Is Seniority a Partial Dynamic Symmetry in the First $\nu g_{9/2}$ Shell?

The low-lying structures of the midshell $\nu g_{9/2}$ Ni isotopes $^{72}$Ni and $^{74}$Ni have been investigated at the RIBF facility in RIKEN within the EURICA collaboration. Previously unobserved low-lying states were accessed for the first time following $\beta$ decay of the mother nuclei $^{72}$Co and $^{74}$Co. As a result, we provide a complete picture in terms of the seniority scheme up to the first $(8^+)$ levels for both nuclei. The experimental results are compared to shell-model calculations in order to define to what extent the seniority quantum number is preserved in the first neutron $g_{9/2}$ shell. We find that the disappearance of the seniority isomerism in the $(8^+_1)$ states can be explained by a lowering of the seniority-four $(6^+)$ levels as predicted years ago. For $^{74}$Ni, the internal de-excitation pattern of the newly observed $(6^+_2)$ state supports a restoration of the normal seniority ordering up to spin $J=4$. This property, unexplained by the shell-model calculations, is in agreement with a dominance of the single-particle spherical regime near $^{78}$Ni.

The pairing interaction, first introduced by Racah for the classification of n electrons in atoms [1], can be treated as a first-order approximation of the strong residual force between identical nucleons [2,3]. The socalled nuclear pairing provides the ground for one of the simplest and most powerful approaches to the nuclear shell model, the seniority scheme [4,5]. The seniority υ refers to the number of protons or neutrons that are not in pairs with total angular momentum J = 0. Using the seniority approach, the j j coupling of nucleons in a single j shell can be classified by two quantum numbers, the total spin J and the seniority υ.
The conservation of seniority results in insightful properties that can be used to identify shell closures far from stability [6,7]. The best known is that the υ = 2 excitation energy spectrum in a single j shell is constant, no matter what the number of valence particles is. Equally important properties imply that M1 transitions can only connect states of the same seniority and the quadrupole operator between states of seniority υ = 2 vanish in midshell, leading to the appearance of seniority isomers arising from the fullyaligned configuration of one neutron or proton pair [8,9,10,11,12,13,14,15].
Seniority is a good quantum number for any twobody interaction in orbitals with j 7/2 [4,5]. The first shell in which it might not be conserved is g 9/2 [7,16], although it can still be valid for a subset of solvable eigenstates [7,16,17,18,19,20]. This property can be seen as an example of a partial dynamic symmetry [21,22]. In particular, for a four-particle (hole) system, two special states with good seniority υ = 4 and total spins J = 4 and 6 are found as eigenstates of any two-body interaction [23].
Excellent test cases for the partial conservation of seniority are the semi-magic Ni isotopes with four neutrons ( 72 Ni) and four neutron holes ( 74 Ni) in the first νg 9/2 shell. While the υ = 2, 8 + seniority isomers in 70 Ni and 76 Ni have lifetimes consistent with the seniority scheme [10,24], their analogs in 72 Ni and 74 Ni were not observed in the expected µs time range [11,25]. This astonishing deviation was attributed to a quasidegeneracy of the 6 + level with seniority υ = 2 and the special 6 + state with seniority υ = 4, being the energy lowering of the latter ascribed to an increased contribution of proton excitations across the Z = 28 gap [8,20,26]. The recent measurement of key experimental observables such as the energy of the yrast 8 + state * Corresponding author.
Email address: Ana.Morales@ific.uv.es. (A.I. Morales ) and the lifetime of the 4 + 1 level in 72 Ni [27,28] supports this explanation. In spite of the great advances, the question on the maintenance of seniority as a good quantum number for close-lying levels with equal total spin J is still under debate [29,30].
In order to give insight into this issue, we have investigated the β decays of 72 Co and 74 Co at the RIBF facility operated by the RIKEN Nishina Center and the Center for Nuclear Study of the University of Tokyo in Japan. We report a wealth set of new spectroscopic information in the daughter nuclei 72 Ni and 74 Ni. For 74 Ni only two γ rays attributed to the decay of the (2 + 1 ) and (4 + 1 ) states were previously known [11,31], while for 72 Ni a comprehensive decay level scheme was reported previously by some of the authors [27]. Here we report for the first time the half-life of the (6 + 1 ) state, which was measured using in-flight β-delayed fast-timing spectroscopy [32,33]. The present experimental results bring to completion the knowledge gained by the EU-RICA collaboration on the decay of neutron-rich oddodd Co isotopes [24,27,34].
