_{36}^{96}Kr_{60}-Low-Z Boundary of the Island of Deformation at N=60.

Prompt γ -ray spectroscopy of the neutron-rich 96 Kr, produced in transfer- and fusion-induced ﬁssion reactions, has been performed using the combination of the AGATA array and the VAMOS++ spectrometer. A second excited state, assigned to J π = 4 + , is observed for the ﬁrst time, and a previously reported level energy of the ﬁrst 2 + excited state is conﬁrmed. The measured energy ratio R 4 / 2 = E (4 + ) /E (2 + ) = 2 . 12(1) indicates that this nucleus does not show a well developed collectivity contrarily to that seen in heavier N = 60 isotones. This new measurement highlights an abrupt transition of the degree of collectivity as a function of the proton number at Z = 36, of similar amplitude to that observed at N = 60 at higher Z values. Possible reason for this abrupt transition could be related to the insuﬃcient proton excitations in the g 9 / 2 , d 5 / 2 , s 1 / 2 orbitals to generate strong quadrupole correlations or to the coexistence of competing diﬀerent shapes. An unexpected continuous decrease of R 4 / 2 as a function of neutron number up to N = 60 is also evidenced. This measurement establishes the Kr isotopic chain as the low- Z boundary of the island of deformation for N = 60 isotones. Comparison with available theoretical predictions using diﬀerent beyond mean-ﬁeld approaches shows that these models fail to reproduce the abrupt transitions at N = 60 and Z = 36.

Prompt γ-ray spectroscopy of the neutron-rich 96 Kr, produced in transfer-and fusion-induced fission reactions, has been performed using the combination of the AGATA array and the VAMOS++ spectrometer. A second excited state, assigned to J π = 4 + , is observed for the first time, and a previously reported level energy of the first 2 + excited state is confirmed. The measured energy ratio R 4/2 = E(4 + )/E(2 + ) = 2.12 (1) indicates that this nucleus does not show a well developed collectivity contrarily to that seen in heavier N = 60 isotones. This new measurement highlights an abrupt transition of the degree of collectivity as a function of the proton number at Z = 36, of similar amplitude to that observed at N = 60 at higher Z values. Possible reason for this abrupt transition could be related to the insufficient proton excitations in the g 9/2 , d 5/2 , s 1/2 orbitals to generate strong quadrupole correlations or to the coexistence of competing different shapes. An unexpected continuous decrease of R 4/2 as a function of neutron number up to N = 60 is also evidenced. This measurement establishes the Kr isotopic chain as the low-Z boundary of the island of deformation for N = 60 isotones. Comparison with available theoretical predictions using different beyond mean-field approaches shows that these models fail to reproduce the abrupt transitions at N = 60 and Z = 36. The systematic evolution and the sudden changes of the properties of a many-body quantum system probe the basic interactions between its constituents. In the atomic nucleus, singularities like: shell closures and their migration [1][2][3], the onset of collectivity [4], shape coexistence [5][6][7] or the rapid transitions among them [8], result from the interplay between the mean field and correlations between individual nucleons [9]. The study of the evolution of these phenomena is crucial to understand the various facets of the nuclear force [2].
In the region of nuclei with A ∼ 100 and N ≥ 60, an large collectivity was reported [10]. In this mass region, it is well established experimentally that Sr, Zr and Mo isotopes exhibit a large prolate deformation in the ground state [11][12][13]. Contrarily, at the N = 50 shell closure, nuclei exhibit a structure typical of the single-particle motion in a spherical potential. The transition, between the spherical and well deformed nuclei, appears to be abrupt at N = 60. This is associated with a shape-coexistence phenomenon where the coexisting spherical and deformed configurations suddenly invert at N = 60 [14][15][16]. This could be compared on one hand, to the "Island of Inversion" associated with a rapid onset of collectivity observed at N = 40 near 64 Cr [17], and on the other hand to the smooth crossing of coexisting configurations in the neutron deficient isotopes of Kr [18]. The low Z-boundary of this island of deformation has attracted much attention in the last years. In Ref. [19], 96 Kr was suggested to be at the critical point of the shape-phase transition near A = 100. Recent predictions from Monte-Carlo Shell-Model (MCSM) calculations support the hypothesis of a quantum phase transi-tion in Zr isotopes [14] and a recent measurement places 97 Rb (Z = 37) as the cornerstone of this island of deformation [20]. Both the transition to large collectivity at N = 60 in Sr-Mo and its sudden disappearance in Kr deserve further investigations [19,21].
