Searching for isovector signatures in the neutron-rich oxygen and calcium isotopes
SSearching for isovector signatures in the neutron-rich oxygen and calcium isotopes
Wei-Chia Chen ∗ and J. Piekarewicz † Department of Physics, Florida State University, Tallahassee, FL 32306 (Dated: September 21, 2018)We search for potential isovector signatures in the neutron-rich oxygen and calcium isotopes withinthe framework of a relativistic mean-field theory with an exact treatment of pairing correlations. Toprobe the isovector sector we calibrate a few relativistic density functionals using the same isoscalarconstraints but with one differing isovector assumption. It is found that under certain conditions,the isotopic chain in oxygen can be made to terminate at the experimentally observed O isotopeand in the case of the calcium isotopes at Ca. To produce such behavior, the resulting symmetryenergy must be soft, with predicted values for the symmetry energy and its slope at saturationdensity being J = (30 . ± .
47) MeV and L = (51 . ± .
5) MeV, respectively. As a consequence,the neutron-skin thickness of
Pb is rather small: R = (0 . ± . “FSUGarnet” —predicts R . = (13 . ± .
1) km for the radius of a “canonical” 1.4 M (cid:12) neutronstar, yet is also able to support a two-solar-mass neutron star. PACS numbers: 21.60.Jz, 21.65.Cd, 21.65.Mn
Density functional theory (DFT) provides the onlyknown tractable framework to describe strongly inter-acting nuclear many-body systems ranging from finitenuclei to neutron stars. In the spirit of DFT, the com-plicated many-body effects are implicitly encoded in theparameters of the model which in turn are determined byfitting directly to experimental data [1]. Thus, the qual-ity of the resultant model depends not only on the formof the functional but, in addition, on the data used for itscalibration. It is widely recognized that the isoscalar sec-tor of the density functional is well constrained by avail-able ground-state observables. This is in sharp contrastto the isovector sector that remains poorly determined;for a recent example see Ref. [2] and references containedtherein. Such a mismatch occurs because physical ob-servables that are dominated by the isoscalar sector—such as binding energies and charge radii of many sta-ble nuclei—have been measured with enormous precision.Instead, data on neutron skins [3, 4] and neutron-starradii [5–8], both highly sensitive to the isovector sector,either lack precision or are still open to debate.Due to the present difficulty in obtaining accurate mea-surements of both neutron skins and neutron-star radii,it seems prudent to seek alternative isovector indicators.A fruitful arena for the search of isovector sensitivity ispure neutron matter whose equation of state is approx-imately equal to that of symmetric nuclear matter plusthe symmetry energy. The behavior of pure neutron mat-ter at low densities is particularly attractive because ofits close resemblance to a resonant Fermi gas. However,although this has stimulated significant amount of the-oretical activity [9–16], one must recognize that neutronmatter remains a purely theoretical construct. A lab-oratory observable that has been identified as a strong ∗ Electronic address: [email protected] † Electronic address: [email protected] isovector indicator is the electric dipole polarizability of
Pb [17–20]. Indeed, the recent high-resolution mea-surement of the electric dipole polarizability of
Pb atthe Research Center for Nuclear Physics [21, 22] has pro-vided a unique constraint on the density dependence ofthe symmetry energy and serves as an ideal complementto measurements of the neutron skin.Given that the symmetry energy accounts for the en-ergy cost in departing from equal number of protons andneutrons, one expects that the evolution of certain nu-clear properties as one moves away from the valley ofstability will become sensitive to the isovector nature ofthe interaction. For example, if the symmetry energy is stiff , namely, if it increases rapidly with density, it be-comes energetically favorable to move neutrons from thecore to the surface, resulting in a thick neutron skin [23].By the same token, a stiff symmetry energy may becomesmall at the dilute nuclear surface which is of particularrelevance to the valence orbitals. As a result, a stiff sym-metry energy predicts a delay in reaching the neutrondrip line relative to their softer counterparts [24].Mapping the precise boundaries of the nuclear land-scape has been identified as one of the most fundamentalproblems in nuclear science; see Refs. [25, 26] and refer-ences contained therein. Although the proton drip linehas been determined up to protactinium (atomic number Z = 91) the neutron drip line remains unknown, except inthe case of a few light nuclei ( Z (cid:46)
