DOI 10.1140/epja/i2014-14020-3

Regular Article – Theoretical Physics

Eur. Phys. J. A (2014) 50: 20 THE EUROPEANPHYSICAL JOURNAL A

Symmetry energy in nuclear density functional theory�

W. Nazarewicz1,2,3,a, P.-G. Reinhard4, W. Satu�la3, and D. Vretenar5

1 Department of Physics and Astronomy, University of Tennessee Knoxville, Tennessee 37996, USA2 Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831, USA3 Faculty of Physics, University of Warsaw, ul. Hoza 69, 00-681 Warsaw, Poland4 Institut fur Theoretische Physik, Staudtstr. 7 D-90158 Universitat Erlangen/Nurnberg, Erlangen, Germany5 Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia

Received: 22 July 2013 / Revised: 22 August 2013Published online: 25 February 2014c© The Author(s) 2013. This article is published with open access at Springerlink.comCommunicated by A. Ramos

Abstract. The nuclear symmetry energy represents a response to the neutron-proton asymmetry. In thispaper we discuss various aspects of symmetry energy in the framework of nuclear density functional theory,considering both non-relativistic and relativistic self-consistent mean-field realizations side by side. Keyobservables pertaining to bulk nucleonic matter and finite nuclei are reviewed. Constraints on the symmetryenergy and correlations between observables and symmetry energy parameters, using statistical covarianceanalysis, are investigated. Perspectives for future work are outlined in the context of ongoing experimentalefforts.

1 Introduction

Density Functional Theory (DFT) is a universal approachused to describe properties of complex, strongly correlatedmany-body systems. Originally developed in the contextof many-electron systems in condensed matter physics andquantum chemistry [1,2] (also known under the name ofKohn-Sham DFT), it is also a tool of choice in microscopicstudies of complex heavy nuclei. The basic implementationof this framework is in terms of self-consistent mean-field(SCMF) models [3–5].

Extending the DFT to atomic nuclei, the nuclear DFT,is not straightforward as nuclei are self-bound, small, su-perfluid aggregations of two kinds of fermions, governedby strong surface effects. Their smallness leads to appre-ciable quantal fluctuations (finite-size effects) which aredifficult to incorporate into the energy density functional(EDF). The lack of external binding potential implies thatthe nuclear DFT must be necessarily formulated in termsof intrinsic normal and anomalous (pairing) densities [6].A density matrix expansion of the effective interactionsuggests that, in addition to the standard local nucleondensity, superior EDFs should also include more involvednucleon aggregates such as the kinetic-energy density andspin-orbit density [3–5]

The commonly used single-reference SCMF methodsinclude the local (Skyrme), non-local (Gogny) and covari-

� Contribution to the Topical Issue “Nuclear Symmetry En-ergy” edited by Bao-An Li, Angels Ramos, Giuseppe Verde,Isaac Vidana.

a e-mail: witek@utk.edu

ant (relativistic) approaches [5,7,8]. All these approachesare thought to be different realizations of an underlyingeffective field theory [9] with the ultraviolet physics hiddenin free parameters adjusted to observations. For that rea-son, predictions for low-energy (infrared) physics shouldbe fairly independent of the particular variant used in cal-culations [10–13]. The underlying EDFs are constructedin a phenomenological way, with coupling constants opti-mized to selected nuclear data and expected properties ofhomogeneous nuclear matter.

In practice, nuclear EDFs differ in their functionalform and are subject to different optimization strategiescausing that their predictions vary even within a singlefamily of EDFs. In particular, large uncertainties remainin the isovector channel, which is poorly constrained byexperiment. A key quantity characterizing the interactionin the isovector channel is the nuclear symmetry energy(NSE) describing the static response of the nucleus to theneutron-proton asymmetry.

As discussed in this Topical Issue, the NSE influences abroad spectrum of phenomena, ranging from subtle isospinmixing effects in N ∼ Z nuclei to particle stability ofneutron-rich nuclei, to nuclear collective modes, and toradii and masses of neutron stars. Various nuclear ob-servables are sensitive probes of NSE, and numerous phe-nomenological indicators can be constructed to probe itsvarious aspects.

It is the aim of this contribution to analyze the re-lations between NSE and measurable observables in fi-nite nuclei. The most promising observables for isovectorproperties that have stimulated vigorous experimental and

Page 2 of 13 Eur. Phys. J. A (2014) 50: 20

theoretical activity include neutron radii, neutron skins,dipole polarizability, and neutron star radii. The ongoingefforts are focused on better constraining the uncertain-ties concerning the equation of state (EOS) of the sym-metric and asymmetric nucleonic matter (NM) and, inparticular, the symmetry energy and its density depen-dence. Parameters that characterize the NSE are not en-tirely independent. They are affected by key nuclear ob-servables in different ways. Thus it is the sine qua non ofa further progress in this area to understand the corre-lation pattern between NSE parameters and finite-nucleiobservables, and to provide uncertainty quantification ontheoretical predictions using the powerful methods of sta-tistical analysis [14].

A second aim is to understand the dependences froma formal perspective and to explore the impact of con-figuration mixing. Within the independent particle pic-ture the isovector response can be described in terms ofa charge-dependent symmetry potential that shifts theneutron well with respect to the proton average poten-tial. The effect can be estimated quantitatively withinthe Fermi-gas model (FGM) augmented by a schematicisospin-isospin interaction [15],

VTT =12κT · T . (1)

In the Hartree approximation this model gives rise to aquadratic dependence of the NSE on the neutron excessI = (N − Z)/A,

Esym/A = asymI2 = (asym,kin + asym,int)I2, (2)

as T = |Tz| = |N − Z|/2 in the ground states of almostall nuclei. The FGM, in spite of its simplicity, has playedan important role in our understanding of the NSE. Inparticular, it separates the NSE strength into kinetic andinteraction (potential) contributions, and predicts a near-equality asym,kin ≈ asym,int of these contributions. It alsoprovides an estimate asym ≈ 25MeV for the NSE coeffi-cient (see ref. [16] for a recent discussion).

Furthermore, we note that the SCMF approach canlead to spontaneous breaking of symmetries. This appar-ent drawback can be turned into an advantage, as the sym-metry breaking mechanism allows to incorporate manyinter-nucleon correlations within a single product stateor, alternatively, within a single-reference DFT sacrific-ing good quantum numbers; broken symmetries have tobe restored a posteriori. We will address this topic usingthe example of isospin mixing which naturally has an im-pact on isovector properties.

This survey is organized as follows. Section 2 outlinesthe SCMF approaches and details various theoretical in-gredients of the models employed in this work. Observ-ables pertaining to bulk NM and finite nuclei that areessential for NSE are discussed in sect. 3. Constraints onNSE and correlations between observables and NSE pa-rameters, using the statistical covariance technique, arepresented in sect. 4. Section 5 summarizes the currentstatus of NSE parameters. The planned extensions of thecurrent DFT work are laid out in sect. 6. Finally, sect. 7contains the conclusions of this survey.

