# Constraining the nuclear equation of state at subsaturation densities

###### Abstract

Only one third of the nucleons in Pb occupy the saturation density area. Consequently nuclear observables related to average properties of nuclei, such as masses or radii, constrain the equation of state (EOS) not at saturation density but rather around the so-called crossing density, localised close to the mean value of the density of nuclei: 0.11 fm. This provides an explanation for the empirical fact that several EOS quantities calculated with various functionals cross at a density significantly lower than the saturation one. The third derivative M of the energy at the crossing density is constrained by the giant monopole resonance (GMR) measurements in an isotopic chain rather than the incompressibility at saturation density. The GMR measurements provide M=1110 70 MeV (6% uncertainty), whose extrapolation gives K=230 40 MeV (17% uncertainty).

###### pacs:

21.10.Re, 21.65.-f, 21.60.JzConstraining the nuclear equation of state (EOS) and reducing the uncertainties on nuclear matter properties in dense stellar objects such as neutron stars and supernovae is one of the major goals of nuclear physics. To do so observables such as masses, radii or energy centroid of the isoscalar Giant Monopole Resonance (GMR) are measured in finite nuclei. Relating the EOS to such observables is usually undertaken in several ways. The liquid drop expansion wei35 is one of the most used methods. In its generalised version, one performs a development of a quantity (the mass for instance) considered at saturation density (volume term), adding several terms such as the surface one. It should be noted that taking the volume term at saturation density is based on the fact that the inner part of the nuclei density is close to the saturation. Another method is the so-called Local Density Approximation (LDA) rin80 . In this approach the global properties in nuclei are obtained from considering nuclei as local pieces of nuclear matter. Finally, another method is based on the microscopic approach, relying on energy density functionals (EDF): using various EDF’s, a correlation diagram is drawn between the predicted observable and a related property of the EOS. The measurement of the observable allows to validate an EDF and the corresponding property of the EOS bro00 ; typ01 ; cen09 ; pie11 . For instance, the neutrons skin measurement is correlated with the slope of the symmetry energy. It should be noted that usually EDF are designed using masses, radii but also the nuclear incompressibility ben03 ; lun03 .

Each of these methods comprise however limitations. The liquid drop expansion is known not to be accurate enough in the case of the incompressibility at the saturation density bla80 ; pea91 , although the inclusion of higher other terms is relevant pat02 . The liquid drop expansion of the mass has been successful, providing an accurate determination of the saturation energy. This quantity is now used in EDF determination, but this is an exceptional case, related to the profusion of data available on masses. In the case of the LDA, its validity is generally questionable for finite size systems. Finally, in the case of the microscopic approach, the EDF employed has usually been adjusted to data on magic nuclei, whereas most applications are aimed to be used for deformed open-shell nuclei. However there may be a general consensus that the microscopic approach should be used in fine, because of its accuracy and reliability.

It seems for now difficult to straightforwardly determine the nuclear incompressibility even with the microscopic method. The earliest microscopic analysis came to a value of K=210 MeV bla80 , but with the advent of microscopic relativistic approaches, a value of K=260 MeV was obtained vre03 . It has been shown that this value is not determined accurately and that the density dependence of the EDF as well as pairing effects (and therefore the shell structure) have an impact on the determination of K jli08 ; kha09 ; col04 . Using K in the design of EDF may not be a sound approach since it cannot be safely determined by the microscopic method. Therefore the method of determination of the nuclear incompressibility has to be rethought and more generally, it is necessary to clarify the link between nuclear matter EOS determination and nuclear observables.

