Evolution of nuclear structure in neutron-rich odd-Zn isotopes and isomers

Collinear laser spectroscopy was performed on Zn ( Z = 30) isotopes at ISOLDE, CERN. The study of hyperﬁne spectra of nuclei across the Zn isotopic chain, N = 33–49, allowed the measurement of nuclear spins for the ground and isomeric states in odd- A neutron-rich nuclei up to N = 50. Exactly one long- lived ( > 10 ms) isomeric state has been established in each 69–79 Zn isotope. The nuclear magnetic dipole moments and spectroscopic quadrupole moments are well reproduced by large-scale shell–model calculations in the f 5 pg 9 and fpg 9 d 5 model spaces, thus establishing the dominant term in their wave function. The magnetic moment of the intruder I π = 1 / 2 + isomer in 79 Zn is reproduced only if the ν s 1 / 2 orbital is added to the valence space, as realized in the recently developed PFSDG-U interaction. The spin and moments of the low-lying isomeric state in 73 Zn suggest a strong onset of deformation at N = 43, while the progression towards 79 Zn points to the stability of the Z = 28 and N = 50 shell gaps, supporting the magicity of 78 Ni.


Introduction
Evaluating the accuracy of large-scale shell-model interactions is dependent on experimental data in regions of shell closures.
For elements with Z ≈ 28, recent experiments have aimed to shed light on nuclear structure in the neutron-rich isotopes and hence assess the reliability of shell-model predictions. This region is known for being rich in nuclear structural change, including the weak sub-shell closure at N = 40 observed in nickel [1] and copper [2], the development of collectivity beyond N = 40 in Ga isotopes [3] and the doubly magic nature of the exotic nucleus 78  which is an important waiting point in the astrophysical r-process [4]. These phenomena can be understood by studies of the nuclear states directly, with laser spectroscopy proving a versatile method for measuring properties of long-lived ground and isomeric states (>10 ms), such as spins, moments and mean-square charge radii.
In particular, the nuclear spins and magnetic dipole moments, μ, give information on the evolution of the single particle (SP) states of nucleons within the shell-model. Additionally, the spectroscopic quadrupole moments, Q s , provide information on the shape of the charge distribution and collectivity in the nucleus.
Isotopes of zinc have two protons outside the Z = 28 shell closure. In a non-interacting shell-model picture these protons occupy the π p 3/2 orbital. However, studies of neighbouring 29 Cu and 31 Ga isotopes show that the filling of the ν g 9/2 orbital after N = 40 induces a reordering of the proton SP levels p 3/2 and f 5/2 , and hence a ground state (g.s.) spin change induced by the tensor force [3,5]. Due to the even proton number of zinc, the effects of this predicted level reordering on the g.s. properties of odd-A Zn isotopes will be more subtle as their moments will be dominated by the unpaired neutron. The tensor interaction decreases the size of the Z = 28 shell gap with increasing N, as the filling of the ν g 9/2 orbit induces a reduction of the spin-orbit splitting between the π f 7/2 and π f 5/2 levels [6]. This interaction therefore has strong implications on shell closures in this region, and most notably at closures far from stability, where the Z = 28 and N = 50 shell gaps provide information on the effectiveness of the magicity of 78 Ni [7,8]. Despite these predictions, mass measurements of 71-81 Zn at ISOLTRAP have highlighted the persistence of the N = 50 shell closure at Z = 30 [9], while similar measurements at JYFLTRAP have indicated an increasing rigidity of the shell gap from Ga to Ni [10]. Studies at RIBF of β-decay half-lives of 76,77 Co, 79,80 Co and 81 Cu, and E(4 + ) of 80 Zn have also pointed towards a doubly magic structure for 78 Ni [11,12].
