Charge radii of exotic potassium isotopes challenge nuclear theory and the magic character of N=32
Á. Koszorús, X. F. Yang, W. G. Jiang, S. J. Novario, S. W. Bai, J. Billowes, C. L. Binnersley, M. L. Bissell, T. E. Cocolios, B. S. Cooper, R. P. de Groote, A. Ekström, K. T. Flanagan, C. Forssén, S. Franchoo, R. F. Garcia Ruiz, F. P. Gustafsson, G. Hagen, G. R. Jansen, A. Kanellakopoulos, M. Kortelainen, W. Nazarewicz, G. Neyens, T. Papenbrock, P.-G. Reinhard, B. K. Sahoo, C. M. Ricketts, A. R. Vernon, S. G. Wilkins
CCharge radii of exotic potassium isotopes challenge nuclear theory and the magic character of N = 32 Á. Koszorús ∗† X. F. Yang , ∗ W. G. Jiang , , S. J. Novario , S. W. Bai J. Billowes C. L. Binnersley M. L. Bissell T. E. Cocolios B. S. Cooper R. P. de Groote , A. Ekström K. T. Flanagan , C. Forssén S. Franchoo R. F. Garcia Ruiz , F. P. Gustafsson G. Hagen , G. R. Jansen A. Kanellakopoulos M. Kortelainen , W. Nazarewicz G. Neyens , T. Papenbrock , P.-G. Reinhard B. K. Sahoo C. M. Ricketts A. R. Vernon , S. G. Wilkins KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium. School of Physics and State Key Laboratory of Nuclear Physicsand Technology, Peking University, Beijing 100871, China. Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee37996, USA. Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. Department of Physics, Chalmers Universityof Technology, SE-412 96 Göteborg, Sweden. School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UnitedKingdom. Department of Physics, University of Jyväskylä, PB 35(YFL) FIN-40351 Jyväskylä, Finland. Helsinki Institute of Physics,Universityof Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland. Photon Science Institute Alan Turing Building, University of Manchester, Manchester M139PY, United Kingdom. Institut de Physique Nucléaire Orsay, IN2P3/CNRS, 91405 Orsay Cedex, France. Experimental Physics Department,CERN, CH-1211 Geneva 23, Switzerland. Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Department of Physics andAstronomy and FRIB Laboratory. Michigan State University, East Lansing, Michigan 48824, USA. Institut für Theoretische Physik, UniversitätErlangen, Erlangen, Germany. Atomic, Molecular and Optical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009,India. Engineering Department, CERN, CH-1211 Geneva 23, Switzerland. ∗ Corresponding authors: Xiaofei Yang ([email protected]),Á. Koszorús ([email protected]) † Present address: Oliver Lodge Laboratory, University of Liverpool, Liverpool, UK.
Nuclear charge radii are sensitive probesof different aspects of the nucleon-nucleoninteraction and the bulk properties of nuclearmatter; thus, they provide a stringent test andchallenge for nuclear theory. The calcium regionhas been of particular interest, as experimentalevidence has suggested a new ‘magic’ numberat
N =
32 [1–3], while the unexpectedly largeincreases in the charge radii [4, 5] open newquestions about the evolution of nuclear size inneutron-rich systems. By combining the collinearresonance ionization spectroscopy method with β -decay detection, we were able to extend thecharge radii measurement of potassium ( Z = K ( t / = 110 ms),produced in minute quantities. Our workprovides the first charge radii measurementbeyond N =
32 in the region, revealing nosignature of the magic character at this neutronnumber. The results are interpreted with twostate-of-the-art nuclear theories. For the firsttime, a long sequence of isotopes could becalculated with coupled-cluster calculations basedon newly developed nuclear interactions. Thestrong increase in the charge radii beyond
N =
The charge radius is a fundamental property ofthe atomic nucleus. Though it globally scales with the nuclear mass as A / , the nuclear charge radiusadditionally exhibits appreciable isotopic variations thatare the result of complex interactions between protonsand neutrons. Indeed, charge radii reflect various nuclearstructure phenomena such as halo structures [6, 7],shape staggering [8] and shape coexistence [9], pairingcorrelations [10, 11], neutron skins[12] and the occurrenceof nuclear magic numbers [5, 13, 14]. The term ‘magicnumber’ refers to the number of protons or neutronscorresponding to completely filled shells, thus resultingin an enhanced stability and a relatively small chargeradius.In the nuclear mass region near potassium, the isotopeswith neutron number N = 32 are proposed to be ‘magic’,based on an observed sudden decrease in binding energybeyond N = 32 [2, 3] and high excitation energy of thefirst excited state in Ca [1]. The nuclear charge radius isalso a sensitive indicator of ‘magicity’: a sudden increasein the charge radii of isotopes is observed after crossingthe classical neutron magic numbers N =
