Pseudospin Symmetry and Microscopic Origin of Shape Coexistence in the ^{78}Ni Region: A Hint from Lifetime Measurements.

Lifetime measurements of excited states of the light N=52 isotones ^{88}Kr, ^{86}Se, and ^{84}Ge have been performed, using the recoil distance Doppler shift method and VAMOS and AGATA spectrometers for particle identification and gamma spectroscopy, respectively. The reduced electric quadrupole transition probabilities B(E2;2^{+}→0^{+}) and B(E2;4^{+}→2^{+}) were obtained for the first time for the hard-to-reach ^{84}Ge. While the B(E2;2^{+}→0^{+}) values of ^{88}Kr, ^{86}Se saturate the maximum quadrupole collectivity offered by the natural valence (3s, 2d, 1g_{7/2}, 1h_{11/2}) space of an inert ^{78}Ni core, the value obtained for ^{84}Ge largely exceeds it, suggesting that shape coexistence phenomena, previously reported at N≲49, extend beyond N=50. The onset of collectivity at Z=32 is understood as due to a pseudo-SU(3) organization of the proton single-particle sequence reflecting a clear manifestation of pseudospin symmetry. It is realized that the latter provides actually reliable guidance for understanding the observed proton and neutron single particle structure in the whole medium-mass region, from Ni to Sn, pointing towards the important role of the isovector-vector ρ field in shell-structure evolution.

the latter provides actually reliable guidance for understanding the observed proton and neutron single particle structure in the whole medium-mass region, from Ni to Sn, pointing towards the important role of the isovector-vector ρ field in shell-structure evolution. DOI: 10.1103/PhysRevLett.121.192502 Spin-orbit (SO) coupling and related effects are universal features of bound fermion systems, whether they are formed by electrons, quarks, nucleons, or hyperons. Ebran et al. [1] recently pointed out that the atomic nucleus occupies a particular position in this quantum landscape. Using the nonrelativistic reduction of the single-particle (SP) Dirac equation, they established a universal relationship between jV LS j=ℏω 0 (the amplitude of the SO splitting relative to the major harmonic oscillator shell gap) and η ¼ m=Δ (the ratio between the mass of the particle and Δ ¼ V − S, the difference between vector and scalar potentials). Because in atomic nuclei η ≈ 1, SO splitting is of the same order as the energy separation of the major shells and nuclear systems naturally stand at the edge of a predicted "giant" SO region [1]. While the coincidence m ≈ Δ could seem to be just a natural curiosity, some more fundamental foundation in terms of quantumchromodynamics (QCD) is suggested by in-medium QCD sum rules [2]. Nucleons experience two almost equivalently large potentials, the short-range repulsive vector potential V ≈ 350 MeV and medium-range attractive scalar potential S ≈ −400 MeV, leading to Δ ≈ 750 MeV of the order of magnitude of the nucleon mass (m ≈ 940 MeV) itself, explaining η ≈ 1 and the large nuclear SO coupling. The counterpart of this QCD-emerging quasiequality S ≈ −V is that the net binding potential Σ ¼ V þ S is inherently small, leading to the far-reaching consequence that pseudospin symmetry (PSS) is approximately realized in nuclei [3][4][5]. Manifestations of this symmetry were early noted [6,7] in the empirical spherical SP-energy sequences where, especially for heavy nuclei, one observed quasidegeneracy of doublets characterized by the sets of quantum numbers (n, l, j ¼ l þ 1=2) and (n − 1, l þ 2, j ¼ l þ 3=2). This suggested a fruitful relabeling to pseudoradial, pseudo-orbital, and pseudospin quantum numbers (ñ ¼ n − 1,l ¼ l þ 1,s ¼ s) thus associating these doublets to pseudospin-orbit (PSO) j ¼l AEs partners (PSOPs) whose splittings could be related to the depth or radial derivative of the Σ potential [3,8]. Spectroscopic manifestations of PSS are far from being restricted to spherical SP degeneracy, since PSS has as a direct consequence (and provides the microscopic foundation for) the realization of a variant of Eliott's SU(3) symmetry called the pseudo-SU(3) symmetry for many-nucleon wave functions. PSS has been indeed recognized as the underlying mechanism leading to a wealth of salient spectroscopic features in the phenomenology of superdeformed nuclei (identical bands for instance [9,10]). However, the full expression and consequences of this symmetry on the structure of exotic nuclei having unusual neutron-to-proton ratios remain largely unexplored.