2 Nuclear DFT

The nuclear EDF constitutes a crucial ingredient for a setof DFT-based theoretical tools that enable an accuratedescription of ground-state properties, collective excita-tions, and large-amplitude dynamics over the entire chartof nuclides, from relatively light systems to superheavynuclei, and from the valley of β-stability to the nucleondrip-lines. In general EDFs are not directly related to anyspecific microscopic inter-nucleon interaction, but ratherrepresent universal functionals of nucleon densities andcurrents. With a small set of global parameters adjustedto empirical properties of nucleonic matter and to selecteddata on finite nuclei [17,18], models based on EDFs enablea consistent description of a variety of nuclear structurephenomena.

The unknown exact and universal nuclear EDF is ap-proximated by simple, mostly analytical, functionals builtfrom powers and gradients of nucleonic densities and cur-rents, representing distributions of matter, spins, momen-tum and kinetic energy. When pairing correlations are in-cluded, they are represented by pair (anomalous) densi-ties. In the field of nuclear structure this method is analo-gous to Kohn-Sham DFT. SCMF models effectively mapthe nuclear many-body problem onto a one-body problemusing auxilliary Kohn-Sham single-particle orbitals. By in-cluding many-body correlations in EDF, the Kohn-Shammethod in principle goes beyond the Hartree-Fock (HF)or Hartree-Fock-Bogolyubov (HFB) approximations and,in addition, it has the advantage of using local potentials.A broad range of nuclear properties have been very suc-cessfully described using SCMF models based on SkyrmeEDFs, relativistic EDFs, and the Gogny interaction [5,7,8,19–22]. (Note that the Gogny model is not strictly lo-cal as the other EDFs.) In the remainder of this sectionwe briefly outline the Skyrme-Hartree-Fock (SHF) methodand the relativistic mean-field (RMF) approach. As bothmethods are widely used and extensively described in theliterature, we keep the presentation short and concentrateon a side-by-side comparison of the models.

The basis of any mean-field approach is a set of single-nucleon canonical (Kohn-Sham) orbitals ψα(r), with oc-cupations amplitudes vα. The ψα denote Dirac four-spinor wave functions in the RMF framework, and two-component-spinor wave functions in the SHF which is aclassical mean-field model. The canonical occupation am-plitudes vα are determined by the pairing interaction. Thestarting point of a particular model is an EDF expressed interms of ψα, vα and the local densities derived therefrom.The energy functional for the SHF method reads

E =∫

d3r (Ekin + Epot) + ECoul + Epair + Ecm, (3)

Ekin =�

2

2mpτp +

�2

2mnτn

Ecm = − 12mA

〈(Pcm

)2〉.

The kinetic energy Ekin is expressed in terms ofsingle-nucleon wave functions. The Skyrme functional is

Eur. Phys. J. A (2014) 50: 20 Page 3 of 13

Table 1. Upper: The basic isoscalar (T = 0) and isovector (T = 1) local densities of SHF (left) and RMF (right). Lower: Thepotential-energy densities in the three considered SCMF models. Model parameters (third row) defining the coupling constantsare indicated by lowercase latin letters. For further explanation see text.

Densities

SHF RMF

T = 0 T = 1 T = 0 T = 1

ρ0(r) =P

α v2α ψ†

αψα ρ1(r) =P

α v2α ψ†

ατ3ψα ρ0(r) =P

α v2α ψ†

αψα ρ1(r) =P

α v2α ψ†

ατ3ψα

τ0(r) =P

α v2α ∇ψ†

α∇ψα τ1(r) =P

α v2α ∇ψ†

ατ3∇ψα ρS(r) =P

α v2α ψ†

αγ0ψα

J0(r) = −iP

α v2α ψ†

α∇ × σψα J1(r) = −iP

α v2α ψ†

ατ3∇×σψα

Potential-energy density

SHF RMF-PC RMF-ME

T = 0 T = 1 T = 0 T = 1 T = 0 T = 1

ρρ Cρ0ρ2

0 Cρ1ρ2

1 Gωρ20 Gρρ2

1 Gωρ01

−Δ+m2ω

ρ0 Gρρ11

−Δ+m2ρρ1

mass Cτ0 ρ0τ0 Cτ

1 ρ1τ1 Gσρ2S GσρS

1−Δ+m2

σρS

� · s C∇J0 ρ0∇·J0 C∇J

1 ρ1∇·J1 “ “

gradient CΔρ0 (∇ρ0)

2 CΔρ1 (∇ρ1)

2 fS(∇ρS)2

dens. dep. Cρ0 = cρ

0 + dρ0ρ

a0 Cρ

1 = cρ1 + dρ

1ρa0 Gi = ai + (bi + cix)e−dix Gi = ai

1+bi(x+di)2

1+ci(x+di)2Gρ = gρe−aρ(x−1)

CτT = cτ

T , CΔρT = cΔρ

T , C∇JT = c∇J

T i ∈ {σ, ω, ρ}, x = ρ0ρsat

i ∈ {σ, ω}, x = ρ0ρsat

contained in the interaction part with the potential-energydensity Epot. The Coulomb energy ECoul consists of thedirect Coulomb term, and the Coulomb exchange that isusually taken into account at the level of the Slater ap-proximation. In most applications the center-of-mass cor-rection Ecm is applied a posteriori because its variationwould considerably complicate the mean-field equations.The pairing functional Epair will be detailed later. TheRMF approach is usually formulated in terms of a La-grangian,

L =∫

d3r (Lkin − Epot) − ECoul − Epair − Ecm, (4)

Lkin =∑α

v2αψ†

αγ0(iγ · ∂ − m)ψα, (5)

where γ is the Dirac matrix. Again, the kinetic part is ex-pressed explicitly in terms of Dirac spinor wave functions,whereas interaction terms are included in the potentialenergy density Epot. Further contributions from Coulomb,pairing and center-of-mass motion are treated similarly asin the SHF approach.

The basic building blocks of an EDF are local densitiesand currents built from single-nucleon wave functions [5,23]. These are summarized in the upper part of table 1. Alldensities appear in two flavors [24,25]: isoscalar (T = 0),or total density (sum of proton and neutron densities),and isovector (T = 1) density (difference between neutronand proton densities). Both can be conveniently expressedusing the isospin operator τ3. The basic ingredients of anEDF are the local densities ρ0 and ρ1. In RMF these can

be associated with the zero-component of the four-vectorcurrent, where ρ0 is often called the vector density and ρ1

the isovector-vector density. RMF uses one more ingredi-ent, the isoscalar-scalar density denoted here as ρS. SHFinstead employs the kinetic-energy densities τ0/1 and thespin-orbit densities J0/1. One can show that τ0 and J0

emerge in the non-relativistic limit of ρS [26]. The princi-pal difference between SHF and RMF is that the quanti-ties τ0 and J0 are independent in SHF, whereas they aretightly related through ρS in RMF. Moreover, the RMFdoes not invoke an isovector counterpart of ρS thus beingmore restricted in the isovector channel.

