Further evidence for shape coexistence in $^{79}$Zn$^{m}$ near doubly-magic $^{78}$Ni

Isomers close to doubly-magic $^{78}_{28}$Ni$_{50}$ provide essential information on the shell evolution and shape coexistence near the ${Z=28}$ and ${N=50}$ double shell closure. We report the excitation energy measurement of the $1/2^{+}$ isomer in $^{79}_{30}$Zn$_{49}$ through independent high-precision mass measurements with the JYFLTRAP double Penning trap and with the ISOLTRAP Multi-Reflection Time-of-Flight Mass Spectrometer. We unambiguously place the $1/2^{+}$ isomer at 942(10) keV, slightly below the $5/2^+$ state at 983(3) keV. With the use of state-of-the-art shell-model diagonalizations, complemented with Discrete Non Orthogonal shell-model calculations which are used here the first time to interpret shape coexistence, we find low-lying deformed intruder states, similar to other ${N=49}$ isotones. The $1/2^{+}$ isomer is interpreted as the band-head of a low-lying deformed structure akin to a predicted low-lying deformed band in $^{80}$Zn, and points to shape coexistence in $^{79,80}$Zn similar to the one observed in $^{78}$Ni. The results make a strong case for confirming the claim of shape coexistence in this key region of the nuclear chart.

The atomic nucleus, a conglomerate of protons and neutrons, is a complex many-body system with unique features.The nuclear shell model has successfully described various nuclear properties, including the emergence of shell closures [1] and magic numbers [2].
Similar to the atomic shell, nuclei can be excited, resulting in a dense level structure.Ground and excited states can show different shapes resulting from the microscopic wave function [3].Deformed excited states often emerge near closed shells, where the excitation of multiple nucleons across the shell gap can be energetically favorable, leading to deformation through the increased number of particles found in the valence space [4].While typically the coexistence of ground states and deformed excited states at low energies are observed, shape inversion can appear when the ground state becomes deformed in coexistence with a spherical excited state [4][5][6] Research on shape coexistence close to the doubly-magic nucleus 78  28 Ni 50 has gained momentum only recently [7].Low-lying intruder states, often indicators of shape coexistence, have been studied in N = 49 isotones through transferreaction experiments [8][9][10][11].First evidence supporting shape coexistence, such as the claimed discovery of a 0 + 2 intruder state in 80  32 Ge 48 from Ref. [12], could, however, not be confirmed in subsequent experiments [13,14].More recently, the doubly-magic nature of 78 Ni was supported through the measurement of its E(2 + 1 ) value [15], as well as γ-ray spectroscopy [16] and mass spectrometry of 79  29 Cu 50 [17], reinforcing the persistence of the Z = 28 gap.The potential appearance of shape coexistence in 78 Ni is furthermore theorized to be a pathway into a new island of inversion at N = 50 [18].
The isomer in 79  30 Zn 49 , a long-lived nuclear state, provides a unique opportunity to further study the interplay between the single-particle and collective degrees of freedom in the close vicinity of 78 Ni.The first spectroscopy of  [20], ME lit is deduced from the ground state mass (weighted average of the measurements from Refs.[27][28][29]) in combination with the excitation energy reported in [19]. 84Kr + from [34] was used as a reference for the ToF-ICR measurements, while 79 Zn + in its ground state from [27][28][29] was used for the MR-ToF MS measurements.12) 943( 14)

