DDo we understand the incompressibility of neutron-rich matter?
J. Piekarewicz ∗ Department of Physics, Florida State University, Tallahassee, FL (Dated: November 1, 2018)
Abstract
The “breathing mode” of neutron-rich nuclei is our window into the incompressibility of neutron-rich matter. After much confusion on the interpretation of the experimental data, consistency wasfinally reached between different models that predicted both the distribution of isoscalar monopolestrength in finite nuclei and the compression modulus of infinite matter. However, a very recentexperiment on the Tin isotopes at the
Research Center for Nuclear Physics (RCNP) in Japanhas again muddled the waters. Self-consistent models that were successful in reproducing theenergy of the giant monopole resonance (GMR) in nuclei with various nucleon asymmetries (suchas Zr,
Sm, and
Pb) overestimate the GMR energies in the Tin isotopes. As important,the discrepancy between theory and experiment appears to grow with neutron excess. This isparticularly problematic as models artificially tuned to reproduce the rapid softening of the GMRin the Tin isotopes become inconsistent with the behavior of dilute neutron matter. Thus, weregard the question of “why is Tin so soft?” as an important open problem in nuclear structure . PACS numbers: 21.60.-n,21.60.Ev,21.60.Jz,21.65.Cd,21.65.Ef ∗ e-mail: [email protected] a r X i v : . [ nu c l - t h ] D ec . INCOMPRESSIBILITY OF SYMMETRIC NUCLEAR MATTER The incompressibility coefficient of infinite nuclear matter—also known as the compres-sion modulus—is a fundamental parameter of the equation of state (EOS). The compressionmodulus controls small density fluctuations around the equilibrium point, thereby providingthe first glimpse into the “stiffness” of the EOS. Whereas existing ground-state observables( e.g., masses and charge radii) have accurately constrained the saturation point of symmet-ric nuclear matter (at a baryon density of ρ (cid:39) .
15 fm − and a binding energy per nucleonof ε (cid:39) −
16 MeV), the extraction of the compression modulus ( K ) is significantly morecomplicated as it requires to probe the response of the nuclear ground state. It is widelyaccepted that the nuclear compressional modes—particularly the quintessential “breathingmode” or isoscalar Giant Monopole Resonance (GMR)—provide the cleanest, most directroute to the nuclear incompressibility [1, 2].Earlier attempts at extracting the compression modulus of symmetric nuclear matterrelied primarily on the distribution of isoscalar monopole strength in
Pb—a heavy, doubly-magic nucleus with a well developed monopole peak [3, 4]. Although such measurementshave existed for some time, the field has enjoyed a renaissance due to new and improvedexperimental facilities and techniques. Indeed, an improved α -scattering experiment foundthe position of the giant monopole resonance in Pb at E GMR = 14 . ± .
28 MeV [5]. Asthis measurement was combined with the distribution of monopole strength in other nuclei—and compared with microscopic calculations—a value of the incompressibility coefficient inthe range of K =220-240 MeV was extracted.As the experimental program reached a high level of maturity and sophistication, the samestrict standards were demanded from the theoretical program. Indeed, calculations of nuclearcompressional modes based on consistent Mean-Field plus Random Phase Approximation (RPA) approaches became routine. Moreover, such consistent models—without any recourseto semi-empirical mass formulas—were able to simultaneously predict the incompressibilityof infinite nuclear matter as well as the distribution of isoscalar monopole strength in finitenuclei [6, 7]. However, as the experimental story was coming to an end, the theoreticalpicture remained unclear. On the one hand, nonrelativistic calculations that reproduced thedistribution of isoscalar-monopole strength in
Pb predicted a nuclear incompressibilitycoefficient in the K =210-230 MeV range [2, 8, 9]. On the other hand, relativistic modelsthat succeeded in reproducing a large body of nuclear observables—including the GMR in Pb—suggested the significant larger value of K (cid:39)
270 MeV [10, 11].The solution to this puzzle was originally proposed by Piekarewicz [12, 13] and hassince been confirmed by several other groups [14, 15, 16]. The solution is based on therealization that the GMR in
Pb does not constrain the compression modulus of symmetric nuclear matter but rather the one of neutron-rich matter. In particular, it was concludedthat the compression modulus of a neutron rich system having the same neutron excessas
Pb is lower than the compression modulus of symmetric nuclear matter. This couldexplain how models with significantly different incompressibility coefficients K may stillreproduce the GMR in Pb [12, 13]. As a result, accurately-calibrated theoretical modelswere built to reproduce simultaneously the distribution of isoscalar-monopole strength inboth Zr and
Pb—nuclei with a well developed monopole peak yet significantly differentnucleon asymmetries [17, 18]. Since then, the large difference in the predicted value of K between nonrelativistic and relativistic models has been reconciled and a “consensus” hasbeen reached that places the value of the incompressibility coefficient of symmetric nuclear2atter at K = 240 ±