First it should be noted that the liquid drop expansion is not a perturbative one since the surface properties of the nucleus are almost as important as the bulk one. Hence it may be a misleading idea to consider the nucleus as mainly composed of nucleons at saturation density with a few at the surface. Let us consider the case of Pb which is usually considered as a benchmark for extracting bulk properties. Fig. 1 shows the total density calculated in the Skyrme Hartree-Fock (HF) approach. The lower part shows the usual representation of the density whereas the upper part displays an equivalent representation with an X axis scaled as r. This allows to take into account the increase of nucleons per volume unit; the total number of nucleons corresponds to the integral of constant steps of the radial density represented on the upper part of Fig. 1. It is now perceptible that about one third of the nucleons of the Pb nucleus lie in the saturation density area, whereas two thirds are localised in the surface at a density lower than the saturation one. Therefore even in heavy nuclei, the contribution of the surface is larger with respect to the volume one, raising the question of the legitimacy of constraining EOS quantities at saturation density by measurements of nuclear observables.

Another illustration is given by calculating the mean density of the Pb nucleus. Using a Skyrme-HF calculation, one obtains = 0.12 fm in Pb with a variance =0.04 fm. Therefore the density value characterising a nucleus is not the saturation one (=0.16 fm) but a significantly lower one, with a range spanning from /2 to . It should be noted that the dependence of the mean density on the nucleus is rather weak: 0.11 fm for Sn. In light nuclei, the mean density drops down as expected: 0.07 fm for Ca.

Consequently the measurement of an averaged observable in a nucleus is more properly related to a correlated EOS quantity defined around the mean density than at the saturation density. This fact is illustrated on Fig. 2, where the density-dependent incompressibility, defined by fet71 ; kha10

(1) |

and obtained from various EDF’s is plotted with several Skyrme, Gogny and relativistic interactions, all designed to describe observables in nuclei such as masses and radii. They intersect around the crossing density 0.11 fm. The existence of a crossing density has been empirically noticed in previous works on the symmetry energy (Fig. 2 of Ref. pie11 ), pairing gap (Fig. 2 of Ref. kha09b ) or the neutron EOS (Fig. 2 of Ref. bro00 and Fig. 1 of Ref. typ01 ), and we provide here an explanation, related to the mean density: when designing EDF with nuclear observables, the corresponding EOS is constrained not at the saturation density but rather around the mean density (the crossing density). In the case of the symmetry energy, the value at a density 0.11 fm is taken to be around 25 MeV, a value close to the symmetry energy coefficient of the liquid drop expansion hor01 ; cen09 ; pie11 , as an empirical prescription. This last value contains both a volume and surface terms, and thus represents the symmetry energy extracted from nuclei observables. For the incompressibility, the GMR is known to be related to the mean square radius of the nucleus by the energy weighted sumrule rin80 . In the design of EDF’s, the considered constraint on nuclear radii induces a constraint on the compression mode, likely explaining the crossing around 0.7 observed on Fig. 2. This shows the universality of the crossing effect, arising from the constraints encoded in the EDF from nuclei observables. Due to this crossing area, a larger K=K() value for a given EDF can be compensated by lower values of K() at sub-crossing densities, so to predict a similar energy of the GMR in nuclei: the GMR centroid is related to the integral of K() over a large density range kha10 . This allows to understand how EDF with different K can predict a similar energy of the GMR, as noted in col04 .

Various EDF’s shall exhibit various density dependencies around the crossing point. At first order the derivative of the pairing gap (or incompressibility or energy) at this point will differ between various EDF’s. Complementary measurements in nuclei are needed to characterise these derivatives. For instance in Ref. bro00 ; typ01 the derivative of the neutron EOS around 0.11 fm is found to be constrained by neutron skin measurements in Pb. The associated interpretation was that the 0.11 fmdensity was considered as the neutron saturation density in nuclei. We provide an alternative and general view: 0.11 fm corresponds to the crossing density, as seen on Fig. 2.