8) [27]. A particularlydramatic example of this mismatch is the case of the flu-orine isotopes, which have F as its only stable member.Whereas F marks the boundary of the proton drip line, F—with 12 neutrons away from stability—remains sta-ble against strong decays. While the Coulomb repulsionis largely responsible for having the proton drip line justa few neutrons away from stability, the basic tenet ofthis letter is that the dynamics of the neutron drip lineis highly sensitive to the nuclear symmetry energy.Among the few isotopic chains with both drip-lineboundaries firmly established, oxygen is perhaps the a r X i v : . [ nu c l - t h ] M a r most intriguing one, as it provides the first clear in-dication of the emergence of a new magic number at N = 16 [28]. Whereas most mean-field calculations (bothnon-relativistic and relativistic) have predicted the sta-bility of the “doubly-magic” nucleus O against strongdecays, experimental efforts have failed to find a stableisotope beyond O [28–31]. This oxygen anomaly hasbeen widely investigated within various formulations and,to date, the most common explanation invokes an extrarepulsion between valence neutrons generated from three-nucleon forces [32–37].The calcium isotopic chain—the next chain after oxy-gen with a magic number of protons—has also received agreat deal of attention due to its rich subshell structurenear N = 32 [38–41]. Particularly exciting is the recentmass determination of various exotic calcium isotopes—up to Ca— at both TRIUMF [40] and CERN [41]. Yet,despite these remarkable achievements, the experimentaldetermination of the neutron drip line in calcium is likelyyears away—especially if the drip line is at or beyond the“doubly-magic” Ca.In order to explore the sensitivity of the neutron-richisotopes to the density dependence of the symmetry en-ergy, we construct theoretical models subject to the sameisoscalar constraints but with a single differing assump-tion on the uncertain isovector sector. In the relativis-tic mean-field (RMF) theory, the nuclear system is com-posed of neutrons and protons interacting via the ex-change of various mesons and the photon. In the versionof the RMF models employed here the interaction amongthe particles is described by an effective Lagrangian den-sity [42–44] whose parameters are determined by fittingmodel predictions to experimental data. In this work weemploy the Lagrangian density given in Ref. [2] and usethe same calibration scheme developed therein to find theoptimal model parameters and their associated theoreti-cal uncertainties [45]. Such a fitting protocol relies exclu-sively on genuine physical observables that can be eithermeasured in the laboratory or extracted from observa-tion. This approach was recently implemented in build-ing the new
FSUGold2 density functional [2]. The datapool of observables is sufficient to constrain the isoscalarsector as evinced by the very small associated theoreticaluncertainties. However, because no inherent isovector bi-ases are incorporated into the fit, FSUGold2 predicts—inaccordance with most relativistic density functionals—a stiff symmetry energy and, as a consequence, a fairlythick neutron skin in
Pb of R = (0 . ± . Pb. That is, we augment the calibration procedureby assuming values of R = 0 .
12 fm, R = 0 .
16 fm,and R = 0 .
28 fm—in all three cases with an associatederror of 0 . R , namely, RMF012, RMF016, and RMF028. Giventhat the data pool of observables involves doubly-magic(or semi-magic) nuclei, pairing correlations are not in-cluded in the calibration procedure. However, once thecalibration is completed, we exploit our recently devel-oped RMF-plus-exact-pairing (RMF+EP) approach [46]to properly describe the mass evolution along both iso-topic chains.
11 12 13-80-70-60
AME2012RMF012RMF016RMF028
10 12 14 16 18 20 22 24 26 28 A -180-160-140-120-100-80-60 G r ound - s t a t e e n e r gy ( M e V ) AME2012RMF012RMF016RMF028 N N O O FIG. 1: (Color online). Evolution of the ground-state en-ergy along the isotopic chain in oxygen—from O to O—as predicted by the three RMF models described in the text.Experimental data are from Ref. [47].
