Laser spectroscopy of neutron-rich ^{207,208}Hg isotopes: Illuminating the kink and odd-even staggering in charge radii across the N=126 shell closure
T. Day Goodacre, A.V. Afanasjev, A.E. Barzakh, B.A. Marsh, S. Sels, P. Ring, H. Nakada, A.N. Andreyev, P. Van Duppen, N.A. Althubiti, B. Andel, D. Atanasov, J. Billowes, K. Blaum, T.E. Cocolios, J.G. Cubiss, G.J. Farooq-Smith, D.V. Fedorov, V.N. Fedosseev, K.T. Flanagan, L.P. Ganey, L. Ghys, M. Huyse, S. Kreim, D. Lunney, K.M. Lynch, V. Manea, Y. Martinez Palenzuela, P.L. Molkanov, M. Rosenbusch, R.E. Rossel, S. Rothe, L. Schweikhard, M.D. Seliverstov, P. Spagnoletti, C. Van Beveren, M. Veinhard, E. Verstraelen, A. Welker, K. Wendt, F. Wienholtz, R.N. Wolf, A. Zadvornaya, K. Zuber
LLaser spectroscopy of neutron-rich , Hg isotopes: Illuminating the kink andodd-even staggering in charge radii across the N = 126 shell closure T. Day Goodacre,
1, 2, 3, ∗ A.V. Afanasjev, A.E. Barzakh, B.A. Marsh, S. Sels,
2, 6
P. Ring, H. Nakada, A.N. Andreyev,
9, 10
P. Van Duppen, N.A. Althubiti,
1, 11
B. Andel,
6, 12
D. Atanasov, † J. Billowes, K. Blaum, T.E. Cocolios,
1, 6
J.G. Cubiss, G.J. Farooq-Smith,
1, 6
D.V. Fedorov, V.N. Fedosseev, K.T. Flanagan,
1, 14
L.P. Gaffney,
6, 15, ‡ L. Ghys,
6, 16
M. Huyse, S. Kreim,
13, 2
D. Lunney, § K.M. Lynch,
1, 2
V. Manea, § Y. Martinez Palenzuela,
6, 2
P.L. Molkanov, M. Rosenbusch, ¶ R.E. Rossel,
2, 19
S. Rothe, L. Schweikhard, M.D. Seliverstov, P. Spagnoletti, C. Van Beveren, M. Veinhard, E. Verstraelen, A. Welker,
2, 20
K. Wendt, F. Wienholtz,
2, 18, ∗∗ R.N. Wolf,
13, 18, †† A. Zadvornaya, and K. Zuber The University of Manchester, School of Physics and Astronomy,Oxford Road, M13 9PL Manchester, United Kingdom CERN, CH-1211 Geneva 23, Switzerland TRIUMF, Vancouver V6T 2A3, Canada Department of Physics and Astronomy, Mississippi State University, MS 39762, USA Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina 188300, Russia KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium Fakult¨at f¨ur Physik, Technische Universit¨at M¨unchen, D-85748 Garching, Germany Department of Physics, Graduate School of Science,Chiba University, Yayoi-cho, I-33, Inage, Chiba 263-8522, Japan Department of Physics, University of York, York, YO10 5DD, United Kingdom Advanced Science Research Center (ASRC), Japan Atomic Energy Agency (JAEA), Tokai-mura, Japan Physics Department, Faculty of Science, Jouf University, Aljouf, Saudi Arabia Department of Nuclear Physics and Biophysics,Comenius University in Bratislava, 84248 Bratislava, Slovakia Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany The Photon Science Institute, The University of Manchester, Manchester, M13 9PL, United Kingdom School of Computing, Engineering, and Physical Sciences,University of the West of Scotland, Paisley PA1 2BE, United Kingdom Belgian Nuclear Research Center SCK • CEN, Boeretang 200, B-2400 Mol, Belgium CSNSM-IN2P3, Universit´e de Paris Sud, Orsay, France Universit¨at Greifswald, Institut f¨ur Physik, 17487 Greifswald, Germany Institut f¨ur Physik, Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany Institut f¨ur Kern- und Teilchenphysik, Technische Universit¨at Dresden, Dresden 01069, Germany (Dated: December 29, 2020)The mean-square charge radii of , Hg ( Z = 80 , N = 127 , , , Hg ( N = 122 , , kink in the charge radii at the N = 126 neutron shell closure has been revealed, providing the first information on its behaviorbelow the Z = 82 proton shell closure. A theoretical analysis has been performed within relativisticHartree-Bogoliubov and non-relativistic Hartree-Fock-Bogoliubov approaches, considering both thenew mercury results and existing lead data. Contrary to previous interpretations, it is demonstratedthat both the kink at N = 126 and the odd-even staggering (OES) in its vicinity can be describedpredominately at the mean-field level, and that pairing does not need to play a crucial role in theirorigin. A new OES mechanism is suggested, related to the staggering in the occupation of thedifferent neutron orbitals in odd- and