# On the nuclear symmetry energy and the neutron skin in neutron rich nuclei

A.E.L. Dieperink KVI, Zernikelaan 25, NL-9747AA Groningen, The Netherlands    Y. Dewulf    D. Van Neck    M. Waroquier Laboratory of Theoretical Physics, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium    V. Rodin Institut für Theoretische Physik der Universität Tübingen D-72076 Tübingen, Germany
March 6, 2022
###### Abstract

The symmetry energy for nuclear matter and its relation to the neutron skin in finite nuclei is discussed. The symmetry energy as a function of density obtained in a self-consistent Green function approach is presented and compared to the results of other recent theoretical approaches. A partial explanation of the linear relation between the symmetry energy and the neutron skin is proposed. The potential of several experimental methods to extract the neutron skin is examined.

Symmetry energy, neutron skin, isovector giant resonances
###### pacs:
21.10.Gv,21.30.Fe,21.65.+f

## I Introduction

The nuclear symmetry energy plays a central role in a variety of nuclear phenomena. It determines to a large extent the equation of state (EoS) and the proton fraction of neutron stars Latt , the neutron skin in heavy nuclei Furn , it enters as an input in the analysis of heavy ion reactions Li ; Bao , etc. Its value at nuclear saturation density, fm 30 MeV, seems reasonably well established, both empirically as well as theoretically; still different parametrizations of relativistic mean field (RMF) models (which fit observables for isospin symmetric nuclei well) lead to a relatively wide range of predictions for the symmetry energy, 24-40 MeV. However, predictions for its density dependence show a substantially larger variation.

Recently it has been pointed out by several authors Brown ; Furn ; Bodmer that there exists a strong correlation between the neutron skin, and the density derivative of the EoS of neutron matter near saturation density. Subsequently in a more detailed analysis in the framework of a mean field approach Furnstahl Furn demonstrated that in heavy nuclei there exists an almost linear empirical correlation between theoretical predictions in terms of various mean field approaches to (i.e., a bulk property) and the neutron skin, (a property of finite nuclei).

This observation has contributed to a renewed interest in an accurate determination of the neutron skin in neutron rich nuclei for several reasons. First precise experimental information on the neutron skin could help to further constrain interaction parameters that play a role in the calculation of the symmetry energy Horo01 . Furthermore a precise value of the neutron skin is required as an input in several processes of physical interest, e.g. the analysis of energy shifts in deeply bound pionic atoms Kolo , and in the analysis of atomic parity violation experiments (weak charge) Poll99 . It has been shown that the calculated symmetry energy is quite insensitive to details of modern realistic NN interactions Engvik . On the other hand the symmetry energy and in particular its density dependence can vary substantially with the many-body approximations employed. For instance the results of lowest order BHF and variational calculations do not seem to agree well.

The aim of this paper is threefold: First we address the sensitivity of the symmetry energy to various many-body approximations. To this end we present results of a calculation of using the self-consistent Green function (SCGF) approach and compare the result with various other theoretical approaches. Secondly we will try to provide some new insight in the origin of the “Furnstahl” relation; for this purpose we use the Landau-Migdal effective interaction in the mean field approximation. Finally in view of the large variety of existing and proposed experimental methods to determine the neutron skin we examine the merits of some recently proposed methods that seem to be of potential interest to provide more accurate information on the neutron skin in the near future.

Section II is devoted to an overview of theoretical approaches to the symmetry energy, and a new calculation in terms of the self-consistent Green function approach is presented. In Section III an interpretation of the Furnstahl relation is presented in terms of the Landau-Migdal approach. Section IV contains an overview of various experimental methods to deduce information on the neutron skin and Section V contains a short discussion of implications for other physical processes where the information on the neutron skin is required as an input.

## Ii The symmetry energy in nuclear matter

The symmetry energy is defined in terms of a Taylor series expansion of the energy per particle for nuclear matter in terms of the asymmetry (or equivalently the proton fraction ),

 E(ρ,α)=E(ρ,0)+S(ρ)α2+O(α4)+…. (1)

It has been shown Zuo ; Zuo02 that deviations from the parabolic law in Eq.(1), i.e. terms in are quite small.