10 MeV [15, 16, 17, 19, 20].
II. INCOMPRESSIBILITY OF NEUTRON-RICH MATTER
To summarize some of the above findings it is convenient to discuss in detail the incom-pressibility coefficient of infinite, neutron-rich matter. On very general grounds—indeed, ina model-independent way—the incompressibility coefficient of neutron-rich matter may bewritten as K ( α ) = K + K τ α + O ( α ) , (1)where α ≡ ( N − Z ) /A is the nucleon asymmetry. From this expression it is immediatelyevident that the GMR in Pb (with a neutron excess of α = 0 .
21) should be sensitive toa linear combination of the incompressibility coefficient of symmetric nuclear matter K and K τ —a quantity that determines the evolution of the incompressibility coefficient withneutron excess. Note that K τ plays the same role in determining the incompressibilitycoefficient as the symmetry energy at saturation density (a quantity often denoted by J or a ) plays in determining the energy-per-nucleon of asymmetric matter.To compute the incompressibility coefficient of neutron-rich matter one proceeds exactlyas in the case of symmetric nuclear matter. First, one determines the equilibrium point ata fixed value of α and then extracts K ( α ) from computing the curvature at the minimum.Having done so for various values of α , one can extract K τ from a simple fit to Eq. (1) [21].An alternative procedure that is highly accurate and significantly more illuminating startsfrom the equation of state of neutron-rich matter parametrized in terms of several bulkparameters defined at normal saturation density. Starting from such a parametrization, itbecomes a simple exercise to compute the equilibrium point (as a function of α ) and thecorresponding curvature at the minimum. In particular, one obtains the following closed-form expression for K τ [21]: K τ = K sym − L − Q K L . (2)where L and K sym represent the slope and curvature of the symmetry energy at saturationdensity [21]. Although often neglected, note that K τ also depends on the third derivativeof the EOS of symmetric nuclear matter Q (a quantity often referred to as the “skew-ness” parameter). Quite generally, as the infinite nuclear system becomes neutron rich, thesaturation density moves to lower densities, the binding energy weakens, and the nuclearincompressibility softens [21]. It is important to stress the dominant role of the symmetrypressure L on these conclusions and in particular on Eq. (2) (because of the large coefficientin front of it). We note that the symmetry pressure L —a quantity that strongly influencesthe neutron-skin thickness of heavy nuclei—is directly proportional to the pressure of pureneutron matter , namely, P nm = 13 ρ L . (3)This connection is important as significant theoretical progress has been made in constrainingthe equation of state of low-density neutron matter. We will draw heavily on this connectionin what follows. 3
II. MEASURING THE BREATHING MODE OF THE TIN ISOTOPES
The realization that the distribution of monopole strength in heavy nuclei is sensitive tothe density dependence of the symmetry energy motivated a recent experimental study ofthe GMR along the isotopic chain in Tin [20, 22]. This important experiment was carried outat the
Research Center for Nuclear Physics (RCNP) in Osaka, Japan. Such an experimentprobed the incompressibility of asymmetric nuclear matter by measuring the distributionof isoscalar strength in a chain of isotopes ranging from
Sn (with α = 0 .
