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arXiv:1304.4721v1 [nucl-th] 17 Apr 2013
Towards a better knowledge of the nuclear equation of state from the isoscalar
breathing mode
E. Khan1and J. Margueron1, 2
1Institut de Physique Nucl´eaire, Universit´e Paris-Sud, IN2P3-CNRS, F-91406 Orsay Cedex, France
2Institut de Physique Nucl´eaire de Lyon, Universit´e de Lyon, IN2P3-CNRS, F-69622 Vil leurbanne, France
(Dated: July 16, 2018)
The measurements of the isoscalar giant monopole resonance (GMR), also called the breathing
mode, are analyzed with respect to their constraints on the quantity Mc, e.g. the density dependence
of the nuclear incompressibility around the so-called crossing density ρc=0.1 fm−3. The correlation
between the centroid of the GMR, EGMR, and Mcis shown to be more accurate than the one
between EGMR and the incompressibility modulus at saturation density, K∞, giving rise to an
improved determination on the nuclear equation of state. The relationship between Mcand K∞
is given as a function of the skewness parameter Q∞associated to the density dependence of the
equation of state. The large variation of Q∞among different energy density functionnals directly
impacts the knowledge of K∞: a better knowledge of Q∞is required to deduce more accurately
K∞. Using the Local Density Approximation, a simple and accurate expression relating EGMR and
the quantity Mcis derived and successfully compared to the fully microscopic predictions.
PACS numbers: 21.10.Re, 21.65.-f, 21.60.Jz
I. INTRODUCTION
The determination of the nuclear incompressibility is
a long standing problem. The earliest microscopic anal-
ysis came to a value of K∞=210 MeV [1], but with the
advent of microscopic relativistic approaches, a value of
K∞=260 MeV was obtained [2]. The fact that K∞can-
not be better determined than 230±40 MeV, taking into
account the whole data on the isoscalar Giant Monopole
Resonance (GMR) as well as the various methods to re-
late the GMR to K∞(see e.g. [1–9]) lead to a recent
effort to reanalyse the method [10].
Pairing effects and similarly the shell structure ef-
fects on the nuclear incompressibility were analyzed along
these lines. Since the first investigation [11], several stud-
ies have shown that pairing effects have an impact on the
determination of K∞[7, 8], and it was considered as a
possible cause of the difficulty to accurately constrain
K∞. This effect of pairing on the incompressibility mod-
ulus has also been analyzed in nuclear matter, showing
that the main effect is occurring at subsaturation densi-
ties [12]. However there is a general consensus between
the various microscopic models that pairing effects on
K∞are not strong enough to explain the lack of accuracy
in the determination of the nuclear incompressibility [7–
9, 13]. Other effects have to be investigated.
Recently, the density dependence of the nuclear incom-
pressibility was re-investigated suggesting that the corre-
lation between the centroid of the GMR and the incom-
pressibility modulus K∞at saturation density is blurred
by the density dependence of the nuclear equation of state
in different models [10]. The observed differences in the
extraction of K∞from the EGMR are based on different
models and attributed to the density dependence of the
equation of state (EoS) which has still to be better con-
strained. The observation of a crossing point provided
a possible path to be investigated. The crossing point
arises from Energy Density Functionnals (EDFs) that
are designed to describe finite-nuclei observables: their
density-dependent incompressibility K(ρ) crosses around
the mean density in nuclei, ρc≃0.1 fm−3. It was there-
fore proposed that the quantity Mc, e.g. the density
dependence of K(ρ) around the crossing density ρcis the
quantity that shall be constrained by measurements of
the GMR, instead of K∞.
The aim of the present article is to further analyze
the correlation method on which is extracted the incom-
pressibility modulus K∞, and to give a better basis on
the alternative method based on the correlation between
EGMR and Mc. A comparison between the two methods
is given in Sec. II for 208 Pb and 120Sn nuclei, showing
the relevance of the new method [10]. In Sec. III the
source of uncertainties in the determination of K∞is di-
rectly related to the skewness parameter Q∞. The skew-
ness parameter gives indeed the present limitation on the
knowledge of the density dependence of the nuclear EoS
between the crossing and the saturation densities. The
origin of the crossing density is also demonstrated in the
case of the Skyrme EDFs. In Sec. IV, the explicit relation
between the centroid of the GMR and the quantity Mc
is derived using the Local Density approximation (LDA),
and keeping as much as possible analytical relations be-
tween the various quantities, in order to facilitate their
interpretation. The results are compared to the fully mi-
croscopic one. Conclusions are given in Sec. V.