Near the saturation density the energy of isospin-symmetric matter and the symmetry energy can be expanded as

 E(ρ,0)=E0+K18ρ20(ρ−ρ0)2+…, (2)

and

 S(ρ)=12∂2E(ρ,α)∂α2|α=0=a4+p0ρ20(ρ−ρ0)+ΔK18ρ20(ρ−ρ0)2+…. (3)

The parameter is the symmetry energy at equilibrium and the slope parameter governs the density dependence.

The relevance of the nuclear matter results in part depends on the question whether there is a surface contribution to the symmetry energy for finite nuclei. In ref.Oya it was found that the latter is of minor importance, which has also been confirmed in Furn .

### ii.1 Self-consistent Green function and Brueckner approach

In this section we describe the calculation of the symmetry energy in the SCGF approach. Since the latter can be considered as a generalization of the lowest order Brueckner-Hartree-Fock (BHF) method we start with a brief discussion of the symmetry energy in the latter case.

#### ii.1.1 Symmetry energy in BHF

In the Brueckner-Hartree-Fock approximation, the Brueckner-Bethe-Goldstone (BBG) hole-line expansion is truncated at the two hole-line level. The short-range NN repulsion is treated by a resummation of the particle-particle ladder diagrams into an effective interaction or -matrix. Self-consistency is required at the level of the BHF single-particle spectrum ,

 ϵBHF(k)=k22m+∑k′. (4)

In the standard choice BHF the self-consistency requirement (4) is restricted to hole states () only, while the free spectrum is kept for particle states . The resulting gap in the s.p. spectrum at is avoided in the continuous-choice BHF (ccBHF), where Eq. (4) is used for both hole and particle states. The continuous choice for the s.p. spectrum is closer in spirit to many-body Green’s function perturbation theory. Moreover, recent results indicate Bal01 ; Bal00 that the contribution of higher-order terms in the hole-line expansion is considerably smaller if the continuous choice is used.

The BHF energy per nucleon can be easily evaluated for both symmetric nuclear matter (SNM) and pure neutron matter (PNM) using the energy sum-rule,

 EA=dρ∫d3k(2π)3(k22m+ϵBHF(k))θ(kF−k), (5)

where is the density and is the isospin degeneracy ( for PNM (SNM)). The symmetry energy is obtained as the difference between PNM and SNM energies for the same density.

We first performed ccBHF calculations with the Reid93 interaction, including partial waves with in the calculation of the G-matrix. The results are presented in Fig.1, where the symmetry energy is decomposed into various contributions as suggested in ref.Zuo and shown as a function of nucleon density . Figure 1: The symmetry energy S (full line) and the contributions to S from the kinetic (dash-dotted line) and potential energy (dashed line), calculated within a ccBHF scheme and using the Reid93 interaction. Also shown (dotted lines) are the T=0 and T=1 components of the potential energy contribution.

The contribution of the kinetic energy to the BHF symmetry energy is given by the free Fermi-gas expression 111This expression differs from the standard one, which is based upon the derivative rather than the finite difference.

 Skin=Ekin,PNM−Ekin,SNM=310m(3π2)23ρ23(1−1223), (6)

and it determines to a large extent the density dependence of . In Fig.1 we also show the symmetry potential , which is much flatter, and the contributions to from both the isoscalar () and isovector () components of the interaction. Over the considered density range is dominated by the positive part. The partial waves, containing the tensor force in the - channel which gives a major contribution to the potential energy in SNM, do not contribute to the PNM energy. The contribution peaks at fm. The decrease of this contribution at higher densities is compensated by the increase of the potential energy, with as a net result a much weaker density-dependence of the total potential energy. (It should be noted that inclusion of a three-nucleon interaction in general leads to a substantial increase for the slope parameter Zuo02 .)

#### ii.1.2 Symmetry energy in SCGF approach

In recent years several groups have considered the replacement of the BBG hole-line expansion with self-consistent Green’s function (SCGF) theory Ramos ; Bozek ; Dick ; Dewulf01 ; Dewulf02 ; Dewulf03 . In Dewulf02 ; Dewulf03 the binding energy for SNM was calculated within the SCGF framework and using the Reid93 potential. In the present paper we have extended these calculations to PNM and calculated the corresponding symmetry energy. Details of a technical nature can be found in Dewulf02 .

