Constraining the density dependence of symmetry energy from nuclear masses
B. K. Agrawal1, J. N. De1 , S. K. Samaddar1 , G. Colò2,3 , and A. Sulaksono4
1
arXiv:1305.5336v1 [nucl-th] 23 May 2013
2
Saha Institute of Nuclear Physics, Kolkata 700064, India
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
3
INFN, sezione di Milano, via Celoria 16, I-20133 Milano, Italy
4
Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424, Indonesia
Empirically determined values of the nuclear volume and surface symmetry energy coefficients
from nuclear masses are expressed in terms of density distributions of nucleons in heavy nuclei in
the local density approximation. This is then used to extract the value of the symmetry energy
slope parameter L. The density distributions in both spherical and well deformed nuclei calculated
within microscopic framework with different energy density functionals give L = 59.0 ± 13.0 MeV.
Application of the method also helps in a precision determination of the neutron skin thickness of
nuclei that are difficult to measure accurately.
PACS numbers: 21.65.Ef, 21.65.Mn, 21.10.Gv
Keywords: Symmetry energy, symmetry energy slope parameter, nuclear matter, neutron skin
The nuclear symmetry energy measures the energy
transfer in converting symmetric nuclear matter to the
asymmetric one. The density dependence gives information on the isospin-dependent part of the equation of
state (EOS) of asymmetric nuclear matter. The density content of symmetry energy is mostly encoded in
the symmetry energy coefficient Cv , the symmetry slope
parameter L and the symmetry incompressibility Ksym ,
all evaluated at the nuclear saturation density ρ0 . Here
Cv (ρ) is the volume symmetry energy per nucleon of homogeneous nuclear matter at density ρ,
Cv (ρ) = [e(ρ, X = 1) − e(ρ, X = 0)],
L = 3ρ0
∂Cv (ρ)
∂ρ
,
(1)
(2)
ρ0
and
Ksym = 9ρ20
∂ 2 Cv (ρ)
∂ρ2
.
(3)
ρ0
In Eq. (1), e is the energy per nucleon and X = (ρn −
ρp )/(ρn +ρp ) is the isospin asymmetry of the system. The
parameters Cv (ρ0 ), L and Ksym deem to be of fundamental importance in both nuclear physics and astrophysics.
The nuclear binding energies, the position of the nuclear
drip lines, the neutron skin thickness or the neutron density distribution in neutron-rich nuclei, − all of these are
known to have affectations from [1–8] the symmetry parameters so mentioned. Many astrophysical phenomena
also depend sensitively on the symmetry slope parameter L. Most notable among them are the dynamical
evolution of the core collapse of a massive star and the
associated explosive nucleosynthesis [9, 10], the cooling
of proto-neutron stars through neutrino convection [11]
or the radii of cold neutron stars [11]. The nature and
stability of phases within a neutron star, its crustal com-
[12] also seem to be strongly influenced by the symmetry
energy and its density dependence. A glimmer of their
import could further be seen in relation to some issues of
new physics beyond the standard model [13, 14].
Attempts on estimates of L have been made in the last
few years from analyses of diverse experimental data, uncertainties still linger, however. Pygmy dipole resonance
[15] in 68 N i and 132 Sn predicts a weighted average in the
range L=64.8 ± 15.7 MeV, but giant dipole resonance in
208
P b [16] points to a value of L ∼ 52 ±7 MeV. Nucleon
emission ratios from heavy ion collisions [17] favor a value
close to it, L ∼ 55 MeV, but isoscaling gives L ∼ 65 MeV
[18] and isospin diffusion shifts the value further up, L =
88 ± 25 MeV [19, 20]. Recently, analyzing nuclear energies within the standard Skyrme-Hartree-Fock approach,
Chen [21] brought down L to 52.5 ± 20 MeV. However,
from the fit [3] of the experimental nuclear masses to
the calculated ones in the finite range droplet model, the
value of L is found to be in the bound L = 70 ± 15 MeV.
Inputs from astrophysical analyses, namely neutron star
masses and radii on the other hand constrain L to 43
< L < 52 MeV [22].
