Shape Coexistence at Zero Spin in 64Ni Driven by the Monopole Tensor Interaction

The low-spin structure of the semimagic 64Ni nucleus has been considerably expanded: combining four experiments, several 0+ and 2+ excited states were identified below 4.5 MeV, and their properties established. The Monte Carlo shell model accounts for the results and unveils an unexpectedly complex landscape of coexisting shapes: a prolate 0+ excitation is located at a surprisingly high energy (3463 keV), with a collective 2+ state 286 keV above it, the first such observation in Ni isotopes. The evolution in excitation energy of the prolate minimum across the neutron N = 40 subshell gap highlights the impact of the monopole interaction and its variation in strength with N.

The low-spin structure of the semi-magic 64 Ni nucleus has been considerably expanded: combining four experiments, several 0 + and 2 + excited states were identified below 4.5 MeV, and their properties established. The Monte Carlo shell model accounts for the results and unveils an unexpectedly-complex landscape of coexisting shapes: a prolate 0 + excitation is located at a surprisingly high energy (3463 keV), with a collective 2 + state 286 keV above it, the first such observation in Ni isotopes. The evolution in excitation energy of the prolate minimum across the neutron N = 40 sub-shell gap highlights the impact of the monopole interaction and its variation in strength with N . associated with quantum-mechanical states (practically) degenerate in energy. The concept was originally introduced by Jahn and Teller who demonstrated that, in non-linear molecules, coupling between degenerate electronic states and collective vibrations can destroy the system's original symmetry [1]. In atomic nuclei, the appearance of ellipsoidal deformation is a realization of this effect with specific superpositions of spherical singleparticle states (e.g., Nilsson model [2]) induced by deformed mean potentials (mean-field approaches) [3,4], or by quadrupole correlations (shell-model descriptions) [5,6], highlighting the interplay between single-particle states and collective modes.
Among the features associated with deformation figures shape coexistence: a phenomenon ubiquitous throughout the nuclear chart [7,8] where different shapes are present at comparable excitation energies. A clearcut signature for its occurrence in even-even systems is the presence of low-lying 0 + excitations residing in local minima of the nuclear potential energy surface (PES) in deformation space.
Over the past two decades, studies of neutron-rich nuclei have highlighted the contribution of the monopole component of the tensor force to the evolution of the structure of exotic nuclei [6,9], especially in the change in single-particle (or shell) structure with neutron excess, with some magic numbers vanishing and other, new ones appearing [10]. Besides such single-particle properties, its role in driving the nuclear shape was subsequently identified [6,[11][12][13][14][15], specifically in connection with shape coexistence.
The present paper focuses on 64 Ni, the heaviest, stable nucleus in the isotopic chain, and reveals a complex landscape in deformation that was not anticipated by mean-field calculations [24][25][26][27][28], which predicted a single, spherical minimum, the development of a secondary prolate one occurring only in heavier isotopes. In contrast, recent Monte Carlo Shell-model (MCSM) calculations [11] indicate coexistence of spherical and deformed oblate and prolate 0 + states already in 62,64 Ni. This coexistence originates from the action of the monopole tensor force which shifts effective single-particle energies, already at the valley of stability, weakening resistance against deformation [6,11,12]. This Letter reports extensive tests of these MCSM predictions. Besides the customary data on level energies, spins and parities, comparisons also extend to state lifetimes, transition probabilities, branching and multipole mixing ratios. Evidence is given for three coexisting shapes, with the prolate 0 + state at ∼3.5 MeV, an excitation energy reproduced only by MCSM calculations incorporating the monopole tensor interaction. To achieve the required experimental sensitivity, four experiments; i.e., transfer reactions, neutron capture, Coulomb excitation and nuclear resonance fluorescence had to be carried out at the IFIN-HH Tandem Laboratory (Bucharest, Romania), the Institut Laue-Langevin (ILL, Grenoble, France), the Argonne National Laboratory (ANL, Argonne, USA) and the Triangle Universities Nuclear Laboratory (TUNL, Duke Univ., USA), respectively. Results from the first three techniques are reported below (for the last one, see Ref. [29]).
Prior to this work, two excited 0 + states had been identified in 64 Ni, at 2867 and 3026 keV, following β-decay and (t,p)-reaction studies [30,31]. These levels were subsequently confirmed in deep-inelastic reaction measurements [32], and their γ decay to the 1346-keV 2 + 1 state was observed. Candidates for other, higher-lying 0 + levels have also been reported [33].