28, 50, 82 and126 [5, 13–15]. Therefore, the experimentally observedstrong increase in the charge radii between N = 28 and N = 32 of calcium [4] and potassium chains [5], and inparticular the large radius of K and Ca (both having32 neutrons) have attracted significant attention.One aim of the present study is therefore to shed lighton several open questions in this region: how does thenuclear size of very neutron-rich nuclei evolve, and isthere any evidence for the ‘magicity’ of N = 32 fromnuclear size measurement? We furthermore provide newdata to test several newly developed nuclear models,which aim at understanding the evolution of nuclearcharge radii of exotic isotopes with large neutron-to-proton imbalance. So far, all ab initio nuclear methods,allowing for systematically improvable calculations based a r X i v : . [ nu c l - e x ] D ec Figure 1. Left: The CRIS setup at ISOLDE, CERN. The nuclei of interest were produced via various nuclear reactions after a1.4-GeV proton beam impinged onto a UCx target. These diffused out of the target, into an ion source, and underwent surfaceionization. The ion beam was then mass separated using a High Resolution Separator (HRS), and subsequently cooled andbunched in a linear Paul trap (ISCOOL). The bunched ion beam was guided towards the CRIS beamline, where the ions werefirst neutralized in a charge-exchange cell filled with potassium vapor. The neutral atoms were then delivered to the interactionregion. Here the bunched beam of atoms was collinearly overlapped with the laser pulses to achieve resonance laser ionization.The ionized radioactive potassium ions could then be detected using either a MagneToF ion detector shown in (a), or plasticscintillator detectors (b). (c) The hfs of K measured with the scintillator detectors. Figures (a), (b) and (c) show the detectedevents as a function of the laser frequency detuning. on realistic Hamiltonians with nucleon-nucleon andthree-nucleon potentials, have failed to explain theenhanced nuclear sizes beyond N = 28 in the calciumisotopes [4, 16]. Meanwhile, nuclear Density FunctionalTheory (DFT) using Fayans functionals has beensuccessful in predicting the increase in the charge radiiof isotopes in the proton-magic calcium chain [11], aswell as the kinks in the proton-magic tin and lead [13].These state-of-the-art theoretical approaches have beenpredominantly used to study the charge radii of even- Z isotopes, and only very recently could charge radii ofodd- Z Cu ( Z = 29 ) isotopes be investigated with A-body in-medium similarity renormalization group (IM-SRG) method and Fayans-DFT [10].Laser spectroscopy techniques yield the most accurateand precise measurements of the charge radius forradioactive nuclei. These highly efficient and sensitiveexperiments at radioactive ion beam facilities haveexpanded our knowledge of nuclear charge radiidistributed throughout the nuclear landscape [17].Laser spectroscopy achieves this in a nuclear-model-independent way by measuring the small perturbationsof the atomic hyperfine energy levels due to theelectromagnetic properties of the nucleus. Althoughthese hyperfine structure (hfs) effects are as smallas one part in a million compared to the totaltransition frequency, they can nowadays be measuredwith remarkable precision and efficiency, even for short-lived, weakly-produced, exotic isotopes [10].To enhance the sensitivity of the high-resolution, optically detected collinear laser spectroscopy methodthat was previously used to measure the mean-squarecharge radii of the potassium isotopes [5, 18, 19], we usedthe Collinear Resonance Ionization Spectroscopy (CRIS)experimental setup at the ISOLDE facility of CERN.This allows very exotic isotopes to be studied with thesame resolution as the optically detected method [10, 20].Relevant details of the ISOLDE radioactive beam facilityand the CRIS setup are depicted in Fig. 1 (see Methodsfor details). The CRIS method relies on the step-wiseresonant laser excitation and ionization of atoms. Forthis experiment, a narrowband laser was used to excitepotassium atoms from one of the hfs components ofthe atomic ground state into a hyperfine energy levelof an excited state. From there, another excitation andsubsequent laser ionization were induced by broadbandhigh-power laser beams, as discussed in Ref. [21]. Theresulting ions were deflected from the remaining (neutral)particles in the beam, and were detected with an iondetector. This method allows nearly background-free iondetection, and thus has very high sensitivity, providedthat the contaminating beam particles are not ionized(through e.g. collisional ionisation or due to the high-power non-resonant laser beams). By counting the ionsas a function of the laser frequency, the energy differencesbetween the atomic hyperfine transitions were measured.If measurements are performed on more than one isotope,the difference in mean-square charge radius of theseisotopes can be obtained from the difference in the hfscentroid frequency of two isotopes (the isotope shift) withmass numbers A and A (cid:48) : δν A,A (cid:48) = ν A − ν A (cid:48) .In order to apply the CRIS method to study a lightelement