Neutron Skin and Giant Resonances in Nuclei
Vadim Rodin
Institute for Theoretical Physics, University of Tübingen
Auf der Morgenstelle 14, D-72076 Tübingen, Germany
November 1, 2018
Abstract
Some aspects, both experimental and theoretical, of extracting the neutron skin ∆R from
properties of isovector giant resonances are discussed. Existing proposals are critically reviewed.
The method relying on the energy difference between the GTR and IAS is shown to lack sensitivity
to ∆R. A simple explanation of the linear relation between the symmetry energy and the neutron
skin is also presented.
1 Introduction
Accurate experimental data on the neutron skin in neutron rich nuclei would allow to further constrain
model parameters involved in the calculations of the nuclear symmetry energy [1]. The latter plays
a central role in a variety of nuclear phenomena. The value a4 ≈ 30 MeV of the nuclear symmetry
energy S(ρ0) = a4+
(ρ− ρ0)+ . . . at nuclear saturation density ρ0 ≈ 0.17 fm
−3 seems reasonably well
established. On the other hand, the density dependence of the symmetry energy can vary substantially
with the many-body approximations employed.
Several authors have pointed out [2, 3] a strong correlation between the neutron skin, ∆R =
〈r2〉n−
〈r2〉p = Rn−Rp, and the symmetry energy of neutron matter near saturation density. In the framework
of a mean field approach Furnstahl [3] demonstrated that in heavy nuclei there exists an almost linear
empirical correlation between theoretical predictions in terms of various mean field approaches to S(ρ)
(i.e., a bulk property) and the neutron skin, ∆R (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. Besides, 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 [4], and
in the analysis of atomic parity violation experiments (weak charge) [5]. It is worth to stress that to
experimentally determine the skin in heavy nuclei is extremely challenging as ∆R is just about few
percents of the nuclear radius.
The present contribution is partially based upon the results published previously in [6].
2 Relationship between the symmetry energy and ∆R
Brown [2] and Furnstahl [3] have pointed out that within the framework of mean field models there
exists an almost linear empirical correlation between theoretical predictions for both a4 and its density
http://arxiv.org/abs/0704.0180v1
dependence, p0, and the neutron skin ∆R 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, following the analysis of [6], this question is addressed from a point of view of the Landau-Migdal
approach.
Let us consider a simple mean-field model with the Hamiltonian consisting of the single-particle
mean field part Ĥ0 and the residual particle-hole interaction Ĥp−h:
Ĥ = Ĥ0 + Ĥp−h, Ĥph =
(F ′ +G′~σa~σb)~τa~τbδ(~ra − ~rb), (1)
Ĥ0 =
(Ta + U(xa)), U(x) = U0(x) + U1(x) + UC(x), (2)
U0(x) = U0(r) + Uso(x); U1(x) =
Spot(r)τ
(3); UC(x) =
UC(r)(1− τ
(3)). (3)
Here, U0(x) is the phenomenological isoscalar part of the mean field potential U(x) (x = {~r, ~σ, ~τ}),
U0(r) and Uso(x) are the central and spin-orbit parts, respectively; F
′ and G′ are the phenomenological
Landau-Migdal parameters. The isovector part U1(x) and the Coulomb mean field UC(x) are both
calculated consistently in the Hartree approximation, Spot(r) is the symmetry potential (r-dependent
symmetry energy in finite nuclei).
The model Hamiltonian Ĥ (1) preserves the isospin symmetry within the RPA if a selfconsistency
relation between the symmetry potential and the Landau-Migdal parameter F ′ is fulfilled:
Spot(r) = 2F
′n(−)(r), (4)
where n(−)(r) = nn(r)− np(r) is the neutron excess density. Thus, in this model the depth of the sym-
metry potential is controlled by the Landau-Migdal parameter F ′ (analogous role plays the parameter
g2ρ in relativistic mean field models). Spot(r) is obtained from Eq.(4) by an iterative procedure; the
resulting dependence of ∆R on the dimensionless parameter f ′ = F ′/(300 MeV fm3) shown in fig. 1
indeed illustrates that ∆R depends almost linearly on f ′. Then with the use of the Migdal relation
(1 + 2f ′) [7] relating the symmetry energy and f ′, a similar, almost linear, correlation between
a4 and ∆R is obtained.
To get more insight in the role of f ′ we consider small variations δF ′. Neglecting the varia-
tion of n(−)(r) with respect to δF ′, the corresponding linear variation of the symmetry potential
is δSpot(r) = 2δF
′n(−)(r). Then in the first order perturbation theory, such a variation of Spot
causes the following variation of the ground-state wave function |δ0〉 = δF ′
〈s|N̂(−)|0〉
E0−Es
|s〉, with “s”
labeling the eigenstates of the nuclear Hamiltonian and a single-particle operator N̂ (−) defined as
N̂ (−) =
n(−)(ra)τ
a . Consequently, the variation of the expectation value 〈0|V̂
(−)|0〉 = NR2n − ZR
of another single-particle operator V̂ (−) =
a can be written as
Rpδ(∆R) = δF
Re〈0|N̂ (−)|s〉〈s|V̂ (−)|0〉
E0 −Es
. (5)
In practice the sum in Eq. (5) is exhausted mainly by the isovector monopole resonance (IMR) which
high excitation energy (about 24 MeV in 208Pb) justifies the perturbative consideration. Eq. (5) is able
to reproduce directly calculated δ(∆R) shown in Fig. 1 with the accuracy of about 10%. As a result,
a simple microscopic interpretation of the linear correlation between ∆R and Landau parameter F ′ is
obtained.
0.6 0.8 1 1.2 1.4
Figure 1: Neutron skin in 208Pb versus the Landau-Migdal parameter f ′.