The lower part of table 1 displays the main componentsof the potential-energy density. The underlying is to takeall bi-linear isoscalar combinations of the local densitiesand to associate a coupling constant with each term [25].The SHF confines the combinations to have at most sec-ond order of derivatives (the term J2 is also dropped).In the RMF approach one keeps only terms that forma Lorentz scalar. Moreover, two bi-linear realizations ofRMF will be considered. First there is the straightforwardpoint-coupling (RMF-PC) realization that corresponds tocontact interactions between nucleons and, second, themeson-exchange folding (RMF-ME). The folding is mo-tivated by the traditional route to RMF as a model ofnucleons coupled to classical meson fields. Of course, atenergies characteristic for nuclear binding meson exchangerepresents just a convenient representation of the effectivenuclear interaction. In practice RMF-PC and RMF-MEpresent equivalent realizations of the relativistic SCMF,differing in the range of effective interactions (zero range

Page 4 of 13 Eur. Phys. J. A (2014) 50: 20

vs. finite range) and the choice of density dependence forthe couplings. In practical applications one restricts thedensity dependence of coupling (vertex) functions to keepthe number of free parameters to a minimum. In SHF,only the leading terms ∝ ρ2

0 and ρ21 are given a (simple)

density dependence as shown in table 1. In RMF-PC andRMF-ME, each term has some density dependence, butnot all of these parameters are actually used. In RMF-PC,in particular, the parameters c1 and a1 (isovector chan-nel) and cv (isoscalar vector channel) are set to zero [27].In RMF-ME, the parameters are correlated by additionalboundary conditions on Gi [28,29]. In total, there are 11adjustable parameters for SHF, 10 for RMF-PC, and 8for RMF-ME. From a formal perspective, SHF and RMF-PC are rather similar, differing mainly in the relativistickinematics, while RMF-ME includes a significantly differ-ent density dependence of the couplings, in addition tothe finite range. These three models thus allow to displayseparately effects of kinematics, density dependence, andrange of the effective nuclear interaction.

As far as particle-particle interaction is concerned, inthe SHF we use the pairing functional derived from adensity-dependent zero-range force,

Epair =14

∑q∈{p,n}

v0,q

∫d3rρ2

q

[1 − ρ(r)

ρpair

], (6a)

ρq(r) =∑α∈q

uαvα

∣∣ψα(r)∣∣2, (6b)

where q runs over protons and neutrons. It involves thepair-density ρq and is usually augmented by some densitydependence. We consider here v0,p, v0,n, and ρpair as freeparameters of the pairing functional in SHF. Note that wedo not recouple to isoscalar and isovector terms becausepairing is considered independently for protons and neu-trons. Actually, the zero-range pairing force works only to-gether with a limited phase space for pairing. We use herea soft cut-off in the space of single-nucleon energies [30]according to ref. [31].

In RMF calculations we use the recently developed sep-arable pairing force [32,33]. It is separable in momentumspace, and is completely determined by two parametersthat are adjusted to reproduce in symmetric nuclear mat-ter the pairing gap of the Gogny force. We have verifiedthat both pairing prescriptions yield comparable resultsfor the pairing gaps.

3 Observables

In this section, we discuss observables pertaining to nu-clear matter (NM) and finite nuclei that are essential fordiscussion of NSE. Those observables can be roughly di-vided [14] into good isovector indicators that correlatevery well with NSE (such as weak-charge form factor, neu-tron skins, dipole polarizability, slope of the symmetry en-ergy, and neutron pressure) and poor isovector indicators(such as nuclear and neutron matter binding energy, giantresonance energies, isoscalar and isovector effective mass,incompressibility, and saturation density).

Table 2. Definitions of NMP used in this work. All derivativesare to be taken at the equilibrium point corresponding to thesaturation density ρeq.

Incompressibility: K∞ = 9 ρ20

d2

dρ20

EA|eq

Symmetry energy: asym = 12ρ20

d2

dρ21

EA|eq

Slope of asym: L = 3ρ0dasymdρ0

|eq

Effective mass – SHF: �2

2m∗ = �2

2m+ ∂

∂τ0

EA|eq

Effective mass – RMF: m∗ = m + GσρS

TRK enhancement: κTRK = mm∗ − 1 + 1

2ρ0

ddρ1

mm∗

n

3.1 Nuclear matter properties

Bulk properties of symmetric nuclear matter, called nu-clear matter properties (NMP), are often used to char-acterize the properties of a model, or functional respec-tively. Starting point for the definition of NMP is thebinding energy per nucleon in the symmetric nuclear mat-ter E/A = E/A(ρ0, ρ1, τ0, τ1). Variation with respect toKohn-Sham wave functions establishes a relation τi =τi(ρ0, ρ1) between kinetic densities τi and densities ρi.This yields the commonly used binding energy at equi-librium, E/A[ρ0, ρ1, τ0(ρ0, ρ1), τ1(ρ0, ρ1)], as a function ofthe densities ρi alone. Table 2 lists the NMP discussed inthis work. We consider τi as independent variables for thepurpose of a formally compact definition of the effectivemass in SHF. This is indicated by the partial derivativeswith respect to τi for m∗/m in SHF and κTRK. Staticproperties are deduced from the binding energy at equi-librium, which depends on ρi only. This is indicated byusing the total derivatives for K∞, asym, and L. Theslope of the symmetry energy L parametrizes the den-sity dependence of asym. This quantity is essential for thecharacterization of the EOS of neutron matter and themass-radius relation in neutron stars [34–39]. The notionof effective mass is ambiguous in RMF. In this survey,we have decided to use what is called the Dirac mass.(Alternatively, one can talk about the effective mass atthe Fermi surface, which is typically 10% larger than theDirac mass.) The Thomas-Reiche-Kuhn (TRK) enhance-ment factor κTRK [40] is merely used to characterize theisovector effective mass. (The alternative value deducedfrom the mass at the Fermi momentum is typically 20%smaller.) The quantity m∗

n is the neutron effective mass. Itis defined in the same way as the isoscalar effective massm∗ but using the derivative with respect to the neutronkinetic density τn.