The present experimental investigation was prompted by the recent, until then unexpected, discovery of evidence for shape coexistence in the 78 Ni region [11,12]. It will be shown that PSS not only provides a natural explanation for the occurrence of shape coexistence around Z ≈ 32, N ≈ 50 but also reliable guidance to understand the observed proton and neutron SP structure in the whole mediummass region, from Ni to Sn, pointing towards the important role of the isovector-vector ρ field [13] in shell evolution.
Gottardo et al. [11] discovered that the first excited state of the N ¼ 48 even-even nucleus 80 Ge is actually a J π ¼ 0 þ state, lying slightly below the normal one-phonon 2 þ state in the excitation spectrum. They interpreted this state as originating from a configuration involving neutron-pair promotions across the N ¼ 50 gap and collecting sufficient proton-neutron quadrupole correlations to counterbalance a SP-promotion cost, apparently also reduced thanks to favorable monopole drifts. The primary objective of the present work was to investigate further this scenario by studying the origin of collective quadrupole effects beyond the N ¼ 50 shell gap, focusing on their evolutions in N ¼ 52 even-even isotones approaching Z ¼ 28 thanks to the measurement of reduced electric quadrupole transition probabilities BðE2; 2 þ → 0 þ Þ. This was an a priori demanding task in view of the accumulation of evidencethough so far only model-dependently inferred-for triaxiality approaching Z ¼ 32 [14,15]. The measurement of 2 þ state lifetimes seemed then the ideal approach, because the extraction of BðE2Þ from Coulomb-excitation cross sections might be distorted either by unmeasured diagonal E2 matrix elements having unexpected values originating from triaxial dynamics or, more importantly, by transitions to yet unobserved or unidentified low-lying 0 þ states originating from shape coexistence.
The experiment was performed at GANIL. The N ¼ 52 88 Kr, 86 Se, and 84 Ge nuclei were produced by fusion and transfer fission (with a cross-section ratio 8=2 for the two processes [16]) of a 238 U beam at 6.2 AMeV on a Be target with a thickness of 2.07 mg=cm 2 . The average intensity of the beam was 0.25 pnA. The fission fragments were identified and their velocities were determined with the magnetic spectrometer VAMOS++ [17], placed at 28°with respect to the beam direction, using the procedure already described in Ref. [18]. A total of ≈1.5 × 10 7 fissionfragment events were identified out of which the 88 Kr, 86 Se, and 84 Ge residues represented 3.0%, 1.7%, and 0.06%, respectively.
The prompt gamma rays were detected by the AGATA array [19,20]. The setup comprised eight triple cluster modules placed at 18.6 cm from the target. The gammaemission spectrum of each nucleus of interest was Doppler corrected event by event using the velocity vector measured in VAMOS and the position of the first gamma interaction in AGATA with a full width half maximum resolution of about 5 mm [21]. The energies of gamma rays were obtained by applying off line the OFT gamma-ray tracking algorithm [22].
Lifetime measurements were performed using the wellestablished recoil distance Doppler shift (RDDS) technique [23]. A Mg degrader foil with a thickness of 5 mg=cm 2 was maintained downstream from the production target at three distances, D ¼ 120, 270, and 520 μm, with a precision of 10 μm, for 75 h, 75 h, and 85 h, respectively, of measuring time by using the Orsay Universal Plunger System (OUPS) [24]. Depending on the lifetime τ of the state of interest and the distance D, gamma rays were emitted partially before and after the degrader at respective fission-product recoil velocities β before and β after . Photopeaks appeared then in the spectra as doublets of Doppler unshifted and shifted components, with intensities I U and I S , respectively. The former corresponds to gamma emission after the degrader corrected for β after precisely determined from VAMOS information [β after ¼ 0.096ð11Þ], the latter to gamma emission before the degrader, at velocity β before > β after , slightly shifted towards lower energies (because AGATA was located at backward angles).