Nuclide
79 Zn from (d,p) transfer reactions found evidence for the presence of intruder states and tentatively assigned a spinparity of 1/2 + to the isomeric state with an excitation energy of 1100(150) keV, as well as a close-lying 5/2 + state with 983(3) keV [19], leaving the exact state ordering uncertain.The spin-parity was confirmed for the 9/2 + ground and the 1/2 + isomeric states through magnetic moment measurements from collinear laser spectroscopy [20,21].More significantly, these works found a large isomer shift in the charge radius but could not connect this increase to a deformation unambiguously.
Recent studies have shown that shape changes may be linked to multiple particle-hole excitations of both protons and neutrons [22][23][24][25][26].However, a precise and accurate excitation energy measurement of the 1/2 + isomer to confirm claims of shape coexistence in 79 Zn is missing from the above-mentioned work.Such measurement will unambiguously determine the state ordering, validate the particle-hole excitation character, and benchmark the binding energy predictions of the employed shell-model interactions near 78 Ni.
While the ground-state mass of 79 Zn is precisely known [27][28][29] via mass measurements, the uncertainty on the excitation energy from the transfer-reaction experiment is rather large.Given the importance of shape coexistence in the immediate vicinity of 78 Ni, we present two independent high-precision mass spectrometry experiments of the isomer 79 Zn m using the JYFLTRAP double Penning trap [30] at the Ion Guide Isotope Separator On-Line (IGISOL) facility [31] in Jyväskylä (Finland), and the Multi-Reflection Time-of-Flight Mass Spectrometer (MR-ToF MS) of ISOLTRAP [32] at ISOLDE/CERN [33] (Switzerland).The experiments were performed at different facilities and with different techniques to ensure the isomer's production and the accuracy of its excitation energy.The results are interpreted by large-scale shell model calculations, utilizing the valence space of interactions used in Refs.[17,18,20,21].While offering a more detailed and accurate picture of the nuclear structure in this critical region, the calculations highlight the relative fragility of the doubly-magic shell strength.
For the Penning-trap measurements, the ions of interest were produced via proton-induced fission using 35-MeV protons from the K130 cyclotron impinging onto a 15 mg cm −2 thick nat U target at IGISOL.The reaction products were stopped and thermalized in the helium gas cell of the fission ion guide [35], leaving a large fraction of the products singly charged.The ions were extracted from the chamber with a sextupole ion guide [36] and accelerated to 30 keV.A 55 • dipole magnet was used to mass-separate the ions based on their mass-to-charge ratio m/q.The mass-separated beam was then stopped in a radiofrequency quadrupole cooler and buncher (RFQ-cb) [37] and released as ion bunches into the double Penning trap JYFLTRAP [30].In the first trap, either the ground-or isomeric-state ions of 79 Zn were selected using mass-selective buffer-gas cooling [38].The selected ions were transferred to the second trap, where the high-precision mass measurements were performed using the Time-of-Flight Ion Cyclotron Resonance (ToF-ICR) method [39].The cyclotron resonance frequency ν c = qB/(2πm) of the 1/2 + state in 79 Zn + was measured using a 100 ms pulse of quadrupolar rf excitation.Altogether four ToF-ICR spectra were measured for the 1/2 + state in 79 Zn + .An example of such a ToF-ICR spectrum is shown in Fig. 1(a).
The magnetic field strength B was determined using 84 Kr + as a mass reference.The mass of the 1/2 + state in 79 Zn was obtained from the measured frequency ratio r = ν c,ref./ν c between the 84 Kr + reference ions and the isomericstate ions of 79 Zn + as m = r(m ref − m e ) + m e , where m ref is the mass of the reference ion and m e is the electron mass.Two sources of systematic uncertainties were taken into account in the analysis, the fluctuation of the magnetic field being 8.18 × 10 −12 × ∆t min −1 [41], where ∆t represents the time between two reference measurements, and mass-dependent uncertainties being 2.2 × 10 −10 × (m − m ref )/u [42].
For the MR-ToF MS measurements, neutron-rich zinc isotopes were produced at the isotope separation online facility ISOLDE at CERN [33] by impinging a 1.4-GeV proton beam onto a solid tungsten block to generate an intense spallation neutron flux [43].The zinc isotopes were then produced through neutron-induced fission processes in an adjacent thick uranium carbide target.Using the tungsten converter resulted in reduced production of isobaric neutron-deficient nuclides.The radioactive fission products then diffused through the target material into a cold quartz transfer line [44,45] which further eliminated contamination of surface-ionized elements, e.g.Ga, Rb, and Sr [46].Darker shades correspond to more ions and lighter shades to fewer ions.The solid red line represents the fit to the data points (black) using the model from Ref. [39].The cyclotron frequencies are indicated with a vertical black dashed line for the ground state (not present in this spectrum) and a vertical blue dash-dotted line for the isomer.(b) Time-of-Flight spectrum for the MR-ToF MS data.The ToF of the ground state is indicated by a vertical black dashed line, and the ToF of the isomer by a vertical blue dash-dotted line.The solid red line represents the fit to the data using the model from Ref. [40].
The remaining radioactive species were then ionized by the Resonance Ionization Laser Ion Source (RILIS) [47], using an element-selective three-step ionization scheme for zinc.The ion beam was then mass-separated using the generalpurpose mass-separator dipole magnet, removing non-isobaric contamination, before being sent to the ISOLTRAP mass spectrometer [32].The quasi-continuous ion beam was cooled and bunched in a linear RFQ-cb [48] with a storage time of 10 ms before being captured in the MR-ToF MS [49] using the in-trap lift method [50].After trapping times of up to 43 ms, the ion bunch was ejected onto a single-ion counting detector using the same in-trap lift.
The excitation energy of the isomeric state E = (∆t/t 0 ) 2 + 2∆t/t 0 m 0 c 2 is related to the time-of-flight difference ∆t between the ground state with mass m 0 and the isomeric state, the absolute flight time of the ground state t 0 , and the speed of light in vacuum c.For long flight times, where ∆t ≪ t 0 , this reduces to E ≈ 2∆t/t 0 × m 0 c 2 .Figure 1 (b) shows the ToF spectrum for the MR-ToF MS data.Zinc was delivered from ISOLDE as a pure beam, thus only the ground state (black dashed line) and the isomeric state (blue dash-dotted line) of 79 Zn were present in the spectrum.A mass resolving power R = t 0 /2∆t FWHM = 300 000 was reached, sufficient to resolve the two states.The asymmetric peak shape was fitted with a multi-component exponentially modified Gaussian ("hyper-EMG") [40] to extract the absolute ToF t 0 = 4 277 871(2) ns of the ground state and the ToF difference ∆t = 274(4) ns between the two states.The excitation energy was then calculated using the ground state mass with ME lit = −53 432.1 (18) keV, which we calculated as the weighted mean of the results from Refs.[27][28][29].The excitation energy was measured for different ion loads to account for ion-ion interactions during the storage time in the MR-ToF MS [51,52].