such as potassium, where the optical transitionexhibits a lower sensitivity to the nuclear properties, thelong-term stability and accurate measurement of the laserfrequency had to be investigated. The details of therelevant developments are presented in Ref. [22], wherethe method was validated by measuring the mean-squarecharge radii of − K isotopes with high precision.For the most exotic isotope, there was an additionalchallenge: a large isobaric contamination at mass A = 52 ,measured to be 2 × times more intense than the Kbeam of interest. The resulting detected backgroundrate was found to be an order of magnitude higher thanthat of the resonantly ionized K ions. In addition,this background rate was found to strongly fluctuatein time, making a measurement with ion detectionimpossible (see the hfs spectrum in Fig. 1(a)). Takingadvantage of the short half-life of K ( t / = 110 ms)and the fact that the isobaric contamination is largelydue to the stable isotope Cr, an alternative detectionsetup was developed, which can distinguish the stablecontamination from the radioactive K. A thin anda thick scintillator detector were installed behind theCRIS setup (Fig. 1). These detectors were used to countthe β -particles emitted though the decay of K. Withthis setup, the fluctuations in the background rate andsignal-to-background ratio were significantly improved,as seen in Fig. 1(b). The obtained hfs spectrum of Kis presented in Fig. 1(c). Note that the hfs spectraof − K were re-measured with the standard CRISmethod, and , K were measured with both ion- and β -detection. This allows a consistent calculation of isotopeshifts of − K (See Tab II in Methods for details).The changes in the mean-square charge radii δ (cid:104) r (cid:105) arecalculated from the isotope shift ν AA (cid:48) via δ (cid:104) r (cid:105) = 1 F (cid:104) ν AA (cid:48) − ( K NMS + K SMS ) m A − m A (cid:48) ( m A + m e ) m A (cid:48) (cid:105) . (1)Here F , K NMS and K SMS are the atomic field shift,specific mass shift and normal mass shift factors,respectively (see Methods for details). Previouslypublished charge radii of potassium isotopes [5, 18, 19, 21,23] have been extracted from the isotope shifts using an F -value calculated with a non-relativistic coupled-clustermethod and an empirically determined K SMS value, asreported in Ref. [24].We employ the recently developed analytic responserelativistic coupled-cluster (ARRCC) theory [25], anadvanced atomic many-body method (see Methods fordetails), to calculate both the F and K SMS constants.The newly calculated atomic field shift factor, F = − . MHz fm is in good agreement with theliterature value F = − MHz fm , and is moreprecise. More importantly, the specific mass shift, ahighly correlated atomic parameter, could be calculatedfrom microscopic atomic theory for the first time.The calculated value, K SMS = − . GHz u,
Table I. Evaluated experimental isotope shifts δν ,A ,differences in mean-square charge radii δ (cid:104) r (cid:105) , and charge radii R ch of nuclei − K. Systematic errors are reported in squarebrackets. The procedure for the evaluation is discussed in theMethods. A N I π δν ,A (MHz) δ (cid:104) r (cid:105) ,A (fm ) R ch (fm)36 17 2 + -403(9) -0.20(8)[4] 3.405(12)[6]37 18 3/2 + -264(6) -0.11(6)[3] 3.419(8)[4]38 19 3 + -126.1(19) -0.075(18)[14] 3.4241(26)[20]39 20 3/2 + − + − + − + − + − + − +
52 33 (2 − ) − ) 1118(5) 1.54(5)[132] 3.652(6)[18]52 33 (3 − ) 1237(5) 0.43(5)[132] 3.497(7)[19] is more precise than the empirical value, K SMS = − . GHz u from Ref. [24], and shows goodagreement. Table I presents the isotope shifts, changesin mean-square charge radii, and absolute charge radiiof − K which were extracted using these new atomicconstants. The isotope shifts and charge radii havebeen re-evaluated using all available data, as describedin the Methods section. In Fig. 2(a) these charge radiiare compared with values obtained using the atomicfactors taken from Ref. [24]. Good agreement is obtained,while the systematic error due to the uncertainty on theatomic factors is clearly reduced. A future measurementof the absolute radii of radioactive potassium isotopesthrough non-optical means (e.g. electron scatteringat the SCRIT facility [26]), would help reduce thesystematic uncertainties.Previously, the nuclear spin and parity of K wastentatively assigned to be I π = (2 − ) , based on thevery weak feeding into the Ca ground state [27].Here, we have analysed our data assuming two otheralternative spin options. Given that the I = 1 and I =3 assumptions produce unrealistically small and largecharge radii (see Fig. 2(a)), our study further supportsan I = 2 assignment.The inset in Fig. 2(a) compares the changes in mean-square charge radii (relative to the radius of isotope withneutron number N = 28 ) of several isotopic chains in thismass region, up to Z = 26 . A remarkable observationis that the charge radii beyond N = 28 follow thesame steep increasing trend, irrespective of the numberof protons in the nucleus. Beyond N = 32 , data are R c h ( f m ) Systematic errorExperiment