The quantities of interest were ratios Variants of the RDDS technique suitable for lowstatistics, low D-number cases developed in Refs. [25,26] were adopted for the present analysis. Seven, five, and two transitions in 88 Kr, 86 Se, and 84 Ge, already identified in Refs. [15,27,28], respectively, could be studied and the effective or intrinsic lifetimes (uncorrected or corrected for the feeding lifetimes, respectively) of the corresponding decaying states were extracted. We report in Table I results relevant for the present discussion. Further analysis details and lifetime results [29] will be given elsewhere. As can be seen in this table, our lifetime and associated BðE2Þ results are in excellent agreement with previous RDDS and Coulomb excitation measurements for 88 Kr and 86 Se. BðE2; 4 þ → 2 þ Þ and BðE2; 2 þ → 0 þ Þ values were obtained for the first time for the hard-to-reach 84 Ge isotope. This result deserves then a few more detailed comments. Because of low statistics, the RDDS-analysis variant developed in Ref. [25] had to be preferred, which consists in summing the statistics obtained over all distances and determining the lifetime as illustrated in Fig. 1. After this summation process, a (so-called cumulative) ratio RðτÞ forming a continuous function of the lifetime τ can be obtained from the relevant set of Bateman equations, including the relative statistics weight at each distance, that intercepts a horizontal line representing the measured cumulative R exp . The central lifetime value stems directly from the projection of the interception point on the τ axis. Following the method of Ref. [25], the complex error propagations were treated by 10 6 Monte Carlo random draws in Gaussian envelops around the central RðτÞ curve and R exp line. Clearly, this procedure leads to inherently asymmetric error bars. In the present case, as no feeding to the 4 þ state was observed, τ 4 þ is an effective lifetime, i.e., an upper intrinsic lifetime [providing a lower BðE2; 4 þ → 2 þ Þ] estimate. On the contrary, τ 2 þ is an intrinsic lifetime, cleared of the time contribution from the apparent lifetime of the   . The shaded area and the two thinner curves materialize the 1σ uncertainty regions. The insets show the relevant regions of the gamma spectrum, with the shifted and unshifted photo-peak components marked with "S" and "U," respectively. 4 þ → 2 þ feeding transition. The observed relative intensities of the 4 þ → 2 þ and 2 þ → 0 þ transitions suggested 10(10)% unobserved feeding to the 2 þ state which was assumed to come from a direct feeding by the reaction. The effect of neglecting the contribution from a hypothetical long-lived unobserved gamma feeding is to overestimate τ 2 þ , i.e., to underestimate the corresponding BðE2; 2 þ → 0 þ Þ probability.
The obtained BðE2Þ values are placed in the systematics of light N ¼ 52 isotones in Fig. 2, where comparison with several calculations is also provided. Shell-model results from Ref. [32] (open circles), assuming an inert 78 Ni core, are in excellent agreement with the experimental central values (closed circles) obtained for 88 Kr and 86 Se. Interestingly enough, both shell-model and experimental values exhaust the limit for pure pseudo-SU(3) symmetry (down triangles) for these two isotones. This limit was obtained [32] by assuming that the (Z − 28) protons lie in ã sd block formed by the f 5=2 p orbits and the two neutrons in ã pf block formed by the g 7=2 ds orbits. This clearly means that the quadrupole coherence offered by this subspace is maximally expressed in these two nuclei. In contrast, both shell-model and pseudo-SU(3) values barely reach the lower tip of the experimental error bar for 84 Ge. Statistically speaking, this means that there is around 85% chance that the actual BðE2Þ value lies higher, i.e., that a fractionpossibly large-of the quadrupole correlations expressed in 84 Ge lies outside the limits of the valence of an inert 78 Ni core. For instance, considering the promotion of a g 9=2 neutron pair across the N ¼ 50 gap to thepf block (up triangles in Fig. 2) one obtains BðE2; 2 þ → 0 þ Þ ¼ 616 e 2 fm 4 , a value strikingly close to the experimental one. Following Ref. [33], one can also understand the situation as originating from 12 neutrons filling the SP sequence ðg 9=2 ; d 5=2 ; s 1=2 Þ considered as forming a quasi-SU(3) subspace. In that case, a similarly close value of 600 e 2 fm 4 is obtained. We note that such an interpretation would imply that inversion of normal and intruder configurations has already occurred in 84 Ge while it seems yet not to be the case in 80 Ge [11]. Because of the large uncertainty, the obtained BðE2Þ cannot be used to determine unambiguously the exact component along the 78 Ni corebreaking neutron (np-nh) configuration of the 84 Ge ground state. Nevertheless, we will try with the following discussion to fully explore the physical consequences of the assumption that the obtained central value is the correct one.