The soundness of the seniority scheme up to the J π = (8 + 1 ) states in 72 Ni and 74 Ni will be discussed in the context of four shell-model calculations [23,26,35,36] that have recently been used to describe the properties of nuclei in the region [27,28,31,34,37,38,39]. The first is the Monte-Carlo Shell Model (henceforth called MCSM) [35,40] based on the A3DA Hamiltonian [41] including the f pg 9/2 d 5/2 model space for both protons and neutrons. The second is based on a phenomenological effective interaction derived for the f 5/2 pg 9/2 neutron model space (referred to as SM Lis) [26]. The third (called SM 2+5) is a realistic shell-model calculation performed with a two-body effective interaction derived within the framework of many-body perturbative theory starting from the CD-Bonn NN potential [42,43] in a valence space consisting of the f 7/2 and p 3/2 orbitals for protons and the f pg 9/2 d 5/2 orbitals for neutrons. The last, called SM SCI, is based on an empirical interaction in the (νg 9/2 ) n shell [23].
A 238 U beam was accelerated up to an energy of 345 MeV/nucleon by the RIBF cyclotron accelerator complex before impinging on a 3-mm beryllium target. Radioactive secondary beams were produced by inflight fission. The primary beam intensity was 10 pnA at the production target. Fission products were separated based on their momenta and mass-to-charge ratios (A/Q) in the BigRIPS separator using a 6-mm-thick Al achromatic degrader placed at the dispersive focal plane. Particle identification was achieved by measuring the energy loss, magnetic rigidity, and time-of-flight of the transmitted fission residues which provided Z and Widths of the arrows are proportional to their absolute intensities (normalized to the number of nuclei produced in the high-and low-spin states, respectively). In both cases the log f t values are calculated assuming that the β-decaying state is the ground state in 74 Co. The Q β = 15640(540) keV is taken from Ref. [44].
A/Q information on an event-by-event basis [45].
The identified nuclei were implanted in the WAS3ABi active stopper after traversing the Ze-roDegree spectrometer [46]. WAS3ABi consisted of a compact stack of 5 double-sided Si strip detectors (DSSSD) divided into 60 vertical (X) and 40 horizontal (Y) strips of 1-mm pitch and 1-mm thickness each [47]. At a distance of 22 cm, the EURICA γ-ray detector array [48] surrounded the active stopper. It was composed of 12 HPGe cluster detectors with 7 crystals each, providing an absolute detection efficiency of about 11% at 662 keV. The setup was implemented with a fast-timing array of 18 LaBr 3 (Ce) detectors for the identification of γ rays and two fast-plastic scintillators for the detection of β particles. The latter were placed at 2-5 mm from the first and last DSSSD detectors, respectively [49].
Implanted nuclei were identified in WAS3ABi as overflow energy signals in both X and Y strips, while β signals were read using high-gain analog electronics. Their time and position (DSSSD, X and Y strips) were recorded event-by-event in order to build spatial and time correlations between implanted nuclei and β particles. The time and energy of γ rays from decay successors were registered by EURICA. Nuclear halflives were obtained from the time differences between β particles and de-exciting γ rays, i.e., the moments of formation and de-excitation of the states, respectively. The β-γ time differences obtained with the fast-timing array were corrected for standard time-walk effects and other dependences related to the use of the β electron as the reference start signal, namely its energy and position. The final time resolution of the system was 300 ps (FWHM). This allowed for the measurement of halflives down to about 100 ps [50]. Almost 9 × 10 5 and 2×10 4 implantation events were registered for 72 Co and 74 Co, respectively.