NNLO GO (450)Fy( r , HFB)NNLO sat (b) Lit. F , K SMS
New F , K SMS (a) I =1 I =3 r , A ( f m ) r , A ( f m ) r
16 20 24 28 320.40.20.00.20.40.60.8
Cr (Z=24)Fe (Z=26)Mn (Z=25)Ti (Z=22)Ca (Z=20)K (Z=19)
16 18 20 22 24 26 28 30 32 34Neutron number0.40.20.00.20.40.6 (c)
Figure 2. (a) Changes in the mean-square charge radii ofpotassium isotopes using the newly calculated atomic fieldshift ( F ) and specific mass shift ( K SMS ) factors, as well asthe values from Ref. [24]. The red and gray bands indicatethe uncertainties originating from these atomic constants,respectively. The inset shows the changes in the mean-squarecharge radii of neighbouring elements. The agreement of thedata for different isotopic chains above N = 28 is striking.(b) Comparison of the measured charge radii of potassiumisotopes with nuclear coupled-cluster calculations using twointeractions (NNLO sat and ∆ NNLO GO (450) ) derived fromthe chiral effective field theory, and with the Fayans-DFTcalculations with the Fy( ∆ r,HFB) energy density functional.(c) Changes in the mean-square charge radii of potassiumrelative to K, in which the systematic uncertainties near N = 28 are largely cancelled. only available for potassium (this work) and manganese( Z = 25 ) [28]. Both charge radius trends are very similar,with no signature of a characteristic kink that wouldindicate ‘magicity’ at N = 32 .Previously, ab initio coupled cluster (CC) calculationsbased on the NNLO sat interaction [29] were used todescribe the nuclear charge radii in calcium isotopes [4,12]. At that time, calculations in this frameworkcould only be performed for spherical isotopes near‘doubly-magic’ nuclei. While these calculations predictedthe absolute charge radii near , Ca very well, theyfailed to reproduce the observed large charge radiiaround neutron number N = 32 . In Fig. 2(b)we compare the experimental data to CC calculationsthat start from a symmetry-breaking reference state,which allows us to compute charge radii of allpotassium isotopes (see Methods for details). Resultsobtained with the NNLO sat interaction significantlyoverestimate the experimental data near stability, wherethe experimental uncertainties on the total radii are thesmallest. This interaction was fitted to experimentalbinding energies and charge radii of selected nucleiup to mass number A = 25 [29]. Therefore, a newlyconstructed ∆NNLO GO (450) interaction was developed,which includes pion-physics and effects of the ∆(1232) isobar. This interaction is constrained by propertiesof only light nuclei with mass numbers A ≤ and bynuclear matter at the saturation point (i.e. its saturationenergy and density, and its symmetry energy, seeMethods for details). By virtue of including saturationproperties, CC calculations using the ∆ NNLO GO (450) interaction yield an improvement in the accuracy ofthe description of potassium charge radii near stability,as shown in Fig. 2(b). However, both NNLO sat and ∆ NNLO GO (450) interactions still underestimate thesteep increase observed beyond N = 28 . This isbetter visualised by plotting the differences in mean-square charge radii relative to K (with neutron number N = 28 ), δ (cid:10) r (cid:11) ,A , shown in Fig. 2(c). The systematicuncertainties near N = 28 are strongly reduced bychoosing this reference. It is also worth noting that,for δ (cid:10) r (cid:11) ,A below N = 28 , both interactions showa good agreement with experimental results within thesystematic uncertainty.What can be the reason for the underestimation ofcharge radii for N > ? The reference state in CCcalculations is the axial Hartree-Fock state. For nearlyspherical nuclei, a general (triaxial) cranked HFB statethat breaks time-reversal and gauge symmetries couldperhaps provide a better reference. Also, since theangular momentum was not restored, the associatedcorrelations are missing as well in CC results.The DFT is the method of choice for heavy systems.Nuclei with Z ≈ cover the region where bothmethods, DFT and CC, can be successfully applied. OurDFT calculations use in particular the Fayans functionalFy( ∆ r,HFB) [30] (see Methods for details) which wasdeveloped with a focus on charge radii. This methodclosely reproduces the absolute charge radii of calciumisotopes, including the steep increase beyond N = 28 [11]. Furthermore, Fy( ∆ r,HFB) reproduces the absoluteradii of the magic tin [13], cadmium [31] as well asthe odd- Z copper isotopes [10]. It is to be notedhowever, that the potassium isotopes are a lighter systemin which the polarization effects are expected to bestronger than in the heavier copper isotopes. To accountfor that, we extended the Fayans-DFT framework toallow for deformed HFB solutions. The isotopic chaincontains odd-odd nuclei and the present DFT and CCtreatment does not allow a clean spin selection forthese. Consequently, the HFB calculations