The results of the two independent measurements are summarized in Tab.I.The extracted excitation energies of the isomer agree very well, resulting in a weighted mean of 942 (10) keV.The isomer energy is lower than 1100(150) keV as given in the NUBASE 2020 evaluation [53], which is based on the transfer reaction experiment from Ref. [19].Our value is significantly more precise and unambiguously sets the isomeric state below the 5/2 + state located at 983(3) keV.We note that the result agrees with the value 943(3) keV, obtained from the beta-decay spectroscopy of 79 Cu [54].Here, we confirm the existence of the isomer and provide a direct measure of its excitation energy.
To interpret the present experimental data, shell-model calculations with the PFSDG-U interaction [15,18] were performed for 79,80 Zn.The valence space is spanned across the full pf shell for protons and full sdg shell for neutrons, with 60 Ca as an inert core.This interaction has been successfully used in the 78 Ni region to describe, among others, the two-neutron separation energies S 2n along the zinc isotopic chain [17], as well as the magnetic g-factor in 79 Zn [21].
The calculated excitation energies for 79 Zn (Tab.II) are in good agreement with the experimental results with 1/2 + and 5/2 + at 0.83 MeV and 0.94 MeV, respectively.We find that the two low-lying excited states in 79 Zn show a one-particle-two-hole configuration, consistent with other N = 49 isotones [8][9][10][11].While the s 1/2 and d 5/2 orbitals lie together at the neutron Fermi surface in the vicinity of 78 Ni, the correlated N = 50 neutron gap S N ( 81 Zn) − S N ( 80 Zn), calculated from the present theoretical values, remains sizeable at about 4.0 MeV and is in agreement with the effective single-particle energies of Ref. [6] and with the experimental value provided in Ref. [34].These states usually recover enough correlation energy (total energy minus the monopole part) to compensate for their energy gap loss.This is observed in our calculations shown in Tab.II, where the total correlation energy is extracted.The excited 1/2 + and 5/2 + states recover correlation energy on the order of ∼ 6.5 MeV − 6.9 MeV compared to the 9/2 + the ground state.
The low excitation energy of these two states can be understood as the balance between an average of 1.3 neutrons excited across the shell gap with respect to the ground state (∼ 5.5 MeV) and the correlation energies (∼ 6.5 MeV − 6.9 MeV) which compensate and result in very low-lying excitation energies for these one-particle-two-holes states.
An inspection of the partial occupancies of the first two excited states (see Tab. II) reveals strong neutron mixing for the orbitals above N = 50, as well as different proton occupancies with respect to the ground state, invalidating the spherical single particle-hole nature of these states and rather suggesting a deformed shape.This neutron mixing and proton reshuffling is visualized in Fig. 2, where the occupancy differences of the excited states with respect to their ground states are plotted.
The structure of the excited states can be interpreted within the Discrete Non-Orthogonal shell-model (DNO-SM) method, newly developed in Ref. [55] and applied recently in Refs.[56,57].This approach expands the shell-model wave functions in the deformed Hartree-Fock basis rather than the usual spherical m-scheme basis.This allows the extraction of the corresponding deformation amplitudes of a given state in the (β, γ) plane.
Figure 3 (top panel) depicts such expansion for the ground state and the first two excited states of 79 Zn.Two clear patterns emerge: in the ground state, the main components of the wave function tend towards the low-deformation region with a small β value, while for the excited states, the wave functions are more fragmented and have, on average, a larger deformation.Also, we find the clustering of the wave-function components consistent with the deformation parameters β ≈ 0.15 (ground state) and β ≈ 0.22 (isomeric state), deduced from the nuclear charge radius in Ref. [20].Moreover, the DNO-SM calculations reveal that both 1/2 + and 5/2 + states belong to the same rotational structure, which is characterized by the K = 1/2 components of 100% and 96%, respectively (see K-component extraction in Refs.[55,57]).
To probe further into the deformed character of the low-lying states in 79 Zn, we have complemented the calculations with the investigation of 80 Zn.We find two low-lying 0 + states: the spherical ground-state and an excited deformed state of two-particle-two-hole nature at 2.16 MeV (see Tab. II).Again, a larger correlation energy (∼ 6.3 MeV) is observed for the excited 0 + state on the same order of magnitude as those of the deformed intruder states in 79 Zn, indicating the same deformation nature of these states.Figure 3 shows the wave function expansions for these two 0 + states (bottom panel).There is a clear similarity between the spherical ground states for 79 Zn(9/2 + )/ 80 Zn(0 + 1 ) and the deformed excited states for 79 Zn(1/2 + , 5/2 + )/ 80 Zn(0 + 2 ), advocating for the deformed nature of the observed isomer in 79 Zn, as well as its 5/2 + companion.
Finally, the present shape coexistence discussed for 79,80 Zn can be put in perspective with the shape coexistence recently observed and discussed for 78 Ni [15,18]: the deformed intruder 0 + 2 of 78 Ni has 2.7 neutron p-h excitations on average, in remarkable agreement with present values quoted in Tab.II for 0 + 2 of 80 Zn.Both 0 + 2 states have ∼ 2.4 − 3 protons on average in the f 5/2 , p 3/2 , p 1/2 shells, leading to close collective structures.Therefore, the shape coexistence in 78 Ni and in the presented 79−80 Zn reveal striking similarities.
To summarize, we have established the level ordering and determined the excitation energy of the isomer in 79 Zn by means of high-precision mass spectrometry.Two measurements were performed independently, using different production methods and measurement techniques at different radioactive ion beam facilities.We show unambiguously that the 1/2 + isomeric state with 942(10) keV lies below the 5/2 + state with 983(3) keV.The new DNO-SM calculations tool provides the theoretical analysis, predicting the occurrence of low-lying deformed intruder states.The 1/2 + isomer is interpreted as the band-head of a low-lying deformed structure of the same nature as a predicted low-lying deformed band in 80 Zn.These findings provide an additional indication for shape coexistence in 79,80 Zn, similar to the one suggested for 78 Ni.
We thank the ISOLDE technical group and the ISOLDE Collaboration for their support.We acknowledge the support of the German Max Planck Society, the French Institut National de Physique Nucléaire et de