even- A nuclei, facilitated by particle-vibration coupling forodd- A nuclei. Experimental investigations of nuclear charge radiihave revealed a rich abundance of regular patterns,abrupt changes and non-linear trends along isotopicchains across the nuclear chart [1, 2]. Two near-universalfeatures include kinks at neutron shell closures and odd-even staggering (OES), where an odd- N isotope has asmaller charge radius than the average of its two even- N neighbors [1, 3]. The commonality of these features indi-cates that their origin is general and independent of localmicroscopic phenomena. As such, measurements of kinks and OES provide a particularly stringent benchmark fornuclear theory [4–9].Historically, the majority of the discussion pertainingto the effect of shell closures on charge radii focused onthe kink in the lead isotopic chain across N = 126 [4–6, 10, 11], which is shown in the inset of Fig. 1. Avariety of theoretical approaches have been employedto investigate this kink. Relativistic mean field studieshave described it with varying degrees of success [5, 12].By contrast, calculations in non-relativistic density func- a r X i v : . [ nu c l - e x ] D ec tional theories (NR-DFTs) based on conventional func-tionals were unable to reproduce the kink [4, 8]. Thedifferences between the descriptions in these theoreticalframeworks are related to the relative occupations of theneutron ν i / and ν g / orbitals (located above the N = 126 shell closure), with the magnitude of the kinkbeing driven by the occupation of the ν i / orbital [11].The extension of non-relativistic functionals, by the ad-dition of gradient terms into the pairing interaction, hasbeen demonstrated as a possible way to improve theirdescription of OES and the kink [6, 8]. An alternativeapproach is the inclusion of a density dependence in thespin-orbit interaction [7, 13, 14], derived from the chiralthree-nucleon interaction by Kohno [15].The optimal method of quantifying the kink at N =126 considers isotopes with N = 124 , , N = 50 and N = 82 [1], thereare limited experimental data on charge radii behavior Ulm et al. 1986Marsh et al. 2018g.s. isomer This workg.s. isomerg.s.0.750.50.250-0.25-0.5-0.75-1-1.25 δ < r > N , ( f m )
95 100 105 110 115 120 125 130
N N =126124 128PbHg0.40.60.8175 180 185 190 195 200 205 210 A FIG. 1. Systematics of the difference in mercury ground state(g.s.) and isomer mean-square charge radii. Data from thepresent work are shown by red symbols, earlier data are takenfrom [16] (black) and [17] (blue). The inset highlights the kinkat N = 126 and the neighboring OES in both the mercuryand lead [18] isotopic chains. The lead isotopes are arbitrarilydisplaced from those of mercury for clarity. The dashed linesthrough N = 124 ,
126 in the inset are added to highlightthe kinks. Statistical uncertainties are smaller than the datapoints. across N = 126 (specifically for N = 128), with corre-sponding measurements available only for Z = 82 [18]and Z = 83 [19].In this Letter we report the first study of charge radiiacross N = 126 in the mercury ( Z = 80) isotopic chain,thus enabling the Z -dependence of the kink at N = 126to be probed and providing the first benchmark for the-ory in the region below the Z = 82 proton shell closure.These new data motivated us to undertake a comparativetheoretical investigation of the kinks and OES in lead andmercury charge radii across N = 126. By applying bothspherical relativistic Hartree-Bogoliubov (RHB) [20] andspherical non-relativistic Hartree-Fock-Bogoliubov (NR-HFB) [21] approaches, we explored whether an alterna-tive explanation of the kink and OES is possible. Thiswas also the first study of OES within a relativistic frame-work.The experimental data originates from the same mea-surement campaign as described in Refs. [17, 22], whereneutron-deficient mercury isotopes were also studied(Fig. 1). Therefore, only a brief experimental overviewis included here. Mercury isotopes were produced at theCERN-ISOLDE facility [23] by impinging a 1.4-GeV pro-ton beam