3 Extracting neutron skin from properties of isovector giant
resonances
Parity violating electron scattering off nuclei is probably the least model dependent approach to probe
the neutron distribution [8]. The weak electron-nucleus potential is Ṽ (r) = V (r) + γ5A(r), where
the axial potential A(r) = GF
ρW (r). The weak charge is mainly determined by neutrons ρW (r) =
(1 − 4 sin2 θW )ρp(r)− ρn(r), with sin
2 θW ≈ 0.23. In a scattering experiment using polarized electrons
one can determine the cross section asymmetry [8] which comes from the interference between the A and
V contributions. Using the measured neutron form factor at small finite value of Q2 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 Rn from a finite Q
2 form factor.
However, the best claimed accuracy of the experimental determination of neutron radii would be on
the level of 1%, that translates to relatively large uncertainty of 20-30% in the neutron skin. On such
accuracy level, some indirect experimental probes of ∆R still can be competitive.
A variety of experimental approaches have been employed to obtain indirect information on ∆R.
To some extent all the analysis contain a certain model dependence, which in many cases is difficult
to estimate quantitatively. For choosing an indirect probe it is very important to address the question
how sensitive is the proposed physical quantity with respect to a variation of ∆R in a single nucleus.
The higher is the sensitivity, the better is the choice of the correlation for the indirect deducing ∆R
from the measured values.
It is not intended here to give a comprehensive review of the existing methods. In particular, the
results from the analysis of the antiprotonic atoms, elastic proton and neutron scattering reactions, and
the pygmy dipole resonance are completely left out. Here, special emphasis will be put on proposals to
provide accurate information on the neutron skin from properties of isovector giant resonances.
3.1 Spin-dipole Giant Resonance
In [9] it has been proposed to utilize the excitation probability of the spin-dipole resonance in charge
exchange reactions for determining the neutron skin. The method has been applied to obtain information
on the variation of the neutron skin in the Sn isotopes [9]. For the relevant operator,
a [~σa ⊗
~ra]JM , (J = 0, 1, 2) the summed ∆L = 1 strength is
S(−) − S(+) = C(NR2n − ZR
p). (6)
Here S(−) and S(+) are the spin-dipole total strengths in β(−) and β(+) channels, respectively; C is the
factor depending on the normalization of the spin-dipole operator (in the definition of Ref. [2] C = 1/4π,
we use here C = 1). Because S(+) could not be measured experimentally, the model-dependent energy-
weighted sum rule was invoked in the analysis of [9] to eliminate S(+). However, the used analytical
representation for the sum rule was oversimplified and led in some cases, e.g. for 208Pb, to absurdly
negative S(+). In [10] another way was proposed, namely, to use for the analysis the ratio S(+)/S(−)
calculated within the pn-RPA. The parameterization of the RPA calculation results for tin isotopes in
the form
S(+)/S(−) = 0.388− 0.012(N − Z)
was used later in [11] to reanalyze the experimental data and led to a marked change in the extracted
∆R’s.
Let us now assess the experimental accuracy for S(−) needed to determine the neutron skin to a
given accuracy. Putting S(+) = 0 (that seems to be a very good approximation for 208Pb) and one has
S(−) = (N − Z)R2p + 2NRp∆R. (7)
The ratio of the second term on the rhs to the first one in case of 208Pb is
2N∆R/((N − Z)Rp) ≈ 5.7∆R/Rp.
Therefore, for Rp = 5.5 fm and ∆R = 0.2 fm the second term is only 25% of the first one and one needs
5% accuracy in S(−) to determine ∆R with 20% accuracy. Because the SD strength is spread out and
probably has a considerable strength at low-energy, the results for the ∆R can be only considered as
qualitative with a relatively large uncertainty (up to 30-50%).
3.2 Isobaric analogue state
The dominant contribution to the energy weighted sum rule (EWSR) for Fermi excitations by the
operator T (−) =
a comes from the Coulomb mean field
(EWSR)F =
UC(r)n
(−)(r)d3r, (8)
The Coulomb mean field UC(r) resembles very much that of the uniformly charged sphere, being inside
a nucleus a quadratic function: UC(r) =
(3 − (r/Rc)
2), r ≤ Rc. It turns out that if one extends
such a quadratic dependence also to the outer region r > Rc (instead of proportionality to Rc/r), it
gives numerically just a very small deviation in (EWSR)F (less than 0.5%, due to the fact, that both
the difference and its first derivative go to zero at r = Rc and n
(−)(r) is exponentially decreasing for
r > Rc). Using such an approximation, one gets:
(EWSR)F ≈ (N − Z)∆C
3(N − Z)R2c
with ∆C =
, and S(−) given in Eq.(7).
Since the IAS exhausts almost 100% of the NEWSR and EWSR, one may hope to extract S(−) from
the IAS energy. However, the term depending on S(−) contributes only about 20% to (EWSR)F , and as
a result, the part of S(−) depending on ∆R contributes only about 4% to (EWSR)F (in
208Pb). ¿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
UC(r) can be calculated rather accurately, and hence one can determine the small difference between
Eqs.(9) and (8). But at the level of 1% accuracy several theoretical effects discarded in Eq.(8) come
into play that makes such an accurate description of the IAS energy very difficult (the Nolen-Schiffer
anomaly).
Also in [12] it was stated that the Coulomb displacement energies (CDE) are sensitive to ∆R. A
gross estimate ∆R = 0.80(5)(N −Z)/A fm was obtained from a four-parameter fit of the experimental
Rp and observed mirror CDE’s. The authors claimed 127 keV to be the rms error of the fit, but they
assumed the nuclear wave functions calculated within the Nuclear Shell Model to be isospin pure. Thus,
the important effect of the Coulomb mixing of the IAS and the IMR was not taken into account, which
is known to decrease the IAS energy by a few percents. Therefore, the Nolen-Schiffer anomaly does not
seem to have been resolved yet.