We start by displaying in Fig. 1 the evolution of theground-state energy along the isotopic chain in oxy-gen. Given that the nearly isospin-symmetric isotopes − O are largely insensitive to the isovector sector, themodel predictions are almost indistinguishable from eachother and are also in good agreement with the 2012Atomic Mass Evaluation (AME2012) [47]. However, asthe neutron-proton asymmetry is increased, the modelpredictions start to differ, indicating that isovector ef-fects are starting to play an increasingly dominant role.Indeed, the models display dramatic differences as theexperimentally determined neutron drip line at O isapproached. Although drip-line nuclei are undoubtedlysensitive to subtle dynamical effects, e.g., mixing to thecontinuum, it appears that the density dependence of thesymmetry energy also plays a critical role. In particular,we find that RMF028 (with the stiffest symmetry en-ergy) overbinds the neutron-rich isotopes, leading to thecommon, yet erroneous, prediction of a drip line at O.In contrast, RMF012 and RMF016 with a softer symme-try energy produce the necessary repulsion to shift theneutron drip line to O. We must underline that suchbehavior is determined by the weakly-bound excess neu-trons that reside in the nuclear surface where the den-sity is low. Thus, it is the low-density component ofthe symmetry energy—which is larger for a soft model—that dictates the physics, rather than the symmetry en-ergy around saturation density. This suggests that mod-els with a small R should be the first ones to reachthe neutron drip line [24], precisely as seen in Fig. 1. Al-though Coulomb effects shift the proton drip line muchcloser to stability, the imprint of the symmetry energyshould also be manifest on the neutron-deficient side ofthe isotopic chain. Indeed, this appears to be the case.As highlighted in the inset of Fig. 1, both RMF012 andRMF016 predict—unlike RMF028—that O is unstableagainst proton emission, in agreement with experiment.Thus, as in the case of the neutron drip line, the twosofter models reach the proton drip line earlier than thestiffer one. Note that the true ground state of the odd-odd nucleus N is a superposition of states with the un-paired proton and neutron being in orbitals that can cou-ple to the ground-state spin of the nucleus (i.e., J π = 1 + ).However, for simplicity we approximate the ground-stateenergy of N by the lowest-energy configuration. Theinset on Fig. 1 seems to validate this approximation. (cid:108) / (cid:108) S ( (cid:108) ) [ M e V ] RMF012RMF016RMF028 (a) N e u t r on d e n s it y (f m - ) RMF012RMF016RMF028 -5 -5 -5 -5 RMF012RMF016RMF028 (b)
FIG. 2: (Color online). (a) Symmetry energy as a function ofdensity in units of ρ = 0 .
148 fm − and (b) neutron density of O as predicted by the three RMF models discussed in thetext.
To further validate this behavior, we display in Fig. 2athe symmetry energy predicted by the three models upto a density slightly above saturation density. The thick-ness of the neutron skin in
Pb is largely determinedby the slope of the symmetry energy at (or near) sat-uration density. In the case of a stiff symmetry energy,such as RMF028, it is energetically advantegeous to moveneutrons from the core (where S is large) to the surface(where S is small), albeit at the expense of an increase insurface tension. Thus, models with a stiff symmetry en-ergy tend to predict thicker neutron skins. However, at adensity of about 2/3 of saturation density, correspondingto a value of the symmetry energy of almost 26 MeV,all three models intersect each other. This well-knownresult emerges from the sensitivity of the binding energyof neutron-rich nuclei to the symmetry energy at a den-sity that is intermediate between that of the core andthe surface [23, 44, 48–52]. As a result, the symmetryenergy below this density becomes larger for the softermodels. This increase in the symmetry energy generates the added repulsion required to shift the neutron dripline from O to O. RMF012 RMF016 RMF028-50-40-30-20-100 S i ng l e - p a r ti c l e e n e r gy ( M e V ) FIG. 3: (Color online). Single-neutron spectrum for O aspredicted by the three RMF models discussed in the text.
