Measurement and microscopic description of odd–even staggering of charge radii of exotic copper isotopes

Nuclear charge radii globally scale with atomic mass number A as A1∕3, and isotopes with an odd number of neutrons are usually slightly smaller in size than their even-neutron neighbours. This odd–even staggering, ubiquitous throughout the nuclear landscape1, varies with the number of protons and neutrons, and poses a substantial challenge for nuclear theory2–4. Here, we report measurements of the charge radii of short-lived copper isotopes up to the very exotic 78Cu (with proton number Z = 29 and neutron number N = 49), produced at only 20 ions s–1, using the collinear resonance ionization spectroscopy method at the Isotope Mass Separator On-Line Device facility (ISOLDE) at CERN. We observe an unexpected reduction in the odd–even staggering for isotopes approaching the N = 50 shell gap. To describe the data, we applied models based on nuclear density functional theory5,6 and A-body valence-space in-medium similarity renormalization group theory7,8. Through these comparisons, we demonstrate a relation between the global behaviour of charge radii and the saturation density of nuclear matter, and show that the local charge radii variations, which reflect the many-body polarization effects, naturally emerge from A-body calculations fitted to properties of A ≤ 4 nuclei. Isotopes with an odd number of neutrons are usually slightly smaller in size than their even-neutron neighbours. In charge radii of short-lived copper isotopes, a reduction of this effect is observed when the neutron number approaches fifty.

Nuclear charge radii globally scale with atomic mass number A as A 1∕3 , and isotopes with an odd number of neutrons are usually slightly smaller in size than their even-neutron neighbours. This odd-even staggering, ubiquitous throughout the nuclear landscape 1 , varies with the number of protons and neutrons, and poses a substantial challenge for nuclear theory [2][3][4] . Here, we report measurements of the charge radii of short-lived copper isotopes up to the very exotic 78 Cu (with proton number Z = 29 and neutron number N = 49), produced at only 20 ions s -1 , using the collinear resonance ionization spectroscopy method at the Isotope Mass Separator On-Line Device facility (ISOLDE) at CERN. We observe an unexpected reduction in the odd-even staggering for isotopes approaching the N = 50 shell gap. To describe the data, we applied models based on nuclear density functional theory 5,6 and A-body valence-space in-medium similarity renormalization group theory 7,8 . Through these comparisons, we demonstrate a relation between the global behaviour of charge radii and the saturation density of nuclear matter, and show that the local charge radii variations, which reflect the many-body polarization effects, naturally emerge from A-body calculations fitted to properties of A ≤ 4 nuclei.
The properties of exotic nuclei, in particular of those close to (doubly) magic systems far from stability, have continually proven pivotal in deepening our understanding of nuclear forces and many-body dynamics. Owing to the presence of the unpaired proton, odd-Z isotopes such as the copper isotopes provide crucial insights into the single-particle proton structure and how this affects the charge radii. However, until now, experimentally accessing charge radii of such isotopes close to exotic doubly closed shells (for example 78 Ni and 100 Sn) has been prohibitively difficult. Extending the existing charge radius measurements 9 beyond 75 Cu has required nearly a decade of developments, culminating in the work presented here.
The first experimental challenge lies in the production of a clean sample of these short-lived species. We produced radioactive ions at the ISOLDE laboratory at CERN. This was done by impinging 1.4-GeV protons onto a neutron converter, producing neutrons that in turn induced fission of 238 U atoms within a thick target, thus minimizing other unwanted nuclear reactions in the target. Several purification steps were nevertheless required to remove contaminants. First, the copper atoms that diffused out of the target were element-selectively laser-ionized by the ISOLDE resonance ionization laser ion source (RILIS) in a hot cavity. The ions were then accelerated to 30 keV for mass separation using the ISOLDE high resolution separator and prepared for high-resolution laser resonance ionization spectroscopy. This required sending the ions into a gas-filled radio-frequency linear Paul trap, ISCOOL, where they were cooled for up to 10 ms. The ions were then released in a short bunch with a length of ~1 μs.
The hyperfine structure of the copper isotopes was measured in the final stage of the experiment using the collinear resonance ionization spectroscopy (CRIS) 10 method. First, the ions were neutralized through a charge-exchange reaction with a potassium vapour. The non-neutralized fraction of the beam was deflected, such that only the neutralized atoms entered into an ultrahigh-vacuum region. Here, they interacted with two pulsed laser beams. The first of these laser systems, tuned to the optical transition at 40,114.01 cm −1 , resonantly excited the atoms, while the second laser further excited these atoms to an auto-ionizing state, chosen for optimal ionization efficiency. Owing to the vacuum of 10 −8 mbar, the collisional ionization rate was less than 1 every 10 min for all except the stable 63,65 Cu, creating a quasi-background-free measurement. As illustrated in the top panel of Fig. 1, by recording the number of ions as a function of the frequency of the first single-mode laser, the hyperfine structure of the copper atoms could be measured. Changes of the charge radius of the nucleus result in small changes in the centroids of these hyperfine structures for each isotope, which is typically a Letters NaTuRE PHySICS 1:10 6 effect. These isotope shifts, while small, were extracted from the data and were used to determine the changes in the meansquared charge radius (see Methods for more details).
