Abstract
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 evenneutron 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 shortlived 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 OnLine 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 Abody valencespace inmedium 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 manybody polarization effects, naturally emerge from Abody calculations fitted to properties of A ≤ 4 nuclei.
Main
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 manybody dynamics. Owing to the presence of the unpaired proton, oddZ isotopes such as the copper isotopes provide crucial insights into the singleparticle 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 shortlived species. We produced radioactive ions at the ISOLDE laboratory at CERN. This was done by impinging 1.4GeV 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 elementselectively laserionized 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 highresolution laser resonance ionization spectroscopy. This required sending the ions into a gasfilled radiofrequency 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 chargeexchange reaction with a potassium vapour. The nonneutralized fraction of the beam was deflected, such that only the neutralized atoms entered into an ultrahighvacuum 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 autoionizing 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 quasibackgroundfree 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 singlemode 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 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 neutronrich 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 ultralow 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 signaltobackground ratio obtained is similar to those achieved in the stateoftheart insource measurements^{12}, but with much narrower linewidths of <100 MHz, typical for conventional fastbeam collinear laser spectroscopy techniques, thus demonstrating a bestofbothworlds performance.
The changes in the meansquared 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, smallerscale effects such as the odd–even staggering (OES) of the charge radii are largely unaffected. Values of the threepoint OES parameter of the radii \({{\mathrm{\Delta}} }_{r}^{(3)}\), defined as
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 groundstate proton configuration. Indeed, as reflected in the groundstate 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 valencespace inmedium similarity renormalization group (VSIMSRG) 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 boxequivalent) radius \({r}_{0}={[3/(4\uppi {\rho }_{0})]}^{1/3}\), 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 manybody correlations. The common interpretation of OES involves various types of polarization exerted by an odd nucleon, occupying a specific shellmodel (or onequasiparticle) orbital^{15}. In particular, the selfconsistent 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 semimagic 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 mediummass region. This region of the nuclear chart represents new territory for Abody theories based on twonucleon (NN) and threenucleon (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 manybody methods^{21,22}. However, the VSIMSRG approach^{7,8} has now sufficiently advanced to study most nuclear properties in essentially all openshell systems below A = 100, including masses, charge radii, spectroscopy and electroweak transitions^{23}. The presence of a potential subshell closure at N = 40 (ref. ^{9}) and the wellevidenced 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^{3,4,5}, has not been tested in this region of the nuclear chart, nor with data on oddZ isotopes in general.
Details on both the DFT and VSIMSRG 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 VSIMSRG 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 fourbody data, with 3Nforces 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 VSIMSRG 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 VSIMSRG 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 DF3a1 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 VSIMSRG binding energies appear to overbind in the midshell region but nevertheless do well in terms of the absolute value as well as the general trend. Figure 2c shows \(\delta {\left\langle {r}^{2}\right\rangle }^{65,A}\) alongside EM1.8/2.0 VSIMSRG and the Fy(Δr) calculations. The agreement of the VSIMSRG calculations with the data is excellent overall except for a small discrepancy near N = 40.
Figure 3 looks at the OES in more detail, by comparing experimental values of \({{\mathrm{\Delta}} }_{E}^{(3)}\) (defined in analogy with \({{\mathrm{\Delta}} }_{r}^{(3)}\)) and \({{\mathrm{\Delta}} }_{r}^{(3)}\) to theoretical values obtained with DFT and VSIMSRG methods. Note how the reduction in experimental \({{\mathrm{\Delta}} }_{r}^{(3)}\) near N = 50 is also visible in \({{\mathrm{\Delta}} }_{E}^{(3)}\), 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 VSIMSRG 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 \({{\mathrm{\Delta}} }_{r}^{(3)}\) 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 VSIMSRG calculations predict an OES of the right order of magnitude close to the neutronshell closures N = 28, 50, but do not reproduce the larger OES in the midshell 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 VSIMSRG 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 manybody correlation effects are under better control in the VSIMSRG 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 fourbody data is encouraging.
In conclusion, we have reported new measurements of the changes in the meansquared charge radii of very neutronrich 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 VSIMSRG methods. Given the intrinsic complexity of mediummass systems with oddZ, 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 VSIMSRG 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 oddZ systems near the heaviest selfconjugate 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.
Methods
Laser systems
Copper atoms were laserionized using a twostep laser ionization scheme. Light for the first step was produced using an injectionlocked Ti:sapphire laser system jointly developed by the Johannes GutenbergUniversität Mainz and the University of Jyväskylä^{30}. This laser cavity is built around a Ti:sapphire crystal that is pumped with 532nm light produced at a repetition rate of 1 kHz by a Lee LDP100MQG pulsed Nd:YAG laser. Through pulsed amplification of continuouswave seed light produced by an Msquared SolsTiS Ti:sapphire laser, narrowband (≈20 MHz) pulsed laser light was produced at a repetition rate of 1 kHz. This laser light was then frequency tripled using two nonlinear crystals to produce the required 249nm light for the resonant excitation step. A maximum of 0.5 μJ of 249nm light was delivered into the CRIS beamline, saturating the resonant step. The wavelength of the scanning laser was recorded by a HighFinesse WSU2 wavemeter every 10 ms, and used in a feedback loop to stabilize the wavelength to a target value. The wavelength of a temperaturestabilized HeNe laser was simultaneously recorded to evaluate the drift of the wavemeter during the measurements. Resonant ionization of these excited atoms was achieved using a 314.2444nm transition to the 3d^{9}4s(^{3}D)4d^{4}P_{3∕2} autoionizing state at 71,927.28 cm^{−1}, using light produced by a frequencydoubled Spectrolase 4000 pulsed dye laser pumped by a Litron LPY 601 50100 PIV Nd:YAG laser, at a repetition rate of 100 Hz. Owing to the 344(20) ns lifetime of the excited state, the 314nm laser pulse could be delayed by 50 ns, removing lineshape distortions and power broadening effects^{10} without appreciable efficiency losses. A small pickoff of the fundamental laser light was sent to a Highfinesse WS6 wavemeter to monitor potential wavelength drifts. The maximum power of this system was 125 μJ, which was not enough to fully saturate the transition.