In a previous publication [13] we reported the laser spectroscopy results of an isomeric state in 79 Zn, which was originally reported in Ref. [14]. A preliminary analysis of this system, which also displays a signature of shape coexistence, indicated the presence of an intruder νs 1/2 state. Here, we report for the first time a full theoretical analysis in the context of new measurements for the entire isotopic chain. The observed nuclear magnetic dipole and spectroscopic quadrupole moments are compared with large-scale shell-model calculations in different model spaces to evaluate the influence of proton excitations across Z = 28 and of neutrons across N = 50, and the evolution of nuclear structure along the isotopic chain. Additionally, this Letter establishes firmly the ground and long-lived (>10 ms) isomeric state spins of odd-A Zn isotopes from N = 33-49. The direct observation of their hyperfine structure (hfs) solves a long-standing discussion about the (non-)existence of several long-lived isomeric states in some of these isotopes [15][16][17][18]14].

Experimental method
The experiment was completed at the collinear laser spectroscopy setup COLLAPS [19] at ISOLDE, CERN. Radioactive fission fragments were produced using a thick UC x target (45 g/cm 2 ) bombarded with pulses of 1.4-GeV protons. A neutron converter [20] suppressed the production of Rb isobars, which contaminate the beam purity of neutron-rich Zn isotopes. The Zn yield was selectively enhanced by a factor of 100 using the Resonant Ionization Laser-Ion Source (RILIS) [21]. Zn ions were accelerated to 30 keV and mass selected using the high-resolution separator (HRS). A gas-filled radio frequency quadrupole, ISCOOL [22], including isomeric structures. See [13] for the full hyperfine spectrum of 79,79m Zn. delivered cooled and bunched ions to the collinear laser spectroscopy setup, with a typical accumulation and release cycle of 200 ms. The ion beam was neutralised in-flight through a charge exchange cell (CEC) filled with Na vapour, quasi-resonantly populating the atomic metastable 4s4p 3 P o 2 level at 32890.3 cm −1 . A co-propagating laser beam was overlapped with the emerging atomic beam in order to resonantly excite the Zn atoms. A tuning potential applied prior to the neutralisation acted to Doppler-shift the laser frequency observed by the atoms, allowing a scan over the hfs resonances. The 481.1873 nm 4s4p 3 P o 2 → 4s5s 3 S 1 transition was studied using a cw frequency-doubled titanium-sapphire laser, locked to a wavelength meter with use of an interferometer which was calibrated by a stabilized HeNe laser. A mass dependent time of flight is assigned to each ion bunch, with a 5 μs gate placed on the photon signal from laser-ion bunch interactions to reduce background from non-resonant scattered photons by a factor of 4 × 10 4 . Due to the relatively fast release of Zn from the target [23], the background was further suppressed by limiting IS-COOL accumulation and release cycles to 600 ms after each proton pulse.

Experimental results
Optical spectra for 63-79 Zn are shown in Fig. 1 for the 481.2 nm line. A full hfs was fitted to each experimental spectrum with a χ 2 -minimisation fitting program to obtain the magnetic dipole, A, and electric quadrupole, B, hfs constants [24]. A slightly asymmetric line shape occurs from energy losses, either by the population of higher levels in the charge exchange process or by additional collisions. The line shape has been modelled using the    high-statistics spectrum of an even-A isotope and was fixed in fitting the spectra of all odd-A isotopes [25]. Unambiguous spin assignments could be made for all nuclei and isomers as only a single spin value reproduces all hfs peak spacings simultaneously. The resulting spin assignments and hyperfine constants are shown in Table 1. Moments are calibrated relative to those of 67 Zn by using the hyperfine constants A( 3 P o nuclear magnetic dipole moment μ = +0.875479 (9) μ N [28] and an updated value for the quadrupole moment, based on new calculations of the electric field gradient (EFG) for the 4s4p 3 P o 1,2 states. Applying both the non-relativistic Hartree-Fock and fully relativistic Dirac-Hartree-Fock multiconfiguration methods [29], using respectively ATSP [30] and GRASP [31] atomic structure codes, different electron correlation models were investigated and their consistency checked between the non-relativistic and relativistic approaches. From this study, a set of A, EFG and Q s values are produced [32] from which we derived a quadrupole moment value of Q s = +0.122(10) b. Recently the EFG of Zn in solid Zn has been recalculated using a hybrid Density Functional Theory approach [33]. Combining this value with the experimental quadrupole coupling constants measured by Potzel et al. [34], and corrected for thermal effects, a new quadrupole moment value for 67 Zn g.s., Q s = +0.122(5) b, is determined in [33]. Although this value perfectly agrees with the present atomic estimation, the claimed error bars appear to be very optimistic, taking the general uncertainty of the DFT functional developments into account. We therefore adopted the reference value Q s = +0.122(10) b for the g.s. of 67 Zn. All extracted moments are shown in Table 2.