Next to NMP come the corresponding bulk surfaceparameter, the (isoscalar) surface energy asurf and the(isovector) surface symmetry energy assym. These sur-face parameters can be determined from the leptoder-mous expansion of the liquid drop model (LDM) energyper nucleon, ELDM = ELDM/A, in terms of inverse radius

Eur. Phys. J. A (2014) 50: 20 Page 5 of 13

(∝ A−1/3) and neutron excess I [41],

ELDM(A, I) = avol + asurfA−1/3 + acurvA

−2/3

+asymI2 + assymA−1/3I2 + a(2)symI4. (7)

The LDM energy ELDM(A, I) is obtained from the DFTcalculation by subtracting the fluctuating shell correc-tion energy. The general strategy behind this correctionand leptodermous expansion is detailed in refs. [41,42]. Inessence, we combine NM calculations (A = ∞) with (shell-corrected) DFT calculations for a huge set of sphericalnuclei and extract the surface parameters by a fit to theexpansion (7). Alternatively and simpler, one can computethe surface energy and surface symmetry energy througha semi-classical approximation (extended Thomas-Fermi)for the semi-infinite nuclear matter [43]. In this paper,we shall apply both strategies, the semi-classical approachwhenever RMF is involved.

An important parameter characterizing the pure neu-tron matter is the neutron pressure

P (ρn) = ρ2n

ddρn

(E

A

)n

, (8)

a quantity that is proportional to the slope of the bind-ing energy of neutron matter at a given neutron density(derivative of neutron EOS). As discussed below, P is ex-cellent isovector indicator.

3.2 Observables from finite nuclei

The total energy of a nucleus E(Z,N) is the most basicobservable described by SCMF. It is also the most impor-tant ingredient for calibrating the functional, see sect. 4.1.We often consider binding energy differences. Of great im-portance for stability analysis are separation energies andQα values. Another energy observable, potentially usefulin the context of NSE, is the indicator

δVpn = −14[E(N,Z) − E(N − 2, Z)

−E(N,Z − 2) + E(N − 2, Z − 2)], (9)

involving the double difference of binding energies [44].Since δVpn approximates the mixed partial derivative ofbinding energy with respect to N and Z, for nuclei withan appreciable neutron excess, the average value of δVpn

probes the symmetry energy term of LDM [45]: δV LDMpn ≈

2(asym +assymA−1/3)/A. That is, the shell-averaged trendof δVpn is determined by the symmetry and surface sym-metry energy coefficients.

It has been shown in [46] that effective SCMF pro-vide a pertinent description of the form factors in themomentum regime q < 2qF where qF is the Fermi mo-mentum. The key features of the nuclear density are re-lated to this low-q range. The basic parameters character-izing nuclear density distributions are: r.m.s. charge radiusrC, diffraction radius RC, and surface thickness σC [47].The diffraction radius RC, also called the box-equivalent

radius, parametrizes the gross diffraction pattern whichresemble those of a hard sphere of radius RC [47]. Theactual charge form factor FC(q) falls off faster than thebox-equivalent form factor Fbox. This is due to the finitesurface thickness σ which, in turn, can be determined bycomparing the height of the first maximum of Fbox withFC from the realistic charge distribution. The charge haloparameter hC is composed from the three basic chargeform parameters and serves as a nuclear halo parameterfound to be a relevant measure of the outer surface dif-fuseness [48].

The charge distribution is basically a measure of theproton distribution. It is only recently that the parity-violating electron scattering experiment PREX has pro-vided some information on the weak-charge form factorFW (q) of 208Pb [49,50]. These unique data gives access toneutron properties, such as the neutron r.m.s. radius rn.Closely related and particularly sensitive to the asymme-try energy is the neutron skin rskin = rn − rp, which isthe difference of neutron and proton r.m.s. radii. (As dis-cussed in ref. [48], it is better to define the neutron skinthrough neutron and proton diffraction radii and surfacethickness. However, for well-bound nuclei, which do notexhibit halo features, the above definition of rskin is prac-tically equivalent.) Neutron radii and skins are excellentisovector indicators [14,36,51–57] that help to check andimprove isovector properties of the nuclear EDF [14].

Nuclear excitations are characterized by the strengthdistributions SJT (E) where J is the angular momentumof the excitation, T its isospin, and E the excitation en-ergy. For example, the cross section for photo-absorptionis proportional to S11(E). The strengths functions can beobtained from the excitation spectrum,

SJT (E) =∑

n

EnBn(EJT )δΔ(E − En), (10)

where En is the excitation energy of state n, Bn(EJT )the corresponding transition matrix element of multi-polarity J and isospin T , and δΔ as finite width fold-ing function —if SJT (E) is calculated theoretically us-ing, e.g., the random phase approximation (RPA). In ourRPA estimates, we use an energy-dependent width Δ =max(Δmin, (En−Ethr)/Eslope) which simulates the broad-ening mechanisms beyond RPA. The parameters for 208Pbare Δmin = 0.2MeV, Ethr = 10MeV, and Eslope = 5MeV.The resulting spectral distributions for heavy nuclei, as208Pb, show one clear giant resonance peak at EGR(JT )for (J, T ) = (0, 0), (1, 1), (2, 0). We will consider these res-onance energies as characteristic observables of dynamicalresponse in heavy nuclei. The strength functions SJT (E)in light nuclei are much more fragmented and cannot bereduced to one single characteristic number.

There are other key observables that can be extractedfrom the strength distributions. In particular, for thedipole case S11(E), a key observable is the electric dipolepolarizability,

αD =∑

n

E−1n Bn(E11). (11)

Page 6 of 13 Eur. Phys. J. A (2014) 50: 20

In the following, we will consider αD for 208Pb. It hasbeen demonstrated [14,57] that αD strongly correlateswith NSE; hence, it can serve as excellent isovector in-dicator that can be precisely extracted from measuredE1 strength [58]. On the other hand, the low-energyE1 strength, sometimes referred to as the pygmy dipolestrength, exhibits weak collectivity. The correlation be-tween the accumulated low-energy strength and the sym-metry energy is weak, and depends on the energy cutoffassumed [14,59,60].

Giant resonances are small amplitude excitations andbelong to the regime of linear response. The low-energybranch of isoscalar quadrupole excitations is often associ-ated with large amplitude collective motion along nuclearshapes with substantial quadrupole deformation. Of par-ticular importance is nuclear fission, which determines ex-istence of heavy and superheavy nuclei. As a simple androbust measure of fission, we shall consider the axial fissionbarrier height in 266Hs. Unlike actinides, most superheavynuclei have one single fission barrier [61–63], which simpli-fies the analysis for our purpose. It has to be kept in mindthat the inner barrier is often lowered by triaxial shapes,but this is not important for the study of large-amplitudenuclear deformability.