Intense experimental efforts have been devoted to investigate the 96 Kr neutron-rich nucleus. Charge radii [22] and mass [19] measurements indicate a smooth development of collectivity in the isotopic chain of Kr. Mǎrginean et al. [23] reported a first excited state at 241 keV in 96 Kr, suggesting a well developed deformation. This measurement was contradicted by Albers et al. [21,24], using Coulomb excitation, reporting a first 2 + excited state at 554 keV, and a correspondingly B(E2; 2 + 1 → 0 + 1 ) value much lower than those of 98 Sr and 100 Zr (see Fig. 1(a)). Several theoretical approaches attempted to describe the structure of Kr isotopes. An oblate shape was proposed for 96 Kr and the possibility of prolate-oblate shape coexistence was argued [24][25][26][27]. However, the sharp transitions at N = 60 and between Kr and Sr are not well reproduced simultaneously by those theories as well as the excited states beyond the first 2 + state highlighting that the underlying mechanisms remain to be understood.
To understand, quantify and characterize the evolution of the nuclear structure along isotopic chains, a systematic study of the energies of the first excited states (E(2 + ), E(4 + )), their ratio (R 4/2 = E(4 + )/E(2 + )) and the corresponding reduced transition probabilities (B(E2)), are often used as primary indicators [4]. Figure 1(a) shows the corresponding experimental information available for isotopic chains of even-even nuclei with 50 ≤ N ≤ 64 and 36 ≤ Z ≤ 42 [28][29][30]. In general, a clear correlation between R 4/2 and B(E2; 2 + 1 → 0 + 1 )/A is observed. In Mo, the smooth increase of R 4/2 as a function of the neutron number, illustrates the evolution from spherical (∼ 1.5) to well deformed (∼ 3) nuclear shapes. Similarly, the corresponding B(E2; 2 + 1 → 0 + 1 )/A continuously increases indicating a larger collectivity. For Sr and Zr, the trend is relatively flat for N ≤ 58 while a sudden increase is observed at N = 60, and both indicators remain well correlated. In contrast, for Kr, R 4/2 increases more steeply than B(E2; 2 + 1 → 0 + 1 )/A for N ≤ 54, and decreases beyond N = 54, while B(E2; 2 + 1 → 0 + 1 )/A continues increasing gradually. This is in contradiction with expectation as the corresponding E(2 + ) and the B(E2; 2 + 1 → 0 + 1 ) are well correlated in 96 Kr [21]. The aim of this work is to investigate whether this singular behaviour of the R 4/2 trend persists at N = 60 in the neutron-rich 96 Kr. In this Letter, we report on a new measurement of 2 + and 4 + excited states in the neutron-rich nucleus 96 Kr populated in transfer-and fusion-induced fission processes.
The measurement was performed at GANIL using a 238 U beam at 6.2 MeV/u, with an intensity of 0.2 pnA, impinging on a 10-micron thick 9 Be target. The advan- tage of the inverse kinematics used in this work is that fission fragments are forward focused at high velocities, resulting in both efficient detection and isotopic identification in the spectrometer. A single magnetic field setting of the large-acceptance spectrometer with its improved detection system VAMOS++ [31], possessing a momentum acceptance of around ± 20%, was used to identify uniquely the fission fragments [32]. The spectrometer was placed at 28 • with respect to the beam axis so as to optimize the acceptance for light fission fragments. The detection system at the focal plane of the spectrometer was composed of a Multi-Wire Parallel Plate Avalanche Counter (MWPPAC), two drift chambers (x, y), and segmented ionization chambers (∆E and E). The Time of Flight (ToF) was obtained using the signals from the two MWPPACs, one located after the target [33] and the other one at the focal plane (flight path ∼7.5 m). The parameters measured at the focal plane [(x, y), ∆E, E, ToF] along with the known magnetic field were used to determine, on an event-by-event basis, the mass number (A), charge state (q), atomic number (Z), and velocity vector ( v) after the reaction for the detected fragments [31]. The prompt γ rays were measured in coincidence with the isotopically identified fragments, using the AGATA γ-ray tracking detector array [34] consisting of 8 triple clusters placed in compact configuration (13.3 cm from the target). The combination of the pulse-shape analysis [35,36] and the OFT γ-ray tracking algorithm [37] allowed to obtain the position of the first γ-ray interaction point. A Doppler correction was applied, using the precise determinations of the v obtained from VAMOS++ [33] and the γ-ray detection angle in AGATA, to calculate the γ-ray energy in the rest frame of the emitting fragment. A γ-ray energy resolution of 5 keV (FWHM) was obtained at 1.2 MeV (for recoiling fragments with v/c ranging between 0.095 to 0.135 and recoil angles varying by ∼ 20 • ). Figure 2 shows the prompt γ-ray spectrum measured in coincidence with the 96 Kr ions detected in VA-MOS++. Three transitions at 554(1) keV, 621(2) keV and 515(2) keV can be seen in the spectrum. The 554 keV transition confirms the excitation energy of the 2 + 1 state of Ref. [21]. The 621 keV transition was observed in coincidence with the 554 keV transition (see inset of Fig. 2). Its relative intensity is 72(15)% as compared to the 554 keV transition. It is interpreted as the transition depopulating the 4 + 1 excited state at 1174 (3) keV. This assignment is supported by the systematic analysis of level energies and relative intensities of the