Nuclear spins of ground and isomeric states
In Fig. 2 we present the experimental ground and isomeric states in 71-79 Zn, for which firm spin-assignments have been made. The assigned parities are based on the measured magnetic moments, as discussed in the next section.
When filling the ν g 9/2 orbital from N = 41 onwards, a g.s. spin I = 9/2 would be expected for the odd-Zn isotopes, in the case that no deformation or correlations are present. This is not the case for any of these isotopes, except for the one-neutron-hole isotope 79 Zn. This is evidence for the magic nature of the N = 50 shell and suggests that the lighter Zn isotopes exhibit significant correlations in their ground states, leading to non-trivial g.s. spins. For 71,73 Zn, the g.s. spin-parity 1/2 − that was tentatively assigned previously [15] is confirmed. The 9/2 + state appears to be isomeric in 71 Zn, and has not yet been observed in 73 Zn. Instead, a 5/2 + isomeric state is observed [18]. In 75,77 Zn, the g.s. spin is 7/2 + and the 1/2 − becomes now a long-lived isomeric state. Finally, in 79 Zn, having 49 neutrons, the g.s. spin is the 9/2 + expected from a noninteracting shell model picture with a hole in the ν g 9/2 orbital.
A long-lived isomeric state with spin-1/2 has been established in this isotope, but its large negative magnetic moment excluded a negative parity for this isomer [13].
In Fig. 2 we compare the lowest energy levels in odd-A Zn isotopes with large-scale shell-model calculations in different model spaces and using different interactions. The simplest model space starts from a 56 Ni core, with protons and neutrons in the f 5/2 , p and g 9/2 orbits. Two effective interactions are available in this model space, JUN45 [35] and jj44b [3]. This model space is extended to include proton excitations from π f 7/2 across Z = 28 and neutron excitations across N = 50 into the νd 5/2 orbital. Two interactions are available in this extended model space: the modified A3DA interaction (A3DA-m) [36] in the Monte Carlo Shell-Model (MCSM) [37] framework and the modified LNPS interaction (LNPSm) that is used with the ANTOINE code [38]. To understand the structure of the newly found positive parity isomer in 79 Zn, we use the recently developed PFSDG-U interaction [39] in the proton pf and neutron sdg valence space.
As can be seen in Fig. 2, none of the calculations reproduces correctly the energy level ordering of all ground and isomeric states in these isotopes. In the calculations with a 56 Ni core (JUN45) the g.s. spin is predicted to be 9/2 + for all isotopes. By opening the proton shell to include excitations from the π f 7/2 orbital across Z = 28, as well as excitations to the νd 5/2 orbital Table 2 Nuclear magnetic dipole moments, μ, and spectroscopic quadrupole moments, Q s , calculated from experimental data are compared to JUN45, LNPS-m and A3DA-m shellmodel calculations.
To calibrate across the isotope chain the nuclear moments of 67 Zn are used as references, with μ and Q s taken from [28] and this work respectively, with details given in [32]. Precise hyperfine constants A( 3 P 2 ) and B( 3 P 2 ) [27] are used.
(A3DA-m and LNPS-m) the agreement becomes better for 77 Zn. However, for the less exotic isotopes the level ordering is still not well reproduced. These interactions predict a positive-parity 1/2 + level in 79 Zn, although it appears at 1.8 and 1.5 MeV respectively, well above the experimental energy of 1.10 (15) MeV [14]. The magnetic moment of this level motivated a further extension of the model space, as realized in the PFSDG-U interaction. In this extended model space an isomeric 1/2 + level is found at the experimental energy.