A SCGF calculation differs in two important ways from a BHF- calculation. Firstly, within SCGF particles and holes are treated on an equal footing, whereas in BHF only intermediate particle states are included in the ladder diagrams. This aspect ensures thermodynamic consistency, e.g. the Fermi energy or chemical potential of the nucleons equals the binding energy at saturation (i.e., it fulfills the Hugenholz-van Hove theorem). In the low-density limit BHF and SCGF coincide. As the density increases the phase space for hole-hole propagation is no longer negligible, and this leads to an important repulsive effect on the total energy. Secondly, the SCGF generates realistic spectral functions, which are used to evaluate the effective interaction and the corresponding nucleon self-energy. The spectral functions include a depletion of the quasi-particle peak and the appearance of single-particle strength at large values of energy and momentum, in agreement with experimental information from reactions. This is in contrast with the BHF approach where all s.p. strength is concentrated at the BHF-energy as determined from Eq. (4).

In a SCGF approach the particle states (), which are absent in the BHF energy sum rule of Eq. (5), do contribute according to the energy sum rule

 EA=dρ∫d3k(2π)3∫εF−∞dω (k22m+ω)Sh(k,ω), (7)

expressed in terms of the nucleon spectral function .

The results for the ccBHF and SCGF calculations for both SNM and PNM are compared in the left and central panels of Fig.2 for the Reid93 interaction. The inclusion of high-momentum nucleons leads roughly to a doubling of the kinetic and potential energy in SNM, as compared to BHF. As seen in Fig.2, the net result for the total energy of SNM is a repulsive effect, increasing with density Dewulf02 . This leads to a stiffer equation of state, and a shift of the SNM saturation density towards lower densities. The above effects are dominated by the tensor force (the isoscalar partial wave). Consequently, the effects are much smaller in PNM.

The corresponding symmetry energy, shown in the right panel of Fig.2, is dominated by the shift in the total energy for SNM, and lies below the ccBHF symmetry energy in the entire density-range. At fm the symmetry energy parameter is reduced from 28.9 MeV to 24.9 MeV, while the slope remains almost the same (from 2.11 MeVfm to 1.99 MeVfm). Figure 2: The total-energy per particle for symmetric nuclear matter (left panel) and pure neutron matter (central panel) for the Reid93-interaction. The dashed line refers to a ccBHF calculation, the full line to a SCGF calculation. The right panel displays the symmetry energy in these two approaches.

### ii.2 Comparison of symmetry energy in other approaches

#### ii.2.1 Calculations with realistic NN forces

Engvik et al. Engvik have performed lowest-order Brueckner-Hartree-Fock calculations in SNM and PNM for all “modern” potentials (CD-Bonn, Argonne v18, Reid93, Nijmegen I and II), which fit the Nijmegen NN scattering database with high accuracy. They concluded that for small and normal densities the symmetry energy is largely independent of the interaction used, e.g. at the values of vary around an average value of =29.83 MeV by about 1 MeV. At larger densities the spread becomes larger; however, the symmetry energy keeps increasing with density, in contrast to some of the older potentials like Argonne v14 and the original Reid interaction (Reid68) for which tended to saturate at densities larger than fm. Some insight into the microscopic origin of the symmetry potential was obtained by Zuo et al. Zuo who decomposed the symmetry energy into contributions from kinetic and potential energy. The BHF calculations in Zuo used the Argonne v14 and the separable Paris interaction. In Fig.1 we showed that the use of the modern Reid93 potential leads to essentially the same conclusions.

Detailed studies for SNM and PNM using variational chain summation techniques were performed by Wiringa et al. Wir for the Argonne Av14 and Urbana Uv14 NN interaction, in combination with the Urbana UVIII three-nucleon interaction (TNI), and by Akmal et al. Akmal for the modern Av18 NN potential in combination with the UIX-TNI. Results for and , extracted from Wir and Akmal , are shown in Table 1. The inclusion of TNI stiffens the EoS for both SNM and PNM, and increases considerably the value of the symmetry energy and its slope at the empirical density . The effect of including a relativistic boost correction (in combination with a refitted TNI) on the values of and is sizeable as well.