The γ-ray energy spectrum following the β decay of 74 Co to 74 Ni is shown in the top left panel of Fig. 1. Sorting conditions included a maximum ionβ time difference of 170 ms and a maximum distance of (X±1,Y±1) strips. The implantation depth was also constrained to the same DSSSD. For the γ-γ coincidence analysis, the maximum time difference between γ rays was set to 300 ns. Coincidence spectra gated on key transitions, namely the proposed (8 γ rays at 226, 1025, and 1150 keV, respectively, are shown in the three bottom panels on the left of Fig. 1. This information was used to build the level schemes following the 74 Co→ 74 Ni decay, which are shown in the central and right panels of Fig. 1. The spin and parity assignments in 74 Ni (see Fig. 1) are proposed based on the strong resemblance with the β decay 72 Co→ 72 Ni [27]. Similarly, we observe two well-defined structures at high and low spins that can be interpreted as arising from two β-decaying states. The half-lives associated to each structure are t 1/2 = 31.6(6) ms for the 403-, 617-, 738-, 744-, 1080-, and 1150-keV transitions and t 1/2 = 28(3) ms for the 2589-and 3615-keV γ rays. It should be noted, though, that in 74 Co the νg 9/2 orbital is well over half-filled and the parabolic splitting of the π f −1 7/2 ⊗ νg 9/2 multiplet, with three neutron holes, may have inverted its orientation with respect to the proton hole-neutron particle system [51,52]. In such a case the lowest-lying members of the multiplet are more probably the 8 − and 1 − states, as in 76 Co [24]. The 8 − level will mainly decay to the 7 − state built on the (ν f −1 5/2 νg 9/2 ) configuration, thus converting a f 5/2 neutron into a f 7/2 proton via an allowed Gamow-Teller transition. The 7 − state is predicted at 4350 keV by the SM Lis calculation. Alternatively, the 8 − level in 74 Co can decay through a νg 9/2 → π f 7/2 firstforbidden transition to the seniority-two 8 + 1 state in 74 Ni, which is predicted at 2728 keV. But, if the multiplet has not flipped yet, the ground state in 74 Co will more likely have J π = 6 − or 7 − as for the lighter Co isotopes [53,27], decaying to the 6 − member of the (ν f −1 5/2 νg 9/2 ) configuration in 74 Ni, which is predicted at 3859 keV by the calculation.
The large feeding to the experimental level at 3658 keV (log f t > 4.32, see Fig. 1) indicates the occurrence of an allowed Gamow-Teller transition. This, together with the fact that the state decays internally to the (6 + ) and (7 + ) candidates, supports a (6 − ) assign-ment for the 3658-keV level in 74 Ni and hence a (6 − ) or (7 − ) nature for the mother state in 74 Co. Further support comes from the SM 2+5 calculation, which predicts the 7 − level in 74 Co to decay by β emission. Indeed, only the J π = 0 − ground state is expected below the 7 − βdecaying isomer located at 103 keV. Based on these arguments, we conclude that the multiplet has not flipped yet in 74 Co.
It is worth noting that the internal decay of the (6 − ) level has a branch to the yrast sequence (8 + ) → (6 + ) → (4 + ) → (2 + ) → (0 + ) through the (7 + ) state, as in 72 Ni [27]. Different from this case, the strongest gamma transition from the (6 − ) level feeds the (6 + 2 ) state. Meanwhile, the low-spin state of 74 Co decays to the 3614-keV level in 74 Ni through a strong Gamow-Teller transition (log f t > 4.38, see Fig. 1). The internal γ de-excitation occurs either through the first (2 + ) level or directly to the ground state. This limits the spin of the 3614-keV level to (1) or (2) and that of the low-spin isomer of 74 Co to J π ≤ (3), which is consistent with the predictions of the SM 2+5 calculation.
For the decay 72 Co→ 72 Ni, we have performed the first in-flight β-delayed fast-timing study. We have applied the same sorting conditions as for 74 Ni, except for the maximum ion-β correlation time, set to 270 ms [27]. The top-left panel of Fig. 2 shows the γ-ray energy signals registered in the LaBr 3 (Ce) detectors as a function of the β-γ time differences measured with the fast-timing system for this decay. Three γ-ray transitions are unambiguously observed as delayed: The (6 + 1 ) → (4 + 1 ) at 454 keV, the (4 + 1 ) → (2 + 1 ) at 843 keV, and the (2 + 1 ) → 0 + 1 at 1095 keV. We conclude, then, that the (6 + 1 ) state in 72 Ni is isomeric. A least-squares fit of the time spectrum gated on the 454-keV transition to a Gaussian plus exponential convolution function yields a half-life of t 6 + 1/2 = 860(60) ps, see the top-right panel of  scription of the electromagnetic transition rates for the Ni isotopes. In Fig. 3, the experimental and theoretical states up to J π = 8 + in 72 Ni (top) and 74 Ni (bottom) are shown. At first sight the SM SCI, MCSM, and SM Lis calculations reproduce well the measured excitation energies, while SM 2+5 underestimates them, most probably due to an excess in the calculated collectivity (see Fig. 2). The arrows connecting states are proportional to the E2 reduced transition probabilities. For the experimental decays (6 + 2 ) → (4 + 1 ) and (6 + 2 ) → (4 + 2 ) in 74 Ni we show the B(E2) ratios normalized to the B(E2;6 + [υ = 2] → 4 + [υ = 2]) strength predicted by SM SCI, which corresponds to the 6 + 2 → 4 + 2 decay in the calculation (see bottom panel in Fig. 3).