neighboring nuclei and isotopic chains populated in fission under the similar experimental conditions and measured in the present experiment. The relative intensity of the 515 keV is 35(15)% as compared to the 554 keV transition. Due to the limited statistics, it was not possible to obtain significant coincidence analysis for the 515 keV transition which was not placed in the level scheme. The non observation of this transition in the Coulomb excitation experiment [21] excludes that this transition depopulates the 2 + 1 state. Figure 3(a) shows the systematics of avail-  able experimental data for 2 + 1 , 4 + 1 , 3 − 1 , and 2 + 2 as a function of neutron number for Kr isotopes. The new level observed in this work follows the systematic trend. The present assignement as 4 + 1 of the newly observed level at 1174 keV results in a R 4/2 = E(4 + )/E(2 + ) = 2.12 (1). The presence of low lying 2 + 2 excited states in the Kr isotopic chain suggests possible assignement of the 515 keV γ-ray to the 2 + 2 → 2 + 1 transition. The newly obtained R 4/2 ratio reported in the present work shows that R 4/2 continuously decreases till N = 60 as shown in Fig. 1(a). This extends the unexpectedly opposite trends of R 4/2 and B(E2; 2 + 1 → 0 + 1 ) for the Kr neutron-rich isotopes and confirms the non-observation of a sudden onset of collectivity at N = 60 as shown in Fig. 1(b). This figure represents the evolution of the R 4/2 ratio as a function of the proton number for isotones between N = 58 and N = 62 and shows the sharp transition between Sr and Kr at N = 60. Another way to investigate the correlation between the above observables is to study their ratio (i.e. ∼B(E2; 2 + 1 → 0 + 1 )/R 4/2 ) as a function of R 4/2 . Figure 3 Gogny D1M + IBM2 [24] Gogny D1S + SCCM [25] PC-PK1 + 5DCH [26] Gogny D1M + IBM2 [38] MCSM [14] Gogny D1S + 5DCH [27] FIG. 4. (color online) Deviation between calculated and experimental R 4/2 ratio against the proton number for N = 58 − 62 isotonic chains and several theoretical predictions from beyond mean-field approach (Gogny D1S+5DCH [27], Gogny D1M+IBM2 [24,38], Gogny D1S+SCCM [25], PC-PK1+5DCH [26]) and MCSM [14].  [39]. Three regions were classified according to the structural properties of the nuclei: (i) spherical vibrator, (ii) rotor, (iii) prolate-oblate shape coexistence with strong mixing [39]. The lines connect the isotopes and the arrows indicate the increasing neutron number.
In the region of spherical vibrator, the characteristic feature of the R 4/2 ratio is a chaotic behaviour (i.e. the absence of clear correlation). In contrast, in the region of rotor, a clear correlation emerges. 96 Kr lies in the region of spherical vibrator. For the lightest isotopes of Kr, the behaviour is chaotic, similarly to Sr, Zr and Mo isotopes. However, for isotopes heavier than 90 Kr, a regular trend can be observed that does not head towards the rotor region, in contrast with other isotopic chains. The neutron-rich Kr isotopes appear therefore to exhibit a different behaviour than neighbouring isotopic chains. To understand these differences from a theoretical point of view, it is necessary to reproduce the sudden transition observed for Sr, Zr and Mo at N = 60, the very abrupt transition observed at Z = 36, and the relation between R 4/2 and B(E2). Different theoretical approaches performed for the 96 Kr nucleus are available in the literature [21,[25][26][27]. The right panel of Fig. 3(a) presents the calculated energies, using different beyond mean-field approaches, for the 2 + 1 , 4 + 1 and 2 + 2 levels in 96 Kr. For this nucleus, Delaroche et al. [27] show the best agreement with the experimental data. Figure 4 shows the deviation of calculated R 4/2 ratio (in percent) from various theoretical works for the isotonic chains N = 58, 60 and 62 as a function of proton number [14,[24][25][26][27]38]. Positive values refer to theoretically underestimated R 4/2 , and could be interpreted as an insufficient degree of collectivity while negative values indicate a too strong degree of collectivity. Only Ref. [27] provides theoretical predictions for all the nuclei in this region. Most of the predictions were obtained from beyond mean-field calculations using Gogny D1S/D1M [24,25,27,38] or PC-PK1 [26] energy density functional and different beyond mean-field mapping procedures. It can be seen from the Fig. 4 that theoretical predictions of beyond mean-field calculations from Refs. [26,27,38] have a similar trend but fail to reproduce the abrupt transition at N = 60 and between Z = 36 and Z = 38. For N ≥ 60, the degree of collectivity is largely insufficient while approaching Z = 38 though it tends to be reasonably large for Z > 40. Results from Refs. [24,25], obtained also from beyond mean-field approaches, show larger disagreement, where available. The continuous decrease of the R 4/2 (see Fig. 1(a)) ratio in Kr beyond N = 54 remains puzzling and could be related to a shape-coexistence phenomenon. Beyond mean-field calculations consistently predict [24,25,27] an oblate ground-state band in 96 Kr and also a second 0 + state at an excitation energy between 1.0 and 1.5 MeV with a prolate character. Rodriguez [25] also suggests a rapid shape change along the first excited band in 94,96 Kr. Similar behaviour was reported for the neutron-deficient 72 Kr isotope where a shape change occurs along the ground-state band at low spin, resulting in a reduced R 4/2 value [40].