The neutron skin thickness Rskin (=Rn − Rp ; Rn and
Rp are the neutron and proton root-mean squared (rms)
radii) of 208 Pb calculated in the droplet model or in selfconsistent quantal calculations with many different interactions was found to have a strong linear correlation
with the density dependence of symmetry energy around
saturation [4–7, 23, 24]. Analyzing the correlation systematics of nuclear isospin with neutron skin thickness
for a series of nuclei in the ambit of droplet model, the
Barcelona Group [23, 24] found a range for as L = 75
± 25 MeV. The neutron skin thicknesses used by them
are measured in antiprotonic atom experiments [25, 26],
they carry the unavoidable strong interaction- related
uncertainties. This had its imprint on their extracted
value of L. The Lead Radius Experiment on Pb (PREX)
promises a model-independent probe of its neutron-skin
2
put the value of Rskin of 208 Pb in more precision than
0.302 ±.175 fm [27, 28]. This large uncertainty mires L
in wide limits.
Given that the neutron skin thickness is a strong indicator of the nuclear isovector properties, the nuclear
dipole polarizability αD has been suggested to provide
a unique constraint on Rskin [29, 30]. The recent high
resolution (p, p′ ) measurement [31] of αD yields the neutron skin thickness of 208 Pb to be 0.168 ± 0.022 fm
[32]. Quadrupole resonance energies may similarly be
exploited to give estimates of Rskin . Microscopic calculations [33] of these observables based on families of non relativistic and covariant energy density functionals (EDF)
point to a value of Rskin in close vicinity, Rskin = 0.14
±0.03 fm for 208 Pb. Consequently, value of the symmetry slope parameter extracted from the GDR and GQR
measurements turns out to be L = 49 ± 11 and 37 ± 18
MeV, respectively.
Nuclear masses are one of the most precisely experimentally determined quantities in nuclear physics. They
provide information, through the liquid-drop mass systematics on the values of the volume and surface symmetry energy coefficients Cv (ρ0 ) and Cs . The symmetry
energy coefficient, asym (A), for a nucleus with mass number A can be expressed in terms of Cv (ρ0 ) and Cs as,
asym (A) = Cv (ρ0 ) − Cs A−1/3 .
(4)
An alternative form for asym (A) has also been suggested
[34], namely,
asym (A) =
Cv (ρ0 )
,
(1 + κA−1/3 )
(5)
where κ = Cs /Cv (ρ0 ). From analyses of ∼ 2000 nuclear
masses, Liu et al . [35] obtained Cv (ρ0 ) =31.1 ± 1.7 MeV
and κ = 2.31 ± 0.38. Equating asym (A) with Cv (ρA ),
where ρA is an equivalent density specific to mass A and
taking an ansatz for the density dependence of Cv (ρ) as
Cv (ρ) = Cv (ρ0 )(
ρ γ
) ,
ρ0
(6)
they obtain, choosing ρA ∼ 0.1 fm−3 for 208 Pb, γ = 0.7 ±
0.1 and thence L = 66 ±13 MeV. The density dependence
of Cv as given by Eq. (6) is quite consistent with the
density dependence obtained from the nuclear EOS with
different interactions [17–20, 36]; the value of γ varies
though from interaction to interaction. The choice for
the value of the equivalent density has no firm theoretical
basis, however.
A meticulous study [37] of the double differences of ’experimental’ symmetry energies were done very recently.
The double differences in symmetry energies of neighboring nuclei has the advantage that effects from pairing
and shell corrections are well canceled out, resulting in a
compact correlation between the double differences and
the mass number of nuclei. This yields values of Cv (ρ0 )
and Cs as 32.1±0.31 MeV and 58.91 ±1.08 MeV, respec-
those found previously. Exploiting the empirical values
of Cv (ρ0 ) and Cs , an exploration was done earlier [38]
with the ansatz given by Eq. (6) to calculate the equivalent density ρA for the ’benchmark’ spherical nucleus
208
Pb in the local density approximation [36]. Hereafter,
we refer to the choice of Eq. (6) for the form of density
dependence of Cv (ρ) as case I and the empirical values
Cv (ρ0 ) and Cs as Cv0 and Cs0 respectively.