3.3 Is the energy spacing between GTR and IAS a good candidate for
determining the neutron skin in isotopic chains?
In a recent paper [13] a proposal has been put forward to use the isotopic dependence of the energy
spacing, ∆E, between the Gamow-Teller resonance (GTR) and the IAS as a tool for determining the
evolution of the neutron skin in nuclei along an isotopic chain. Here, we would like to present some
physical arguments which question the physical relevance of this method.
The authors of [13] have used the fact that both functions, ∆R and ∆E, are monotonic functions
(increasing and decreasing, respectively) of the neutron excess (N−Z) to state that “isotopic dependence
of the energy spacings between the GTR and IAS provides direct information on the evolution of neutron
skin-thickness along the Sn isotopic chain”. Arguing in such a way one can find a correlation between
any two monotonic functions of a single physical parameter and plot them as a function of one another
like is done in Fig. 2 of [13] 1. However, it does not imply automatically a real physical correlation
between the functions which are determined also by many other model parameters which are kept fixed
while performing calculations (the calculations in [13] have been performed within the relativistic mean
field (RMF) and relativistic QRPA (RQRPA) approaches).
Again, the relevant question to be addressed is how sensitive is one physical quantity with respect to
a variation of another in a single nucleus? In other words, one has to evaluate what variation of ∆E
is produced by varying ∆R in a single nucleus. Imaging an extreme situation (which is actually not
far from reality) that ∆E were not sensitive to ∆R at all, one would get by varying ∆R a family of
different calculated dependences (like shown in the upper panel of Fig. 2 of [13]) which would give no
clue about the real dependence seen in nature.
Thus, it is quite important to understand the physical reasons which cause the energy splitting
between the GTR and the IAS. It is well-known that if the nuclear Hamiltonian possessed Wigner
SU(4) symmetry then the GTR and the IAS would be degenerate, ∆E = 0. In such a case any
variation of ∆R, not violating the symmetry, would not affect ∆E at all. However, it is also known
that the spin-isospin SU(4) symmetry is broken in nuclei. Hence, ∆E is determined by those terms in
1note that, to avoid confusion in comparing the measured and calculated dependences, the authors should have plotted
the experimental points in the upper panel as the function of the measured ∆R rather than calculated ∆R and should
have added the horizontal error bars to them reflecting the experimental uncertainty in ∆R shown in the lower panel.
the nuclear Hamiltonian which violate the symmetry. Their qualitative and semi-quantitative estimates
in terms of the energy weighted sum rules for the Gamow-Teller (EWSRGT ) and the Fermi (EWSRF )
excitations have been already known for more than 20 years (see, e.g., [14]). The analysis of these
authors as well as a quantitative analysis performed recently in [15] has shown that there are three
basic sources in the Hamiltonian which violate SU(4) symmetry and contribute to the difference of the
sum rules: spin-orbit mean field and both particle-particle and particle-hole residual charge-exchange
interactions. One sees that none of the sources explicitly refers to the symmetry potential, to which
∆R is especially sensitive.
An estimate of ∆E as ∆E =
EWSRGT − EWSRF
N − Z
can be calculated according to [15] in the
Sn isotopes. From the sources violating SU(4) symmetry, spin-orbit mean field represents the major
one and contributes about 5 MeV to the splitting. The contribution of the particle-hole interaction is
negative and about 1–2 MeV in the absolute value. The contribution of the particle-particle interaction
is rather difficult to evaluate (due to uncertainty in the strength of the spin-dependent particle-particle
interaction) but it seems to be of minor importance (very probably no more than 0.5 MeV, especially
for large (N − Z)) and can safely be neglected.
Now let us turn to the discussion of the sensitivity of the contributions to the variation of ∆R. We
could reproduce the corresponding analytical expressions from [15] explicitly, but it is enough for our
purpose just to mention that the dominating contribution to ∆E from the spin-orbit mean field is given
by its expectation value in the ground state and is determined basically only by the unfilled spin-orbit
doublets. This expectation value is completely insensitive to the variation of ∆R.
Within the Landau-Migdal approach described above, the particle-hole contribution
∆Eph =
2(G′ − F ′)
N − Z
(n(−)(r))2d3r (10)
is given by the product of the volume integral of the neutron excess density squared and the difference
of the p-h strengths G′ and F ′ [15]. In the SU(4)-symmetric limit one has G′ = F ′ and ∆Eph = 0
explicitly. Still, in this limit one has a freedom to choose different F ′ that produces a variation in ∆R,
similar to shown in fig 1. Therefore, as already mentioned, one can get no clue about the actual ∆R
from ∆E = 0 in the SU(4)-symmetric limit.
In a realistic situation G′ 6= F ′ (f ′ = 1.0 and g′ = 0.8 were taken in [15]), but ∆Eph depends only on
the difference G′ − F ′. One usually fixes G′ in order to reproduce the GTR energy in some nuclei (the
authors of [13] have followed this way, too) and possible information from ∆E about the absolute value
of F ′ is lost. Furthermore, one can a priori think that a degree of violation of the SU(4) symmetry
should be a sort of a fundamental property of the residual interaction.Therefore, the difference G′ − F ′
should stay more stable in different models as compared to some possible variation of F ′ producing
different ∆R.