4 Symmetry energy: constraints andcorrelations

4.1 Brief review of χ2 technique and correlationanalysis

As discussed in sect. 2, the nuclear EDF is characterizedby about a dozen of coupling constants p = (p1, . . . , pF )that are determined by confronting DFT predictions withexperiment. The standard procedure is to adjust the pa-rameters p to a large set of nuclear observables in carefullyselected nuclei [5,17,18,64,65]. This is usually done by thestandard least-squares optimization technique. The start-ing point is the χ2 objective function

χ2(p) =∑O

(O(th)(p) −O(exp)

ΔO

)2

, (12)

where “th” stands for the calculated values, “exp” for ex-perimental data, and ΔO for adopted errors. The opti-mum parametrization p0 is the one which minimizes χ2

with the minimum value χ20 = χ2(p0). Around the min-

imum p0, there is a range of “reasonable” parametriza-tions p that can be considered as delivering a good fit,i.e., χ2(p) ≤ χ2

0 +1. As this range is usually rather small,we can expand χ2 as

χ2(p)−χ20 ≈

F∑i,j=1

(pi−pi,0)Mij(pj−pj,0), (13)

Mij = 12∂pi

∂pjχ2|p0 . (14)

The reasonable parametrizations thus fill the confidenceellipsoid given by

(p − p0)M(p − p0) ≤ 1, (15)

see sect. 9.8 of [66]. Given a set of parameters p, any ob-servable A = 〈A〉 can be uniquely computed. In this way,A = A(p). The value A thus varies within the confidenceellipsoid, and this results in some uncertainty ΔA. Let usassume for simplicity that the observable varies weaklywith p such that one can linearize in the relevant rangeA(p) = A0 + (p − p0) · ∂pA. Let us, furthermore, asso-ciate a weight ∝ exp (−χ2(p)) with each parameter set.A weighted average over the parameter space yields thecovariance between two observables A and B, which rep-resents their combined uncertainty,

ΔAΔB =∑ij

∂piA(M−1)ij∂pj

B. (16)

For A = B, eq. (16) gives the variance Δ2A that defines astatistical uncertainty of an observable. Variance and co-variance are useful quantities that allow to estimate theimpact of an observable on the model and its parametriza-tion. We shall explore the covariance analysis in three dif-ferent ways,

1) We perform a constrained fit during which the observ-able of interest is kept fixed at a desired value. Inthe present paper, we consider the symmetry energyasym as constraining observable. Comparing uncertain-ties from a constrained fit with those from an uncon-strained fit provides a first indicator on the impact ofthe constrained observable on other observables.

2) The next step is a trend analysis, in which one per-forms a series of constrained fits with systematicallyvaried values of the constraining observable. One thenstudies other observables as a function of the con-strained quantity. This provides valuable informationon possible inter-dependences.

3) Finally, we compute correlation (16) between asym andother observables. Here, a useful dimensionless mea-sure is given by the Pearson product-moment correla-tion coefficient [66],

cAB =|ΔAΔB|√ΔA2 ΔB2

. (17)

A value cAB = 1 means fully correlated and cAB = 0– uncorrelated.

In the following, we will apply these three ways of studyingcorrelations with asym to different groups of observables.To this end, we have produced a series of parametrizationswith systematically varied asym for the SV Skyrme familyand for the RMF-ME and RMF-PC models.

The optimization and covariance analysis carried outin this survey for all three EDFs (SHF-SV, RMF-PC, andRMF-ME) is based on the same standard set of data onspherical nuclei (masses, diffraction radii, surface thick-ness, charge radii, separation energies, isotope shifts, and

Eur. Phys. J. A (2014) 50: 20 Page 7 of 13

odd-even mass differences) that has originally been pro-posed in ref. [17] and recently employed in refs. [67,38].We wish to emphasize that this is the first time that oneconsistent phenomenological input has been used to con-strain SHF and RMF EDFs. A slightly modified variant ofthe fitting protocol has been used for RMF-ME. This EDFdid not lead to stable results in the fits which were uncon-strained by NMP. Consequently, we included the nuclearmatter information on (E/A)eq into the dataset. This isstill much less than in the previously published optimiza-tion protocols of RMF-ME, in which all NMP were con-strained [28,29,68].

4.2 Correlations with nuclear matter properties

The NMP corresponding to unconstrained optimization ofSHF-SV, RMF-PC, and RMF-ME EDFs —using the samestandard dataset— are shown in table 3. They are com-pared with NMP of SHF-RD [67] (employing a modifieddensity dependence and the standard dataset) and SHF-TOV [38] (using neutron star data in addition the stan-dard dataset in the optimization process). As expected,isoscalar effective mass is significantly lowered in RMF ascompared to SHF, and the opposite holds for κTRK. Theslope parameter L is predicted to be very different in allfive models. In particular, RMF-ME has very low value ofL, and —at the same time— the uncertainty on asym inthis model is very small.

Figure 1 shows the trends for selected properties ofsymmetric nuclear matter with asym. The purpose of thisanalysis is to relate systematic variations with asym to sta-tistical uncertainties. The isoscalar properties K∞, m∗/mas well as the isovector dynamical response κTRK are fairlyinsensitive to asym. Their variation with asym are muchsmaller than the typical statistical uncertainties. This in-dependency is also indicated by the fact that the un-certainty obtained in the unconstrained fit is not visiblylarger than those from the constrained optimizations. Thetrend is markedly different for the density dependence ofthe symmetry energy L: variations with asym well exceedthe statistical error bars and the uncertainties from uncon-strained fits are larger than those from constrained calcu-lations. It is to be noted that the dedicated variations ofasym stay within the uncertainty of asym in the uncon-strained optimization. The uncertainty of L in the freefit thus covers nicely the uncertainty of the constrainedcalculations plus the variation of L with asym. Anyway,the results shows that L cannot be used as independentNMP although the formal structure of the EDF would al-low that. There seems to be a strong link established bythe data which yet has to be worked out.

4.3 Correlations with properties of finite nuclei

Figure 2 illustrates the trends with asym and extrapola-tion uncertainties for three observables in 208Pb: weak-charge form factor at q = 0.475 fm−1 (q-value of PREX),neutron skin, and dipole polarizability. These observables

150

200

250

300

28 30 32 34 36

K∞

(M

eV)

asym (MeV)

RMF MERMF PCSHF SV

0.6

0.8

1.0

m*/

m

-50

0

50

100

150

L (

MeV

)

-0.4

0

0.4

0.8

κ TR

K

Fig. 1. Behavior of selected nuclear matter properties withsymmetry energy asym for the SV Skyrme family and for theME and PC RMF model families. The statistical uncertaintiesare indicated by error bars. The result of the unconstrainedfits are shown by large open symbols with corresponding errorbars.

are all known to be sensitive to isovector properties ofEDF [14,57]. This is confirmed by the trends in the presentresults. The comparison of uncertainties shows a largegrowth when going from constrained to unconstrained op-timizations. This corroborates the close relation betweenthe symmetry energy and the three isovectors indicatorsshown in fig. 2. It is, furthermore, interesting to note thatSHF and RMF-PC stay safely within the bands given byexperimental data and RMF-ME is not far away. A betterdiscrimination between models requires more precise data,a task on which presently many experimental groups areheavily engaged.