from the Proton Synchrotron Booster on to amolten-lead target. The neutral reaction products ef-fused into the anode cavity of a Versatile Arc Dischargeand Laser Ion Source (VADLIS) [24]. Laser light from theISOLDE Resonance Ionization Laser Ion Source (RILIS)complex [25] was used to excite three sequential atomictransitions, for the resonance ionization of the mercuryisotopes [26]. The photo-ions were extracted and massseparated by the ISOLDE general purpose separator andthen directed to either a Faraday cup for direct photo-ion detection, or to the ISOLTRAP Paul trap [27] andmulti-reflection time-of-flight mass spectrometer (MR-ToF MS) [28] for single-ion counting and discriminationfrom isobaric contamination.The first of the three atomic transitions(5 d s S → d s p P ◦ , 253.65 nm) was probedby scanning a frequency-tripled titanium-sapphire laser(full width at half maximum bandwidth < δν A,A (cid:48) = ν A − ν A (cid:48) ,in the frequency of this transition were measured formercury nuclei with A = 202 , , , ,
208 relativeto the stable reference isotope with A (cid:48) = 198. Samplespectra are presented in Fig. 2. Details of the scanningand fitting procedures can be found in Refs. [22, 30]and Refs. [22, 31], respectively, with further informationon the data analysis in Refs. [32, 33]. The relativechanges in the mean-square charge radii, δ (cid:104) r (cid:105) A,A (cid:48) , wereextracted from the measured δν A,A (cid:48) values via standardmethods described in the supplemental material [34].The extracted δν A,A (cid:48) and δ (cid:104) r (cid:105) A,A (cid:48) values are pre-sented in Table I and the δ (cid:104) r (cid:105) A,A (cid:48) data are plotted inFig. 1. There is a visible kink at N = 126, with a mag-nitude similar to that in the lead isotopic chain. The IS -50 -40 -30 100-10-20 C o u n t s ( A r b . u n i t s ) δν A,198 (GHz) Hg Hg Hg FIG. 2. Sample hyperfine spectra for − Hg, with thephoto-ion rate measured using the MR-ToF MS. The cen-troids are indicated with black lines, red lines represent fittingwith Voigt profiles. for , Hg, and the remeasured neutron-deficient iso-topes [17, 22] from the same experimental campaign, arein good agreement with literature. For
Hg there is a ≈ δν , value from this work is used for thefollowing discussion.To interpret the data, a new RHB code was developedwhich enables the blocking of selected single-particle or-bitals and allows for fully self-consistent calculations ofthe ground and excited states in odd- A nuclei. A sepa-rable version of the Gogny pairing is used [36], with thepairing strength of Ref. [37]. The NL3*, DD-PC1, DD-ME2 and DD-ME δ covariant energy density functionals(CEDF) were employed, the global performance of whichwas tested in Ref. [37]. The functionals achieved a com-parable description of the kink and the OES, thus onlythe DD-ME2 results are discussed below. The results TABLE I. δν A, values in the 253.65 nm line from this workand literature. δ (cid:104) r (cid:105) A, values are calculated as described inthe supplemental material [34]. Statistical uncertainties areshown in parentheses and systematic uncertainties in curlybrackets.Isotope I π δν A, δ (cid:104) r (cid:105) A, Ref.A (MHz) (fm )202 0 -10 100(180) 0.197(3) { } This work-10 102.4(42) 0.1973(1) { } [16]203 5/2 − -11 870(200) 0.232(4) { } This work-11 750(180) 0.2295(35) { } [16]206 0 -20 930(160) 0.409(3) { } This work-20 420(80) 0.3986(16) { } [16, 35]207 (9/2 − ) -25 790(190) 0.503(4) { } This work208 -32 030(160) 0.624(3) { } This work for the other functionals will be included in a follow-uppaper [38].The NR-HFB calculations were performed assumingspherical symmetry with the semi-realistic M3Y-P6a in-teraction, the spin-orbit properties of which were modi-fied [13] to improve the description of the charge radii ofproton-magic nuclei [7, 13, 14]. We applied it here for thefirst time to the mercury isotopic chain. For N ≤ (cid:104) β (cid:105) / < . (cid:104) β (cid:105) is the mean-square deformation deduced from δ (cid:104) r (cid:105) usingthe droplet model [3, 39]. This restriction corresponds to N ≥