Ground and isomeric state g-factors and wave functions
In the f 5 pg 9 model space used for JUN45 and jj44b, we use g eff s = 0.7g free s for magnetic moment calculations, and effective charges e eff p = 1.5e, e eff n = 1.1e [35]. These interactions were used in our previous work on the nuclear moments and spins of the Cu and Ga ground states [40,3], reproducing these observables rather well. The LNPS-m and A3DA-m interactions start from a 40 Ca core and include also the d 5/2 orbital, thus using a fpg 9 d 5 model space. The neutron f 7/2 orbit is blocked for the LNPS-m calculations, but this has no influence on the spectroscopy of the neutron-rich (N > 38) isotopes. Free g-factors can be applied in this extended model space. Furthermore, the effective neutron charge can be reduced to e eff n = 0.46e because of the inclusion of the νd 5/2 orbital, while the proton charge is taken as e eff p = 1.31e [38]. The nuclear moments have been calculated for the lowest lying energy level with the confirmed spin assignment. The nuclear magnetic moment provides a sensitive probe of the wave function of the state. By comparing the measured magnetic moments, and more specifically the corresponding g-factor (g = μ/I) to the effective SP values of nearby orbitals, the leading contributions to the wave functions can be deduced (Fig. 3). These values are also compared to the predictions of the shellmodel interactions, from which we can extract the calculated main contribution in the wave function.
The experimental g-factors for the 1/2 − ground states of 69,71,73 Zn and the isomeric states in 75,77 Zn are in good agreement with the effective SP value for the ν p 1/2 orbit. The (ν p 1/2 ) 1 led wave function configuration for these states is confirmed by the calculations in the f 5 pg 9 model space that predict a >50% contribution from a ν p 1/2 hole configuration for the ground states and >60% for the isomers. The calculated magnetic moments appear to systematically underestimate the measured values of the 1/2 − states. Further theoretical investigations are needed to understand this. The ground states of 65,67 Zn (N = 35, 37) both have spin 5/2 + and their g-factor is in good agreement with that for an unpaired neutron configuration in the ν f 5/2 orbital. The high-spin states in the 69-79 Zn isotopes, having spins 9/2 + , 5/2 + and 7/2 + , all have a g-factor that is very close to that for an unpaired neutron configuration in the ν g 9/2 orbital, suggesting this is the leading term in their wave function.
For the isomeric 9/2 + states in 69,71 Zn and the 9/2 + g.s. in 79 Zn, a single unpaired g 9/2 neutron configuration is expected to dominate the wave function. That is confirmed by the large-scale shell-model calculations from JUN45 and jj44b, which predict this configuration to have the largest contribution indeed: about 50% in 69 Zn (N = 39), 40% in 71 Zn (N = 41) and nearly 100% in 79 Zn (N = 49). The excellent agreement of the calculated magnetic moments with the observed value for the 79 Zn g.s. with all interactions (Table 2), illustrates the persistence of N = 50 as a shell gap.
Also the values for the 7/2 + ground states in 75,77 Zn are well reproduced by all large scale shell-model calculations. The leading term in their wave function is a seniority-3 ν(g 9/2 ) 3 7/2 configuration, which makes up roughly half of the wave function in 75 Zn and 77 Zn.
The 5/2 + isomeric state in 73 Zn also has a g-factor that agrees well with the value for an unpaired g 9/2 neutron configuration.
Note that this 5/2 + g-factor does not agree with that of a νd 5/2  SP configuration. This excludes that this 5/2 + isomeric state is an intruder isomer, dominated by a neutron excited into the νd 5/2 orbital. The fact that its g-factor is somewhat lower than that of the other high-spin states suggests that some admixture with such a configuration cannot be excluded. In Fig. 4a, the high-spin g-factors are presented on a smaller scale, in order to better see how well each of the calculations agrees with the data. The divergence of predictions from the LNPS-m and A3DA-m interactions from experiment towards N = 50 while JUN45 and jj44b converge hints at the persistence of the Z = 28 and N = 50 shell gaps. All Table 3 Calculated energies (MeV) and electromagnetic moments μ (μ N ) and Q s (b) for 79 Zn and 81 Ge using the PFSDG-U interaction. The same effective charges are used as for LNPS-m. interactions predict a very fragmented wave function for the 5/2 + isomeric state in 73 Zn, with the largest contribution in JUN45 being less than 10%. For the isomeric state in 79 Zn, the model space limitations of the JUN45 and jj44b interactions prevent any predictions for the positive parity 1/2 + state. The spin and positive parity of this level were tentatively assigned in [14] and firmly established by [13], based on its strong negative g-factor, which is incompatible with a p 1/2 hole configuration (see Fig. 3). The larger model space of the LNPS-m and A3DA-m interactions consider excitations across the N = 50 shell closure into the νd 5/2 orbit only. In this model space, a 1/2 + level is predicted at 1.8 and 1.5 MeV respectively, with g-factors g A3DA−m = −1.206 and g LNPS−m = −1.482, closer to the observed value, g exp = −2.038, but still not in agreement.