providean averaged description for the odd-odd isotopes. Asseen in Fig. 2(b), except for the neutron-poor side,Fy( ∆ r,HFB) calculations reproduce the average globaltrend rather well, in particular the steep increase above N = 28 . However, this model grossly overestimates theodd-even staggering. The odd-even staggering in thepotassium isotopes is significantly reduced with respectto calcium isotopes. This is very well captured by the CCcalculations, due to the fact that these describe in detailthe many-body correlations (see Methods for details). Innuclear DFT, local many-body correlations are treatedless precisely.In summary, this work presents the first measurementof nuclear charge radii beyond the proposed ‘magic’number N = 32 in the calcium region. This wasachieved by combining collinear resonance ionizationspectroscopy with β -decay detection, enabling the exoticisotope K to be studied, despite its short half-life,low production rate and poor purity. Taking advantageof recent developments in atomic calculations, precisecharge radii of the potassium chain were extracted. Nosudden change in the K charge radius is observed,thus no signature for ‘magicity’ at N = 32 is found.The comparison with nuclear theory predictions for thedemanding case of potassium isotopes helps to uncovermore about the strengths and open problems in currentstate-of-the-art nuclear models. CC calculations basedon the new nucleon-nucleon potentials derived fromchiral effective field theory, optimized with few-bodynuclei properties as well as nuclear saturation properties,describe very well the absolute nuclear charge radiiof the potassium isotopes near stability, and also thesmall odd-even staggering. However, the steep rise incharge radii above N = 28 remains underestimated.The similarity in the performance of ∆ NNLO GO (450) and NNLO sat interactions suggests that the charge radiibeyond N = 28 are insensitive to the details of chiralinteractions at next-to-next-to-leading-order, and somecrucial ingredient is lacking in these many-body methods.The Fayans-DFT model captures the general trend acrossthe measured isotopes and reproduces the absolute radiirather well, including the steep increase up to N =33 . However, this model overestimates the odd-evenstaggering significantly. These findings highlight ourlimited understanding on the size of neutron-rich nuclei, and will undoubtedly trigger further developments innuclear theory as demanding nuclear data on charge radiikeeps uncovering problems with the best current models. Acknowledgments
We acknowledge the support ofthe ISOLDE collaboration and technical teams. Thiswork was supported in part by the National Key R&DProgram of China (Contract No: 2018YFA0404403);the National Natural Science Foundation of China(No:11875073); the BriX Research Program No.P7/12, FWO-Vlaanderen (Belgium), GOA 15/010from KU Leuven; ERC Consolidator Grant no.648381 (FNPMLS); the STFC consolidated grantsST/L005794/1 and ST/L005786/1 and ErnestRutherford Grant No. ST/L002868/1; the EUHorizon2020 research and innovation programmethrough ENSAR2 (no. 654002); the U.S. Departmentof Energy, Office of Science, Office of Nuclear Physicsunder grants DE-00249237, DE-FG02-96ER40963 andDE-SC0018223 (SciDAC-4 NUCLEI collaboration).This work received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon2020 research and innovation programme (Grantagreement No. 758027), the Swedish Research Councilgrant 2017-04234, and the Swedish Foundation forInternational Cooperation in Research and HigherEducation (STINT) grant IG2012-5158. BKSacknowledges use of Vikram-100 HPC cluster ofPhysical Research Laboratory, Ahmedabad for atomiccalculations. Computer time was provided by theInnovative and Novel Computational Impact on Theoryand Experiment (INCITE) program. This researchused resources of the Oak Ridge Leadership ComputingFacility and of the Compute and Data Environmentfor Science (CADES) located at Oak Ridge NationalLaboratory, which is supported by the Office of Scienceof the Department of Energy under Contract No.DE-AC05-00OR22725.
Author contribution
A.K., X.F.Y., S.W.B. J.B.,C.L.B., M.L.B., T.E.C.,B.S.C., R.P.d.G„ K.T.F., S.F.,R.F.G.R., F.P.G., A.K., G.N., C.M.R, A.R.V., andS.G.W. performed the experiment. A.K., X.Y. ledthe experiment and A.K., X.Y., R.F.G.R. and S.W.B.designed, simulated and installed the β -detection system.A.K., X.F.Y. performed the data analysis. W.G.J., G.H.,T.P., A.E., G.R.J, S.N. C.F. developed the ∆ NNLO GO interactions and performed the CC calculation. W.N.,P.-G.R., M.K. performed the DFT calculation. B.K.S.performed the atomic physics calculations. A.K., X.F.Y.,G.N., W.N., P-G.R., G.H. and T.P. prepared themanuscript. A.K., X.F.Y., G.N., W.N., P.-G.R. and G.H.prepared the figures. All authors discussed the resultsand contributed to the manuscript at all stages. A.K.,X.F.Y contributed equally to this work. Ethics declarations
The authors declare no competing interests.