The trend drawn by the central values in Fig. 2 is indeed clear and it is the one of a shape transition from Z ¼ 34 to 32 at N ¼ 52. One must then investigate the following question: What is the microscopic origin of such an onset of quadrupole coherence specifically at Z ¼ 32? A fully microscopic approach, free from valence space limitations, should show the way to an explanation. A self-consistent mean field calculation using a relativistic functional was chosen in order to highlight the possible PSS roots of this phenomenon. Following Refs. [35,36], constrained meanfield (CMF) calculations of energy surfaces as functions of quadrupole ðβ; γÞ deformation parameters were performed using the relativistic functional DD-PC1 [37]. The collective Hamiltonian was subsequently diagonalized to obtain the spectroscopy of the 30 ≤ A ≤ 38 N ¼ 52 even-even isotones. The resulting BðE2Þ values are reported in Fig. 2 (open diamonds). The obtained trend from Z ¼ 36 to 32 is an almost ideal linear increase as indeed observed experimentally (it is the only calculation in Fig. 2 showing this feature). The slope, however, is lower than the experimental one. This is expected: it is well known that the description of the quadrupole properties from beyond mean-field treatments deteriorates (for other functionals as well, see e.g., [38]) while approaching shell closures. From Z ¼ 36 to 32 one gets closer to both Z ¼ 28 and N ¼ 50 magic numbers and this effect is further enhanced. Nevertheless, because the trend is correct, the obtained proton mean-field SP sequence can now be used to explain an increased quadrupole collectivity in 84 Ge. As can be seen in the inset of Fig. 3, the K-orbital sequence stemming from the 1f 5=2 2p (orsd) system actually forms a textbook example of an emerging sequence of deformed pseudospin ½Ñn 3Λ K ¼Λ AE 1=2 doublets as described in Ref. [39]. In the central panel, the Fermi level (dashed-dotted line) crosses a pronounced triaxial gap maximally opened at maximum triaxiality, γ ≈ 30°. Starting from the prolate side, for (β ¼ 0.2, γ ≳ 0°), where the memory of the intrinsic-axis projection number K is not completely lost, one observes an apparent repulsion of the ½ g 211 K ¼ 1=2; 3=2 components ofsd parentage, as if due to strong ΔK ¼ AE1 Coriolis mixing (thus opening the gap) in a way reminiscent of the famous mixing of pseudospin aligned and unaligned bands based on neutron K ¼ 1=2, 3=2 Nilsson orbits of 2d origin in 187;189 Os [40]. Therefore, it becomes clear that it is this specific π1f 5=2 2p SP arrangement that triggers quadrupole coherence in 84 Ge-and it finds its roots in PSS as we will now demonstrate. The evolution with isospin of this 1f 5=2 2p system is most clearly manifested in the structure of the neutron-rich Cu isotopes and its experimental investigation has been pursued for exactly two decades (see Ref. [41] and references therein). The faster increase of the 1f 5=2 binding compared to the one of 2p 3=2 , with increasing neutron number, has generally been ascribed to Ostuka's tensor mechanism [42]. However, the necessary counterpart is a loss of 1f 7=2 binding energy and the closing of the Z ¼ 28 gap (Fig. 4 in Ref. [42]) which should have led to the observation of increasingly large 1f 7=2 strength in the low lying 7=2 − Cu excited states approaching N ¼ 50. This is at odds with any of the recent experimental findings [41,43]. PSS, on the contrary, explains nicely this feature because 1f 7=2 is a "pseudospin-unpaired state" [4] (it has no PSOP) while 1f 5=2 2p 3=2 are PSOPs. The empirical proton 1d reduced PSO splitting Δ PSO is reported as a function of isospin T in Fig. 4, and compared to the well known [4] proton 2p and 1f doublets of the Sb (Z ¼ 51) chain: the behavior is strikingly similar. The loss of PSS (increase of Δ PSO ) while approaching the N ¼ 50 shell closure (T ¼ 21=2) is probably related to the nodal structure of the proton and neutron SP states involved, as a comparison with the Ce case (approaching N ¼ 126) [44] would suggest.