At IFIN-HH, 64 Ni was populated by 62 Ni( 18 O, 16 O) two-neutron (2n) transfer on a 5 mg/cm 2 -thick target, with a 39-MeV beam energy; i.e., just below the Coulomb barrier in order to reduce competition from fusionevaporation. Transitions of interest were measured with ROSPHERE, an array of 25 Compton-suppressed HPGe detectors with ∼2% total efficiency at 1.3 MeV [34]. The same reaction, but with a thinner, 0.92 mg/cm 2 target and a 5 mg/cm 2 -Ta stopper, placed at six distances from the target (10,17,25,45, 100 and 150 µm), was employed for lifetime measurements via the recoil-distance technique. The sub-barrier one-proton (1p) transfer reaction 65 Cu( 11 B, 12 C) 64 Ni at 26 MeV on a 7.22 mg/cm 2thick target was performed as well [35].
A twenty-day experiment was then conducted at ILL [36], where 64 Ni was populated via thermal-neutron capture on a 2 GBq 63 Ni sample (extracted from a larger CERN-nTOF sample [37]), where 12.1 mg of NiO grains were glued between two 6 µm-thick Al foils and contained ≈ 8% 63 Ni (T 1/2 = 101.2 (15)  clover detectors (8 on loan from IFIN-HH) arranged in a 4π geometry, particularly suitable for angular correlation studies in view of the large number (71) of relative angles between the germanium crystals. The 1/2 − spin-parity of the 63 Ni ground state results in a 0 − or 1 − 64 Ni capture state at 9657.47 keV, and strong population of 0 + , 1 + and 2 + levels through E1 primary γ rays is expected. Direct feeding of the known 0 + 2 and 0 + 3 states, at 2866.9 and 3025.5 keV, is clearly visible in coincidence spectra gated on respective 6791-and 6632-keV primary transitions. Figure 2(a) displays the 6632-keV gated spectrum where both the 0 + 3 → 2 + 1 , 1680-keV transition and a new, weaker (3.6(2)% of the former) 749-keV, 0 + 3 → 2 + 2 decay branch are observed. The angular correlation for the 0 Fig.  2(a)) agrees with the 0 + spin-parity assignment to the 3026-keV level. A search for additional 0 + excited states was undertaken by considering every primary transition in coincidence with the 1346-keV ground-state transition, and also feeding levels in the 3-6 MeV excitation energy range. Five such high-energy transitions, at 6194, 5801, 5389, 4954 and 3889 keV, populating states at 3463.1, 3856.0, 4268.1, 4703.9 and 5768.6 keV were found to exhibit decay patterns only consistent with 0 + spin-parity assignments [48]. Further, the angular correlation analysis yielded firm 0 + assignments for the states at 3463.1, 4268.1, 4703.9 and 5768.6 keV by considering in each case pairs of γ rays composed, on the one hand, of the decay branch to the 2 + 1 state and, on the other, of the 2 + 1 →0 + 1 , 1346-keV transition. The relevant analysis for the 0 + 4 , 3463.1-keV level is illustrated in Fig. 2(b).
It should be emphasized that this level is not populated in 64 Co β-decay [30], in contrast to all other 0 + states, up to 0 + 6 , which are fed in this process. This observation already points to a marked difference in structure for this excitation, and is reminiscent of that occurring in 66 Ni [23], where the prolate-deformed 0 + 4 state at 2974 keV was also the only 0 + excitation not fed in the βdecay of the spherical 66 Co ground state [39]. Further inspection of the ILL data revealed three 2 + states (firmly established in this work) at 3647.9, 3749.1 and 3798.7 keV, which complement four such excitations, at 1345.8, 2276.6, 2972.1 and 3276.0 keV, reported in Refs.
The left part of Fig. 3 provides a 64 Ni level scheme encompassing states of positive parity with spins 0, 1, 2,   [11,23]. The model space includes protons and neutrons in the full f p shell with, in addition, the g 9/2 and d 5/2 orbitals, and the Hamiltonian is based on the A3DA-m effective interaction [11]. The transition probabilities were obtained with standard effective charges (e p = 1.5 e, e n = 0.5 e), a spin quenching factor of 0.7 and an isovector orbital g-factor of 0.1 [46]. State energies are reproduced satisfactorily − the rms deviation is ∼300 keV, commensurate with expectations for shell-model calculations. For the first three 0 + excitations, the computed decay patterns mirror the data: the decay to the 2 + 1 level dominates the deexcitation from 0 + 2 and 0 + 3 states, and the branching ratios between the four transitions from the 0 + 4 state are qualitatively reproduced, with the 0 + 4 → 2 + 2 one being strongest. The relative B(E2) strengths calculated for the 0 + 2,3,4 → 2 + 1 decays (i.e., 12, 2.4 and 5 10 −4 W.u.) are consistent with the data, even though the absolute strengths are larger. Finally, the MCSM calculations also reproduce the lack of feeding of the 0 + 4 state in β decay, when compared to that of the other 0 + levels.