116 and N ≥
121 for lead and mercury isotopes,respectively.The importance of a simultaneous agreement of ener-getic and geometric nuclear observables in such investi-gations has been highlighted previously [6]. Thus, in ad-dition to calculating the charge radii, we also checked thequality of the binding energy description. There is a goodagreement between the DD-ME2 and experimental bind-ing energies for both lead and mercury isotopes [40]. Therms deviation is 1.3 MeV and 1.1 MeV for − Pb and − Hg, respectively, a comparable performance to thewidely used NR-DFT functional UNEDF1 [41] (1.4 MeVand 1.0 MeV for lead and mercury, respectively [42]).The binding and separation energy descriptions of allof the employed CEDFs will be discussed in detail inRef. [38].Two different procedures labeled as “LES” and “EGS”are used for the blocking in odd- A nuclei and the resultsof respective calculations are labeled by the “Functional-Procedure” labels (for example, DD-ME2-EGS). In theLES procedure, the lowest in energy configuration isused: this is similar to all earlier calculations of OES [6,43]. In the EGS procedure, the configuration with thespin and parity of the blocked state corresponding tothose of the experimental ground state is employed, al-though it is not necessarily the lowest in energy. For ex-ample, in the RHB(DD-ME2) calculations the EGS con-figurations with a blocked ν g / state are located at ex-citation energies of 137 keV, 122 keV and 96 keV abovethe ground state configurations with a blocked ν i / state in , , Pb. At first glance, this contradictsexperimental findings that the ground state is based onthe ν g / orbital in odd- A lead isotopes with N > ν i / one (see Fig.5 in Ref. [44]) so that it becomes the lowest in energy inthe PVC calculations. Note that PVC significantly im-proves the accuracy of the description of the energies ofexperimental states in model calculations [44, 45]. How-ever, it is neglected in the present study since its impacton charge radii is still an open theoretical question.The results of the RHB and NR-HFB calculations arepresented in Fig. 3, together with the experimental re-sults for the lead and mercury chains. In both cases, thekink at N = 126 is visibly better reproduced in the RHB(DD-ME2) calculations. To facilitate a quantitative com-parison of the experimental and theoretical results, twoindicators are employed. OES is quantified consideringthe isotope’s nearest neighbors via the commonly usedthree-point indicator∆ (cid:104) r (cid:105) (3) ( A ) = 12 (cid:2) (cid:104) r ( A − (cid:105) + (cid:104) r ( A + 1) (cid:105) − (cid:104) r ( A ) (cid:105) (cid:3) . (1)To quantify the shell effect at N = 126, the kink indicatorof Ref. [8] is used which considers the isotope’s next-to-nearest neighbors and is defined as∆ R (3) ( A ) = 12 [ R ( A −
2) + R ( A + 2) − R ( A )] , (2)where R ( A ) = (cid:104) r (cid:105) / ( A ) is the charge radius of the iso-tope with mass A of the element under consideration.Note that the kink indicator is independent of the block-ing procedure in odd- A nuclei and therefore we omit thecorresponding specifications (LES or EGS) in the discus-sion of the kink.∆ (cid:104) r (cid:105) (3) ( A ) and ∆ R (3) ( A ) values calculated from theexperimental results and theoretical calculations for bothlead and mercury are presented in Fig. 4, and the∆ R (3) ( A ) values are listed in the supplemental mate-rial [34]. It is evident in Figs. 4 (a) and 4 (b) that themagnitude of the kinks in the isotopic chains are com-parable, suggesting that the kink at N = 126 is broadlyinsensitive to the change of the occupied proton stateswhen crossing Z = 82 ( π d / in mercury and π s / inlead). In addition, the RHB(DD-ME2) calculations bestreproduce the kink, while the NR-HFB(M3Y-P6a) andNR-HFB(Fy(∆r) [8]) results underestimate and overes-timate its magnitude, respectively. It is worth notingthat both the RHB(DD-ME2) and NR-HFB(M3Y-P6a) (b)(a) δ < r > N , ( f m ) N = N = N Expt.