Page 8 of 13 Eur. Phys. J. A (2014) 50: 20

Table 3. Nuclear matter parameters of SHF-SV, RMF-PC, and RMF-ME EDFs used in this paper (with error bars) obtainedby means of unconstrained optimization. Also shown are the values of NMP of SHF-RD [67] and SHF-TOV [38].

Model ρeq E/A K∞ m∗/m asym L κTRK

(fm−3) (MeV) (MeV) (MeV) (MeV)

SHF-SV 0.161(1) −15.91(4) 222(9) 0.95(7) 31(2) 45(26) 0.08(29)

RMF-PC 0.159(1) −16.14(3) 185(18) 0.57(1) 35(2) 82(17) 0.75(2)

RMF-ME 0.159(3) −16.2(2) 250(19) 0.56(1) 32.4(1) 6(7) 0.79(2)

SHF-RD 0.161 −15.93 231 0.90 32(2) 60(32) 0.04(32)

SHF-TOV 0.161 −15.93 222 0.94 32(1) 76(15) 0.21(26)

0.22

0.24

0.26

28 30 32 34 36

FW

(0.4

75)/F

W(0

)

10

12

14

16

18

0.1

0.2

0.3

asym (MeV)

α D[2

08Pb

] (f

m2 /

MeV

)r sk

in[2

08Pb

] (f

m) RMF ME

RMF PCSHF SV

Fig. 2. Similar to fig. 1, but for selected properties of 208Pb:neutron skin (top), dipole polarizability (middle), and weak-charge form factor (bottom). The current experimental rangesare shaded grey: rskin = 0.33+0.16

−0.18 fm [49], αD = 14.0 ±0.4 fm2/MeV [58], and FW (0.475)/FW (0) = 0.204± 0.028 [50].

To explore the usefulness of δVpn as an isovector indi-cator, we choose the heavy deformed nucleus 168Er, as itseven-even neighbors have similar structure and the calcu-lated values of δVpn for even-even Er isotopes show littlevariations around N = 100. The results displayed in fig. 3show a gradual decrease of this quantity with asym, butthe magnitude of the variation is very small and cannot ac-count for the deviation from experiment (around 50 keV).It is apparent that this quantity is too strongly influencedby shell effects (given by the deviation from the LDM es-

0.25

0.30

0.35

28 30 32 34

exp

LDM

asym (MeV)δV

pn (

MeV

)

168Er

SHF SV

Fig. 3. Behavior of δVpn in 168Er with symmetry energy asym

for SHF-SV (solid line) as compared to experiment (dashedline) and the LDM value (filled square). The result of the un-constrained fit is marked by a large open square with corre-sponding error bars.

10

11

12

13

14

28 30 32 34

EG

R in

208 Pb

(M

eV)

GDR

GMR

GQR

asym (MeV)

SHF SV

Fig. 4. Behavior of giant resonance energies in 208Pb withsymmetry energy asym for the SV Skyrme family [17]. In ordernot to make the graph too busy the uncertainties from theunconstrained fit are not shown; they have the same size asthose from the constrained fits.

timate; also around 50 keV) to probe NSE, see refs. [45,69] and sect. 4.4 below.

Figure 4 shows the trends of the three major gi-ant resonances in 208Pb: isoscalar monopole resonance(GMR), isovector dipole resonance (GDR), and isoscalarquadrupole resonance (GQR). For technical reasons, weonly show results obtained with the SV Skyrme family.The isoscalar resonances show no dependence on asym atall; this is understandable for the symmetry energy be-longs to the isovector sector. Somewhat surprisingly, theGDR exhibits very little dependence on asym as well, with

Eur. Phys. J. A (2014) 50: 20 Page 9 of 13

4

6

8

10

12

28 29 30 31 32 33 34 35 36

17.0

17.2

17.4

17.6

asym (MeV)

a surf (

MeV

)B

f [26

6 Hs]

(M

eV)

RMF MERMF PCSHF SV

Fig. 5. Similar to fig. 1, but for surface energy (top) and fissionbarrier (bottom) in 266Hs. The surface energy from RMF-MEis not shown.

the magnitude of variations well below the statistical un-certainties. As demonstrated earlier [17,57], it is the sum-rule enhancement factor κTRK that has the dominant im-pact on the GDR peak frequency rather than asym. Thecovariance analysis of fig. 4 confirms that the energies ofGMR, GDR, and GQR do not obviously relate to asym.

Figure 5 shows behavior of surface energy asurf and theinner fission barrier Bf in 266Hs with asym. The surfaceenergy was computed by means of the extended Thomas-Fermi method. (The surface energy from RMF-ME is notshown: we do not have at our disposal a code for semi-infinite matter which could handle this particular RMFfunctional.) The trends of asurf predicted by SHF andRMF are similar. An offset of about 2MeV is most likelydue to very different effective masses in both models. Muchlarger differences are seen for the fission barriers. The basicdifference between SHF and RMF can again be explainedpredominantly in terms of effective masses. Barriers areproduced by shell effects and shell effects are larger forlower effective masses. There is also a difference betweenthe two RMF models. This could be due to a differenthandling of gradient terms (only RMF-PC contains such)and a much different parametrization of density depen-dence. All three models show not only different values assuch, but also different trends.

The statistical errors differ substantially between themodels. RMF-ME shows a small uncertainty in Bf . Thismay be due to the missing gradient term in this modelwhich would also restrict the uncertainty in the surfaceenergy. We note, however, that the gradient term in RMF-PC is to a certain extent equivalent the mass term of thesigma meson in RMF-ME, which is considered a free pa-rameter. The plot of the Bf demonstrates nicely the rel-ative role of statistical and systematic errors, with the

0 0.2 0.4 0.6 0.8 1.0

K∞

m*/m

L

κTRK

Corr(asym,Y)

asurf

P

FW(0.475)rskin

αD

E(302120)

E(148Sn)Bf(266Hs)

Qα(302120)

SHF SVRMF PCRMF ME

Fig. 6. The correlation (17) between symmetry energy andselected observables (Y ) for three models: SHF-SV, RMF-PCand RMF-ME. Results correspond to unconstrained optimiza-tion employing the same strategy in all three cases. For RMF-ME no reliable numbers could be obtained for asurf ; this isindicated by an open circle.

statistical errors being much smaller than inter-model dif-ferences. As discussed in refs. [70,42], fission barriers arestrongly affected by asurf and assym of EDF. In particular,the recently developed EDF UNEDF1, suitable for stud-ies of strongly elongated nuclei, has relatively low valuesof asurf and assym (see fig. 7 below) that reflect the con-straints on the fission isomer data. The reduced surfaceenergy coefficients result in a reduced effective surface co-efficient a

(eff)surf = asurf + assymI2, which has profound con-

sequences for the description of fission barriers, especiallyin the neutron-rich nuclei that are expected to play a roleat the final stages of the r-process through the recyclingmechanism [71].

