells (see, e.g., Ref. [3]). For a closed-shell configuration, the B(E2) value is at a minimum and governed by the individual single-particle degrees of freedom. As the nucleon number deviates from a "magic" number, quadrupole surface vibrations around spherical symmetry first develop. These are generally followed by the gradual evolution of increasingly deformed shapes towards the well-developed axiallysymmetric shapes and their associated rotational excitations and maximal B(E2) values when the Fermi level is situated at midshell.
The B(E2) values usually increase with spin for the lowlying yrast states within a collective (rotational or vibrational) band structure [2]. As a consequence, the B(E2; J + 4 → J + 2)/B(E2; J + 2 → J ) ratio (hereafter designated as B 4/2 ), J being the angular momentum of the lowest state in the band, is strictly larger than unity for collective excitations. For an ideal quantum rotor, a value of B 4/2 = 10/7 = 1.43 known as the Alaga rule is obtained, whereas for a harmonic vibrator, B 4/2 = 2, reflecting the ratio between the number of phonons in the initial state for each transition. The ratio B 4/2 > 1 is also true for descriptions of collective behaviour within algebraic models such as the Interacting Boson Model (IBM) [4]. In principle, the only conceivable exceptions to this rule are structures exhibiting seniority symmetry near "magic" nucleon numbers or shape coexistence. While seniority conservation has previously been considered to be the most likely, albeit unexpected, scenario [5], shape coexistence requires special attention since neutron deficient isotopes of the transition metals are well-known examples of nuclei exhibiting low-lying coexisting structures built on different quadrupole shapes [6,7].
We here report on the first measurement of lifetimes of excited states in 169,171,173 Os. Partial level schemes of these nuclei are shown in Fig. 1. The deduced B(E2) values are in qualitative agreement with the trends previously observed for members of the even-N tungsten, osmium and platinum isotopes [5,[13][14][15][16][17] with similar neutron numbers.

Experimental details
Excited states in 169 Os, 171 Os, and 173 Os were populated in the 92 Mo( 83 Kr, 2pxn) fusion-evaporation reaction at the Accelerator Laboratory of the University of Jyväskylä (JYFL), Finland. The 83 Kr beam was produced and accelerated to 383 MeV by the K130 cyclotron, then bombarded a 0.52 mg/cm 2 thick, isotopically enriched 92 Mo target foil, which was stretched and mounted in  169,171,173 Os and the decay paths to the ground states for 169,173 Os. The figure has been reproduced from the data of Refs. [8][9][10][11][12]. The widths of the arrows above the 13/2 + state are proportional to the relative intensities of the transitions. The states in red depict the isomeric states and the transitions in red were used here for tagging, see details in the text. the differential plunger for unbound nuclear states (DPUNS) device [18]. The fusion-evaporation residues were slowed down in the 1 mg/cm 2 thick Mg degrader foil of DPUNS to approximately 80% of the initial recoil velocity before they entered the gas-filled electromagnetic ion separator RITU [19]. RITU was used to separate the velocity-degraded recoiling fusion products from unreacted beam particles and unwanted reaction residues and to transport them to the GREAT spectrometer [20] located at its focal plane.
Prompt γ rays at the target position were detected by the JUROGAM II germanium detector array consisting of 15 Eurogam Phase I-type [21] and 24 Euroball clover [22] escape-suppressed detectors, with a total photopeak efficiency of ∼6% at 1.3 MeV.
The clover detectors were arranged symmetrically relative to the direction perpendicular to the beam (twelve at 75.5 • and twelve at 104.5 • ), while the Phase I detectors were placed at backward angles with respect to the beam direction (five at 157.6 • and ten at 133.6 • ). The lateral position and kinetic energy loss of the recoils were measured when passing through the multi-wire proportional counter (MWPC) and they were subsequently implanted into two double-sided silicon strip detectors (DSSDs) of the GREAT spectrometer. Meanwhile, the time-of-flight (TOF) between the MWPC and the DSSDs was measured, which was combined with the energy loss in the MWPC to select the recoils from the scattered beam. Located downstream of the DSSDs within the vacuum chamber was a double-sided planar germanium detector [23], which was used to measure predominantly low-energy γ and X rays.
The planar detector has an absolute efficiency of up to 30% at 100 keV [20], making it very suitable for the measurement of lowenergy delayed or isomeric γ rays.
Using the recoil distance Doppler shift (RDDS) technique [24], lifetimes of excited states were measured at nine different targetto-degrader distances ranging from 11 μm to 2 mm. The data streaming from the JUROGAM II and the GREAT spectrometer were collected and merged using the triggerless Total Data Readout (TDR) data acquisition system with a global 100 MHz clock [25]. Offline sorting and analysis of the measured data were performed using the GRAIN [26] and RADWARE [27] software packages.