FIG. 1 .
FIG. 1. (a)A typical ToF-ICR spectrum for the 1/2 + state in 79 Zn + .Colored bins indicate the number of detected ions.Darker shades correspond to more ions and lighter shades to fewer ions.The solid red line represents the fit to the data points (black) using the model from Ref.[39].The cyclotron frequencies are indicated with a vertical black dashed line for the ground state (not present in this spectrum) and a vertical blue dash-dotted line for the isomer.(b) Time-of-Flight spectrum for the MR-ToF MS data.The ToF of the ground state is indicated by a vertical black dashed line, and the ToF of the isomer by a vertical blue dash-dotted line.The solid red line represents the fit to the data using the model from Ref.[40].

2 FIG. 2 .
FIG. 2. Occupancy differences between excited states in79,80 Zn, and 78 Ni with their respective ground states.The data for 78 Ni is taken from[18].

TABLE I .
Frequency ratio r or time-of-flight difference ∆t, mass-excess values ME, and excitation energy of the isomer E determined in this work.The values for J π and T 1/2 are from Ref.

TABLE II .
[18]pation of orbitals in the full proton pf and neutron sdg valence space for low-lying states in79,80Zn and 78 Ni (the latter taken from Ref.[18]).Eexp and E theo (in MeV) are the experimental and theoretical excitation energies.Ecorr and E * corr (in MeV) are the total correlation energy and the correlation energy difference of an excited state with respect to its ground state.n * ν and n * π are the total numbers of protons and neutrons above Z=28 and N=50, respectively.
Physique des Particules (IN2P3), the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreements No. 682841 'ASTRUm', 654002 'ENSAR2', and 101020842 'EUSTRONG'), as well as the German Federal Ministry of Education and Research (BMBF; Grants No. 05P18HGCIA, 05P21HGCI1, and 05P21RDFNB).L.N. acknowledges support from the Wolfgang Gentner Programme of the German Federal Ministry of Education and Research (Grant No. 13E18CHA).This work has been supported by the Academy of Finland under the Finnish Centre of Excellence Program (Nuclear and Accelerator Based Physics Research at JYFL 2012-2017), and under Academy of Finland grants No. 275389, 284516, 312544, 295207, and 306980.We acknowledge