4.4 Correlations summary

The summary of our correlation analysis for asym is givenin fig. 6. The first four entries concern the same nuclearmatter properties as in fig. 1. It is only for L, the slope ofsymmetry energy, that a strong correlation with asym isseen. This complies nicely with the findings of the trendanalysis in fig. 1. The next entry concerns the neutronpressure (8) at ρn = 0.08 neutrons/fm3. It is also stronglycorrelated with asym, which is no surprise because it isan excellent isovector indicator [14,36,53,54,72,73]. Thediagram shows, furthermore, the (isoscalar) surface en-ergy asurf computed in semi-classical approximation. Thisquantity is well correlated with asym for SHF and practi-cally uncorrelated for RMF.

Page 10 of 13 Eur. Phys. J. A (2014) 50: 20

The next three entries are observables in 208Pb: weak-charge form factor, neutron skin, and dipole polarizability.All three are known to be strong isovector indicators [14,57,36]. This is confirmed here for all three models.

The remaining four entries deal with exotic nuclei.These are: binding energy and α-decay energy in yet-to-be-measured superheavy nucleus Z = 120, N = 182, bind-ing energy in an extremely neutron rich 148Sn, and thefission barrier in 266Hs (for which trends had been shownalready in fig. 5). The data on Z = 120, N = 182 con-sistently do not correlate with asym. The binding energyof 148Sn shows some correlation with asym, about equallystrong in the three models. This is expected as a large neu-tron excess surely explores the static isovector sector. Fi-nally, the correlation with fission barrier in 266Hs exhibitsan appreciable model dependence with some correlationin SHF and practically none in RMF.

We also studied correlations between δVpn in 168Er andother observables for finite nuclei and NM. We did notfind a single observable that would correlate well with thisbinding-energy indicator. In particular, the correlation co-efficient (17) with asym is 0.41, with αD in 208Pb is 0.6,and with rskin in 208Pb is 0.54. This results demonstratesthat δVpn in one single nucleus is too strongly influencedby shell effects to be used as an isovector indicator.

5 Symmetry energy parameters of EDFs

The actual values of symmetry energy parameters dependon i) the form of EDF and ii) the optimization strat-egy used. The first point is nicely illustrated in table 3,which compares NMP for different functional forms (SHF-SV, SHF-RD, RMF-PC, and RMF-ME) using the samedataset and the same optimization technique. As far asthe second point, it is instructive to compare SHF-SVand SHF-TOV NMP; namely, the inclusion of additionaldata on neutron stars in SHF-TOV has significantly im-pacted L and κTRK. Many other examples can be foundin refs. [19,74] that demonstrate divergent predictions ofSkyrme EDFs for neutron and nuclear matter.

The range of asym is fairly narrowly constrained byvarious data and ab initio theory [34]; it is 28MeV <asym < 34MeV. The recent Finite-Range Droplet Model(FRDM) result [75] is asym = 32.5 ± 0.5MeV. All EDFslisted in table 3 are consistent with these expectations.

The values of L are less precisely determined [34–39,76–79]; there is more dependence on specific observablesor methodology used. Recent surveys [34,39] suggest thata reasonable range of L is 40MeV < L < 80MeV, andFRDM gives L = 70± 15MeV [75]. Except for RMF-ME,all models shown in table 3 are consistent with these es-timates. The low value of L in RMF-ME is troublesome;here we note that while SHF-SV and RMF-PC EDFs fallwithin the error bars of the current experimental data infig. 2, RMF-ME (as defined by the present optimizationprotocol) does not. In this context, it is worth mention-ing that in the case of RMF-ME the objective function isvery steep in the direction of asym. This is seen in table 3,which shows the uncertainties of asym for the three models

24 26 28 30 32 34 36−120

−100

−80

−60

−40

−20

0

a sym (MeV)

ass

ym (M

eV)

Fig. 7. Correlation between the symmetry and surface symme-try coefficients taken from ref. [70] (Skyrme EDFs, dots; LDMvalues, stars) and ref. [81] (Skyrme EDFs, circles). The UN-EDF1 values [42] are marked by a square. (Adopted from [70].)

employed. Consequently, the RMF-ME results are mostlyin a sub-optimal range of the model in a broad range ofasym. For instance, at asym = 30MeV, the average devia-tion for binding energies with RMF-ME is about twice aslarge as for the optimal value asym = 32.4MeV.

As discussed in ref. [41], the leading surface and sym-metry terms appear relatively similar within each fam-ily of EDFs, with a clear difference for asym betweenSHF and RMF. By averaging over Skyrme-EDF resultsof refs. [41,80], one obtains: asym ≈ 30.9 ± 1.7MeV,assym ≈ −48 ± 10MeV. Older relativistic models providesystematically larger values [41]: asym ≈ 40.4 ± 2.7MeVand assym ≈ −103 ± 18MeV. (Codes for a leptodermousexpansion of the recent RMF-PC and RMF-ME modelshave yet to be developed.)

The coefficient assym is poorly constrained in the cur-rent EDF parameterizations and there are large differ-ences between models, see fig. 7. In addition, the val-ues of asym and assym have been shown to be systemati-cally (anti)correlated [51,70,81,82]. Figure 7, displays thepairs (asym, assym) for various Skyrme EDFs and LDMparametrizations. While a correlation between asym andassym is apparent, a very large spread of values is seen thatdemonstrates that the data on g.s. nuclear properties arenot able to constrain assym. It is interesting to note thatthe LDM values and phenomenological estimates clusteraround asym = 30MeV and assym = −45MeV. The val-ues for UNEDF1 functional, additionally constrained bythe data on very deformed fission isomers (thus probingthe surface-isospin sector of EDF) are asym = 29MeV andassym = −29MeV.

6 Isospin physics and symmetry energy

The emergence of NSE is rooted in the isobaric symme-try and its breaking as a function of neutron excess andmass. Single-reference DFT is essentially the only frame-work allowing for understanding global behavior of isospin

Eur. Phys. J. A (2014) 50: 20 Page 11 of 13

effects throughout the entire nuclear landscape. While thenuclear interaction part of the nuclear EDF is constructedto be an isoscalar [24,25], the Coulomb interaction breaksisospin manifestly. There are, therefore, two differentsources of isospin symmetry breaking in the nuclear DFT:spontaneous isospin breaking associated with the self-consistent response to the neutron excess, and the explicitbreaking due to the electric charge of the protons [83].

Effects related to isospin breaking and restoration aredifficult to treat theoretically within the nuclear DFT. Be-low, we discuss two ways of dealing with this problem:isocranking and isospin projection.