The saturation density was chosen as ρ0 = 0.155
±0.008 fm−3 ; this covers the saturation densities that
come out from the EOSs of different Skyrme and relativistic mean-field (RMF) models. The empirical information of the proton rms radius in 208 Pb and the correlation of the symmetry slope parameter with the neutron skin thickness of 208 Pb, all deduced from a family
of RMF energy density functionals served to give a value
of γ ∼ 0.664 ± 0.051, and hence L = 64 ± 5 MeV [38].
The neutron skin thickness for 208 Pb is found to be 0.188
±0.014 fm, the equivalent density ρA , comes out to be
0.089 fm−3 from the model.
To confirm the robustness of the procedure adopted
in our recent work [38], it is imperative to see whether
choice of forms of density dependence of symmetry energy other than that given by Eq. (6) produce nearly the
same results. It is also important to check if other heavy
nuclei, spherical or otherwise, collate the conclusions arrived with this procedure. In the present communication
we use different functional forms for the Cv (ρ), employed
very recently in Ref. [39, 40] in checking the validation
of the relationship between Cv (ρ0 ), L and Ksym using
different effective interactions. Our calculations are performed for both the spherical and well deformed 208 Pb
and 238 U nuclei, respectively. The density distribution of
nucleons corresponding to different neutron-skin thickness, required for our calculations, are obtained using
different energy density functionals based on the RMF
model. The choices for these energy density functionals
are exactly the same as in Ref. [38].
The form of Cv (ρ) given by Eq. (6) (case I) has some
limitations [39]. With this functional form, Cv (ρ0 ), L and
Ksym , when calculated from Eqs. (1) -(3) can be correlated as,
Cv (ρ0 ) =
L
.
3 + Ksym /L
(7)
The results for the relative error ∆Cv /Cv , obtained using different RMF models, are displayed in Fig. 1. The
∆Cv for the case I (black squares) is the difference in the
lhs and rhs of Eq. (7). The values of Cv , L and Ksym
are calculated at ρ0 using Eqs. (1), (2) and (3), respectively. The departure of ∆Cv /Cv from zero is indicative
of the inaccuracy involved in expressing the functional
dependence of symmetry energy as given by Eq. (6).
Another functional form for Cv (ρ) seems more pertinent [40]:
C (ρ) = C (
ρ
)2/3 + (C (ρ ) − C )(
ρ
)γ .
(8)
3
0.4
150
I
II
III
0.3
NL3
Cv(ρ) (MeV)
100
∆Cv/Cv
0.2
0.1
BSR10
FSU
50
DDM3Y
FSU
TM1
0
NL3
BSR14
BSR13
BSR12
BSR11
BSR10
BSR9
-0.1
BSR8
0
FIG. 1: (Color online) The relative error ∆Cv /Cv calculated
at ρ0 are shown for the various interactions pertaining to the
cases I, II, and III. The meaning of the three cases are explained in the text.
The first term in the rhs of the above equation takes care
of the kinetic energy contribution to the Cv (ρ) , the last
term comes from interactions. The value of Ck is Ck =
2
PF,0
(22/3 − 1) × 35 2m
∗ where PF,0 is the Fermi momentum
corresponding to the saturation density ρ0 and m∗ is the
nucleon effective mass.
Inspired by the density-dependent M3Y (DDM3Y) interaction, the density dependence of Cv (ρ) has also been
shaped as [41],
Cv (ρ) = Ck (
ρ 2/3
ρ
ρ
) + C1 ( ) + C2 ( )5/3 ,
ρ0
ρ0
ρ0
(9)
where C2 = Cv (ρ0 )−C1 −Ck . The density dependence of
Cv (ρ) in the DDM3Y interaction, however, shows a different behavior as compared to the RMF models. Never
the less, we use this functional form also to check the
robustness of our calculations. The forms for Cv (ρ) depicted by Eqs. (8) and (9) are hereafter referred to as
case II and case III. The three symmetry parameters are
correlated as,
Cv (ρ0 ) = Ck +
(L − 2Ck )2
,
3L + Ksym − 4Ck
(10)
and
Cv (ρ0 ) =
L Ksym
Ck
−
+
,
3
15
5
0
0.1
0.2
0.3
-3
ρ (fm )
for the cases II and III respectively. As seen in Fig. 1, the
values of ∆Cv (ρ0 ) for cases II (red circles) and III (blue
triangles) are quite similar for the interactions we have
chosen; they lie closer to zero in comparison to those in
0.5
FIG. 2: (Color online) Symmetry energy plotted as a function
of density for a few representative interactions. The lines in
violet, blue, green and magenta colors correspond to NL3,
BSR-10, FSU and DDM3Y interactions, respectively. The
solid lines refer to Cv (ρ) calculated from Eq. (1), the dotted,
dashed and lines with square symbols refer to case I, II and
III, respectively. For the BSR10 and FSU, results for cases I
and II overlap on each other. For the DDM3Y, result for case
III is almost indistinguishable from the one calculated using
Eq. (1).