In [13], a 1p-2h neutron excitation to a positive parity spin-1/2 state is suggested, with a large part of the wave function dominated by a single neutron in the s 1/2 orbit. A new shell model interaction has been developed, suitable to calculate levels in isotopes around N = 50 with protons limited to the pf shell and neutrons to the sdg space [39], including all spinorbit partners (allowing the use of free g-factors). Thus the interaction is not suited to calculate levels in which neutrons in the pf orbits play an important role (such as the 1/2 − isomeric states in 75,77 Zn). Therefore, we limit the calculations to the N = 49 isotones with protons in the pf shell, where positive-parity intruder orbits have been observed between Z = 38 and Z = 30. In Fig. 5 we compare the calculated lowest positive parity 9/2 + , 1/2 + and 5/2 + levels in 85 Kr, 83 Se, 81 Ge, 79 Zn and 77 Ni to the experimental data. The energies for these intruder levels are well reproduced for 79 Zn, given typical shell-model uncertainties (a few 100 keV) on energy predictions. For 77 Ni, these intruder levels are predicted close to 2 MeV, suggesting a rather good doubly-magic nature for 78 Ni. For the heavier isotones they appear 200 to 500 keV too low, which needs some further investigation. The 1/2 + state appears to be isomeric in 79 Zn and 81 Ge [41], and the intruder nature of this state is firmly established via its magnetic moment. Indeed, excellent agreement is observed between the calculated and observed magnetic moment of the intruder state in 79 Zn (Table 3). In Fig. 6 we represent the normal- ized neutron occupancy in all sdg levels. The g.s. wave function is dominated by a single neutron hole in the g 9/2 orbit (50%), but a significant amount of neutron excitations into the sd space are observed for the isomeric state. That leads to a better agreement with the experimental magnetic and quadrupole moment compared to the LNPS-m calculations. The isomeric level has a large occupancy of the s 1/2 level, but also more np-nh excitations into the d 5/2 orbital as well as the higher gd orbits. The main component in the isomeric wave function is found to be indeed of a 1p-2h nature (40%).

Quadrupole moments of ground and isomeric states
The quadrupole moments of the high-spin states, shown in Fig. 4b, are able to shed further light on the single particle structure as well as collectivity and structural changes across the isotope chain as the ν g 9/2 orbit is filled. In Table 2, the A3DA-m interaction is shown to most accurately predict the measured quadrupole moments of neutron rich isotopes from 69-79 Zn, except for the g.s. of 75 Zn.
Already starting from 69 Zn, the neutrons gradually fill the g 9/2 orbital. In general the quadrupole moment should reflect the nature of a single g 9/2 neutron particle in 69 Zn and single g 9/2 neutron hole in 79 Zn, as discussed for the magnetic moments measured in this work. In Fig. 4b it can be seen that the 1p configuration for the isomeric state in 69 Zn has indeed the opposite quadrupole moment of the 1h configuration for the g.s. of 79 Zn. The quadrupole moments for the seniority-1 (ν g 9/2 ) n configurations with spin-9/2 ( 69,71,79 Zn) follow the expected linear increase, crossing zero in the middle of the shell [42] (red dashed line of Fig. 4b) which corresponds to A = 74 in the present case.