Data Availability Statement
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The schematic layout of theCRIS setup is presented in Fig. 1. The mass selectedions were cooled and bunched in the ISCOOL devicewhich operated at 100 Hz, the duty cycle of the CRISexperiment. The ion bunches typically have a 6 µs temporal width corresponding to a spatial length ofaround 1 m length. First, the bunched ion beam wasneutralized in the CRIS beamline through collisions withpotassium atoms in the charge-exchange cell (CEC). Theremaining ions were deflected just after the CEC withan electrostatic deflector plate. In addition, atoms thatwere produced in highly excited states through the chargeexchange process were field ionized and deflected out ofthe beam. The beam of neutral atoms passed throughthe differential pumping region and arrived in the 1.2-minteraction region (IR) maintained at a pressure of − mbar. Here the atom bunch was collinearlyoverlapped with 3 laser pulses, which were used to step-wise excite and ionize the potassium atoms. A detailedstudy of this particular resonance ionization scheme canbe found in Ref. [21]. In the IR, ions can also beproduced in non-resonant processes, introducing higherbackground rates [22]. Normally, the ions created in theIR are guided towards an ion detector (MagneToF). Thistechnique was used for the measurement of − K. The K isotope was produced at a rate of less than 2000particles per second and its hfs spectrum was measuredin less than 2 hours. The study of K, however, requireda still more selective detection method. This isotope,produced at about 360 particles per second rate, is anisobar of the most abundant stable chromium isotope,which is the main contaminating species in the A = 52 beam, with an intensity of 6 × particles per second.In order to avoid the detection of the non-resonantlyionized Cr, the CRIS setup was equipped with a decaydetection station, placed behind the end flange of thebeamline. The MagneToF detector was removed fromthe path of the ion beam, and the ionized bunches wereimplanted into a thin aluminium window with 1 mmthickness allowing the transmission of β particles withenergies larger than 0.6 MeV. The decay station behindthis window consisted of a thin and thick scintillatordetector (A and B in Fig. 1) for a coincidence detectionof β -particles. The dimensions of the detectors were mm × cm × cm and cm × cm × cm. The β counts detected in coincidence were recorded by thedata acquisition system (DAQ) together with the laserfrequency detuning. The DAQ recorded the number ofevents in the detectors with a timestamp. The timestampof the proton bunches impinging into the target ofISOLDE was also recorded and used to define the timegates in the data analysis. Laser system:
A three-step resonance ionizationscheme was used in this experiment. The laserlight for the first excitation step was produced by acontinuous wave (cw) titanium-sapphire (Ti:Sa) laser(M-Squared SolsTiS) pumped by an 18-W laser at532 nm (Lighthouse Photonics). In order to avoid opticalpumping to dark states due to long interaction times,this cw light was “chopped” into 50-ns pulses at arepetition rate of 100 Hz by using a Pockels cell [20]. Thewavelength of this narrowband laser was tuned to probethe hfs of the 4s S / → P / transition at 769 nm.Atoms in the excited 4p P / state were subsequentlyfurther excited to the 6s S / atomic state by apulsed dye laser (Spectron PDL SL4000) with a spectralbandwidth of 10 GHz. This dye laser was pumped by a532 nm Nd:YAG laser (Litron TRLi 250-100) at a 100-Hz repetition rate. The fundamental output of the sameNd:YAG laser (1064 nm) was used for the final non-resonant ionization. The arrival of ion bunches andlaser pulses in the interaction region were synchronizedand controlled using a multi-channel pulse generator(Quantum Composers 9520 Series). Charge radii extraction:
The perturbation of theatomic states caused by the different nuclear chargedistribution in isotopes leads to small differences in theatomic transition frequency, δν AA (cid:48) , between centroids( ν A , ν A (cid:48) ) of the hfs of two isotopes with mass number A and A (cid:48) . The isotope shifts of − K were extracted fromthe hfs spectra of − K analyzed using the SATLAS [32]Python package, as displayed in third column of TableII, along with all available results from literature. Moredetails on the analysis process can be found in Ref.[21].The changes in the nuclear mean-square charge radii of − K can then be extracted from the isotope shiftsusing: δν AA (cid:48) = K MS m A − m A (cid:48) m A (cid:48) ( m A + m e ) + F δ (cid:104) r (cid:105) AA (cid:48) , (2)where K MS and F are the atomic mass shift andfield shift, m stands for the nuclear mass of isotopes A , A (cid:48) , and an electron. The nuclear mass wasobtained by subtracting the mass of the electrons fromthe experimentally measured atomic mass reported inRef.[33]. The atomic constants, K MS and F , were calculated using the atomic ARRCC method as describedbelow. The root-mean-square charge radii of theseisotopes are: R = (cid:113) δ (cid:104) r (cid:105) + R , (3)where R is the charge radius of K taken from Ref.[34].