11) to
Sn(with α = 0 . K τ to the pressure of pure neutron matter [seeEqs. (2) and (3)], this experiment represents an attractive hadronic complement to the purelyelectroweak Parity Radius Experiment (PREx) at the Jefferson Laboratory that aims tomeasure the neutron radius of
Pb accurately and model independently via parity-violatingelectron scattering [23, 24]. We note that such a fundamental measurement will have far-reaching implications in areas as diverse as nuclear structure [25], heavy-ion collisions [26,27, 28, 29, 30], atomic parity violation [25, 31, 32] and nuclear astrophysics [33, 34, 35, 36,37, 38, 39, 40].Shortly after the completion of the RCNP experiment a serious discrepancy was revealed:accurately calibrated models that reproduce the GMR in Zr,
Sm, and
Pb overestimatethe distribution of isoscalar monopole strength in the Tin isotopes [41, 42, 43]. Moreover,the discrepancy between theory and experiment appears to grow with neutron excess α , sug-gesting that the models significantly underestimate the value of | K τ | . We have colloquiallyreferred to this problem as “why is Tin so soft?” . To illustrate this situation we display inFig. 1 a comparison between the experimental centroid energies [20, 22] for the neutron-even Sn-
Sn isotopes and three theoretical calculations that have been extended up to thedoubly magic nucleus
Sn. All theoretical predictions were generated using a consistentRPA approach. That is, the linear response of the system was computed using the same interaction employed to generate the mean-field ground state. A detailed description of thisapproach may be found in Refs. [44, 45].The results depicted with the blue triangles were generated using the accurately calibratedFSUGold parametrization [17]. This relativistic model is characterized by a soft behaviorfor both symmetric nuclear matter and the symmetry energy. (Note that the terms “soft” and “stiff” refer to whether the energy increases slowly or rapidly with density.) Such asoft behavior is reflected in the relatively small values of K , L , and | K τ | listed in Table I.Clearly, the model overestimates the experimental data (black squares). Moreover, thediscrepancy increases with neutron number: from about 0 . Sn to 0 . Sn. Such a serious discrepancy is particularly troublesome given that the same FSUGoldmodel successfully reproduces the centroid energy of the GMR in Zr,
Sm, and
Pb, asshown in the inset of Fig. 2. Figure 2 also displays the distribution of isoscalar monopolestrength from which the centroid energies depicted in the inset were computed (as the ratioof the first to the zeroth moment). In addition to the four nuclei— Zr,
Sn,
Sm,and
Pb—of Ref. [5], we display the distribution of monopole strength for the doubly-magic nuclei
Sn and
Sn. We note that the GMR predictions for all six nuclei fallnicely within the “liquid-drop” inspired curve E GMR (cid:39) A − / [46, 47]. Moreover, thesepredictions reproduce the experimentally extracted GMR energies [5]—except for the caseof Sn. Figures 1 and 2 capture the essence of the problem of why is Tin so soft?