Recently, a microscopic description of the shape coexistence in the Zr isotopic chain using the MCSM approach was reported [14,16], which reproduces quite well the experimental data. These calculations provide for the first time a microscopic description of the quantum shapephase transition in Zr isotopes at N = 60. Figure 4 shows that these predictions are in a very good agreement the experimental data for N = 62, while display a different trend than the mean-field results for N = 58, 60. The MCSM calculations, performed in a very large model space, describe the ground state of 98 Zr as spherical, while the deformed low-lying 0 + 2 state is predicted as multiparticle-multihole excitation of protons from the f pshell to the intruder g 9/2 and d 5/2 orbitals and neutrons into the g 7/2 and intruder h 11/2 orbitals. The two 0 + states cross in excitation energy in 100 Zr, having a deformed ground state. The development of deformation in this mass region is explained in Ref. [14] as resulting from the change in the shell structure induced by the monopole central and tensor components of the protonneutron interaction for specific particle-hole excitations. As a consequence intruder orbits are lowered, thus favoring the cross-shell excitations. The combination of the monopole and the quadrupole components of the nuclear interaction results in the development of deformation in Zr isotopes. It will be crucial to pursue theoretical efforts, within the microscopical framework of MCSM calculations to understand the sudden transition at Z = 36 as well as the unexpected trend of R 4/2 , highlighted in this work.
Similar phenomena have been observed in other regions of neutron-rich nuclei, the so-called Islands of Inversion and can be understood in terms of the dynamical symmetries pseudo-and quasi-SU3 that apply in shell model spaces that include the relevant degrees of freedom for the development of quadrupole collectivity [41,42]. In this mass region, the relevant valence space for protons consists of the 1f 5/2 , 2p 3/2 , 2p 1/2 , 1g 9/2 , 0g 7/2 , 2d 5/2 , 3s 1/2 orbitals while the neutrons move in the space spanned by 2d 5/2 , 2d 3/2 , 3s 1/2 , 1g 7/2 , 1h 11/2 , 2f 7/2 , 3p 3/2 . The development of deformation in this valence space requires a critical number of protons and neutrons. An analogy between the Z = 40, N ∼ 60 region and the Island of Inversion at Z ∼ 24, N = 40 [17] could be drawn. The onset of deformation develops at N = 38 in Cr (Z = 24) and at N = 40 in Fe (Z = 26) isotopes. Isotones with N = 36 remain transitional, as it is for 96 Kr with Z = 36 in the Z = 40, N ∼ 60 deformation region. This similarity highlights possible analogue microscopic mechanisms where insufficient proton occupation of the quasi-SU3 orbits (g 9/2 , d 5/2 , s 1/2 ) prevent the development of strong quadrupole correlations in Kr at N = 60.
In summary, the present work establishes for the first time the second excited J π = 4 + state in the neutron rich 96 Kr. The deduced ratio R 4/2 = 2.12(1) confirms that Kr represents the low-Z edge of the island of deformation at N = 60. Its unexpectedly low value is in contradiction with the previously established smooth development of collectivity along the Kr isotopic chain. The value of R 4/2 and the rapid onset of deformation in the region are not reproduced by available beyond mean-field theoretical models. Recent advances of Monte-Carlo Shell-Model calculations in this region should allow progressing in understanding the microscopic nuclear structure of 96 Kr and the moderate development of collectivity in this isotope compared to heavier N = 60 isotones. Further experimental investigations in even more neutron-rich Kr isotopes shall reveal if the decreasing R 4/2 trend does persist. In parallel to multi-step Coulomb excitation and spectroscopy of higher excited states, direct reactions using neutron-rich radioactive ion beams will allow to probe the orbital occupation of relevant states in this mass region.