II. THE MICROSCOPIC APPROACH
In this section, we first summarize the Constrained
Hartree-Fock-Bogoliubov approach (CHFB) used to ac-
curately predict the Isoscalar Giant Monopole Resonance
(GMR) energy. We then provide the definition of the
parameter Mc, driving the density dependence of the in-
2
compressibility around the crossing point. Finally, using
these two quantities, the correlation analysis between the
GMR energy and the incompressibility modulus K∞on
one hand, and the GMR energy and the parameter Mc
at the crossing density on the other hand, are compared.
A. Microscopic calculation of the GMR energy
We first recall how to predict the EGMR. We use the
sum rule approach in order to microscopically calculate
the centroid energy of the GMR. In such a microscopic
approach, we calculate the energy as
EGMR =rm1
m−1
.(1)
where the k-th energy weighted sum rule is defined as
mk=X
i
(Ei)k|hi|ˆ
Q|0i|2,(2)
with the RPA excitation energy Eiand the isoscalar
monopole transition operator,
ˆ
Q=
A
X
i=1
r2
i.(3)
The calculations using fully microscopic approaches
based on EDF are usually performed using CHFB or the
RPA approach [14]. In the present case we calculate the
GMR centroid for the Skyrme EDF with the CHFB ap-
proach. For completeness, results using other function-
nals such as Gogny and relativistic functionnals will also
be given. The CHFB method is known to provide an
accurate prediction of the GMR centroid.
In the following the energy weighted moment m1and
the m−1moment are directly evaluated from the ground-
state obtained from Skyrme CHFB calculations. The
moment m1is evaluated by the double commutator using
the Thouless theorem [15]:
m1=2~2A
mhr2i.(4)
where Ais the number of nucleons, mthe nucleon mass
and hr2iis the rms radius evaluated on the ground-state
density given by Skyrme HFB.
Concerning the evaluation of the moment m−1, the
constrained-HFB approach is used. It should be noted
that the extension of the constrained HF method [4, 16]
to the CHFB case has been demonstrated in Ref. [17]
and employed in [8]. The CHFB Hamiltonian is built
by adding the constraint associated with the isoscalar
monopole operator, namely
ˆ
Hconstr. =ˆ
H+λˆ
Q, (5)
and the m−1moment is obtained from the derivative of
the expectation value of the monopole operator on the
CHFB solution |λi,
m−1=−1
2∂
∂λ hλ|ˆ
Q|λiλ=0
.(6)
B. Constraints on the equation of state deduced
from EGMR
Next, the parameter Mcis defined. Instead of corre-
lating EGMR and K∞, it was proposed that the energy
of the GMR (1) gives a strong constraint on the quan-
tity Mcdefined, at the crossing density ρc≃0.1 fm−3,
as [10],
Mc≡3ρcK′(ρ)|ρ=ρc,(7)
where the density-dependent incompressibility K(ρ) is
derived from the thermodynamical compressibility χ(ρ)
as [18],
K(ρ) = 9ρ
χ(ρ)=18
ρP(ρ) + 9ρ2∂2E(ρ)/A
∂ρ2,(8)
and the pressure is
P(ρ)≡ρ2∂E(ρ)/A
∂ρ (9)
The parameter Mcwas introduced instead of K∞≡
K(ρ0) (where ρ0is the saturation density) in the cor-
relation analysis based on EGMR since i) the crossing
density ρcis closer to the average density in finite nuclei
than the saturation density ρ0, and ii) the crossing of the
incompressibility at ρcmakes the EGMR mostly sensitive
to the derivative of the incompressibility at the crossing
density [10]. It should be noted that the existence of a
crossing density for other EoS quantities, such as for in-
stance the symmetry energy [19], the neutron EoS [20, 21]
and the pairing gap in nuclear matter [22], was also ob-
served. It might reveal the general trend that the experi-
mental constraints drive these quantities towards a cross-
ing point at around the average density of finite nuclei.
Various EDF’s shall however exhibit various density de-
pendencies around the crossing point. At first order the
derivative of the incompressibility (or symmetry energy
or pairing gap) at this point will differ between various
EDF’s and additional measurements in nuclei shall char-
acterize these derivatives. For instance, the derivative of
the neutron EoS around ρc≃0.11 fm−3was found to be
strongly correlated to the neutron skin in 208Pb [20, 21],
giving a strong support to improved experimental mea-
surements of this quantity [23].