Also shown in Figure 1 are the predictions from the NL3 parameter set [10, 48]. TheNL3 set has been remarkably successful in reproducing a myriad of nuclear ground-state4
12 116 120 124 128 132 A E G M R ( M e V ) FSUNL3HybridRCNP
Sn-Isotopes ! =10 MeV ! =20 MeV FIG. 1: (Color online) Comparison between the GMR centroid energies ( m /m ) of all neutron-even Sn-
Sn isotopes extracted from experiment [20, 22] (black solid squares) and the theoreticalpredictions from the FSUGold (blue up-triangles), NL3 (green down-triangles), and Hybrid (reddiamonds) models. The corresponding dashed lines were obtained from a fit to the centroid energiesof the form E GMR = E A − λ [see Eq. (4)]. properties (such as masses, charge-radii, and deformations) throughout the periodic table.Although it now seems likely that the stiff behavior predicted by NL3 may be unrealistic,at the time of its inception most of the information in favor of a softer equation of statewas unavailable. Thus, although NL3 may reproduce the GMR in Pb, the data on theTin isotopes provides ample evidence that such a stiff behavior is inconsistent with data.Note, however, that although NL3 significantly overestimates the GMR energies in the Tinisotopes, the softening of the incompressibility coefficient (namely, its dependence with A )appears consistent with data. That is, Figure 1 suggests that whereas NL3 has too large avalue of K , its value for K τ may be consistent with experiment (see Eq. (1) and Table I).Motivated by the above facts, we have built a “Hybrid ” model with a low incompress-ibility coefficient and a stiff symmetry energy [21]. However, unlike the NL3 and FSUGoldparametrizations, the Hybrid model was not accurately calibrated. Thus, the Hybrid modelshould be simply regarded as a “test” model that illustrates how surprisingly soft are theexperimental GMR energies of the Tin isotopes relative to the theoretical predictions. Weobserve in Fig. 1 that the Hybrid model yields a significant improvement in the descriptionof the experimental data. Indeed, the predictions of the Hybrid model fall within 0 . odel ρ ε K Q J L ( P nm ) K sym K τ FSU 0.148 − − − − − − − − − ρ . The quantities ε , K , and Q represent the binding energy per nucleon, incompress-ibility coefficient, and third derivative (or “skewness” coefficient) of symmetric nuclear matter at ρ . Similarly, J , L , and K sym represent the energy, slope, and curvature of the symmetry energyat saturation density. All quantities are in MeV, except for ρ which is given in fm − and the pres-sure of pure neutron matter at saturation density ( P nm ) which is given in MeV/fm . A detailedexplanation of all these quantities may be found in Ref. [21]. ! (MeV) S L ( q , ! ) ( M e V - ) Zr Sn Sn Sn Sm Pb
90 116 144 208 A E G M R ( M e V ) ExperimentFSUGoldFit q=0.23 fm -1 E fit =69A -0.3 MeV
FIG. 2: (color online) Distribution of isoscalar monopole strength predicted by the FSUGold modelof Ref. [17]. The inset includes a comparison against the experimental centroid energies reportedin Ref. [5], with the solid line providing the best fit to the theoretical predictions. of the experimental data for the full isotopic chain. Note that relative to FSUGold, theimproved description provided by the Hybrid model is entirely due to its large negativeasymmetric term K τ , as they shared the same value of K (see Eq. (1) and Table I). Indeed,we can capture the A dependence of the E GMR predicted by all the models with a liquid-drop6nspired formula of the form E GMR (cid:39) E A − λ . We obtain, E GMR = . A − . for FSU,102 . A − . for NL3,112 . A − . for Hybrid. (4)The Hybrid model suggests a falloff with A that is significantly faster than the λ = − / underestimates the GMR centroid energy in Pb by almost 1 MeV [21, 43].This suggests that the rapid softening with neutron excess predicted by the Hybrid modelmay be unrealistic.