The symmetry energy has also been computed in the Dirac-Brueckner-Hartree-Fock (DBHF) approach Lee ; Toki . In relativistic approaches the symmetry energy generally is found to increase almost linearly with density, and more rapidly than in the non-relativistic case. This difference can be attributed to two effects. First the covariant kinetic energy which is inversely proportional to is larger because of the decreasing Dirac mass, , with increasing density. Secondly the contribution from rho-exchange appears to be larger than in the non-relativistic case Lee .

#### ii.2.2 Mean-field approach using effective interactions

Since the Furnstahl relation has been verified mainly in terms of mean-field models we discuss some results obtained in these approaches, which in general are based upon a parameterized effective interaction.

Brown Brown has investigated proton and neutron radii in terms of the non-relativistic Skyrme Hartree-Fock (SHF) model. First he noted that a certain combination of parameters in the SHF is not determined well by a fit to ground state binding energies, and that a wide range of predictions for the EoS for PNM is obtained. He also pointed out a direct correlation between the derivative of the neutron matter EoS (i.e., basically the symmetry energy coefficient ) and the neutron skin in Pb.

Covariant approaches are in general based upon either a covariant lagrangian with and exchange (and possibly other mesons) Ring ; Horo02 , or on the use of contact interactions Furn , solved as an energy density functional (EDF) in the Hartree-Fock approximation. Sets of model parameters are determined by fitting bound state properties of nuclei. Specifically the isovector degree of freedom is determined by the exchange of isovector mesons; in case of rho-meson exchange the (positive definite) contribution to is given by

 a4=k2F6√m∗2+k2F+g2ρ8m2ρρ0, (8)

and its potential energy contribution to , which scales with that for , is Furn . Typical values obtained for are around 4-6 MeV fm, and 30-36 MeV, i.e. considerably larger than in non-relativistic approaches (a large part of the enhancement can be ascribed to the fact that the kinetic contribution is larger, because ). Recently in Liu02 ; Greco this approach was extended by inclusion of the isovector-scalar partner, of the isoscalar scalar meson. Because of the presence of the Lorentz factor in the scalar potential contribution, which decreases with increasing density its inclusion leads to an even larger net value for Liu02 ; Greco .

Since explicit pion exchange is usually not included in the mean field approaches it is difficult to make a meaningful comparison with microscopic ones. In fact it can be argued that in contrast to isoscalar properties the long-range pion exchange could play an essential role in determining the isovector properties Furn .

#### ii.2.3 Effective field theory

Recently the density dependence of the symmetry energy has been computed in chiral perturbation effective field theory, described by pions plus one cutoff parameter, to simulate the short distance behavior Kais . The nuclear matter calculations have been performed up to three-loop order; the resulting EoS is expressed as an expansion in powers of and the value of GeV is adjusted to the empirical binding energy per nucleon. The value obtained in this approach for MeV is in remarkable agreement with the empirical one; at higher densities ( fm) a downward bending is predicted. However, in its present form the validity of this approach is clearly confined to relatively small values of the Fermi momentum, i.e. rather low densities. It is interesting to note that there are relatively small (large) contributions to coming from one-pion exchange Fock diagram (three-loop diagrams with either two or three medium insertions).

#### ii.2.4 Comparison

To summarize the present status in Fig.(3) various results of the approaches for discussed above are compared. As noted above one sees that the covariant models yield a much larger increase of with the density than the non-relativistic approaches. The LOBHF leads to a higher value of than both variational and the SCGF method which include more correlations; that the SCGF result is close to the variational approach may be fortuitous. Effects from three-body forces are not included. Figure 3: Overview of several theoretical predictions for the symmetry energy S: Brueckner-Hartree-Fock (continuous choice) with Reid93 potential (circles), self-consistent Green function theory with Reid93 potential (full line), variational calculation from Wir with Argonne Av14 potential (dashed line), Dirac-Brueckner-Hartree-Fock calculation from Lee (triangles), relativistic mean-field model from Liu02 (squares), effective field theory from Kais (dash-dotted line).