Fig. 1 depicts K(ρ), between half of the saturation
density and the saturation density, for several Gogny,
Skyrme and relativistic EDFs. A large dispersion is
observed at saturation density (ρ/ρ0= 1) whereas at
ρ/ρ0≃0.71 there is a much more focused area, defining
the crossing density ρc.
3
0.5 0.6 0.7 0.8 0.9 1
ρ/ρ0
-100
-50
0
50
100
150
200
250
300
K(ρ) (MeV)
SKM*
SLy5
Sk272
Sk255
DDME2
FSUGold
NL3
D1S
FIG. 1. (Color online) EoS incompressibility K(ρ) calcu-
lated with various relativistic and non-relativistic function-
nals. The Skyrme EDF are in solid lines
We now analyze the two contributions to the density-
dependent incompressibility, as depicted by Eq. (8). The
first term of the r.h.s in Eq. (8) is proportional to the
pressure P(ρ), which is indeed related to the first deriva-
tive of E/A and the second term is the second derivative
of the binding energy E/A with respect to the density.
The former vanishes at saturation density, by definition.
Table I displays the expectation values of these two con-
tributions to the incompressibility K(ρ) (Eq. (8)) at the
crossing density ρc= 0.71ρ0and at the saturation den-
sity for several EDFs. The total value of K(ρc)≡Kc
at the crossing density is also displayed, while at ρ0,
K∞= 9ρ2
0
∂2E(ρ)/A
∂ρ2|ρ0. The very weak dispersion of Kc
as a function of the EDFs is striking, whereas the incom-
pressibility modulus at the saturation density K∞is more
scattered. At the crossing density, the incompressibility
Kcis given as the sum of the first and second derivatives
of the energy per particle E/A, which act in opposite
signs, see Table I. The contribution of the pressure at ρc
is not negligible, at variance with its contribution at ρ0,
and largely contributes to the stabilisation of Kc. The
correlations between EGMR and the solely second deriva-
tive of E/A at the saturation density might not be the
most appropriate one and the EDF-invariant property of
the crossing point (ρc,Kc) shall be useful.
C. (EGMR,K∞) versus (EGMR,Mc) correlation
analysis
Using EGMR and Mcdiscussed in the previous sections,
it is possible to determine if Mcis better constrained by
EGMR than K∞. The correlation diagrams (EGMR ,K∞)
and (EGMR,Mc) are compared on Fig. 2 in the case of
208Pb. The relativistic DDME2 interaction in the cor-
Crossing Saturation
Kc18
ρcP(ρc) 9ρ2
c∂2E(ρ)/A
∂ρ2|ρcK∞P(ρ0) 9ρ2
0∂2E(ρ)/A
∂ρ2|ρ0
MeV MeV MeV MeV MeV MeV
SLy5 36 -103 139 230 0 230
SkM∗34 -99 133 217 0 217
Sk255 36 -113 149 255 0 255
Sk272 35 -119 154 272 0 272
SGII 34 -98 132 215 0 215
TABLE I. Evaluation of Kcand of the two terms defining
the incompressibility K(ρ) (Eq. (8)), at the crossing density
ρc= 0.71ρ0and at the saturation density ρ0for a set of
different Skyrme EDFs.
relation graph (EGMR ,K∞) is largely deviating from the
others, as it is well known [2–4] while it is much more
compatible with the others in the (EGMR,Mc) correlation
graph [10]. On the contrary, restricting to the Skyrme in-
teractions, the quantities (EGMR,K∞) and (EGMR ,Mc)
are equally well correlated. This is directly related to
the good correlation between (Mc,K∞) due to a simi-
lar density dependence among the Skyrme EDFs (in ρα),
which will be discussed in section III.C. However, consid-
ering various models with different density dependencies,
a better correlation is observed between EGMR and Mc,
compared to the one between EGMR and K∞. It should
be noted that in the (EGMR ,Mc) correlation graph, an
ordering between Skyrme and relativistic models is also
observed.