Such unique behavior of the symmetry energy leavesa distinct imprint on the neutron density of O; seeFig. 2b. First, we note that for a stiff symmetry energy,as in the case of RMF028, more neutrons are pushed tothe surface resulting in both a depletion of the densityin the interior and a larger neutron radius. Second, ata distance of about 3 fm, corresponding to a density ofabout 0.05 fm − , the neutron density predicted by theRMF028 model now becomes the largest, as this is theregion that dominates the contribution to the neutronradius. Finally, at even larger distances where the den-sity is dominated by the weakly-bound valence neutrons,the density is again lowest for the stiffest model (see theinset in Fig. 2b). That is, the smaller symmetry energyat low density of the RMF028 model yields less repulsionfor the valence orbitals and consequently a faster falloffof the density. The single-neutron spectrum displayed inFig. 3 serves to reaffirm these trends. In particular, wenotice a “compression” of the single-particle spectrum asthe symmetry energy becomes stiffer. Indeed, whereasthe “core” sp -orbitals become less bound with increasingstiffness, the valence sd -orbitals are more strongly bound.Particularly, the neutron 1d / orbital becomes unboundfor the softer models—a critical requirement for the dripline in oxygen to be found at O.We continue by displaying in Fig. 4 ground-state ener-gies for calcium—the next isotope with a fully closed pro-ton shell. Predictions have been made for a wide rangeof neutron-proton asymmetries starting with Ca andending with the very neutron-rich Ca isotope. The cal-culation for the neutron-rich isotopes was done using theaugmented f pg / valence space. It is found that in-cluding the 1 g / orbital enhances the binding energy inthe − Ca region bringing the predictions from bothRMF012 and RMF016 into closer agreement with exper-
32 33 34 35-280-260-240-220AME2012RMF012RMF016RMF028
32 36 40 44 48 52 56 60 64 68 A -500-450-400-350-300-250-200 G r ound - s t a t e e n e r gy ( M e V ) AME2012RMF012RMF016RMF028 Ca K Ca K Ca K FIG. 4: (Color online). Evolution of the ground-state energyalong the isotopic chain in calcium—from Ca to Ca—aspredicted by the three RMF models described in the text.Experimental data are from Ref. [47]. iment. Also shown in the figure are experimental datafrom the latest AME2012 compilation [47]. Note that theAME2012 results quoted for Ca and beyond were “de-rived not from purely experimental data” [47]. Contraryto the isotopic chain in oxygen where the neutron dripline has been firmly established, the experimental datashow no evidence that the neutron drip line is withinreach. Given that all three models were calibrated usingground-state energies for both Ca and Ca, it is notsurprising that the agreement among them—and withexperiment—is very good. However, beyond Ca whereisovector effects start to play a critical role, significantdifferences emerge. In particular, and fully consistentwith the results obtained along the isotopic chain in oxy-gen, the stiff RMF028 model predicts an overbinding thatis inconsistent with experiment. Although all three mod-els agree that Ca is particle bound, a subtle odd-evenstaggering emerges thereafter. Nevertheless, upon closerexamination we found that for RMF012 and RMF016 theneutron drip line is reached at Ca. On the other hand,the very flat plateau displayed by RMF028 makes it diffi-cult to identify the exact location of the neutron drip line.This observation is also supported by the single-neutronenergies of Ca displayed in Fig. 5. Indeed, compared toits softer counterparts, the barely unbound 1 g / orbitalin RMF028 makes the identification of the drip line am-biguous. We stress that a more accurate description ofthe neutron drip line remains a serious theoretical chal-lenge. For example, whereas Holt and collaborators [38]predict—like we do—that the neutron drip line will bereached at or beyond Ca, Hagen et al. find Ca to beparticle unbound relative to Ca [39]. However, whilethe results of Holt et al. depend critically on the roleof three-nucleon forces, Ekstr¨om and collaborators have recently found that a properly optimized chiral nucleon-nucleon interaction can describe many aspects of nu-clear structure without explicitly invoking three-nucleonforces [53].Finally, we turn to the neutron-deficient side of thecalcium isotopes. As shown in Fig. 4 and highlightedin the inset, models with a soft symmetry energy reachboth drip lines earlier than their stiffer counterparts. In-deed, whereas the proton drip line in both RMF012 andRMF016 can be placed at Ca in agreement with exper-iment, RMF028 predicts its location at or beyond Ca.(Again, for the odd-odd nuclei, K and K, we used thesame approximation as for N.)