Considering value of G′ − F ′ fixed, one can employ a simple model varying only ρn(r) to see how a
change of ∆R affects ∆Eph via variation of the neutron excess density n
(−)(r). A small variation of ρn(r)
can be approximately represented as δρn(r) = −
(3ρn(r) +R
dρn(r)
), where δRn is a change of the
rms neutron radius Rn, R is the nuclear radius (with R
n ≈ 0.6R
2). Assuming the proton and neutron
densities be constant inside a nucleus, the final estimate is
δ∆Eph
N + Z − 2γN
N − Z
), where
γ = n(−)(R)/n(−)(0).
Thus, in Sn isotopes with the experimental charge radii about Rp =4.6 fm a rather significant
variation of ∆R about 0.1 fm, that is of the order of magnitude of δR, would cause
δ∆Eph
= 0.3 and
δ∆Eph
= 0.15 for 112Sn and 132Sn, respectively (γ = 0.5), that corresponds to the absolute change about
0.3 MeV in ∆E, to be compared with the experimental uncertainties in ∆E of the same order. It
is clear that to draw any conclusion about ∆R from the measured ∆E would be premature. Even if
the experimental errors in ∆E were exactly zero, the accuracy of the theoretical model itself would be
hardly believed to be of the necessary level. For instance, apart from the obvious uncertainties in the
isotopic dependence of the spin-orbit potential, the GTR does not exhaust 100% of the corresponding
sum rules and the shell-structure effects such as configurational and isospin splitting of the GTR can
have some effect on the calculated GTR energy.
It is also noteworthy that, in spite of the claimed self-consistency of the calculations, the slope of the
calculated isotopic dependence of the IAS energy is about 3 times larger than the experimental one (see
inset in Fig. 1 of [13]). Note, that the isospin self-consistent continuum-QRPA calculations of [15] were
able to nicely reproduce the slope (while overall underestimated the IAS energy by about 0.5 MeV, the
well-known Nolen-Schiffer anomaly).
To conclude, we believe that the suggested in [13] method to deduce the neutron skin from the
energy spacing between GTR and IAS is rather questionable in its origin and does not fairly provide
“direct information on the evolution of neutron skin-thickness”.
4 Some implications of ∆R
In several processes of physical interest knowledge of ∆R plays a crucial role and in fact a more accurate
value could lead to more stringent tests:
(i) The pion polarization operator [4] (the s-wave optical potential) in a heavy nucleus Π(ω, ρp, ρn) =
−T+(ω)ρ−T−(ω)(ρn−ρp) has mainly an isovector character (T
+(mπ) ∼ 0). Parameterizing the densities
by Fermi shapes for the case of 208Pb the main nuclear model dependence in the analysis comes from
the uncertainty in the value of ∆R multiplying T−.
(ii) The parity violation in atoms is dominated by Z−boson exchange between the electrons and the
neutrons [5]. Taking the proton distribution as a reference there is a small so-called neutron skin (ns)
correction to the parity non-conserving amplitude, δEnspnc, for, say, a 6s1/2 → 7s1/2 transition, which is
related to ∆R as (independent of the electronic structure)
δEnspnc
(αZ)2
. (11)
In 133Cs it amounts to a δE/E ≈ −(0.1−0.4)% depending on whether the non-relativistic or relativistic
estimates for ∆R are used [5]. The corresponding uncertainty in the weak charge QW is −(0.2− 0.8)σ.
(iii) The pressure in a neutron star matter can be expressed as in terms of symmetry energy and its
density dependence
P (ρ, x) = ρ2
∂E(ρ, x)
= ρ2[E ′(ρ, 1/2) + S ′(ρ)(1− 2x)2 + . . .]. (12)
By using beta equilibrium in a neutron star, µe = µn − µp = −
∂E(ρ,x)
, and the result for the electron
chemical potential, µe = 3/4h̄cx(3π
2ρx)1/3, one finds the proton fraction at saturation density, ρ0, to
be quite small, x0 ∼ 0.04. Hence, the pressure at saturation density can be approximated as
P (ρ0) = ρs(1− 2x0)(ρ0S
′(ρ0)(1− 2x0) + S(ρ0)x0) ∼ ρ
′(ρ0). (13)
At higher densities the proton fraction increases; this increase is more rapid in case of larger p0 [1]. While
for the pressure at higher densities contributions from other nuclear quantities like compressibility will
play a role in 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, RM ∼ C(ρ,M)(
P (ρ)
MeVfm−3
)0.25 km, where C(ρ = 1.5ρs,M = 1.4Msolar) ∼ 7.
This shows that a determination of a neutron star radius would provide some constraint on the symmetry
properties of nuclear matter.
5 Conclusion
In this contribution we discuss some aspects of extracting the neutron skin from properties of isovector
giant resonances and critically review existing proposals. The theoretical method relying on the energy
difference between the GTR and IAS is shown to lack sensitivity to ∆R. It is also shown that the phe-
nomenological, almost linear, relationship between the symmetry energy and the neutron skin in finite
nuclei, observed in mean field calculations, can be understood in terms the Landau-Migdal approach.
Acknowledgments
The work is supported in part by the Deutsche Forschungsgemeinschaft (grant FA67/28-2) and by the
EU ILIAS project (contract RII3-CT-2004-506222). The author would like to thank Profs. L. Dieperink
and M. Urin for useful discussions.
References
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[5] S. J. Pollock and M. C. Welliver, Phys. Lett. B 464 (1998) 177 .
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064307 .
[7] A.B. Migdal, Theory of finite Fermi-systems and properties of atomic nuclei (Moscow, Nauka,
1983) (in Russian).
[8] C. J. Horowitz, S. J. Pollock, P. A. Souder and R. Michaels, Phys. Rev. C 63 (2001) 025501 ;
http://hallaweb.jlab.org/parity/prex.
[9] A. Krasznohorkay et al., Phys. Rev. Lett. 82 (1999) 3216 .
[10] V. Rodin, M. Urin, KVI annual report (2000).