6.1 1D and 3D isocranking

The isocranking model [84,80] attributes the kinetic co-efficient asym,kin contribution to the mean level spacingat the Fermi energy ε(A) rather than to the total kineticenergy itself. The SHF calculations also revealed that theisovector mean potential of the Skyrme EDF can be quitewell characterized by an effective VTT interaction (1) char-acterized by a strength parameter κ(A). The actual isovec-tor part of the Skyrme mean-field potential is composedof several terms [24,25]. As can be seen from table 1, inthe uniform NM limit, two terms contribute in SHF, Cρ

1ρ21

and Cτ1 ρ1τ1, and the NSE strength reads

asym =18

m

m∗ εFG +

[(3π2

2

)2/3

Cτ1 ρ

5/30 + Cρ

1ρ0

], (18)

where εFG is the average level splitting in FGM. There-fore, within this scenario, asym is non-trivially modifiedby momentum-dependent effects introducing, in the lead-ing order, the dependence of asym,kin and asym,int on theisoscalar and isovector effective mass, respectively.

Within the nuclear shell model, NSE appears througha contribution to the binding energy proportional toT (T + 1) [85]. However, the local enhancement ofbinding around N = Z (the Wigner energy) suggestan enhancement of the linear term to T (T + λ) withλ ≈ 1.26 [86–88]. Since the Wigner energy is neitherfully understood nor included properly within the SCMFmodels [89], the microscopic origin of λ is still a matter ofdebate. Within the isocranking model, the Fock exchange(isovector) potential gives rise to λ ≈ 0.5, at variancewith enhancement seen in experimental data. The Wignerenergy can be explained by shell-model calculations [89]in terms of configuration mixing. The Wigner term isusually associated with the isoscalar neutron-proton (np)pairing [90,25], but its understanding is poor as realisticcalculations involving simultaneous np mixing in both theparticle-hole (p-h) and particle-particle (p-p) channelshave not been carried out. It is only very recently that3D isocranking calculations including np mixing in thep-h channel have been reported [91]. This is the first steptowards developing the nuclear superfluid DFT includingnp mixing in both p-h and p-p channels. An improvedtreatment of isospin within the 3D isocranking will opennew opportunities for quantitative studies of isobaricanalogue states and, in turn, the NSE.

theory experiment

3

2

0

1

4

E (

MeV

)

1

1

1

0

0

4622, 4636

T

32S

Fig. 8. Energies of Iπ = 1+ states in 32S normalized to the iso-baric analogue state Iπ = 1+, T = 1. The results of projectedSHF-SkV calculations involving configuration mixing [94] (left)are compared to experiment (right). The calculations are basedon 24 I = 1+ states projected from 6 HF determinants repre-senting low-lying 1p-1h configurations.

6.2 Isospin projected DFT

The isospin and isospin-plus-angular-momentum pro-jected DFT models have been developed recently to de-scribe isospin mixing effects. These new tools open newavenues to probe NSE. To gain insight on this line of mod-els, it is instructive to to consider the spontaneous isospinsymmetry breaking effect in the so-called anti-aligned p-h configurations in N = Z nuclei, which are mixtures ofT = 0 and T = 1 states [92]. Restoration of the isospinsymmetry results in the energy splitting, ΔET , betweenthe actual T = 0 and T = 1 configurations. Since thesestates are projected from a single mean-field determinant,the splitting is believed to be insensitive to kinematics,and the method can be used to probe dynamical effectsgiving rise to the interaction term asym,int. The results ofSHF calculations [92] performed in finite nuclei confirmthat asym,int is indeed correlated with the isoscalar effec-tive mass in agreement with the NM relation (18).

The isospin and isospin plus angular momentum pro-jected DFT were designed and applied to study the isospinimpurities [83] and isospin symmetry breaking correctionsto the superallowed 0+ → 0+ β-decay rates [93]. Unfor-tunately, the calculations show that these two observablesare not directly correlated with the symmetry energy. Am-biguities associated with these calculations stimulated fur-ther development of the formalism in the direction of theResonating-group method. The scheme proceeds in threesteps. i) First, a set of low-lying (multi)p-(multi)h SHFstates {Φi} is calculated. These states form a basis fora subsequent projection. ii) Next, the projection tech-niques are applied to calculate a family {Ψ (α)

I } of goodangular momentum states with properly treated K-mixingand isospin mixing. iii) Finally, a configuration mixing of{Ψ (α)

I } states is performed using techniques suitable fornon-orthogonal ensambles.

Page 12 of 13 Eur. Phys. J. A (2014) 50: 20

Although at present the calculations can be realizedonly for the SkV EDF, the preliminary results [94] are en-couraging, as shown in fig. 8. Since the projected approachtreats rigorously the angular momentum conservation andthe long-range polarization due to the Coulomb force, itopens up a possibility of detailed studies of the isovectorterms of the nuclear EDF that are sources of the NSE.

7 Conclusions

This paper surveys various aspects of NSE within the nu-clear DFT represented by non-relativistic and relativisticself-consistent mean-field frameworks. After defining themodels and statistical tools, we reviewed key observablespertaining to bulk nucleonic matter and finite nuclei. Us-ing the statistical covariance technique, constraints on thesymmetry energy were studied, together with correlationsbetween observables and symmetry energy parameters.

Through the systematic correlation analysis, we scruti-nized various observables from finite nuclei that are acces-sible by current and future experiments. We confirm thatby far the most sensitive isovector indicators are observ-ables related to the neutron skin (neutron radius, diffrac-tion radius, weak charge form factor) and the dipole polar-izability [14,57]. In this context, PREX-II measurementof the neutron skin in 208Pb [95] (a follow-up measure-ment to PREX [49] designed to improve the experimen-tal precision), CREX measurement of the neutron skin in48Ca [96], and on-going measurements of αD in neutron-rich nuclei [97] are indispensable.

The masses of heavy neutron-rich nuclei also seem tocorrelate well with NSE parameters. Other observables,such as Qα-values, δVpn, barrier heights, and low-energydipole strength [14,59,60] are too strongly impacted byshell effects to be useful as global isovector indicators.

A major challenge is to develop the universal nuclearEDF with improved isovector properties. Various improve-ments are anticipated in the near future. Those includeconstraining the EDF at sub-saturation densities usingab initio models [98,99] and using the density matrix ex-pansion to develop an EDF based on microscopic nuclearinteractions [100]. This work will be carried out underthe Nuclear Low Energy Computational Initiative (NU-CLEI) [101]. Other exciting avenues are related to multi-reference isospin projected DFT, which will enable us tomake reliable predictions for isobaric analogues, isospinmixing, and mirror energy differences.

This work was supported by the U.S. Department of Energyunder Contract No. DE-FG02-96ER40963 (University of Ten-nessee), No. DE-SC0008499 (NUCLEI SciDAC Collaboration);by BMBF under Contract No. 06 ER 142D; and by NCN underContract No. 2012/07/B/ST2/03907

Open Access This is an open access article distributedunder the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/4.0), whichpermits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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