RMF012 RMF016 RMF028-60-50-40-30-20-100 S i ng l e - p a r ti c l e e n e r gy ( M e V ) FIG. 5: (Color online). Single-neutron spectrum for Ca aspredicted by the three RMF models discussed in the text.Note that the 2 s / and 1 d / orbitals are predicted to benearly degenerate in all three models. The results obtained so far suggest that by adopt-ing certain reasonable assumptions one can reproducethe observed experimental trends in the isotopic chainsof both oxygen and calcium. In our particular case,such a “plausible assumption” implies the adoption ofan ad-hoc value of R . This finding is significant as ithas been previously shown that calibrating RMF func-tionals by relying exclusively on well-measured physi-cal observables invariably results in the prediction offairly large neutron skins [2, 45, 54]. Instead, one im-mediate consequence of the present analysis is that theneutron-skin thickness of neutron-rich nuclei can not beoverly large. Indeed, the model that can best reproduceground-state energies along the oxygen and calcium iso-topic chains is RMF016—a model that henceforth willbe referred to as “FSUGarnet” . This model predicts R = (0 . ± . R and neutron-star radii leads to thefollowing prediction for the radius of a 1.4 M (cid:12) neutronstar: R . = (13 . ± .
1) km. Note that although thesymmetry energy is relatively soft, the overall equation ofstate is stiff enough to support a two-solar-mass neutronstar [55, 56]. Indeed, the maximum neutron-star masssupported by FSUGarnet is M max = (2 . ± . M (cid:12) .Finally, given that no property of infinite nuclear mat-ter was incorporated into the fit, the symmetry energy J = (30 . ± .
47) MeV and its slope L = (51 . ± .
5) MeVat saturation density represent legitimate model predic-tions.Although all our results were obtained from the cal-ibration of a relativistic density functional constrainedexclusively from experimental and observational data—plus a critical assumption on R —it is instructiveto compare them against the predictions from variousother analyses. In the particular case of the neutron-skin thickness of Pb, it falls safely within the R =(0 . − .
23) fm range suggested by a myriad of differ-ent analyses [21, 57–64]. In regard to the symmetry en-ergy and its slope at saturation density, many of thesesame publications are consistent with the predictionsfrom FSUGarnet. This is not overly surprising given thatthe value of the symmetry energy J is largely constrainedby nuclear masses and its slope L by the value of R .In summary, we have explored sensitivity to isovectoreffects in the neutron-rich oxygen and calcium isotopesby calibrating RMF models with the same isoscalar con-straints but with a single differing assumption on the isovector sector: the neutron-skin thickness of Pb.We found that in these neutron-rich isotopes isovectoreffects play a critical role in reproducing the correct ex-perimental trends along both isotopic chains. In par-ticular, FSUGarnet—a newly calibrated relativistic den-sity functional—displays a soft symmetry energy thatcan provide the extra repulsion required to terminate theoxygen chain at O and predicts the neutron drip linein calcium to be reached at Ca. The same isovectortrends were also found on the neutron-deficient side ofboth isotopic chains. Indeed, FSUGarnet predicts theproton drip line in oxygen and calcium to be reached at O and Ca, in agreement with experiment. Althoughwe have established the critical role that the symmetryenergy plays in the delineation of the drip lines, we rec-ognize that our results may be model dependent (seeRef. [65]). Yet, we are confident that our findings areof sufficient interest to motivate alternative studies withother classes of density functionals.This material is based upon work supported by theU.S. Department of Energy Office of Science, Officeof Nuclear Physics under Award Number DE-FD05-92ER40750. [1] W. Kohn, Rev. Mod. 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