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http://hallaweb.jlab.org/parity/prex
http://arxiv.org/abs/nucl-th/0201065
	Introduction
	Relationship between the symmetry energy and R
	Extracting neutron skin from properties of isovector giant resonances 
	Spin-dipole Giant Resonance
	Isobaric analogue state
	Is the energy spacing between GTR and IAS a good candidate for determining the neutron skin in isotopic chains?
	Some implications of R
	Conclusion

  Some aspects, both experimental and theoretical, of extracting the neutron
skin $\Delta R$ from properties of isovector giant resonances are discussed.
Existing proposals are critically reviewed. The method relying on the energy
difference between the GTR and IAS is shown to lack sensitivity to $\Delta R$.
A simple explanation of the linear relation between the symmetry energy and the
neutron skin is also given.

Introduction
Accurate experimental data on the neutron skin in neutron rich nuclei would allow to further constrain
model parameters involved in the calculations of the nuclear symmetry energy [1]. The latter plays
a central role in a variety of nuclear phenomena. The value a4 ≈ 30 MeV of the nuclear symmetry
energy S(ρ0) = a4+
(ρ− ρ0)+ . . . at nuclear saturation density ρ0 ≈ 0.17 fm
−3 seems reasonably well
established. On the other hand, the density dependence of the symmetry energy can vary substantially
with the many-body approximations employed.
Several authors have pointed out [2, 3] a strong correlation between the neutron skin, ∆R =
〈r2〉n−
〈r2〉p = Rn−Rp, and the symmetry energy of neutron matter near saturation density. In the framework
of a mean field approach Furnstahl [3] demonstrated that in heavy nuclei there exists an almost linear
empirical correlation between theoretical predictions in terms of various mean field approaches to S(ρ)
(i.e., a bulk property) and the neutron skin, ∆R (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. Besides, 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 [4], and
in the analysis of atomic parity violation experiments (weak charge) [5]. It is worth to stress that to
experimentally determine the skin in heavy nuclei is extremely challenging as ∆R is just about few
percents of the nuclear radius.
The present contribution is partially based upon the results published previously in [6].
2 Relationship between the symmetry energy and ∆R
Brown [2] and Furnstahl [3] have pointed out that within the framework of mean field models there
exists an almost linear empirical correlation between theoretical predictions for both a4 and its density
http://arxiv.org/abs/0704.0180v1
dependence, p0, and the neutron skin ∆R 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, following the analysis of [6], this question is addressed from a point of view of the Landau-Migdal
approach.
Let us consider a simple mean-field model with the Hamiltonian consisting of the single-particle
mean field part Ĥ0 and the residual particle-hole interaction Ĥp−h:
Ĥ = Ĥ0 + Ĥp−h, Ĥph =
(F ′ +G′~σa~σb)~τa~τbδ(~ra − ~rb), (1)
Ĥ0 =
(Ta + U(xa)), U(x) = U0(x) + U1(x) + UC(x), (2)
U0(x) = U0(r) + Uso(x); U1(x) =
Spot(r)τ
(3); UC(x) =
UC(r)(1− τ
(3)). (3)
Here, U0(x) is the phenomenological isoscalar part of the mean field potential U(x) (x = {~r, ~σ, ~τ}),
U0(r) and Uso(x) are the central and spin-orbit parts, respectively; F
′ and G′ are the phenomenological
Landau-Migdal parameters. The isovector part U1(x) and the Coulomb mean field UC(x) are both
calculated consistently in the Hartree approximation, Spot(r) is the symmetry potential (r-dependent
symmetry energy in finite nuclei).
The model Hamiltonian Ĥ (1) preserves the isospin symmetry within the RPA if a selfconsistency
relation between the symmetry potential and the Landau-Migdal parameter F ′ is fulfilled:
Spot(r) = 2F
′n(−)(r), (4)
where n(−)(r) = nn(r)− np(r) is the neutron excess density. Thus, in this model the depth of the sym-
metry potential is controlled by the Landau-Migdal parameter F ′ (analogous role plays the parameter
g2ρ in relativistic mean field models). Spot(r) is obtained from Eq.(4) by an iterative procedure; the
resulting dependence of ∆R on the dimensionless parameter f ′ = F ′/(300 MeV fm3) shown in fig. 1
indeed illustrates that ∆R depends almost linearly on f ′. Then with the use of the Migdal relation
(1 + 2f ′) [7] relating the symmetry energy and f ′, a similar, almost linear, correlation between
a4 and ∆R is obtained.
To get more insight in the role of f ′ we consider small variations δF ′. Neglecting the varia-
tion of n(−)(r) with respect to δF ′, the corresponding linear variation of the symmetry potential
is δSpot(r) = 2δF
′n(−)(r). Then in the first order perturbation theory, such a variation of Spot
causes the following variation of the ground-state wave function |δ0〉 = δF ′
〈s|N̂(−)|0〉
E0−Es
|s〉, with “s”
labeling the eigenstates of the nuclear Hamiltonian and a single-particle operator N̂ (−) defined as
N̂ (−) =
n(−)(ra)τ
a . Consequently, the variation of the expectation value 〈0|V̂
(−)|0〉 = NR2n − ZR
of another single-particle operator V̂ (−) =
a can be written as
Rpδ(∆R) = δF
Re〈0|N̂ (−)|s〉〈s|V̂ (−)|0〉
E0 −Es
. (5)
In practice the sum in Eq. (5) is exhausted mainly by the isovector monopole resonance (IMR) which
high excitation energy (about 24 MeV in 208Pb) justifies the perturbative consideration. Eq. (5) is able
to reproduce directly calculated δ(∆R) shown in Fig. 1 with the accuracy of about 10%. As a result,
a simple microscopic interpretation of the linear correlation between ∆R and Landau parameter F ′ is
obtained.