Extraction of charge radii from isotope shifts
The hyperfine parameters and the centroids ν_{A} are extracted from the hyperfine spectra by fitting them with correlated Voigt profiles centred at the resonance transition frequencies. This analysis was performed using the SATLAS analysis library^{31}. More details on the analysis procedure can be found in ref. ^{32}. The isotope shift, \(\delta {\nu }^{65,A^{\prime} }={\nu }_{A}{\nu }_{65}\), can be determined from the shift of the centroid of the hyperfine spectrum of one isotope relative to that of a reference isotope. Such reference measurements of ^{65}Cu were used to take into account possible changes in the beam energy or drifts in wavemeter calibration during the fourday experiment. Table 1 displays the isotope shifts obtained in this work. From these isotope shifts, the meansquared charge radius difference \(\delta {\left\langle {r}^{2}\right\rangle }^{65,A^{\prime} }\) is obtained from:
with m the mass of the isotope, and M and F being the mass and fieldshift factor, respectively. The atomic factors for the 249nm line were determined through a King plot analysis^{32} using the data for the 324.7540nm line^{9}. Using values of M_{324} = 1,413(27) GHz u (where u is the unified atomic mass unit) and F_{324} = −779(78) MHz fm^{−2} obtained from the same reference, we find M_{249} = 2,284(24) GHz u and F_{249} = 439(80) MHz fm^{−2}. The extracted radii are in excellent agreement with those measured in the 324nm transition (Table 1), although a factor of 24 less precise due to the lower value of F_{249} as compared with F_{324}.
DFT calculations
In this work, we employed the recently developed parameterizations Fy(std)^{5} and Fy(Δr)^{4,6}, which differ in the pool of data used for the optimization protocol. Fy(std) uses the standard set of groundstate data (binding energies and charge/matter radii of even–even semimagic isotopes) from ref. ^{33} complemented with the OES of energies, defined in an analogous way as equation (1). Fy(Δr), optimized at the Hartree–Fock–Bogolyubov level, additionally uses the charge radii of the even–odd calcium isotopes. The inclusion of this information substantially increases the strength of the gradient term in the pairing functional (which is absent in other DFT forms and practically inactive in Fy(std)). This was found to be crucial in reproducing the OES of charge radii in the calcium isotopes^{5}. However, this parameterization overestimates the OES in the heavier masses^{3} and the kink in charge radii around ^{132}Sn (ref. ^{4}). For all practical details pertaining to our FayansDFT calculations, we refer the reader to ref. ^{6}.
VSIMSRG calculations
Working in an initial harmonicoscillator basis of 13 major shells, we first transformed to the Hartree–Fock basis, then used the Magnus prescription^{34} of the IMSRG to generate approximate unitary transformations to decouple first the given core energy, then as well as a specific valencespace Hamiltonian from the full Abody Hamiltonian. In the IMSRG(2) approximation used in this work here, all operators are truncated at the twobody level. In addition, we capture effects of 3N forces between valence nucleons via the ensemble normal ordering procedure of ref. ^{35}. The same transformation is then applied to the intrinsic proton meansquared radius operator, with appropriate corrections to obtain core and valencespace (hence absolute) charge radii. We take ^{56}Ni as our core reference state, with a valence space consisting of the proton and neutron p_{3∕2}, p_{1∕2}, f_{5∕2} and g_{9∕2} singleparticle orbits. Finally, using the NUSHELLX@MSU shell model code^{36}, we diagonalize the valencespace Hamiltonian to obtain ground (and excited) state energies, as well as expectation values for the charge radius operator, where induced twobody corrections are included naturally in the VSIMSRG approach.
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Acknowledgements
We acknowledge the support of the ISOLDE collaboration and technical teams, and the University of Jyväskylä for the use of the injectionlocked cavity. This work was supported by the BriX Research Program no. P7/12 and FWOVlaanderen (Belgium) and GOA 15/010 from KU Leuven, FNPMLS ERC Consolidator Grant no. 648381, the Science and Technology Facilities Council consolidated grant ST/P004423/1 and continuation grant ST/L005794/1, the EU Seventh Framework through ENSAR2 (654002), the NSERC and the National Research Council of Canada, and by the Office of Science, US Department of Energy under award numbers DESC0013365 and DESC0018083 (NUCLEI SciDAC4 collaboration). We acknowledge the financial aid of the Ed Schneiderman Fund at New York University.
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R.P.d.G., J.B., C.L.B., M.L.B., T.E.C., T.D.G., G.J.F.S., D.V.F., K.T.F., S.F., R.F.G.R., W.G., Á.K., K.M.L., G.N., S.R., H.H.S., A.R.V., K.D.A.W., S.G.W., Z.Y.X. and X.F.Y. performed the experiment. R.P.d.G. and C.L.B. performed the data analysis and R.P.d.G. prepared the figures. J.D.H. and T.M. performed the VSIMSRG calculations. W.N. and P.G.R. performed the DFT calculations. R.P.d.G., W.N., P.G.R. and J.H. prepared the manuscript. All authors discussed the results and contributed to the manuscript at all stages.
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de Groote, R.P., Billowes, J., Binnersley, C.L. et al. Measurement and microscopic description of odd–even staggering of charge radii of exotic copper isotopes. Nat. Phys. 16, 620–624 (2020). https://doi.org/10.1038/s415670200868y
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