Using this sequence of techniques, we could extend our knowledge of the charge radii of the copper isotopes by another three neutron numbers, up to 78 Cu (N = 49). Our new data thus represent an important step towards understanding the nuclear sizes in close proximity to the very neutron-rich doubly magic system of 78 Ni (ref. 11 ). The high efficiency and selectivity of the CRIS technique allows the observation of signals with detection rates of less than 0.05 ions s -1 on resonance (Fig. 1). Thanks to the ultra-low background rates inherent to the method, these detection rates were sufficient for a successful measurement of the charge radius of 78 Cu in less than one day, while other isotopes typically only require a few hours of beam time. The signal-to-background ratio obtained is similar to those achieved in the state-of-the-art in-source measurements 12 , but with much narrower linewidths of <100 MHz, typical for conventional fast-beam collinear laser spectroscopy techniques, thus demonstrating a best-of-both-worlds performance.
The changes in the mean-squared charge radii extracted from the hyperfine structure spectra are plotted in the bottom panel of Fig. 1 (white dots), complemented by literature values for 58−62,67 Cu (ref. 9 ) (white diamonds). The radii of the isomeric states are shown with black markers. The shaded area shows the uncertainty due to the atomic parameters (see Methods for more details). While these atomic uncertainties influence the slope of the charge radii curve, smaller-scale effects such as the odd-even staggering (OES) of the charge radii are largely unaffected. Values of the three-point OES parameter of the radii Δ ð3Þ r I , defined as   9 ). The inset shows the OES of the radii defined by equation (1). Error bars are statistical (one standard deviation). In the main plot, these errors are too small to be seen except for 78 Cu. The blue shaded area represents the systematic uncertainty due to the atomic parameters.

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are shown in the inset of Fig. 1. The OES of the radii is quite pronounced near N = 40, but our new data points reveal a reduction of the OES towards N = 50, starting at 74 Cu. This is likely to be attributed to the change in the ground-state proton configuration. Indeed, as reflected in the ground-state spins and moments 13,14 , up to 73 Cu, the odd proton resides predominantly in the πp 3∕2 orbital, while from 74 Cu onwards it occupies the πf 5∕2 shell.
We will now demonstrate that modern density functional theory (DFT) and the valence-space in-medium similarity renormalization group (VS-IMSRG) frameworks can both provide a satisfactory understanding of changes in the charge radii and binding energies of the copper isotopic chain between neutron numbers N = 29 and N = 49, down to the scale of the small OES. In the context of the following discussion, it is important to remember that the global (bulk) behaviour of nuclear charge radii is governed by the Wigner-Seitz (or box-equivalent) radius r 0 ¼ ½3=ð4πρ 0 Þ 1=3 I , which is given by the nuclear saturation density ρ 0 . On the other hand, the local fluctuations in charge radii, including OES, are primarily impacted by the shell structure and many-body correlations. The common interpretation of OES involves various types of polarization exerted by an odd nucleon, occupying a specific shell-model (or one-quasiparticle) orbital 15 . In particular, the self-consistent coupling between the neutron pairing field and the proton density provides a coherent understanding of the OES of charge radii of spherical nuclei such as semi-magic isotopic chains 5, [16][17][18] .
With measurements now spanning all isotopes between the two exotic doubly magic systems 56,78 Ni, the copper isotopes represent an ideal laboratory for testing novel theoretical approaches in the medium-mass region. This region of the nuclear chart represents new territory for A-body theories based on two-nucleon (NN) and three-nucleon (3N) forces derived from chiral effective field theory 19,20 . In general, OES of masses has only been sparsely studied within the context of nuclear forces and many-body methods 21,22 . However, the VS-IMSRG approach 7,8 has now sufficiently advanced to study most nuclear properties in essentially all open-shell systems below A = 100, including masses, charge radii, spectroscopy and electroweak transitions 23 . The presence of a potential sub-shell closure at N = 40 (ref. 9 ) and the well-evidenced structural changes due to shell evolution as N = 50 is approached 13 all serve to test such calculations even further. From the side of the DFT calculations, the recently developed Fayans functional, successful in describing the global trends of charge radii in the Sn (Z = 50) and Ca (Z = 20) mass regions 3-5 , has not been tested in this region of the nuclear chart, nor with data on odd-Z isotopes in general.