IV. WHY IS TIN SO SOFT?
So why is Tin so soft and why does it become even softer as the nucleon asymmetryincreases? Are we any closer to the answer now than we were then [20, 22] ? Unfortunatelynot! As we elaborate below, we will assume that the experimental extraction of the GMRenergies is without error—although the large discrepancy between the RCNP [20, 22] andthe Texas A&M [49] results should be resolved.To date, only two plausible scenarios have been advanced to explain why accurately-calibrated models that reproduce the GMR energies in Zr,
Sm, and
Pb fail to do sofor the Tin isotopes. One of them is encapsulated in the Hybrid model discussed above [21]whereas the other one suggests that pairing correlations are responsible for the softeningof the monopole response [50, 51]. As discussed earlier, the Hybrid model is based on aneffective interaction that generates a soft EOS for symmetric nuclear matter (a small K )but a stiff symmetry energy (a large | K τ | ). Any such model should generate soft monopoleexcitations for symmetric ( N = Z ) nuclei and significantly softer ones for the neutron-richisotopes (see Fig. 1). Unfortunately, the Hybrid model—and others like it [43]—can onlyreproduce the GMR energies in Tin at the expense of significantly underestimating the GMRenergy in Pb. In the case of pairing correlations, the explanation is based on the conjecturethat a superfluid—such as the open-shell Tin isotopes—may be easier to compress than anormal fluid [51]. Whereas the validity of this statement is both interesting and presentlyunknown, recent Quasiparticle RPA (QRPA) calculations seem to support the conjecture—at least in part [50, 51]. “At least in part” because although pairing correlations yield anappreciable softening for the lighter isotopes (from
Sn to
Sn), the discrepancy for theheavier ones (
Sn and
Sn) remains large [50, 51]. This indicates that pairing correlationscan not account for the observed softening of the GMR energies with nucleon asymmetry. Tomake matters worse, we now argue that the rapid softening displayed by the experimentalGMR energies in the Tin isotopes may be even harder to explain as one incorporates thephysics of dilute neutron matter.
V. LOW-DENSITY NEUTRON MATTER
By building on the universal behavior of dilute Fermi gases with an infinite scatteringlength [57, 58, 59, 60], significant progress has been made in constraining the equation7 .2 0.4 0.6 0.8 1.0 1.2 k F (fm -1 ) E / N ( M e V ) Schwenk-PethickFriedman-PandharipandeHF-V low-k (S-P)Hebeler-SchwenkGandolfi et al.
Gezerlis-CarlsonRMF-FSURMF-NL3RMF-Hybrid
Pure Neutron Matter F e r m i G a s FIG. 3: (color online) Equation of state of pure neutron matter as predicted from a variety ofmicroscopic models [52, 53, 54, 55, 56]. Also shown are predictions from the relativistic mean-fieldmodels FSUGold [17], NL3 [10, 48], and Hybrid [21]. of state of pure neutron matter. One of the biggest challenges in understanding diluteneutron matter arises from the non-negligible effective range of the neutron-neutron ( nn )interaction ( r e = +2 . | a | = 18 . k F (cid:39) /r e (cid:46) . − ). To date, a host of models using different nn interactions anda variety of many-body techniques have been employed to compute the EOS of dilute neutronmatter (see Fig. 3). These models range from the venerated equation of state of Friedmanand Pandharipande [52], to those based on modern effective field theory approaches [53, 54],to those using sophisticated “ab-initio” Monte Carlo techniques [55, 56], to name just a few(for a more comprehensive list see Ref. [56]). Remarkably, all these microscopic models arefairly consistent with each other.Also shown in Fig. 3 are the predictions from three relativistic mean-field models whoseparameters have been fitted directly to various properties of finite nuclei. By directly fittingto the experimental data, the parameters of these models encode physics (such as few- andmany-body correlations) that goes beyond a simple single-particle picture. In this regard,the underlying parameters of the model may have, at best, a tenuous connection to thoseappearing in microscopic descriptions of the nucleon-nucleon (
N N ) interaction. As a result,if the mean-field models are not sufficiently constrained by experimental data, they canpredict behavior that is inconsistent with microscopic approaches—and with nature. Thisis clearly the case for the NL3 and Hybrid models displayed in Fig. 3, and for most of8he older relativistic parametrizations. (Note that the energy of pure neutron matter is to avery good approximation equal to the energy of symmetric nuclear matter plus the symmetryenergy). Given that existing ground-state observables do not place stringent constraints onthe isovector