## Iii Relationship between symmetry energy and ΔR

Brown Brown and Furnstahl Furn have pointed out that within the framework of mean field models there exists an almost linear empirical correlation between theoretical predictions for both and its density dependence, and the neutron skin, in heavy nuclei. This observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems.
Here we wish to address this question from a different point of view, namely in the spirit of Landau-Migdal approach. Let us consider a simple mean-field model (see, e.g., Gor00 ) with the Hamiltonian consisting of the single-particle mean field part and the residual particle-hole interaction :

 ^H=^H0+^Hp−h, ^H0=∑a(Ta+U(xa)), (9) U(x)=U0(x)+U1(x)+UC(x), (10) U0(x)=U0(r)+Uso(x); U1(x)=12Spot(r)τ(3); UC(x)=12UC(r)(1−τ(3)),x=(r,σ,τ)

Here, the mean field potential includes the phenomenological isoscalar part along with the isovector and the Coulomb parts calculated consistently in the Hartree approximation; and are the central and spin-orbit parts of the isoscalar mean field, respectively; is the symmetry potential (the potential part of the symmetry energy).
In the Landau-Migdal approach the effective isovector particle-hole interaction is given by

 ^Hph=∑a>b(F′+G′→σa→σb)→τa→τbδ(→ra−→rb), (11)

where and are the phenomenological Landau-Migdal parameters.

The model Hamiltonian in Eq.(10) preserves isospin symmetry if the condition

 [^H,^T(−)]=^U(−)C, (12)

is fulfilled, where With the use of Eqs. (10),(11),(12) the condition eq.(12) in the RPA formalism leads to a selfconsistency relation between the symmetry potential and the Landau parameter Bir74 :

 Spot(r)=2F′n(−)(r), (13)

where is the neutron excess density. Thus, in this model the depth of the symmetry potential is controlled by the Landau-Migdal parameter ( analogously to the role played by the parameter in relativistic mean field models).

is obtained from Eq.(13) by an iterative procedure; the resulting dependence of on the dimensionless parameter MeV fm) shown in fig. 4 indeed illustrates that depends almost linearly on Then with the use of the Migdal relation Migd which relates symmetry energy and

 a4=ϵF3(1+2f′), (14)

a similar almost linear correlation between the symmetry energy, and the neutron skin is obtained. Figure 4: Neutron skin in 208Pb vs. the Landau-Migdal parameter f′.

To get more insight in the role of we consider small variations . Neglecting the variation of with respect to one has a linear variation of the symmetry potential: . Then in first order perturbation theory, such a variation of causes the following variation of the ground-state wave function

 |δ0⟩=δF′∑s⟨s|^N(−)|0⟩E0−Es|s⟩, (15)

with labeling the eigenstates of the nuclear Hamiltonian and a single-particle operator . Consequently the variation of the expectation value of the single-particle operator with can be written as

 Rpδ(ΔR)=δF′2A∑sRe⟨0|^N(−)|s⟩⟨s|^V(−)|0⟩E0−Es. (16)

In practice the sum in Eq. (16) is exhausted mainly by the isovector monopole resonance of which the high excitation energy (about 24 MeV in Pb) justifies the perturbative consideration. We checked that Eq. (16) is able to reproduce directly calculated shown in fig. 4 with the accuracy of about 10%. As a result a simple microscopic interpretation of the linear correlation between the neutron skin thickness and Landau parameter is obtained.

## Iv Experimental methods to determine ΔR

A variety of experimental approaches have been explored in the past to obtain information on To a certain extent all analysis contain a certain model dependence, which is difficult to estimate quantitatively. It is not our intention to present a full overview of existing methods for the special case of Pb. In particular the results obtained in the past from the analysis of elastic scattering of protons and neutrons have varied depending upon specifics of the analysis employed. At present the most accurate value comes from a recent detailed analysis of the elastic proton scattering reaction at GeV Clark03 , and of neutron and proton scattering at MeV Karat02 . For details we refer to these papers. Here we restrict ourselves to a discussion of the some less well known methods that have the potential to provide more accurate information on the neutron skin in the future. In particular the excitation of isovector giant resonances through the restoring force contain information about the symmetry energy.

### iv.1 Isovector giant resonances

We begin with a brief overview of the study of excitation of isovector giant resonances. Sum rules for the latter contain direct information on .