We apply this method to the determination of the nuclear incompressibility. When the Giant Monopole Resonance is measured and well reproduced by a given EDF, it shall therefore not be correlated with the incompressibility of EOS at saturation density but rather with its first derivative M around the crossing density. It should be recalled that the crossing density exists because the EDF are determined by including nuclear observables such as masses and radii in their fit. On Fig. 2 the crossing point at 0.7 yields K() 40 MeV for the incompressibility (this is analogous to the fixed symmetry energy taken to be 25 MeV, as discussed above). To be consistent with a generalised liquid drop expansion duc11 , the derivative M of the incompressibility at this point is defined by:

(2) |

where K() is the derivative of the incompressibility density dependent term defined in Eq. (1).

Most of the GMR measurements have been performed on Pb and data on other nuclei like the Sn isotopic chain is relevant tli07 . In Fig. 3 the GMR prediction using the constrained Hartree-Fock (CHF) method as a function of M is shown for various Skyrme EDF’s in Pb and Sn. The CHF method is a sum rule approach to calculate the centroid energy of the isoscalar GMR:

(3) |

The moment is evaluated by the double commutator using the Thouless theorem rin80 and for the moment, the CHF approach is used boh79 ; cap09 ; kha09 : the CHF Hamiltonian is built by adding the constraint associated with the IS monopole operator. The CHF method has the advantage to very precisely predict the centroid of the GMR using the sumrule. To be comprehensive, Fig. 3 also displays values for several relativistic and the Gogny D1S EDF’s in the case of Pb.

The value of M deduced from both nuclei is compatible within few percents. In order to derive a sound value of M, we have further performed similar calculations in the Sn isotopes, as well as in Zr and Sm nuclei, leading to M1100 MeV 70 MeV. The contributions to the uncertainty shall come from the small variance between the mean density in the respective nuclei compared to the crossing density, the isospin dependence of the incompressibility and the pairing effects. The linear correlation observed on Fig. 3 is striking, since the interactions employed can have a symmetry energy spanning from 30 to 37 MeV. This shows that the present results moderately depend on the isospin asymmetry of the system studied. The neutron vs. proton asymmetry parameter =(N-Z)/A also remains rather constant in the considered nuclei for the GMR such as Pb and stable Tin isotopes: , validating the present isospin independent approach.

It should be noted that the present method clarifies the issue of determining different incompressibility values at saturation when using either the Pb or the Sn data pie07 ; kha09 : a close M value is found with these two data sets. However the remaining discrepancy of the M values deduced from the Sn and the Pb measurements shows that the proper density dependence of the EDF’s has not been revealed yet. Another striking feature is that the Gogny and relativistic EDF’s are found on the same linear correlation than the Skyrme one, showing the universality of the E vs. M correlation, contrary to the E vs. K one: it is well-known that relativistic EDF’s can predict a similar E in Pb but with a significantly larger value of K vre03 .

Let us now investigate why a clear correlation exists between the centroid of the GMR and K in the specific case of the Skyrme functionals. col04 . Fig. 4 displays the relative contribution of the kinetic, central (t), finite range (t,t) and density (t) terms of the Skyrme functionals to the EOS and its derivative values at the crossing density. A striking regularity is observed among the functionals used. The dominant terms are the central and the density ones, but the central term vanishes from the second derivative of the EOS and beyond, allowing the density term (in ) to dominate alone. Therefore the second derivative terms and beyond are correlated to each other, implying that the correlation between M and E is propagated to the one between K and E: a linear correlation is preserved between M and K due to the vanishing of the central term from the second derivative and beyond of the equation of state. In other words the density dependence of the incompressibility, driven by these derivatives, is correlated to M. Therefore K is correlated to M, and this outcome is similar for all Skyrme functionals as seen on Fig. 4: the (E,K) linear correlation may be due to an artefact from the specific density dependence of the Skyrme EDF’s.