169 Os
The first observation of excited states in 169 Os was reported by Joss et al. and a collective band was attributed to a singleneutron i 13/2 configuration [8]. Thornthwaite [9] later proposed the bandhead 13/2 + state to be an isomeric state with a halflife of 17.3 (12) μs and tentatively assigned a cascade of 80 keV and 112 keV γ -ray transitions to connect the νi 13/2 band to the 7/2 − ground state. The rotational sequence in the yrast band and its decay path to the 7/2 − ground state is displayed in Fig. 1. In the present experiment, the 112 keV γ -ray transition was clearly observed by the planar detector at the focal plane, while the 80 keV transition was extremely weak and its correlation with 112 keV was barely visible above the background, presumably due to its large internal conversion coefficient. The strong presence of Os K X-rays in coincidence with the 112 keV line in the focal plane germanium detectors supports this assumption. In this work, recoil-isomer tagging (RIT) [28] was employed in order to select the γ -ray transitions of interest from the vast background of γrays from more prolific reaction channels. Isomer-delayed 112 keV transitions detected at the focal plane within 30 μs after the recoil implantation were used to select the temporally correlated prompt γ rays of 169 Os emitted at the target position. The alternative approach of using recoil-decay tagging (RDT) [29] technique proved in this case to yield insufficient statistics and selectivity due to the low α-decay branching ratio of 11(1)% and the relatively long halflife of 3.6(2) s [30].
In the present study, the lifetimes of the I π = 17/2 + and 21/2 + yrast states were measured using the RDDS method applied to the detectors at 157.6 • and 133.6 • . Typical spectra and the time-behaviour evolution with the target-to-degrader distance at 157.6 • of the 280 keV and 479 keV γ -ray transitions are illustrated in Fig. 2 (a) and (b), respectively. The lifetimes were extracted by means of the Differential Decay Curve Method (DDCM) for the singles case [24,31] which is a variation of the RDDS approach. For a level of interest, i, with a depopulating transition feeding the level, j, and populated directly from the level h, namely h → i → j, the lifetime is given by: graded and fully Doppler-shifted components of the transitions under investigation, respectively. And b ij is the branching ratio of level i depopulating to j, v is the average recoil velocity and J hi , J ij are the relative intensities of the feeding and depopulating transitions, respectively. The derivative of Q ij (x) as a function of the distance, d dx Q ij (x), was extracted using the APATHIE software [32]. The measured mean lifetimes of the 17/2 + and 21/2 + states were deduced from the statistical weighted average of the individual lifetimes at the different target-to-degrader distances within the sensitive region, as shown in Fig. 3 (a) and (b). In a singles RDDS analysis, the lifetime of the level of interest can be influenced by unobserved feeding. As has been discussed in Refs. [31,33,34], in certain cases where the observed feeding times are not particularly long compared with the depopulating transition and the majority of the feeding is observed, it is reasonable to assume that the unobserved feeding does not significantly perturb the measured lifetime values. This assumption has been tested and validated in several measurements [35][36][37] where both singles and coincidence RDDS analysis were carried out, and has been applied in the present analysis. The final lifetime, τ , for each state was where α is the internal conversion coefficient taken from Ref. [38],  Table 1 assuming K = 1/2 (i.e. that the unpaired neutron mostly occupies the 1/2 + [660] Nilsson orbital).

171 Os
The relatively large reaction cross-section for the 2p2n evaporation channel leading to the nucleus 171 Os made it suitable for a γ γ -coincidence analysis. The two separate coincident matrices for each target-to-degrader distance were obtained by sorting the recoil-gated γ -rays recorded with the Phase-I detectors at 157.6 • or 133.6 • on one axis and the coincident γ -rays recorded with the whole JUROGAM array on the other axis. In the present RDDS study of 171 Os, two types of gating procedures were used, namely gating on a depopulating transition or a feeding transition. In the latter case, unobserved side-feedings can be effectively ruled out [24].
A gate on a depopulating transition was used for the 29/2 + state where the gate was set on the full line shape of the 253.8 keV transition (17/2 + → 13/2 + ). The method employed here to determine the lifetime is the same as the one used for the singles case (Eq. (1)), where the same assumption was made concerning the unobserved feeding. Gating on a feeding transition can eliminate the uncertainties originating from the unobserved feeding transitions effectively [24], which was used for the analysis of the other three lower yrast states. To produce the gated spectra of the 17/2 + and 21/2 + states, the gate was set on the full line shape of the 627.7 keV (29/2 + → 25/2 + ) transition. For the 25/2 + state, the gate was set on the 644.0 keV (33/2 + → 29/2 + ) transition.