A sequence of relatively close-lying 2 + levels is also predicted with deexcitation patterns and transition probabilities agreeing with observations, at least when the calculated 2 + 7 state is associated with the 2 + 6 experimental one − the 2 + 6 and 2 + 7 levels are computed to lie only 235 keV apart; i.e., within the expected accuracy of the A3DA-m interaction. Theory also reproduces (i) relative variations in B(E2) values between the 2 + levels (including the retardation for the transitions out of the 2 + 6 state, which agrees with the observed small B(E2) upper limits), and (ii) the strong E2 component in the ∆I = 1, 2 + 6 → 2 + 1 transition, where the measured mixing ratio δ(E2/M 1) = +1.23(10) (vs. δ MCSM =2.6) contrasts those for similar transitions from the other 2 + excitations ( Fig. 2 (c)).
According to the MCSM calculations, the first four 0 + states reside in spherical, oblate, spherical and prolate minima, respectively, in the PES obtained for the A3DA-m Hamiltonian by the constrained Hartree-Fock method [11,12]. Thus, the 0 + 4 →2 + 1 decay is a prolate-tospherical shape-changing transition, resulting in significant retardation, in line with the B(E2) limit of <0.08 W.u. The same picture applies to the computed 2 + 7 level, which theory also locates in the prolate minimum. The observed decay pattern, the limits on the decay strengths and the dominant E2 character of the 2 + 6 → 2 + 1 transition argue in favor of this interpretation for the observed 2 + 6 state. Hence, based on the consistency between data and theory, this 2 + 6 level represents the first observation in the Ni isotopes of a 2 + excited state in a well-isolated prolate potential minimum. The "shape-isomer"-like properties of the 0 + 4 excitation in 64 Ni mirror closely those found in 66 Ni [23] with, in addition, the observation of the first element of a rotational sequence. Using the Raman systematics, linking 2 + energies to transition strengths [47], this results in a computed β 2 deformation of ∼0.4, in agreement with the MCSM result ( Fig. 4(a)). The lowenergy 286-keV in-band 2 + 6 → 0 + 4 transition, even with a calculated B(E2) strength of ∼40 W.u., cannot be observed: the flux proceeds through high-energy (>1 MeV) E2 γ rays due to the E 5 γ factor. [MeV] With these new, extensive data in 64 Ni, the evolution in energy of the prolate minimum with N can now be traced in the Ni isotopes, revealing a sharp contrast with that exhibited by the 2 + 1 levels of spherical nature. The latter are all in the 1250 -1450 keV range, with the exception of the 2033-keV value for 68 Ni due to the N = 40 sub-shell closure. In contrast, the prolate 0 + excitation rises from 1567 keV in 70 Ni, to 2511 and 2905 keV in 68 Ni and 66 Ni, and 3463 keV in 64 Ni. This behavior for N < 40 differs markedly from the lowering of deformed intruder states when moving away from a (sub)shell closure, observed in the Hg and Pb nuclei [7,8], for example. Low-lying prolate intruder states in the aforementioned neutron-rich Ni isotopes reflect the action of the monopole tensor force which is often referred to as Type II shell evolution [12,13,23], and involves particle-hole excitations of neutrons to the g 9/2 uniqueparity orbital from the f p shell. Extra binding for such intruder states is provided largely by the monopole tensor part of the nucleon-nucleon force (the proton f 5/2f 7/2 spin-orbit splitting is reduced, favoring proton excitations across the Z = 28 shell gap), which stabilizes isolated, deformed local minima in the PES (Fig. 4(a)). This additional binding is reduced for lower N values as there are progressively fewer neutrons which can be excited to the g 9/2 orbital. The deformed minimum rises in excitation energy as a result. As demonstrated in Fig.  4(b), by deactivating components of the monopole interaction (i.e., monopole frozen [6]), a nearly vanishing prolate minimum would reside at even higher excitation, in line with mean field predictions [24][25][26][27][28].
The present work has unveiled an unexpectedly complex landscape of nuclear deformation at zero spin in stable, semi-magic 64 Ni. This includes the first identification, in Ni isotopes, of a 2 + excitation in the prolate minimum. The new results provide, for the first time, a complete picture of the mechanisms underlying the appearance of deformation and shape coexistence in the Ni  [11], and (b) monopole-frozen interaction (i.e., the monopole component is subtracted from the proton-neutron interaction, and singleparticle energies are adjusted to original effective values of the spherical minimum [6]).
isotopes. They highlight the impact of the monopole tensor interaction in driving deformation at zero spin, even in 64 Ni, a nucleus within the valley of stability.
This work was partially supported by the European project for Nuclear Physics ENSAR2 (project number: 654002), by the Italian Istituto Nazionale