#### iv.1.1 Giant Dipole Resonance (GDR)

In the past the excitation of the isovector giant dipole resonance (GDR) with isoscalar probes has been used to extract Kras94 . In the DWBA optical model analysis of the cross section the neutron and proton transition densities are needed as an input. In the Goldhaber-Teller picture

with the oscillation amplitude and . Assuming ground state neutron and proton distributions of the form

 ρi(r)=12(1±x∓γx)ρ(r−c(1±γx/3)). (18)

While for the transition density vanishes, for the isovector transition density is finite

 Δg(r)=κγN−ZA(dρdr+c3d2ρdr2),

where is related to
Excitation of the GDR by alpha particle scattering (isoscalar probe) the corresponding transition optical potential is given by

 ΔUtr=κγN−ZA(dUdr+R0d2U03dr2). (19)

By comparing the experimental cross section with the theoretical one (calculated as a function of the ratio ) the value of can be deduced.

It is difficult to make a quantitative estimate of the uncertainty in the result coming from the model dependence of the approach. In the analysis several assumptions must be made, such as the radial shape of the density oscillations and about the actual values of the optical model parameters.

#### iv.1.2 Spin-dipole Giant Resonance

Recently it has been proposed to utilize the excitation of the spin-dipole resonance, excited in charge exchange reactions, to determine the neutron skin; in fact the method has been applied to obtain information on the variation of the neutron skin in the Sn isotopes with isotope number Kras99 . For the relevant operator, the summed strength is

 S(−)−S(+)=C(NR2n−ZR2p). (20)

Here and are the spin-dipole total strengths in and channels, respectively; is the factor depending on the definition of the spin-dipole operator (in the definition of Ref.  , we use here ). Because could not be measured experimentally, the model-dependent energy-weighted sum rule was invoked in the analysis to eliminate . Let put (that seems to be a very good approximation for Pb) and ask the question what experimental accuracy for is needed to determine the neutron skin to a given accuracy. With

 S(−)=(N−Z)R2p+2NRpΔR (21)

the ratio of the second term on the rhs to the first one in case of Pb is

 2NΔR/((N−Z)Rp)≈5.7ΔR/Rp.

Therefore, for fm and fm the second term is only 25% of the first one and one needs 3% accuracy in to determine with 10% accuracy. Because the SDR strength is spread out and probably has a considerable strength at low-energy the results for the can be only considered as qualitative with a relatively large uncertainty (of order 30-50%).

#### iv.1.3 Isobaric analogue state

The dominant contribution to the energy weighted sum rule (EWSR) for Fermi excitations by the operator comes from the Coulomb mean field Auer72

 (EWSR)F=∫UC(r)n(−)(r)d3r, (22)

The Coulomb mean field resembles very much that of the uniformly charged sphere, being inside a nucleus a quadratic function: . It turns out that if one extends such a quadratic dependence also to the outer region (instead of proportionality to ), it gives numerically just very small deviation in (less than 0.5%, due to the fact, that the difference and its first derivative go to 0 at and is exponentially decreasing at ). Using such an approximation, one gets:

 (EWSR)F≈(N−Z)ΔC(1−S(−)3(N−Z)R2c) (23)

with , and given in Eq.(21).

Since the IAS exhausts almost 100% of the NEWSR and EWSR, one may hope to extract from the IAS energy. However, the term depending on contributes only about 20% to , and as a result, the part of depending on contributes only about 4% to (in Pb). ¿From the experimental side, the IAS energy can be determined with unprecendently high accuracy, better than 0.1%. Also, from the experimentally known charge density distribution the Coulomb mean field can be calculated rather accurately, and hence one can determine the small difference between Eqs.(23) and (22). But at the level of 1% accuracy several theoretical effects discarded in Eq.(22) come into play (see, e.g., Auer72 ) which makes the reliability of such a method questionable. On the other hand in a forth coming paper we will show that for an isotopic chain the excitation of the IAS can be used as a quantitative tool to obtain the variation of with neutron number.

### iv.2 Anti-protonic atoms

Recently neutron density distributions were deduced from anti-protonic atoms Trzc01 . The basic method determines the ratio of neutron and proton distributions at large differences by means of a measurement of the annihilation products which indicates whether the antiproton was captured on a neutron or a proton. In the analysis two assumptions are made. First a best fit value for the ratio of the imaginary parts of the free space and scattering lengths equal to unity is adopted. Second in order to reduce the density ratio at the annihilation side to a a ratio of rms radii a two-parameter Fermi distribution is assumed. The model dependence introduced by this assumptions is difficult to judge. Since a large number of nuclei have been measured one may argue that the value of is fixed empirically.