0.6 0.8 1 1.2 1.4
Figure 1: Neutron skin in 208Pb versus the Landau-Migdal parameter f ′.
3 Extracting neutron skin from properties of isovector giant
resonances
Parity violating electron scattering off nuclei is probably the least model dependent approach to probe
the neutron distribution [8]. The weak electron-nucleus potential is Ṽ (r) = V (r) + γ5A(r), where
the axial potential A(r) = GF
ρW (r). The weak charge is mainly determined by neutrons ρW (r) =
(1 − 4 sin2 θW )ρp(r)− ρn(r), with sin
2 θW ≈ 0.23. In a scattering experiment using polarized electrons
one can determine the cross section asymmetry [8] which comes from the interference between the A and
V contributions. Using the measured neutron form factor at small finite value of Q2 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 Rn from a finite Q
2 form factor.
However, the best claimed accuracy of the experimental determination of neutron radii would be on
the level of 1%, that translates to relatively large uncertainty of 20-30% in the neutron skin. On such
accuracy level, some indirect experimental probes of ∆R still can be competitive.
A variety of experimental approaches have been employed to obtain indirect information on ∆R.
To some extent all the analysis contain a certain model dependence, which in many cases is difficult
to estimate quantitatively. For choosing an indirect probe it is very important to address the question
how sensitive is the proposed physical quantity with respect to a variation of ∆R in a single nucleus.
The higher is the sensitivity, the better is the choice of the correlation for the indirect deducing ∆R
from the measured values.
It is not intended here to give a comprehensive review of the existing methods. In particular, the
results from the analysis of the antiprotonic atoms, elastic proton and neutron scattering reactions, and
the pygmy dipole resonance are completely left out. Here, special emphasis will be put on proposals to
provide accurate information on the neutron skin from properties of isovector giant resonances.
3.1 Spin-dipole Giant Resonance
In [9] it has been proposed to utilize the excitation probability of the spin-dipole resonance in charge
exchange reactions for determining the neutron skin. The method has been applied to obtain information
on the variation of the neutron skin in the Sn isotopes [9]. For the relevant operator,
a [~σa ⊗
~ra]JM , (J = 0, 1, 2) the summed ∆L = 1 strength is
S(−) − S(+) = C(NR2n − ZR
p). (6)
Here S(−) and S(+) are the spin-dipole total strengths in β(−) and β(+) channels, respectively; C is the
factor depending on the normalization of the spin-dipole operator (in the definition of Ref. [2] C = 1/4π,
we use here C = 1). Because S(+) could not be measured experimentally, the model-dependent energy-
weighted sum rule was invoked in the analysis of [9] to eliminate S(+). However, the used analytical
representation for the sum rule was oversimplified and led in some cases, e.g. for 208Pb, to absurdly
negative S(+). In [10] another way was proposed, namely, to use for the analysis the ratio S(+)/S(−)
calculated within the pn-RPA. The parameterization of the RPA calculation results for tin isotopes in
the form
S(+)/S(−) = 0.388− 0.012(N − Z)
was used later in [11] to reanalyze the experimental data and led to a marked change in the extracted
∆R’s.
Let us now assess the experimental accuracy for S(−) needed to determine the neutron skin to a
given accuracy. Putting S(+) = 0 (that seems to be a very good approximation for 208Pb) and one has
S(−) = (N − Z)R2p + 2NRp∆R. (7)
The ratio of the second term on the rhs to the first one in case of 208Pb is
2N∆R/((N − Z)Rp) ≈ 5.7∆R/Rp.
Therefore, for Rp = 5.5 fm and ∆R = 0.2 fm the second term is only 25% of the first one and one needs
5% accuracy in S(−) to determine ∆R with 20% accuracy. Because the SD strength is spread out and
probably has a considerable strength at low-energy, the results for the ∆R can be only considered as
qualitative with a relatively large uncertainty (up to 30-50%).
3.2 Isobaric analogue state
The dominant contribution to the energy weighted sum rule (EWSR) for Fermi excitations by the
operator T (−) =
a comes from the Coulomb mean field
(EWSR)F =
UC(r)n
(−)(r)d3r, (8)
The Coulomb mean field UC(r) resembles very much that of the uniformly charged sphere, being inside
a nucleus a quadratic function: UC(r) =
(3 − (r/Rc)
2), r ≤ Rc. It turns out that if one extends
such a quadratic dependence also to the outer region r > Rc (instead of proportionality to Rc/r), it
gives numerically just a very small deviation in (EWSR)F (less than 0.5%, due to the fact, that both
the difference and its first derivative go to zero at r = Rc and n
(−)(r) is exponentially decreasing for
r > Rc). Using such an approximation, one gets:
(EWSR)F ≈ (N − Z)∆C
3(N − Z)R2c
with ∆C =
, and S(−) given in Eq.(7).
Since the IAS exhausts almost 100% of the NEWSR and EWSR, one may hope to extract S(−) from
the IAS energy. However, the term depending on S(−) contributes only about 20% to (EWSR)F , and as
a result, the part of S(−) depending on ∆R contributes only about 4% to (EWSR)F (in
208Pb). ¿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
UC(r) can be calculated rather accurately, and hence one can determine the small difference between
Eqs.(9) and (8). But at the level of 1% accuracy several theoretical effects discarded in Eq.(8) come
into play that makes such an accurate description of the IAS energy very difficult (the Nolen-Schiffer
anomaly).
