Charge radius of the short-lived Ni and correlation with the dipole polarizability

S. Kaufmann,1 J. Simonis,2 S. Bacca,2, 3 J. Billowes,4 M. L. Bissell,4 K. Blaum,5 B. Cheal,6 R. F. Garcia Ruiz,4, 7 W. Gins,8 C. Gorges,1 G. Hagen,9 H. Heylen,5, 7 A. Kanellakopoulos,8 S. Malbrunot-Ettenauer,7 M. Miorelli,10 R. Neugart,5, 11 G. Neyens,7, 8 W. Nörtershäuser,1, ∗ R. Sánchez,12 S. Sailer,13 A. Schwenk,1, 5, 14 T. Ratajczyk,1 L. V. Rodríguez,15 L. Wehner,16 C. Wraith,6 L. Xie,4 Z. Y. Xu,8 X. F. Yang,8, 17 and D. T. Yordanov15 Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany Institut für Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany Helmholtz-Institut Mainz, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, United Kingdom Max-Planck-Institut für Kernphysik, D-69117 Heidelberg, Germany Oliver Lodge Laboratory, Oxford Street, University of Liverpool, Liverpool, L69 7ZE, United Kingdom Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland KU Leuven, Instituut voor Kernen Stralingsfysica, B-3001 Leuven, Belgium Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA and Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada Institut für Kernchemie, Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany Technische Universität München, D-80333 München, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany Institut de Physique Nucléaire, CNRS-IN2P3, Université Paris-Sud, Université Paris-Saclay, 91406 Orsay, France Institut für Kernchemie, Universität Mainz, D-55128 Mainz, Germany School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China


INTRODUCTION
The nuclear equation of state (EOS) plays a key role in supernova explosions and compact object mergers. In fact, the gravitational wave signal from the neutron star merger GW170817 has recently lead to constraints on the EOS of neutron-rich matter [1], which is consistent with our knowledge of nuclear physics. While the EOS of symmetric nuclear matter is well constrained around saturation density [2], the properties of neutron-rich matter are still rather uncertain. These properties are encoded in the nuclear symmetry energy S(n) as a function of density n and the slope parameter L = 3n 0 ∂S(n 0 )/∂n at saturation density n 0 . Studies on atomic nuclei can provide information on the L parameter [3] through a nuclide's neutron-skin thickness R skin = R n − R p defined as the difference between the point-neutron and pointproton radii. The neutron skin is a consequence of the competition between the surface tension and the pressure of neutron matter, which is determined by the L parameter. In the heavy nucleus 208 Pb energy density functional (EDF) calculations confirmed this strong cor-relation between R skin and L with a correlation coefficient of 0.979. This allows one to further constrain L based on R skin [4]. Unfortunately, the direct measurement of R skin is experimentally very challenging. In recent measurements it was extracted by its correlation to the dipole polarizability α D , which can be explored, e.g., with proton inelastic scattering, as it was the case for 48 Ca [5], 120 Sn [6] and 208 Pb [7]. Here mostly EDFs were used to extract the neutron skin from the dipole polarizability, but in the case of 48 Ca the neutron skin was predicted from first principles coupled-cluster calculations to be surprisingly small, only 0.12-0.15 fm [8]. These ab initio calculations starting from two-and three-nucleon interactions based on chiral effective field theory (EFT) [9][10][11] further revealed a correlation between the charge radius and the dipole polarizability, which was predicted to be in the range 2.19-2.60 fm 3 . Recent measurements of 48 Ca by Birkhan et al. [5] yielded a dipole polarizability of α D = 2.07 (22) fm 3 in agreement with the chiral EFT predictions. The only short-lived nucleus for which α D has been experimentally determined is 68 Ni, using Coulomb excitation in inverse kinematics. The pygmy arXiv:2003.06353v1 [nucl-ex] 13 Mar 2020 and the giant dipole resonances were observed and α D was extracted [12]. In this Letter, we focus on the charge radius of 68 Ni, determined by collinear laser spectroscopy. It is the first laser spectroscopy result on a short-lived nickel isotope, since access to this element at ISOL facilities is limited due to the slow release from the target. On the theory side, we report on the first coupled-cluster calculation including triples of R c and α D of 68 Ni based on chiral EFT interactions, being now the heaviest system for which this has been achieved. We study the correlation of the charge radius with the dipole polarizability in novel ab initio calculations including triples contributions. The measured charge radius in combination with the experimental dipole polarizability enables the first test of this correlation in a neutron-rich medium-mass nucleus.