Typical projections of recoil-gated γ γ -coincidence spectra and the time-behaviour evolution with the target-to-degrader distance for the first two excited states are shown in Fig. 4. The lifetimes of the yrast 17/2 + , 21/2 + and 25/2 + excited states were extracted using the following equation [24]: where the quantities Q ij (x) and Q hi (x) have been described in Eq. (1).
In Fig. 5 the lifetime values are shown for the yrast 17/2 + , 21/2 + , 25/2 + , and 29/2 + states in 171 Os. The results were obtained within the sensitive target-to-degrader distance region using the germanium detectors placed at an angle 133.6 • relative to the beam direction and the normalized decay curves I d The lifetimes measured in different conditions, the final weighted average lifetimes, as well as the corresponding transition probabilities B(E2 ↓) and quadrupole moments |Q t | are summarized in Table 1. With the K = 3/2 assumption for the yrast band of 171 Os, quadrupole moments |Q t | = 4.7(4) eb and |Q t | = 4.4(6) eb are deduced for the first two transitions, respectively. Assuming K = 5/2, the corresponding quadrupole moments 5.3(4) eb 4.7(6) eb for the first two transitions are obtained.

173 Os
The isomeric 9/2 + state in 173 Os, shown in Fig. 1, was firstly observed and estimated to have a lifetime of several microseconds by Bark et al. [11]. An unobserved transition with an estimated energy of less than 60 keV was postulated to depopulate from the 13/2 + state to the 9/2 + state and a series of transitions were assigned as members of the favoured sequence of the i 13/2  band [11]. Subsequently, the i 13/2 band was confirmed and extended by Kalfas et al. [12]. In the present work, by employing the RIT technique, isomer-delayed 49.5 keV and 91.8 keV transitions, which were detected at the focal plane within 30 μs after the recoil implantation, were used for tagging to select the correlated prompt γ rays of 173 Os emitted at the target position. Due to the low production cross section via the 2p evaporation channel, the lifetimes of the first two excited states 17/2 + and 21/2 + were measured only with the detectors placed at 133.6 • and then extracted by using Eq. (1).
Examples of typical spectra for the 232.7 keV and 390.2 keV transitions are illustrated in Fig. 6 and the lifetime determination of the two states is shown in Fig. 7. Assuming K = 5/2 for the i 13/2 band of 173 Os as introduced in Ref. [12], the transition quadrupole moments |Q t | = 5.2(5) eb and |Q t | = 6.2(12) eb, Table 1 Measured lifetime values and extracted electromagnetic properties for excited states under investigation in 169 Os, 171 Os and 173 Os. K = 1/2, K = 3/2, and K = 5/2 are assumed for the i 13/2 bands of 169 Os, 171 Os, and 173 Os, respectively. for the 17/2 + → 13/2 + and 21/2 + → 17/2 + transitions, respectively, are obtained within the rotational model [39] and given in Table 1 together with the mean lifetimes and the corresponding transition probabilities B(E2 ↓). With the K = 3/2 assumption, the corresponding quadrupole moments for these two transitions |Q t | = 4.6(4) eb and |Q t | = 5.8(11) eb are deduced.