Also in [12] it was stated that the Coulomb displacement energies (CDE) are sensitive to ∆R. A
gross estimate ∆R = 0.80(5)(N −Z)/A fm was obtained from a four-parameter fit of the experimental
Rp and observed mirror CDE’s. The authors claimed 127 keV to be the rms error of the fit, but they
assumed the nuclear wave functions calculated within the Nuclear Shell Model to be isospin pure. Thus,
the important effect of the Coulomb mixing of the IAS and the IMR was not taken into account, which
is known to decrease the IAS energy by a few percents. Therefore, the Nolen-Schiffer anomaly does not
seem to have been resolved yet.
3.3 Is the energy spacing between GTR and IAS a good candidate for
determining the neutron skin in isotopic chains?
In a recent paper [13] a proposal has been put forward to use the isotopic dependence of the energy
spacing, ∆E, between the Gamow-Teller resonance (GTR) and the IAS as a tool for determining the
evolution of the neutron skin in nuclei along an isotopic chain. Here, we would like to present some
physical arguments which question the physical relevance of this method.
The authors of [13] have used the fact that both functions, ∆R and ∆E, are monotonic functions
(increasing and decreasing, respectively) of the neutron excess (N−Z) to state that “isotopic dependence
of the energy spacings between the GTR and IAS provides direct information on the evolution of neutron
skin-thickness along the Sn isotopic chain”. Arguing in such a way one can find a correlation between
any two monotonic functions of a single physical parameter and plot them as a function of one another
like is done in Fig. 2 of [13] 1. However, it does not imply automatically a real physical correlation
between the functions which are determined also by many other model parameters which are kept fixed
while performing calculations (the calculations in [13] have been performed within the relativistic mean
field (RMF) and relativistic QRPA (RQRPA) approaches).
Again, the relevant question to be addressed is how sensitive is one physical quantity with respect to
a variation of another in a single nucleus? In other words, one has to evaluate what variation of ∆E
is produced by varying ∆R in a single nucleus. Imaging an extreme situation (which is actually not
far from reality) that ∆E were not sensitive to ∆R at all, one would get by varying ∆R a family of
different calculated dependences (like shown in the upper panel of Fig. 2 of [13]) which would give no
clue about the real dependence seen in nature.
Thus, it is quite important to understand the physical reasons which cause the energy splitting
between the GTR and the IAS. It is well-known that if the nuclear Hamiltonian possessed Wigner
SU(4) symmetry then the GTR and the IAS would be degenerate, ∆E = 0. In such a case any
variation of ∆R, not violating the symmetry, would not affect ∆E at all. However, it is also known
that the spin-isospin SU(4) symmetry is broken in nuclei. Hence, ∆E is determined by those terms in
1note that, to avoid confusion in comparing the measured and calculated dependences, the authors should have plotted
the experimental points in the upper panel as the function of the measured ∆R rather than calculated ∆R and should
have added the horizontal error bars to them reflecting the experimental uncertainty in ∆R shown in the lower panel.
the nuclear Hamiltonian which violate the symmetry. Their qualitative and semi-quantitative estimates
in terms of the energy weighted sum rules for the Gamow-Teller (EWSRGT ) and the Fermi (EWSRF )
excitations have been already known for more than 20 years (see, e.g., [14]). The analysis of these
authors as well as a quantitative analysis performed recently in [15] has shown that there are three
basic sources in the Hamiltonian which violate SU(4) symmetry and contribute to the difference of the
sum rules: spin-orbit mean field and both particle-particle and particle-hole residual charge-exchange
interactions. One sees that none of the sources explicitly refers to the symmetry potential, to which
∆R is especially sensitive.
An estimate of ∆E as ∆E =
EWSRGT − EWSRF
N − Z
can be calculated according to [15] in the
Sn isotopes. From the sources violating SU(4) symmetry, spin-orbit mean field represents the major
one and contributes about 5 MeV to the splitting. The contribution of the particle-hole interaction is
negative and about 1–2 MeV in the absolute value. The contribution of the particle-particle interaction
is rather difficult to evaluate (due to uncertainty in the strength of the spin-dependent particle-particle
interaction) but it seems to be of minor importance (very probably no more than 0.5 MeV, especially
for large (N − Z)) and can safely be neglected.
Now let us turn to the discussion of the sensitivity of the contributions to the variation of ∆R. We
could reproduce the corresponding analytical expressions from [15] explicitly, but it is enough for our
purpose just to mention that the dominating contribution to ∆E from the spin-orbit mean field is given
by its expectation value in the ground state and is determined basically only by the unfilled spin-orbit
doublets. This expectation value is completely insensitive to the variation of ∆R.
Within the Landau-Migdal approach described above, the particle-hole contribution
∆Eph =
2(G′ − F ′)
N − Z
(n(−)(r))2d3r (10)
is given by the product of the volume integral of the neutron excess density squared and the difference
of the p-h strengths G′ and F ′ [15]. In the SU(4)-symmetric limit one has G′ = F ′ and ∆Eph = 0
explicitly. Still, in this limit one has a freedom to choose different F ′ that produces a variation in ∆R,
similar to shown in fig 1. Therefore, as already mentioned, one can get no clue about the actual ∆R
from ∆E = 0 in the SU(4)-symmetric limit.
In a realistic situation G′ 6= F ′ (f ′ = 1.0 and g′ = 0.8 were taken in [15]), but ∆Eph depends only on
the difference G′ − F ′. One usually fixes G′ in order to reproduce the GTR energy in some nuclei (the
authors of [13] have followed this way, too) and possible information from ∆E about the absolute value
of F ′ is lost. Furthermore, one can a priori think that a degree of violation of the SU(4) symmetry
should be a sort of a fundamental property of the residual interaction.Therefore, the difference G′ − F ′
should stay more stable in different models as compared to some possible variation of F ′ producing
different ∆R.