But this linear correlation vanishes when including relativistic EDF predictions, as mentioned above. M shall be the exclusive quantity which can be deduced from GMR measurements, whereas the K determination relies on additional density dependence assumptions. Using the M value at the crossing point, a linear expansion provides K=230 MeV. An uncertainty of 40 MeV can be inferred from the spreading of K values on Fig. 2 obtained with the various functionals, which is of the order of 17%. It is therefore argued that measurements in nuclei constrain the EOS around the crossing density (namely the derivative of the EOS quantity at the crossing density) and deducing values at saturation density remains a mainly model-dependent extrapolation. In the case of nuclear incompressibility, it is proposed to change the usual E vs. K correlation plot (only valid in the specific density dependence of Skyrme EDF’s) and to replace it by a more reliable and universal E vs. M plot (Fig. 3), where M is the derivative of the incompressibility at the crossing density. The measurement of the GMR in more neutron-rich nuclei mon08 ; van10 will certainly open the possibility to study a part of the isospin dependence of the incompressibility.

This work has been partly supported by the ANR NExEN and SN2NS contracts, the Institut Universitaire de France, by CompStar, a Research Networking Programme of the European Science Foundation and by the initiative QREN financed by the UE/FEDER through the Programme COMPETE under the projects, PTDC/FIS/113292/2009, CERN/FP/109316/2009 and CERN/FP/116366/2010. J.M. would like to thank K. Bennaceur and J. Meyer for useful discussions during the completion of this work.

## References

- (1) C.F. Von Weizsäcker, Zeitschrift für Physik 96 (1935) 431; H.A. Bethe and R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82.
- (2) P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag, Heidelberg, (1980).
- (3) B.A. Brown, Phys. Rev. Lett. 85, 5296 (2000).
- (4) S. Typel and B.A. Brown, Phys. Rev. C64, 027302 (2001).
- (5) M. Centelles, X. Roca-Maza, X. Viñas and M. Warda, Phys. Rev. Lett. 102, 122502 (2009).
- (6) J. Piekarewicz, Phys. Rev. C83, 034319 (2011).
- (7) M. Bender, P.-H. Heenen, and P.-G. Reinhard Rev. Mod. Phys. 75, 121 (2003).
- (8) D. Lunney, J.M. Pearson, and C. Thibault, Rev. Mod. Phys. 75, 1021 (2003).
- (9) J.-P. Blaizot, Phys. Rep. 64, 171 (1980).
- (10) J.M. Pearson, Phys. Lett. B271, 12 (1991).
- (11) S.K. Patra, M. Centelles, X. Viñas and M. Del Estal, Phys. Rev. C65 (2002) 044304.
- (12) D. Vretenar, T. Niksic and P. Ring, Phys. Rev. C68 (2003) 024310.
- (13) J. Li, G. Colò and J. Meng, Phys. Rev. C78, 064304 (2008).
- (14) E. Khan, Phys. Rev. C80, 011307(R) (2009).
- (15) G. Colò, N.V. Giai, J. Meyer, K. Bennaceur and P. Bonche, Phys. Rev. C70, 024307 (2004).
- (16) A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971).
- (17) E. Khan, J. Margueron, G. Colò, K. Hagino and H. Sagawa, Phys. Rev. C82, 024322 (2010).
- (18) E. Khan, M. Grasso and J. Margueron, Phys. Rev. C80, 044328 (2009).
- (19) C.J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86, 5647 (2001).
- (20) C. Ducoin, J. Margueron, C. Providẽncia and I. Vidaña, Phys. Rev. C83, 045810 (2011).
- (21) T. Li et al., Phys. Rev. Lett. 99, 162503 (2007).
- (22) O. Bohigas, A.M. Lane and J. Martorell, Phys. Rep. 51, 267 (1979).
- (23) L. Capelli, G. Colò and J. Li, Phys. Rev. C79, 054329 (2009).
- (24) D.H. Youngblood et al., Phys. Rev. C69 034315 (2004).
- (25) J. Piekarewicz, Phys. Rev. C76, 031301 (2007).
- (26) C. Monrozeau et al., Phys. Rev. Lett. 100, 042501 (2008).
- (27) M. Vandebrouck et al., Exp. E456a performed at GANIL (2010).