Evaluation of the isotope shifts and chargeradii:
The isotope shifts of potassium isotopes weremeasured using several different techniques over manyyears, ranging from magneto-optical trap experiments[23] to laser spectroscopy of thermal [18, 35, 36] andaccelerated beams [5, 19, 21], relying on photon and iondetection. The available results in Refs. [18, 19, 23,35, 36] are referenced to the stable K isotope, andare presented in the second column of Table II. Theisotope shifts in Refs. [5, 21] and this work, shown inthe third column of Table II, were extracted with respectto K. The systematic error from the experimentsis given in curly brackets. Note that the systematicuncertainties in collinear laser spectroscopy experimentsare mostly related to the inaccuracy of the accelerationvoltage. In this work, the systematic uncertainty wasnegligible by using the laser scanning approach [21] anda well-calibrated high-precision voltage divider (with arelative uncertainty of × − ) from PTB. In order tocompile a consistent data set with reliable evaluation ofuncertainties, the following steps were taken:1) The isotope shifts obtained with respect to Kwere recalculated relative to K, in order to link alldata to the same reference. For this, the weightedaverage of all available δν , isotope shifts fromRefs. [5, 21] is used as a reference. These re-referenced values are listed in the fifth column ofTable II and their uncertainty is increased due tothe additional error associated with δν , (boldvalue in column six). Note that the systematicerrors are always taken into account using the linearmodel [37], σ = σ sys + σ stat .2) Next, the final isotope shift of each potassiumisotope, δν ,A (shown in the sixth column of TableII), was calculated as the weighted average ( ˆ x ) ofthe available results ( x i ) using: ˆ x = Σ ni =1 ( x i σ − i )Σ ni =1 σ − i , (4)where σ i is the total uncertainty of i th measurement. The error of the weighted mean wasobtained using: σ ˆ x = (cid:115) ni =1 σ − i , (5) Table II. Experimental re-evaluated isotope shifts, changes in mean-square charge radii and the absolute charge radii of − K. Experimental and theoretical systematic errors are reported in curly and square brackets, respectively. The statisticaluncertainties are presented in brackets. The reference value for the charge radius of K is taken from Ref. [34], where it isgiven without uncertainties.A δν ,A (MHz) δν ,A (MHz) Reference δν ,A re (MHz) δν ,A δ (cid:104) r (cid:105) ,A (fm ) R ch (fm)36 -403(5){4} Ref. [19] -403(9) -0.20(8)[4] 3.405(12)[6]37 -264(3){3} Ref. [19] -264(6) -0.11(6)[3] 3.419(8)[4]38 -127.0(53) Ref. [18] -126.1(19) -0.075(18)[14] 3.4241(26)[20]38 -985.9(4){34} Ref. [5] -127.5(39)38 -983.8(4){18} Ref. [21] -125.4(24)39 -862.5(9){30} Ref. [5] 0 0 3.43539 -858.4(6){5} Ref. [21]40 125.58(26) Ref. [35] 125.63(9) 0.025(1)[13] 3.4386(1)[19]40 125.64(10) Ref. [36]41 235.25(75) Ref. [18] 235.47(9) 0.135(1)[26] 3.4546(1)[37]41 235.27(33) Ref. [35]41 235.49(9) Ref. [36]42 351.7(19) Ref. [18] 352.4(10) 0.128(10)[38] 3.4536(15)[55]42 -506.7(7){17} Ref. [5] 351.7(26)42 -505.5(6){3} Ref. [21] 352.9(13)43 459.0(12) Ref. [18] 459.0(12) 0.165(11)[49] 3.4590(16)[71]44 564.3(14) Ref. [18] 565.1(8) 0.163(7)[60] 3.4586(11)[87]44 -292.1(5){10} Ref. [5] 566.3(17)44 -293.19(56){23} Ref. [21] 565.2(12)45 661.7(16) Ref. [18] 661.7(16) 0.203(15)[70] 3.4644(22)[102]46 762.8(15) Ref. [18] 764.1(14) 0.150(13)[80] 3.4568(19)[116]46 -91.6(5){3} Ref. [5] 766.8(12)46 -95.81(55){6} Ref. [21] 762.6(11)47 857.5(17) Ref. [18] . accounting for possible over-, or under-dispersionusing: ˆ σ x = σ x χ , (6)where χ is the reduced chi-squared.3) These evaluated isotope shifts were used to extractthe changes in mean-square charge radii (column 7of Table II) using the new theoretical values for theatomic field and mass shifts, obtained in this work.The absolute charge radii of all potassium isotopes(last column of Table II) were then calculated byusing the above mentioned Eq.3, relative to theabsolute radius of the stable K [34].
Atomic coupled-cluster calculations:
The wavefunction of an atomic state with a closed-shell and avalence orbital electronic configuration can be expressedusing the CC theory ansatz as | Ψ v (cid:105) = e ˜ S | Φ v (cid:105) = e T { S v }| Φ v (cid:105) , (7)where | Φ v (cid:105) is the mean-field wave function and ˜ S is theCC excitation operator. We further divide as ˜ S = T + S v to distinguish electron correlations without involving thevalence electron ( T ) and involving the valence electron( S v ). In the analytic response procedure, the first-orderenergy of the atomic state is obtained by solving the0equation ( H at − E (0) v ) | Ψ (0) v (cid:105) = ( E (1) v − H int ) | Ψ (1) v (cid:105) , (8)where H at is the atomic Hamiltonian, H int is theinteraction Hamiltonian, | Ψ (0) v (cid:105) is the unperturbed wavefunction with energy E (0) v , and | Ψ (1) v (cid:105) is the first-orderperturbed wave function with the first-order energy E (1) v . Here, H at involves the Dirac terms, the nuclearpotential, the lower-order quantum electrodynamicscorrections, and the electron-electron interactions dueto the longitudinal and transverse photon exchanges,while H int is either the FS operator due to the Ferminuclear charge distribution in the evaluation of F or the