N N interaction, most relativistic mean-field models predict a stiff symmetryenergy, namely, one that increased rapidly with density for ρ (cid:38) . − (see the large valuesof L and K sym listed in Table I). In contrast, the FSUGold parametrization incorporatescollective modes directly into the fit, including GMR energies for both Zr (with α = 0 . Pb (with α = 0 . a-priori guarantee, it is gratifying to observe that the softening of thesymmetry energy displayed by the FSUGold model is consistent with the EOS predicted bythe various microscopic approaches (see Fig. 3). We suggest that the equation of state ofpure neutron matter provides a powerful constraint that should be routinely and explicitlyincorporated into future determinations of energy density functionals.So what is the connection between the largely model-independent equation of state ofdilute neutron matter and the incompressibility coefficient of neutron-rich matter? To ap-preciate this connection we first combine Eqs. (1) and (2) to write the incompressibilitycoefficient of asymmetric matter as K ( α ) = K + (cid:18) K sym − L − Q K L (cid:19) α + O ( α ) . (5)Then, as the energy of pure neutron matter is to an excellent approximation equal to the sumof the energy of symmetric matter plus the symmetry energy, the EOS of pure neutron matteraround saturation density may be expressed in terms of a conveniently defined dimensionlessparameter x = ( ρ − ρ ) / ρ and the same bulk parameters appearing in Eq. (5). That is, E nm /N = ( ε + J ) + Lx + 12 ( K + K sym ) x + 16 ( Q + Q sym ) x + . . . (6)Thus, the evolution of the incompressibility coefficient with nucleon asymmetry may beparametrized in terms of two bulk parameters of the EOS of symmetric nuclear matter ( K and Q ) and the slope and curvature of either the symmetry energy ( L and K sym ) or of pureneutron matter ( L nm ≡ L and K nm = K + K sym ). That is, K τ = K sym − L − Q K L = K nm − L nm − K + Q L nm K . (7)This establishes a strong correlation between K τ and the density dependence of the EOSof pure neutron matter (through L nm and K nm ). In most accurately-calibrated models thedominant contribution to K τ comes from the slope of the symmetry energy L = L nm [61, 62].Indeed, in the three relativistic mean-field models considered here the dominant term (6 L )accounts for at least 75% of the value of K τ . Given this fact, we believe that values as largeas | K τ | (cid:39)
550 MeV—as seem to be suggested by experiment [20, 22]—are inconsistent withthe behavior of dilute neutron matter.
VI. CONCLUSIONS
The present contribution centered around the recently measured distribution of isoscalarmonopole strength in the Tin isotopes [20, 22]. This critical experiment suggests a significant9oftening of the GMR energies that is unexplained by existing theoretical models. Beforethe publication of the experimental data in 2007 [20, 22], there was strong evidence insupport of a value of the incompressibility coefficient of symmetric nuclear matter around K = 240 ±
10 MeV. However, the measurement on the Tin isotopes has forced us to pauseand re-examine our models. Particularly confusing is the fact that some of these accurately-calibrated models are successful in reproducing the GMR energies in Zr,
Sm, and
Pb.
So why is Tin so soft and why does it become even softer with an increase in the neutronexcess?
One possible explanation relies on the open-shell structure of the Tin isotopes andits assumed superfluid character [50, 51]. Although this approach has met with some success,the impact of pairing correlation on the heavy Tin isotopes (
Sn to
Sn) is modest sothe discrepancy remains. Another approach—encapsulated in the Hybrid model discussedabove and introduced in Ref. [21]—adopts a small value for the incompressibility coefficientof symmetric matter and a large value for the slope of the symmetry energy. This “soft-stiff” combination is fairly successful in describing the rapid softening of the GMR energies in theTin isotopes. However, the same model significantly underestimates the GMR energy in
Pb. Moreover, the EOS of neutron matter generated by the Hybrid model—and essentialto reproduce the rapid softening of the GMR in the Tin isotopes—is inconsistent withmicroscopic models that based their predictions in the universality of dilute Fermi gases.In conclusion, the distribution of isoscalar monopole strength in the Tin isotopes remainsan important open problem in nuclear structure. As one attempts to solve this difficultproblem, one must remember that the challenge is not solely to describe the distribution ofmonopole strength in the Tin isotopes, but rather, to do so while simultaneously describinga host of ground-state observables, collective modes, and the equation of state of low-densityneutron matter.
Acknowledgments
This work was supported in part by a grant from the U.S. Department of Energy DE-FD05-92ER40750. 10
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