UNEDF1M3Y-P6a-EGSDD-ME2-EGS
Pb Hg
FIG. 3. δ (cid:104) r (cid:105) A,A (cid:48) of lead (a) and mercury (b) isotopesrelative to
Pb and
Hg ( N = 126), respectively. Ex-perimental mercury data: this work and [16], experimentallead data: [18], the statistical uncertainties are smaller thanthe data points. RHB(DD-ME2) results: this work, NR-HFB(M3Y-P6a) results: this work (mercury) and [14] (lead),NR-HFB(UNEDF1) results: [42]. approaches are reasonably successful in the reproductionof the absolute charge radius values for Hg and
Pb,the details are included in the supplemental material [34].In both approaches, the OES is best reproduced if theEGS procedure is applied (see Figs. 4 (c), (d) and Fig. 5).If the LES procedure is applied, the experimental OESis significantly underestimated for all nuclei under studyin the RHB calculations and for
N <
126 nuclei in theNR-HFB calculations. Note that for simplicity we showonly NR-HFB results with both procedures in Fig. 4 (c)and (d).For a better understanding of the underlying mecha-nisms of both the kink and OES, we also performed RHBcalculations without pairing for lead isotopes. The la-bels identifying such results contain “np”. Significantly, akink is still present in the results as depicted in Fig. 4 (a),due to the occupation of the ν i / orbital. This indi-cates a mechanism alternative to the one based on gra-dient terms in pairing interactions [6, 8].The RHB results with and without pairing are com-pared via ∆ (cid:104) r (cid:105) (3) ( A ) in Fig. 5. OES appears in these U N E D F M Y - P a D D - M E - n p D D - M E F y ( Δ r ) U N E D F M Y - P a D D - M E ∆ R ( ) ( A ) ( - f m ) (b)(a) Pb Hg (d)(c) Pb Hg ∆ < r > ( ) ( A ) ( f m ) Expt. N = N = M3Y-P6a-EGS122 124 126 128
FIG. 4. Comparison of experimental and theoretical∆ R (3) ( A ) and ∆ (cid:104) r (cid:105) (3) ( A ) values for isotopes of lead, (a) and(c), respectively, and mercury (b) and (d), respectively. Ex-perimental values are taken from this work and from Ref. [16](mercury) and Ref. [18] (lead). The RHB(DD-ME2) and NR-HFB(M3Y-P6a) results are obtained in this work and theNR-HFB results with Fy(∆ r ) and UNEDF1 are taken fromRefs. [8] and [42], respectively. Experimental uncertainty isdepicted as translucent gray bars in (a) and (b), and as errorbars in (c) and (d). N
118 128 130 132
Expt. DD-ME2-EGS DD-ME2-np-EGSDD-ME2-np-LES ∆ < r > ( ) ( A ) ( f m ) N = FIG. 5. Comparison of experimental and theoretical∆ (cid:104) r (cid:105) (3) ( A ) values for lead isotopes. Experimental datafrom [18], theoretical results are from this work. See textfor details. calculations (the curves labeled as “DD-ME2-EGS” and“DD-ME2-np-EGS”) under the condition that, in odd- A nuclei, the EGS procedure is used. One can see that theinclusion of pairing somewhat reduces this effect. How-ever, OES is mostly absent if the LES procedure is usedin odd- A nuclei.Let us consider the lead isotopes with N ≥
126 fora more detailed discussion of the origin of OES in thecalculations without pairing. By designating the groundstate of
Pb as a “core” and noting that PVC low-ers the energy of the ν g / state below ν i / in odd- A nuclei [44], the sequence of the ground states in the N ≥
126 nuclei can be described as “core” (
Pb),“core” ⊗ ν (2 g / ) ( Pb), “core” ⊗ ν (2 i / ) ( Pb),“core” ⊗ ν (2 i / ) (2 g / ) ( Pb) and so on in the rela-tivistic calculations without pairing. The ν i / orbitalhas a smaller rms-radius than the ν g / orbital. How-ever, because of the isovector nature of nuclear forces itsoccupation leads to a larger charge radius as comparedwith the occupation of ν g / orbital. Thus, the stagger-ing in their occupation between odd and even isotopesresults in the OES seen in Fig. 5.On the contrary, in the majority of conventional non-relativistic functionals, the ν g / orbital is lower in en-ergy than the ν i / orbital. This is in agreement withexperimental data on the structure of the ground statesin odd-mass nuclei, but creates a problem in the descrip-tion of the kinks. In addition, in calculations with andwithout pairing this leads to the sole or predominant oc-cupation of the ν g / state in even-even and odd-evennuclei with N >