relativistic form of the SMS operator for thedetermination of K SMS . In the ARRCC theory, theunperturbed and the perturbed wave functions areobtained by expanding T (cid:39) T (0) + λT (1) and S v (cid:39) S (0) v + λS (1) v (9)with λ representing the perturbation parameter. Aftersolving the amplitudes of both the unperturbed andperturbed CC operators, as described in Ref. [25], weevaluate the first-order energy as E (1) v = (cid:104) Φ v | ( H at e T (0) ) c { S (1) v + T (1) }| Φ v (cid:105) + (cid:104) Φ v | ( H int e T (0) ) c { S (0) v }| Φ v (cid:105) , (10)in which the subscript c means the connected terms.We have considered all possible single, double andtriple electronic excitation configurations in our ARRCCmethod for performing the atomic calculations. Nuclear coupled cluster calculations:
The nuclearCC calculations start from the intrinsic Hamiltonianincluding two- and three-nucleon forces, ˆ H = (cid:88) i 36 38 40 42 44 46 48 50 52 A K < r > , A ( f m ) NNLO GO (394)NNLO GO (450)NNLO sat Experiment 36 38 40 42 44 46 48 50 52 A K < r > , A ( f m ) MF [ NNLO GO (394)]CC [ NNLO GO (394)]MF [1.8/2.0 (EM)]CC [1.8/2.0 (EM)] Figure 3. Changes in mean-square charge radii of potassiumisotopes calculated based on newly developed interactions andNNLO sat are compared with (a) experimental data and (b)that from deformed mean-field (MF) calculations with CCSDcomputations for two different interactions. major harmonic oscillator shells ( N max = 12 ) with theoscillator frequency (cid:126) Ω = 16 MeV. The three-bodyinteraction has the additional cutoff on allowed three-particle configurations E max = N + N + N ≤ , with N i = 2 n i + l i . This model-space is sufficient to convergethe radii of all the potassium isotopes considered in thiswork to within ∼ 1% . In this work we calculate theexpectation value of the squared intrinsic point protonradius, i.e. (cid:104) O (cid:105) = (cid:104) /Z (cid:80) i 36 38 40 42 44 46 48 50 52Neutron Number3.403.453.503.55 R c h ( f m ) Fy( r,HFB) deformedFy( r,HFB) sphericalSystematic errorExperiment Figure 4. Charge radii along the chain of potassiumisotopes from spherical as well as deformed calculations withthe functional Fy( ∆ r ,HFB) and compared with experimentaldata. [29], which has been constrained by nucleon-nucleonproperties, and binding energies and charge radii of nucleiup to oxygen. It includes terms up to next-to-next-to leading order in the Weinberg power counting. Thenewly constructed ∆NNLO GO (450) interaction includes ∆ isobar degrees of freedom, exhibits a cutoff of450 MeV, and is also limited to next-to-next-to-leadingorder contributions. Its construction starts from theinteraction of Ref. [40] and its low-energy constantsare constrained by the saturation density, energy andsymmetry energy of nuclear matter, by pion-nucleonscattering [41], nucleon-nucleon scattering, and by the A ≤ nuclei. A second interaction, ∆NNLO GO (394) ,was similarly constructed but with a cutoff of 394 MeV.To look at the sensitivities in the changes in the mean-square change radii, Fig. 3a compares results fromthe newly developed interactions with NNLO sat . Whilethere are differences below N = 28 , all interactionsyield essentially identical results beyond N = 28 .This suggests that charge radii beyond N = 28 areinsensitive to details of chiral interactions at next-to-next-to-leading-order.To shed more light onto this finding, Fig. 3b comparesresults from deformed mean-field (MF) calculations withCCSD computations for two different interactions. The1.8/2.0(EM) interaction [42] contains contributions atnext-to-next-to-next-to-leading order and thereby differsfrom the interactions used in this work. First, for the ∆NNLO GO (394) interaction, MF and CC yieldessentially the same results for N > , though CCincludes many more wave-function correlations. Second,for the 1.8/2.0(EM) interaction, MF and CC results differsignificantly by the strong odd-even staggering, which isa correlation effect. None of the interactions explain thedramatic increase of the charge radii beyond N = 28 . DFT calculations: For the DFT part of this work,we use the non-relativistic Fayans functional in the formof Ref.[43]. This functional is distinguished from othercommonly used nuclear DFT in that it has additionalgradient terms at two places, namely in the pairingfunctional and in the surface energy. The gradient termsallow, among other features, a better reproduction of theisotopic trends of charge radii [44]. This motivated arefit of the Fayans functional to a broad basis of nuclearground state data with additional information on changesin mean-square charge radii in the calcium chain [4, 30].We use here Fy( ∆ r ,HFB) from Refs. [4, 30] whichemployed the latest data on calcium radii. It is only withrather strong gradient terms that one is able to reproducethe trends of radii in calcium at all, in particular itspronounced odd-even staggering, however, with a slighttendency to exaggerate the staggering. It was found laterthat Fy( ∆ r ,HFB) performs very well in describing thetrends of radii in cadmium and tin isotopes [13, 31]. Herewe test it again for the potassium chain next to calcium.For all practical details pertaining to our Fayans-DFTcalculations, we refer the reader to Ref. [30].All above mentioned calculations with Fy( ∆ rr