EXPERIMENT
Nickel isotopes were produced at ISOLDE/CERN using proton pulses at an energy of 1.4 GeV to cause fragmentation, spallation, and fission inside a uranium carbide target. The target was heated beyond standard operation temperatures up to ∼ 2200 • C to enhance the release of chemically reactive nickel isotopes that have generally quite long release times from the target. Nickel atoms were then ionized by resonant laser ionization using RILIS [13] and accelerated towards the highresolution mass separator on ground potential by applying an electrostatic potential of approximately 30 kV and 40 kV to the ion source in a first and a second beamtime, respectively. The mass separated ions were injected into the radio-frequency ion beam cooler and buncher IS-COOL [14] and accumulated for typically 10 − 100 ms. Extracted ion bunches of 5 µs duration were transported to the collinear laser spectroscopy beam line COLLAPS where the ions were superimposed with a co-propagating laser beam. Potassium vapor in a charge-exchange cell [15,16] was used to neutralize the ions. Various excited states of the nickel atoms were populated within this non-resonant process, among them the metastable 3d 9 4s 3 D 3 level [17] that served as the starting point for laser spectroscopy, performed in the 352.45 nm transition to the 3d 9 4p 3 P 2 level. Fluorescence photons from spontaneous emission were detected by four photomultiplier tubes and the individual events were recorded with a new time-resolving data acquisition system. The laser light was produced using a frequency-doubled singlemode continuous-wave titanium sapphire laser stabilized on a high-resolution wavemeter, which was calibrated regularly with a stabilized helium-neon laser. Typical spectra of the isotopes 58,60,61,62,64,68 Ni are shown in Fig. 1. All isotopes were measured alternating with the reference isotope 60 Ni to compensate for any remaining long-term drifts in the ion velocity or the laser frequency. Ni with their center frequency indicated with a vertical line. The count rate is normalized for each isotope, and the frequency is given relative to the center frequency of the reference isotope 60 Ni.

ANALYSIS
Isotope shifts δν 60,A = ν A − ν 60 for the stable isotopes 58,61,62,64 Ni and the radioactive 68 Ni were calculated from their respective center frequency ν A with respect to the center frequency ν 60 of the reference isotope 60 Ni. Results are listed in Table I. Isotope shifts are related to differences in mean-square charge radii δ r 2 c 60,A via a field shift factor F and a mass shift factor M according to with µ = (m A − m 60 )/(m A · m 60 ) and m A being the respective atomic masses. A King-fit analysis was performed using the procedure described in [18] and based on the known rms charge radii of the stable nickel isotopes extracted from the combined analysis of muonic atom data and elastic electron scattering provided in [19]. An x-axis offset of α = 397 u fm 2 was used to remove the correlation between M and F and the result of the fit is depicted in Fig. 2 The results are listed in Table I. With the rms charge radius of R c ( 60 Ni) = 3.806(2) fm taken from [19], the charge radius of 68 Ni is obtained as R c ( 68 Ni) = 3.887(3) fm.