From the above results, one can infer about the adequacy of the three functional forms for modeling the
density dependence of the symmetry energy only at the
saturation density. In Fig. 2, we plot the values of Cv (ρ)
over a wide density range 0 ≤ ρ ≤ 3ρ0 which are obtained using Eq. (1) for a few representative interactions. These values of Cv (ρ) are also compared with the
ones obtained by fitting them to the cases I, II and III.
In this density range, one sees that the symmetry energy has a stiff density dependence for NL3 [42] interaction, BSR10 [43] and FSU [7] display a softer dependence. All the three functional forms (Eqs. (6), (8), and
(9)) are seen to give a moderatly good representation of
the density dependence of symmetry energy for the specific RMF interactions except at supranormal densities.
For the DDM3Y interaction, it is seen that only case III
(Eq. (9))compares extremely well with Cv (ρ). This is so
because of construction.
To a good approximation, Cv (ρ) is given as [23]
Cv (ρ) = Cv (ρ0 ) − Lǫ +
(11)
0.4
Ksym 2
ǫ
2
(12)
where ǫ = (ρ0 − ρ)/3ρ0 . Since asym (A) = Cv (ρA ), one
immediately gets, from Eqs. (4) and (12),
Cs = A1/3 [LǫA −
Ksym 2
ǫ ],
2 A
(13)
4
208
100
Pb
238
I
U
50
L (MeV)
(a)
(d)
II
100
50
(b)
(e)
III
100
50
(c)
0.15
(f)
0.2
0.25
0.15
0.2
0.25
Rskin (fm)
FIG. 3: (Color online) Symmetry slope parameter L calculated using Eq. (2) plotted as a function of Rskin of 208 Pb
and 238 U evaluated with the different RMF interactions are
shown by the blue triangles. The green lines with envelopes of
slanted black lines refer to the least-squared fits to them with
the spread-out error. The shaded regions in the left and right
panels represent the envelope of possible L values calculated
in case I, II and III. The intersection of the slanted envelopes
and the shaded regions depict the acceptable window for the
values of L and Rskin for the cases concerned.
asym (A)(≡ Cv (ρA )), in the local density approximation
can be calculated as,
Z
1
Cv (ρA ) =
drρ(r)Cv (ρ(r))[X(r)]2 ,
(14)
AX02
−Z
) is the isospin asymmetry of the nuwhere X0 (= NA
cleus, ρ(r) is the sum of its neutron and proton densities and X(r) is the local isospin asymmetry. Once
the neutron-proton density profiles in the nucleus are
known, for case II say, with Cv (ρ) as given by Eq. (8),
a chosen value of γ gives ρA and hence ǫA . The symmetry slope parameter L and Ksym are obtained from the
density derivative of Cv (ρ) as L = 2Ck + 3(Cv0 − Ck )γ,
the symmetry incompressibility Ksym is determined from
Eq. (10) using this value of L and the input value of
Cv (ρ0 ) (=Cv0 ). Only when Eq. (13) is satisfied (with
Cs = Cs0 ), one obtains the desired solution γ, thence
L, Ksym and ρA . These can, likewise be determined for
Case I. In a similar fashion, for case III, C1 of Eq. (9) can
be determined iteratively to give L = 5Cv0 − 3Ck − 2C1
and Ksym = 10Cv0 − 12Ck − 10C1 .