In the following we will discuss the nature of the states shown in Fig. 3 within the seniority scheme. We start by the assertion that the short-lived character of the (8 + 1 ) → (6 + 1 ) transition in 72 Ni (see Fig. 2) and the nonobservation of the competing (8 + 1 ) → (6 + 2 ) γ branch [27] support the inversion of seniorities υ = 2 and 4 in the (6 + ) states. These arguments apply also to 74 Ni, pointing to a main υ = 4 configuration for the (6 + 1 ) state and a υ = 2 character for the (6 + 2 ) level. However, the two nuclei show a different decay pattern for the (6 + 2 ) state. While in 72 Ni the main deexcitation branch feeds the (4 + 1 ) state [27], in 74 Ni the (4 + 1 ) and (4 + 2 ) levels are fed with equal intensity, Assuming exact conservation of seniority, the following B(E2) ratios are obtained [18,20,23,30]: One sees from Fig. 3 that all calculations predict the B(E2; 4 + 1 → 2 + ) value greater than the B(E2; 4 + 2 → 2 + ) one. Hence, according to eq. (1), the calculated yrast 4 + states in 72 Ni and 74 Ni can be interpreted as the special J π = 4 + , υ = 4 levels that conserve seniority for any two-body interaction [23]. While the measured B(E2) strengths in 72 Ni [including the present B(E2,6 + 1 → 4 + 1 ) result] are in agreement with the theoretical predictions, the discrepancies found between the experimental and theoretical B(E2,6 + 2 →4 + 2 ) B(E2,6 + 2 →4 + The B(E2,6 + 2 →4 + 2 ) B(E2,6 + 2 →4 + 1 ) ratio extracted here for 74 Ni, R ∼ 21, provides experimental evidence for the hindrance of the (6 + 2 ) → (4 + 1 ) transition. This can only be explained within the seniority scheme if the corresponding E2 quadrupole operator connects levels of equal seniority. Based on the above argument and the non-observation of isomerism of the (8 + 1 ) state, we propose a υ = 2 character for both the (6 + 2 ) and the (4 + 1 ) levels. This results in a recovery of the normal seniority ordering for the (4 + ) states in 74 Ni, which is also supported by the reasonable agreement with the B(E2;6 B(E2;6 + [υ=2]→4 + [υ=2]) ratio expected for a seniority-conserving interaction, R S CI ≈ 7.6 (see eq. 2). The deviation of a factor ∼3 between the experimental result and the SM SCI prediction can be explained by the existence of a γ-soft deformed minimum in 74 Ni stabilized by the so-called "type-II" shell evolution, which is related to many particle-hole excitations across Z = 28 and N = 40 [35]. This type of shell evolution has been shown to be crucial for the development of a deep prolate minimum in close-lying nuclei [34,37,54,55]. In 74 Ni, the 0 + 2 bandhead of the γsoft shape is predicted by the MCSM at an energy of 2301 keV, comparable to those of the (4 + 2 ), (6 + 1 ), and (6 + 2 ) levels at 2104, 2380, and 2507 keV, respectively. This proximity might favor the appearance of protonneutron non-conserving interactions [29] that result in a larger experimental B(E2;6 + 2 →4 + 2 ) B(E2;6 + 2 →4 + 1 ) ratio. Except for this small difference, the present experimental results are in agreement with the restoration of the single-particle spherical regime near 78 Ni, in agreement with the recently observed smoother reduction of the Z = 28 shell gap from N = 40 to 50 [38].
Summarizing, the low-lying seniority structure of the midshell νg 9/2 72 Ni and 74 Ni isotopes has been investigated following β decay at the RIKEN Nishina Center. We have observed for the first time the sought-after 8 + seniority-two states in both nuclei. The disappearance of the seniority isomerism in these levels can be understood, as foreseen many years ago [8], in terms of a strong reduction in the excitation energy of the special (6 + ) states with seniority υ = 4, which have an even lower excitation energy than the (6 + ) states with seniority υ = 2. In 74 Ni, the recovery of the normal seniority ordering up to spin J = 4 has been observed, a feature that the present shell-model calculations do not reproduce. Further quantitative knowledge on the electromagnetic transition rates between the yrast and yrare (6 + ) and (4 + ) states is necessary to discern the underlying cause of the discrepancies found.
The excellent work of the RIKEN accelerator staff for providing a stable and high intensity 238 U beam is acknowledged. We acknowledge the EUROBALL Owners Committee for the loan of germanium detectors and the PreSpec Collaboration for the readout electronics of the cluster detectors. Part of the WAS3ABi was supported by the Rare Isotope Science Project