### iv.3 Parity violating electron scattering

Recently it has been proposed to use the (parity violating) weak interaction to probe the neutron distribution. This is probably the least model dependent approach Horopv . The weak potential between electron and a nucleus

 ~V(r)=V(r)+γ5A(r), (24)

where the axial potential The weak charge is mainly determined by neutrons

 ρW(r)=(1−4sin2θW)ρp(r)−ρn(r), (25)

with In a scattering experiment using polarised electrons one can determine the cross section asymmetry Horopv which comes from the interference between the and contributions. Using the measured neutron form factor at small finite value of and the existing information on the charge distribution one can uniquely extract the neutron skin. Some slight model dependence comes from the need to assume a certain radial dependence for the neutron density, to extract from a finite form factor.

## V Discussion of ΔR for 208Pb and some implications

In table 2 we present a summary of some recent results on in Pb. One sees that (with the exception of the analysis of proton and neutron pickup reactions in terms of mean field orbitals in Mairle ) all recent results are consistent with fm. Therefore it appears that the data agree with the result of conventional Skyrme model approach but seem to disagree with the results of the RMF models considered in Furn . On the basis of the correlation plot between and shown in Furn on would then conclude that a small value for MeV/fm is preferred over larger values prdedicted in RNF approaches.

In several processes of physical interest knowledge of plays a crucial role and in fact a more accurate value could lead to more stringent tests:

(i) The pion polarization operator Kolo (the s-wave optical potential) in a heavy nucleus has mainly an isovector character ( Parameterizing the densities by Fermi shapes for the case of Pb the main nuclear model dependence in the analysis comes from the uncertainty in the value of multiplying

(ii) Parity violation in atoms is dominated by boson exchange between the electrons and the neutrons Poll99 ; Derev . Taking the proton distribution as a reference there is a small socalled neutron skin (ns) correction to the parity non-conserving amplitude, for, say, a transition, which is related to as Derev (independent of the electronic structure)

 δEnspncEpnc=−37(αZ)2ΔRRp. (26)

In Cs it amounts to a depending on whether the non-relativistic or relativistic estimates for are used Poll99 . The corresponding uncertainty in the weak charge is

(iii) The pressure in neutron star matter can be expressed as in terms of the symmetry energy and its density dependence Latt

 P(ρ,x)=ρ2∂E(ρ,x)∂ρ=ρ2[E′(ρ,1/2)+S′(ρ)(1−2x)2+…]. (27)

By using the beta equilibrium condition in a neutron star, and the result for the electron chemical potential, one finds the proton fraction near saturation density, to be quite small, Hence the pressure at saturation density can be approximated as

 P(ρ0)=ρ0(1−2x0)(ρ0S′(ρ0)(1−2x0)+S(ρ0)x0)∼ρ20S′(ρ0). (28)

At higher densities the proton fraction increases; this increase is more rapid in case of larger Horo02 . While for the pressure at higher densities contributions from other nuclear quantities like compressibility will play a role in Latt it was argued that that there is a correlation of the neutron star radius and the pressure which does not depend on the EoS at the highest densities. Numerically the correlation can be expressed in the form of a power law, km, where This shows that a determination of a neutron star radius would provide some constraint on the symmetry properties of nuclear matter.

## Vi Conclusion

In this paper we have discussed the bulk symmetry energy, and compared various approaches to compute it as a function of density. Because the tensor interaction plays an important role the symmetry energy is sensitive to details of the treatment of the many-body correlations. It was shown that the self-consistent Green function approach in which more correlations are included than in lowest order BHF leads to a smaller value of the symmetry energy. The relatively large values for obtained in the relativistic mean field approach can be associated with an effective mass effect.

We showed that the phenomenological almost linear relationship between symmetry energy and neutron skin in finite nuclei observed in mean field calculations could be understood in terms the Landau-Migdal approach. Finally we compared several experimental tools of potential interest for the determination of the neutron skin.

Acknowledgement
This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) with financial support from the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO). Y. Dewulf acknowledges support from FWO-Vlaanderen.
The authors would like to thank Prof. M.Urin for several clarifying remarks.
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