Concerning now the origin of the neutron SP structure of 84 Ge, the 2p and 1f doublets are actually also recognizable in the spectroscopy of the heaviest N ¼ 51 isotones, and their Δ PSO are represented in Fig. 4. The evolution with T is the opposite of the one for protons, this is a well-known result of relativistic mean field and originates from the ρ-meson interaction which is repulsive for neutrons and attractive for protons (see Refs. [4,5] and references therein, in particular Ref. [45]). As discussed in the introduction, PSS is restored (Δ PSO is reduced) for vanishing binding Σ potential. But, following the conclusions of Ref. [45], decrease and increase of the nuclear radius and diffusivity, respectively, both lead to a reduction of Δ PSO which can become inverted for sufficiently weak Σ. For the N ¼ 51 2p and 1f doublets, starting from a neutron pseudospin symmetric 101 Sn (see Fig. 4) and increasing T (decreasing mass number), all effects contribute jointly to increase an inverted Δ PSO . This not only elegantly explains the otherwise puzzling observation of the 3s 1=2 (2p doublet member) lowering [46], but also implies a 2d 5=2 (1f doublet member) lowering towards 78 Ni. The latter orbit, by getting closer to the pseudospin-unpaired state 1g 9=2 , reduces the N ¼ 50 gap: precisely the second ingredient needed [11] to explain the emergence of shape coexistence phenomena in the 78 Ni region and its realization in 84 Ge. The Δ PSO slope breaks and increases around T ¼ 4 (Fig. 4), further suggesting an increased neutron diffusivity towards 78 Ni. This may well happen due to the formation of a neutron skin, actually predicted by relativistic Hartree-Fock-Bogoliubov (HFB) calculations [44], whose spectroscopic manifestations have just been experimentally noticed [47].
In conclusion, thanks to the superior resolving power of AGATA and precise gamma-emission angle determination offered by the advent of gamma tracking, the RDDS lifetime measurement of excited states in light N ¼ 52 isotones could be extended down to the exotic 84 Ge. Our data suggest for the first time a shape transition from  Z ¼ 34 (soft triaxial) to Z ¼ 32 (prolate deformed), a result all the more unexpected as the shell model predicts a "fifth island of inversion" only for much lighter (Z < 28) systems [48]. Isospin asymmetry of the PSS is the hidden architect of the proton and neutron SP organization, preparing the fertile ground for its occurrence. It is a manifestation of the tensor ρ-exchange potential, but as PSS is a dynamical symmetry in nature, originating from the cancellation of several terms contributing to the SP energy [49], it contains richer spin-isospin properties than can be exhaustively captured in the usual tensor schematic picture. In particular, since this short-range potential is essentially manifested through exchange terms it naturally links shell evolution and formation of giant halos or neutron skins [44] involving large s and p orbits-whose spectroscopic expression has incidentally long passed unnoticed in the 78 Ni region [47]. Thus, while quantitative relativistic calculations and more precise experimental results for this mass sector are certainly called for, it can already be conjectured that PSS concepts and the related role of the ρ potential are to become more and more useful for understanding observed shell evolutions in very neutron-rich, medium-to-heavy mass nuclei.
D. Ve. expresses his gratitude to K. Sieja for fruitful discussions, to J.-P. Ebran