Discussion
Excitation energy ratios, E 4 + 1 /E 2 + 1 for even-mass Os isotopes and (E 21/2 + − E 13/2 + )/(E 17/2 + − E 13/2 + ) for the νi 13/2 + bands in odd-mass Os isotopes, are shown in Fig. 8(a). The energy ratios are compared with predictions for a harmonic vibrator and the collective model for rigid axial and gamma-soft triaxial rotors [2,44]. While the energy ratios for the even and odd-N isotopes follow each other rather closely as a function of neutron number until N = 98, a bifurcation between the even and odd-mass isotopes appears for N > 98. After this point, the E 4 + 1 /E 2 + 1 ratios for the even- mass isotopes start deviating from the (E 21/2 + − E 13/2 + )/(E 17/2 + − E 13/2 + ) ratios for the νi 13/2 + bands, demonstrating the entry into a regime where the odd valence quasineutron is strongly coupled to the even-even core. For neutron numbers N < 98, the odd i 13/2 + valence neutron is decoupled from the core and does not essentially affect the relative excitation energies within the νi 13/2 + bands. With the exception of 163 W [45], a similar trend is found in the neighbouring W isotopic chain (see below). In Fig. 8 values for the even-mass Os isotopes. In Fig. 8(c)    ues that cannot be reproduced in terms of the collective rotational model [5,[13][14][15]. Of the presently obtained new B 4/2 ratios in 169,171,173 Os, the B 4/2 values for 169 Os and 171 Os deviate clearly from the expectations for collective excitations. The ratio B 4/2 ( 169 Os) = 0.79(16) might also be considered "anomalous", even though it, interestingly, seems to be significantly larger than for its even-N nearest neighbours 168 Os and 170 Os. Turning to the Interacting Boson Model, "microscopic" predictions of collective behaviour in the W, Os, and Pt nuclei of interest can be studied. and also reveal the expected tendency of increasing collectivity with the neutron-proton valence number product N N × N P [54] which in this case for a given N b is successively larger for smaller Z , i.e. larger for tungsten than osmium and larger for osmium than platinum. The B 4/2 ratios in Fig. 9 (b) are compared with the predictions for the U(5) (vibrational), O(6) (γ -soft triaxial rotor), and SU(3) (axial rotor) limits of the IBM. The B 4/2 ratios exhibit the previously noted and unexpected "phase transition" for N b ≈ 8 − 10 between an apparently non-collective and a collective regime [5,[13][14][15]. The results obtained in the present work show that both the B(E2; 17/2 + → 13/2 + ) values and the B(E2; 21/2 + → 17/2 + )/B(E2; 17/2 + → 13/2 + ) ratios for 169,171,173 Os follow, within experimental uncertainties, those of the corresponding B(E2) values and B 4/2 ratios in the neighbouring even-N isotopes, as expected from the similar energy ratios and consistent with the decoupling of the odd i 13/2 + valence neutron from the core. This accordingly indicates that the odd i 13/2 + valence neutron has little effect on the electric quadrupole strength.
In the recent work of Lewis et al. [45], lifetime measurements for the 21/2 + and 17/2 + states in the νi 13/2 + band of 163 W were reported. As can be seen in Fig. 9 (b), the B 4/2 ratio reported for 163 W [45], although with a relatively large experimental uncertainty, stands out and deserves further attention. Furthermore, the reported B 4/2 ratio for 163 W is in better agreement with the collective vibrational limit rather than a collectively rotating system as was proposed in Ref. [45]. Lewis et al. [45] also argued that the core polarising effect of the i 13/2 + valence neutron would influence the B 4/2 ratio in 163 W by making it adopt an axial prolate shape.
This, however, is not likely since its B(E2; 17/2 + → 13/2 + ) value lies right in between the B(E2; 2 + respectively. In the osmium isotopes, manifestations of coexistence of different quadrupole-deformed shapes have been associated with an early upbend in the yrast band. This happens at spin I ≈ 6h in the yrast ground-state band of 172 Os [58], which was successfully interpreted based on a phenomenological band mixing model [11,[61][62][63]. In the more neutron deficient Os isotopes 171 Os [10], 170 Os [61], and 168 Os [64], band mixing calculations suggested that as the neutron number decreases toward the N = 82 closed shell, the deformed prolate intruder band is shifted to increasingly higher excitation energies relative to the weakly deformed ground-state band. Based on this systematic trend it is not anticipated that shape coexistence will play an important role in the low-lying yrast structures in Os isotopes with N < 96 and even less likely that it will lead to the ratio B 4/2 < 1. We conclude that the enigma of anomalous B 4/2 ratios in the region of heavy, neutron deficient transitional nuclei appears to remain unexplained by theory. Furthermore, the present work indicates that the addition of an odd i 13/2 + valence neutron has little effect on either the B(E2) value or the B 4/2 ratio of these even-N systems.

Summary
The lifetimes of the first excited 17/2 + and 21/2 + states in the neutron-deficient osmium isotopes 169,171,173 Os as well as the 25/2 + and 29/2 + states in 171 Os have been measured using the values for the 21/2 + → 17/2 + and 17/2 + → 13/2 + electromagnetic transitions follow the same unusual trend as observed in the neighbouring even-N osmium isotopes. The deduced low value for the ratio B 4/2 = 0.79 (16) in 169 Os might also be considered to join the group of "anomalous" B(E2) ratios observed for members of the even-N tungsten, osmium and platinum isotopes [5,[13][14][15] with similar neutron numbers. It is concluded that the decoupled i 13/2 + valence neutron appears to act mainly as a "spectator", also with respect to the appearance of anomalous B 4/2 ratios in the most neutron deficient osmium isotopes.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.