The procedure as described works best for heavy nuclei
where volume effects predominate over surface effects. In
Ref. [38], for case I, 208 Pb was chosen as a representative
nucleus. It usually serves as a benchmark for comparison with extracted bulk nuclear properties. The present
calculations along with the spherical nucleus 208 Pb are
to see if the extracted informations from the two nuclei
are in consonance.
We have used the interactions BSR8-BSR14 [43], FSU
[7], NL3 [42], and TM1 [44] to generate the neutron and
proton density profiles of the two nuclei in the RMF
model. These interactions in general reproduce many
experimental bulk properties of finite nuclei and nuclear
matter. The binding energies of 208 Pb and 238 U are obtained in close experimental agreement. The calculated
values of the quadrupole deformation β2 = 0.26−0.28 for
238
U from the various RMF models considered here agree
very well with the experimental value ∼ 0.29. The proton rms radii are also reproduced nearly exactly. There
is a wide variation in the neutron rms radii, however; the
calculated neutron skin thickness varies from ∼ 0.17 fm
to ∼ 0.28 fm. In Fig. 3(a)-3(f), the symmetry energy
slope parameters evaluated with these interactions using
Eq. (2) are displayed as a function of the corresponding calculated Rskin (blue filled triangles) for 208 Pb and
238
U. For both the nuclei, an almost linear correlation
of L with Rskin is observed. We also use Eqs. (6), (8),
(9), (13), and (14) to calculate L from these functional
forms (case I-III) by employing the microscopic nuclear
densities for 208 Pb and 238 U obtained within the RMF
models using the empirical constraints on the parameter
sets Cv0 (=32.1±0.31 MeV), Cs0 (=58.91 ±1.08 MeV) and
ρ0 (=0.155 ±0.008 fm−3 ). The so-calculated L values
are depicted in the figures by a shaded region, the spread
coming from the uncertainties in the values of Cv0 , Cs0 and
ρ0 . These values display an altogether different type of
correlation of L with Rskin , the weak dependence coming
from the imposed empirical constraints.
The change in L with Rskin as displayed by the filled
triangles, after least-squares fitting, gives an almost linear correlation shown by the green straight lines passing
through the triangles. The correlation coefficient for both
208
Pb and 238 U nuclei is 0.93. The small deviations from
complete linearity yields the rms error on L values ∼ 7.2
MeV. The spread in the L values for a given neutron-skin
thickness within the RMF models is contained within
the slanted black lines in the Fig. 3. Their intersection
with the shaded region projects out those value of neutron skin thickness for the nuclei concerned and also the
density slope parameter L that are commensurate with
the empirical windows for Cv0 , Cs0 and ρ0 . The values of
the symmetry slope parameter L and the neutron skin
thickness Rskin for the two nuclei and the three cases are
summarized in Table I. It is seen that for a particular
functional form for Cv (ρ) (case I-III) chosen, the value of
L is nearly independent of the heavy system. If results
for all the three cases and the two nuclear systems are
put together, the value of L spans the range 59.0 ± 13.0
MeV. Case III, designed specifically for M3Y interaction
shows a distinctly different behavior for Cv (ρ) at somewhat higher density beyond saturation as shown in Fig. 2.
Remembering that in case II (as opposed to case I), the
density dependence of the symmetry energy originating
5
TABLE I: The values of symmetry energy slope parameter
L and the neutron-skin thickness Rskin obtained for different
forms for the density dependence of Cv (ρ), labelled as case I,
II and III.
208
L (MeV)
Case I 64.8±7.2
Case II 64.1±6.8
Case III 58.2±11.7
Pb
Rskin (fm)
0.195±0.022
0.196±0.021
0.193±0.018
238
U
L (MeV) Rskin (fm)
62.5±6.2 0.192±0.022
62.2±6.8 0.191±0.021
56.0±10.0 0.184 ±0.014
it appears that case II is the most suitable choice for the
functional form of Cv (ρ). There is not much difference in
the values of L deduced from the two cases though, this
is because the exponent γ in case I is not much different
from the density exponent in the symmetry kinetic energy. There is some confusion as to whether the effective
nucleon mass m∗ or the bare mass m should be used in
evaluating Ck , we find that effects of interaction implicitly hidden in m∗ are compensated by the corresponding
change in γ so that the value of L is nearly unchanged.
In summary, building on the precise knowledge of the
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