Considering value of G′ − F ′ fixed, one can employ a simple model varying only ρn(r) to see how a
change of ∆R affects ∆Eph via variation of the neutron excess density n
(−)(r). A small variation of ρn(r)
can be approximately represented as δρn(r) = −
(3ρn(r) +R
dρn(r)
), where δRn is a change of the
rms neutron radius Rn, R is the nuclear radius (with R
n ≈ 0.6R
2). Assuming the proton and neutron
densities be constant inside a nucleus, the final estimate is
δ∆Eph
N + Z − 2γN
N − Z
), where
γ = n(−)(R)/n(−)(0).
Thus, in Sn isotopes with the experimental charge radii about Rp =4.6 fm a rather significant
variation of ∆R about 0.1 fm, that is of the order of magnitude of δR, would cause
δ∆Eph
= 0.3 and
δ∆Eph
= 0.15 for 112Sn and 132Sn, respectively (γ = 0.5), that corresponds to the absolute change about
0.3 MeV in ∆E, to be compared with the experimental uncertainties in ∆E of the same order. It
is clear that to draw any conclusion about ∆R from the measured ∆E would be premature. Even if
the experimental errors in ∆E were exactly zero, the accuracy of the theoretical model itself would be
hardly believed to be of the necessary level. For instance, apart from the obvious uncertainties in the
isotopic dependence of the spin-orbit potential, the GTR does not exhaust 100% of the corresponding
sum rules and the shell-structure effects such as configurational and isospin splitting of the GTR can
have some effect on the calculated GTR energy.
It is also noteworthy that, in spite of the claimed self-consistency of the calculations, the slope of the
calculated isotopic dependence of the IAS energy is about 3 times larger than the experimental one (see
inset in Fig. 1 of [13]). Note, that the isospin self-consistent continuum-QRPA calculations of [15] were
able to nicely reproduce the slope (while overall underestimated the IAS energy by about 0.5 MeV, the
well-known Nolen-Schiffer anomaly).
To conclude, we believe that the suggested in [13] method to deduce the neutron skin from the
energy spacing between GTR and IAS is rather questionable in its origin and does not fairly provide
“direct information on the evolution of neutron skin-thickness”.
4 Some implications of ∆R
In several processes of physical interest knowledge of ∆R plays a crucial role and in fact a more accurate
value could lead to more stringent tests:
(i) The pion polarization operator [4] (the s-wave optical potential) in a heavy nucleus Π(ω, ρp, ρn) =
−T+(ω)ρ−T−(ω)(ρn−ρp) has mainly an isovector character (T
+(mπ) ∼ 0). Parameterizing the densities
by Fermi shapes for the case of 208Pb the main nuclear model dependence in the analysis comes from
the uncertainty in the value of ∆R multiplying T−.
(ii) The parity violation in atoms is dominated by Z−boson exchange between the electrons and the
neutrons [5]. Taking the proton distribution as a reference there is a small so-called neutron skin (ns)
correction to the parity non-conserving amplitude, δEnspnc, for, say, a 6s1/2 → 7s1/2 transition, which is
related to ∆R as (independent of the electronic structure)
δEnspnc
(αZ)2
. (11)
In 133Cs it amounts to a δE/E ≈ −(0.1−0.4)% depending on whether the non-relativistic or relativistic
estimates for ∆R are used [5]. The corresponding uncertainty in the weak charge QW is −(0.2− 0.8)σ.
(iii) The pressure in a neutron star matter can be expressed as in terms of symmetry energy and its
density dependence
P (ρ, x) = ρ2
∂E(ρ, x)
= ρ2[E ′(ρ, 1/2) + S ′(ρ)(1− 2x)2 + . . .]. (12)
By using beta equilibrium in a neutron star, µe = µn − µp = −
∂E(ρ,x)
, and the result for the electron
chemical potential, µe = 3/4h̄cx(3π
2ρx)1/3, one finds the proton fraction at saturation density, ρ0, to
be quite small, x0 ∼ 0.04. Hence, the pressure at saturation density can be approximated as
P (ρ0) = ρs(1− 2x0)(ρ0S
′(ρ0)(1− 2x0) + S(ρ0)x0) ∼ ρ
′(ρ0). (13)
At higher densities the proton fraction increases; this increase is more rapid in case of larger p0 [1]. While
for the pressure at higher densities contributions from other nuclear quantities like compressibility will
play a role in 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, RM ∼ C(ρ,M)(
P (ρ)
MeVfm−3
)0.25 km, where C(ρ = 1.5ρs,M = 1.4Msolar) ∼ 7.
This shows that a determination of a neutron star radius would provide some constraint on the symmetry
properties of nuclear matter.
5 Conclusion
In this contribution we discuss some aspects of extracting the neutron skin from properties of isovector
giant resonances and critically review existing proposals. The theoretical method relying on the energy
difference between the GTR and IAS is shown to lack sensitivity to ∆R. It is also shown that the phe-
nomenological, almost linear, relationship between the symmetry energy and the neutron skin in finite
nuclei, observed in mean field calculations, can be understood in terms the Landau-Migdal approach.
Acknowledgments
The work is supported in part by the Deutsche Forschungsgemeinschaft (grant FA67/28-2) and by the
EU ILIAS project (contract RII3-CT-2004-506222). The author would like to thank Profs. L. Dieperink
and M. Urin for useful discussions.
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http://hallaweb.jlab.org/parity/prex
http://arxiv.org/abs/nucl-th/0201065
	Introduction
	Relationship between the symmetry energy and R
	Extracting neutron skin from properties of isovector giant resonances 
	Spin-dipole Giant Resonance
	Isobaric analogue state
	Is the energy spacing between GTR and IAS a good candidate for determining the neutron skin in isotopic chains?
	Some implications of R
	Conclusion