The experimental magnetic moments of the 7/2 + states in 75,77 Zn show a seniority-3 ν(g 9/2 ) 3 7/2 + configuration, explaining why the their quadrupole moments do not follow the straight line of the seniority-1 cases.
An estimate of the quadrupole moments of the 7/2 + seniority-3 states can be obtained using the effective single-particle quadrupole moment observed in the seniority-1 cases, Q sp = J |Q | J , the applicable coefficients of fractional parentage [43] and the relation [44,45], in which the quadrupole moment of a state with n particles in an orbit and seniority ν is linked to the Q sp via the coefficients of fractional parentage, cfp = [ J n−1 (ν 1 , J 1 ), J ; I|} J n ν I] [44]. Predictions for the seniority-3 I = 7/2 + (violet dashed line) and I = 5/2 + states (green dashed line) based on eq. (1) are presented in Fig. 4b.
The experimental values for 75,77 Zn are observed to follow this trend.
The larger quadrupole moment of 5/2 + state of 73m Zn does not coincide with the seniority-3 5/2 + prediction, indicating substantial deformation. This confirms the conclusions from earlier beta-decay studies [15] and transfer reaction studies [46], where the observed low-energy spectra of 73 Zn were interpreted as a more strongly deformed prolate structures (β ≈ +0.2) in 73 Zn. This result, along with the increase of configuration mixing at 73m Zn mentioned above, witnessed both in its magnetic moment and its quadrupole moment, therefore signals a rapid shape transition at N = 43 in the Zn isotopes. A low-energy Coulomb excitation study of zinc isotopes at REX-ISOLDE supports this conclusion. Here a sudden lowering of 2 + states at N = 40 and an increase in B(E2↓) strength towards N = 44 was associated with an increase in collectivity due to proton-neutron correlations and a weakening of the sub-shell closure [47,48]. An onset of collectivity is also observed in the quadrupole moments of odd-A Ga isotopes when N > 40 [3], while no such increase in collectivity is observed in the quadrupole moments of Cu isotones [40]. Therefore, Zn isotopes are considered to lie within a transitional region between spherical Ni and deformed Ge nuclei [49].

Conclusion
In summary, the nuclear spins, magnetic dipole moments and electric quadrupole moments have been determined for the ground and isomeric states in 63-79 Zn by means of collinear laser spectroscopy. Exactly one long-lived (t 1/2 > 10 ms) isomeric state has been observed in all odd-A Zn isotopes from N = 39-49. This has provided an insight into the neutron level systematics as the N = 50 shell closure is approached.
The magnetic dipole moments of ground and isomeric states have been compared to a variety of large-scale shell-model calculations in different model spaces. All states up to 79 Zn (except for the 1/2 + isomeric level) are well described with interactions assuming a 56 Ni core and neutrons limited to the pf and g 9/2 orbits.
Extending the model space both for proton excitations across Z = 28 and neutron excitations across N = 50 does not significantly improve the agreement with experiment, with their high-spin gfactors diverging from the experimental values for N = 45-49 suggesting the preservation of these shell gaps when approaching 78 Ni.
For the 1/2 + intruder isomer, a newly developed shell-model interaction in the pf +sdg model space is needed to reproduce the magnetic dipole moment of 79m Zn, which lies outside the f 5 pg 9 and fpg 9 d 5 model spaces. The PFSDG-U interaction suggests a leading wave function configuration formed by a 1p-2h excitation from ν g 9/2 to ν3s 1/2 , consistent with the spin-parity of the isomeric state as 1/2 + . A similar isomeric intruder level has been suggested in 81 Ge [41] and a future magnetic moment measurement should confirm its 1p-2h intruder nature. Also in 80 Ga a low-lying short-lived intruder 0 + state has been inferred from β-decay studies [50]. Thus further studies to establish the deformation of these proposed shape-coexisting states are needed.
The quadrupole moments reveal a strong onset of collectivity from 71m Zn to 73m Zn, where the deviation of the quadrupole moment from the seniority-3 5/2 + prediction indicates substantial deformation at N = 43. The measured quadrupole moments are described most accurately by the A3DA-m interaction, but in general they are well reproduced by all shell-model interactions.