126 and thus to a negligible or compar-atively small OES. To address this, several prescriptionshave been suggested over the years to increase the occu-pation of the ν i / orbital in the N >
126 lead nuclei.One approach includes a modification of the spin-orbitinteraction leading either to the inversion of the relativeenergies of these two states or to their proximity in en-ergy [11, 46–50]. The NR-HFB results with M3Y-P6a shown in Figs. 3 and 4 are also based on a modifica-tion of the spin-orbit interaction, with the inclusion of adensity dependent term in the spin-orbit channel. Alter-natively, the so-called Fayans functionals employ a spe-cific form of the pairing interaction containing a gradientterm [6, 8, 43, 51]. Although this improves the generaldescription of experimental data, discrepancies betweentheory and experiment still exist in the lead and tin iso-topic chains [8]. Moreover, pairing becomes a dominantcontributor to the kink and OES [8].The present RHB interpretation of the kinks and OESdiffers from that suggested in [8], which is based onnon-relativistic Skyrme and Fayans functionals. In theRHB approach, the kink and OES are already present inthe calculations without pairing. Thus, the evolution ofcharge radii with neutron number depends significantlyon the mean-field properties. Pairing acts only as anadditional tool which mixes different configurations andsomewhat softens the evolution of charge radii as a func-tion of neutron number.In conclusion, the determination of the δ (cid:104) r (cid:105) A,A (cid:48) of , Hg has revealed a kink at N = 126 in the mer-cury nuclear charge radii systematics, with a magnitudecomparable to that in the lead isotopic chain. Thesenew data have been analyzed via both RHB and NR-HFB approaches, together with the traditional magic- Z theoretical benchmark of the lead isotopic chain. Wedemonstrate that the kinks at the N = 126 shell closureand the OES in the vicinity, are currently best describedin the RHB approach without any readjustment of theparameters defined in Ref. [52]. According to the RHBcalculations, the kink at N = 126 in δ (cid:104) r (cid:105) A,A (cid:48) originatesfrom the occupation of the ν i / orbital located abovethe N = 126 shell gap. A new mechanism for OES issuggested, related to the staggering in the occupation ofneutron orbitals between odd and even isotopes and fa-cilitated by PVC in odd-mass nuclei. Thus, contrary toprevious interpretations, it is determined that both thekink and OES in charge radii can be defined predomi-nantly in the particle-hole channel.This project has received funding through the Eu-ropean Union’s Seventh Framework Programme forResearch and Technological Development under grantagreements 267194 (COFUND), 262010 (ENSAR),289191 (LA ∗ [email protected] † Present address: CERN, 1211, Geneva 23, Switzerland. ‡ Present address: Department of Physics, University ofLiverpool, Liverpool, L69 7ZE, United Kingdom § Present address: Universit´e Paris-Saclay, CNRS/IN2P3,IJCLab, 91405 Orsay, France. ¶ Present address: Wako Nuclear Science Center (WNSC),Institute of Particle and Nuclear Studies (IPNS),High Energy Accelerator Research Organization (KEK),Wako, Saitama 351-0198, Japan ∗∗ Present address: Institut f¨ur Kernphysik, TechnischeUniversit¨at Darmstadt, 64289 Darmstadt, Germany. †† Present address: ARC Centre of Excellence for Engi-neered Quantum Systems, School of Physics, The Uni-versity of Sydney, NSW 2006, Australia.[1] I. 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