Details on both the DFT and VS-IMSRG calculations can be found in the Methods, but a few key aspects will be mentioned. The DFT calculations were carried out with the Fayans energy density functional 24 , which-importantly-reproduces the microscopic equations of state of symmetric nuclear matter and neutron matter. The inclusion of surface and pairing terms dependent on density gradients has been shown to be crucial for reproducing (the OES of) the calcium charge radii 5 . The VS-IMSRG calculations were performed with two sets of NN+3N forces derived from chiral effective field theory, the PWA and 1.8/2.0(EM) interactions of ref. 25 . Both are constrained by only two-, three-and four-body data, with 3N-forces specifically fit to reproduce the 3 H binding energy and 4 He charge radius.
The absolute charge radii of the copper isotopes are compared to the theoretical calculations in Fig. 2a. These total charge radii are obtained using the reference radius 1 r 65 = 3.9022(14) fm. Excellent overall agreement was obtained with the two sets of DFT calculations, whereas the VS-IMSRG calculations either generated absolute radii that are too small (EM1.8/2.0) or too large (PWA). This confirms earlier findings 2,26 that the reproduction of the absolute radii requires a correct prediction of the nuclear saturation density ρ 0 . While both Fayans functionals used here meet this condition, this is not the case for the interactions used in the VS-IMSRG calculations: the EM1.8/2.0 interaction saturates at too high of a density, and the PWA interaction saturates at a lower density.
The mismatch between DFT and experiment for the neutrondeficient isotopes is primarily related to the pairing in the πf 5∕2 pshell region: as discussed in the Methods, the pairing gradient term that was adjusted using data from the calcium region is too strong in heavier nuclei. We note that a similar reduction around N = 30 was predicted by the Fayans DF3-a1 DFT calculations for the Ni chain 27 . For the binding energies per nucleon 28,29 , shown in Fig. 2b, the DFT calculations matched particularly well with experiment, with both interactions yielding practically identical results. The VS-IMSRG binding energies appear to overbind in the mid-shell region but nevertheless do well in terms of the absolute value as well as the general trend. Figure  near N = 50 is also visible in Δ ð3Þ E I , although less pronounced. This demonstrates that charge radii are more sensitive to the underlying nuclear structure changes in this region, such as to the spin change due to the inversion of the p 3∕2 and f 5∕2 proton orbitals from N = 45 onwards. The OES in binding energies is reproduced very well with the VS-IMSRG calculations, while it is overestimated with DFT. For the charge radii, the functional Fy(Δr) that includes the OES in calcium isotopes as fitting parameters, generates substantially more OES than Fy(std), as expected. Near N = 40, the agreement for Δ ð3Þ r I is particularly good. Including the systematic uncertainty of the uniform blocking approximation employed, the total uncertainty of the DFT calculations on the OES is estimated to be less than 5 × 10 −3 fm. This uncertainty covers the small quantitative mismatch of Fy(Δr) at small N, though the deviation at large N remains notable and provides a helpful benchmark for further development.
The VS-IMSRG calculations predict an OES of the right order of magnitude close to the neutron-shell closures N = 28, 50, but do not reproduce the larger OES in the mid-shell and close to N = 40. This is very likely related to the missing proton excitations from the πf 7∕2 shell 14 , which were shown to be important from N = 41 onwards, but reduced in 77,78 Cu. The observation that the VS-IMSRG calculations perform as well or even better than DFT when it comes to predicting the local features of the radii and energies illustrates that the many-body correlation effects are under better control in the VS-IMSRG approach. This provides an important clue to the microscopic origins of the OES. In particular, the natural emergence of the OES of both radii and binding energies from an interaction that is only constrained with four-body data is encouraging.
In conclusion, we have reported new measurements of the changes in the mean-squared charge radii of very neutron-rich copper isotopes, which were made possible thanks to the high resolution and high sensitivity of the newly developed CRIS method at ISOLDE/CERN. This technique is now available to study exotic isotopes, approaching the nuclear driplines, thus accessing for the first time the anchor points in the nuclear chart that benchmark nuclear theories. We demonstrated good agreement between our measurements and results from DFT and VS-IMSRG methods. Given the intrinsic complexity of medium-mass systems with odd-Z, this represents also a major achievement in nuclear theory, and an important step forward in our global understanding of the nuclear binding energy and charge radius of exotic isotopes. The interplay between the bulk nuclear properties (better captured by DFT) and local variations (better captured by VS-IMSRG calculations) was shown to be crucial in revealing the microscopic description of the OES effect in radii and binding energies. The OES emerges naturally from NN+3N interactions derived from chiral effective field theory, constrained to the properties of isotopes with up to four nucleons only, which presents an important step forward towards a predictive nuclear theory. The comparison with heavier odd-Z systems near the heaviest self-conjugate isotope 100 Sn, which can now be studied experimentally over a range of more than 30 isotopes, is expected to provide the next challenge for nuclear theory.

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