DISCUSSION
The extracted R c can be used to benchmark theoretical calculations, to test and expand their reliability and predictive power away from stable nuclei. First principle calculations were recently performed for 48 Ca [8], which lead to an improved understanding of the neutron and proton distributions in nuclei, as well as their difference encoded in R skin . The observed correlation between the dipole polarizability α D and the rms charge radius of 48 Ca allowed to narrow down constraints on the dipole polarizability, α D = 2.19 − 2.60 fm 3 , and on the neutron skin, R skin = 0.12 − 0.15 fm. The latter was found to be considerably smaller than previously thought [8]. The recent Darmstadt-Osaka experimental determination of the dipole polarizability α D = 2.07(22) fm 3 [5] is indeed in good agreement with the theoretical predictions. Subsequently, new calculations have included higher-order coupled-cluster correlations [20], so-called linearized 3 particles-3 holes (3p-3h) correlations. This leads to a reduction of the dipole polarizability and to an improved agreement with the experimental data for 48 Ca, while the charge radius is found to not depend sensitively on 3p-3h correlations [21].
Coupled-cluster calculations of α D based on chiral EFT interactions, initiated in Refs. [8,[20][21][22][23][24], have progressed towards heavier, more complex nuclei and have now reached the short-lived 68 Ni. Contrary to the stable isotopes, for which inelastic proton scattering was used to experimentally access the dipole polarizability, α D of 68 Ni was determined using Coulomb excitation in inverse kinematics by measuring the invariant mass in the oneand two-neutron decay channels [12]. This result, subsequently refined in Ref. [25], is shown together with our first experimental determination of R c in Fig. 3. Figure 3 also shows our theoretical results using four different chiral nucleon-nucleon (NN) and three-nucleon (3N) interactions from Ref. [26] (with the same labeling used here: 1.8/2.0, 2.0/2.0, 2.2/2.0 (EM) and 2.0/2.0 (PWA)) as well as the NNLO sat interaction from Ref. [27]. The Hamiltonians of Ref. [26] are based on a chiral N 3 LO NN potential evolved to low resolution using the similarity renormalization group combined with N 2 LO 3N interactions fit to the 3 H binding energy and the 4 He charge radius. These interactions have been successfully used to study the structure of medium-mass nuclei up to 100 Sn (see, e.g., Refs. [28][29][30][31]). Figure 3 shows two sets of coupled-cluster calculations: one with singles and doubles correlations (dashed points, line and light blue band) and another one where the leading 3p-3h correlations are included (solid points and line with darker blue band). We observe that triples corrections lead to a sizable reduction of α D (from 8% for the softest NN+3N interactions 1.8/2.0 (EM) to 15% for the hardest NNLO sat interaction), while R c is changed only mildly (maximally 0.7% for the hardest interaction). Each theoretical point is shown with a corresponding estimate of the theoretical uncertainty, which includes both the residual model-space dependence and the coupledcluster truncation error, following the protocol explained in Ref. [21]. As expected, the uncertainties are smaller (larger) for soft (hard) interactions. For completeness, the charge radius R c is obtained from the point-proton radius R p by where r 2 p = 0.7080(32) fm 2 [32] and r 2 n = −0.117(4) fm 2 [33] are the rms charge radii of a proton and a neutron, respectively, (3/4M 2 ) = 0.033 fm 2 [34] is the relativistic Darwin-Foldy correction and r 2 so is the spin-orbit correction for 68 Ni, which we calculate consistently for each Hamiltonian.
It is interesting to note that the behavior is very similar to that observed for the stable nucleus 48 Ca [5,8,20,21]. The theoretical results exhibit a clear correlation between the dipole polarizability and the charge radius both at the singles-and-doubles and triples excitations level. The inclusion of triples excitations, however, alters the slope of this correlation. In Fig. 3, the correlation is highlighted by the linear fit to the calculations and the corresponding blue uncertainty bands, which are chosen to include the full error bars of the individual calculations. Most notably, for the results with singles and doubles excitations, the band does not overlap with the intersection region of the measured R c and α D . When including 3p-3h correlations, the theoretical band nicely overlaps with the experimental constraints. This shows that 3p-3h correlations are not negligible, and that state-of-the-art coupledcluster computations are reliable for this first test of the charge radius and α D of the neutron-rich nucleus 68 Ni.
Compared to the results α D = 3.60 fm 3 obtained recently by Raimondi and Barbieri [35] with the selfconsistent Green's function method using the NNLO sat interaction, we obtain a considerably larger value of α D = 4.65(49) fm 3 using the same interaction. The reason of the discrepancy could either be due to the different method used, or more likely related to the larger Ω value in Ref. [35]. The central value of our coupled-cluster results are obtained with Ω = 12 MeV, which shows a very nice convergence pattern as a function of the model-space size; while we have observed that larger Ω values exhibit a much slower convergence.
In addition to our ab initio calculations, the dipole polarizability of 68 Ni was also studied with nuclear EDFs, which suggested that α D is strongly correlated with the neutron skin and this correlation is even stronger when information on the symmetry energy is taken into account [25]. This led to a prediction of the neutron skin R skin = 0.16(4) fm [25] of 68 Ni. In Table II we list the coupled-cluster results including 3p-3h corrections for the optimal Ω for the different chiral NN and 3N interactions studied. Taking as best interactions for the correlation plot, Fig. 3, the ones closest to the intersection region, 2.0/2.0 (EM), 2.2/2.0 (EM), and NNLO sat , we predict in Table II a range for the point-neutron radius R n = 3.9 − 4.1 fm of 68 Ni and its neutron skin R skin = 0.18 − 0.20 fm, in very good agreement with the EDF correlation prediction of Ref. [25]. SUMMARY We have presented the first measurement of the isotope shift of the neutron-rich 68 Ni isotope by collinear laser spectroscopy. This enabled the extraction of the rms charge radius to R c = 3.887(3) fm based on a Kingplot analysis and the known charge radii of the stable nickel isotopes. This radius is used to benchmark coupled-cluster calculations including novel triples corrections for a range of chiral NN and 3N interactions. A strong correlation between the charge radius and the dipole polarizability is shown by the theoretical calculations. Our results including the leading 3p-3h contributions agree much better with the experimental data compared to the case when triples corrections are neglected. In particular the theoretical correlation band intersects nicely with the measured R c and α D bands. This correlation combined with coupled-cluster calculations of the point-neutron radius and neutron skin of 68 Ni allows these to be constrained to R n = 3.9 − 4.1 fm and R skin = 0.18 − 0.20 fm.