Fig. 3 displays the same correlations in the case of the
120Sn nucleus. In this case pairing effects are known to
slightly impact the position of the GMR [12], leading to
a larger dispersion compared to the 208 Pb case. How-
ever similar conclusions can be drawn, namely i) Mcis
a better correlated quantity with EGMR than K∞, ii) a
good (EGMR ,K∞) correlation is also observed among the
Skyrme interactions and iii) the (EGMR ,Mc) correlation
exhibits an ordered correlation between the Skyrme and
the relativistic EDFs.
To conclude, these results on Mcprovide a step to-
wards compatible results between Skyrme, Gogny and
relativistic approaches [10]. The extracted value for the
quantity Mcin 120Sn and 208Pb nuclei are also in bet-
ter agreement between each other than the corresponding
K∞values: considering the various EDFs as well as the
120Sn and the 208Pb data, one gets Mc=1100 ±70 MeV
(6% uncertainty), and K∞=230 ±40 MeV (17% uncer-
tainty) [10]. It should be noted that these considerations
on the slope of the incompressibility Mcat the crossing
point have recently been used in Ref. [29] where a good
linear correlation between EGMR and Mcis also found,
including the so-called BCPM functionnal.
In summary, using microscopic approaches, it is ob-
served that the correlation between Mcand the centroid
4
200 210 220 230 240 250 260 270 280
K (MeV)
13
13.5
14
14.5
15
15.5
16
EGMR(MeV)
1000 1100 1200 1300 1400 1500
Mc (MeV)
13
13.5
14
14.5
15
15.5
16
EGMR(MeV)
D1S
DDME2
SkM*
FSUGold NL3
SLy5
Sk255
Sk272
D1S SkM* SLy5
FSUGold
DDME2
Sk255
NL3
Sk272
8
BCPM
BCPM
FIG. 2. Centroid of the GMR in 208Pb calculated with the
microscopic method (see text) vs. the value of K∞(top) and
Mc(bottom) for various functionals [3, 5, 10, 24–29]. The
solid and dashed lines correspond to fit on the Skyrme and
relativistic EDF values, respectively.
EGMR is less dispersive, and therefore more universal
among various models, than the one between K∞and
EGMR [10]. In the next section, we shall provide a more
quantitative understanding of the differences between the
quantities Mcand K∞, explaining the role of the density
dependence of the equation of state between the crossing
and the saturation densities.
III. DENSITY EXPANSION OF THE
EQUATION OF STATE
The striking stability of Kcamong the various Skyrme
EDFs (Table I) deserves an investigation. In this sec-
tion, the density dependence of the equation of state is
discussed in terms of the derivatives of the EoS with re-
spect to the density. The stability of Kcas well as the
relation between the slope of the incompressibility mod-
ulus Mcand the parameters K∞and Q∞are derived,
providing an explanation for the difficulty to constrain
K∞.
A. Density dependence of the equation of state
around ρ0
We start from a systematic expansion around the sat-
uration density ρ0of the binding energy, such as in the
Generalized Liquid Drop Model (GLDM) [30, 31] where,
in symmetric matter, the energy per particle reads
E(x) = E∞+1
2K∞x2+1
6Q∞x3... (10)
200 210 220 230 240 250 260 270 280
K (MeV)
15
15.5
16
16.5
17
17.5
18
EGMR(MeV)
1000 1100 1200 1300 1400 1500
Mc (MeV)
15
15.5
16
16.5
17
17.5
18
EGMR(MeV)
D1S
DDME2 SkM*
FSUGold NL3
SLy5
Sk255
Sk272
D1S
SkM*
SLy5
FSUGold DDME2
Sk255
NL3
Sk272
8
FIG. 3. Same as Fig. 2 for 120Sn.
with x= (ρ−ρ0)/(3ρ0), ρ0being the saturation density
of symmetric nuclear matter. Q∞is the third derivative
of the energy per particle.
Applying Eqs. (8) and (9) to the expansion Eq. (10),
one obtains the pressure,
P(x) = 1
3(1 + 3x)2hK∞x+1
2Q∞x2+...i,(11)
and the incompressibility,
K(x) = (1 + 3x)hK∞+ (9K∞+Q∞)x+ 6Q∞x2
+...i.(12)
Fig. 4 displays the binding energy Eq. (10), pressure
Eq. (11) and incompressibility Eq. (12) as function of
the density ρgoing from 0 to 0.2 fm−3, for typical values
for the quantities : E∞=-16 MeV, K∞=240 MeV and
Q∞=-350 MeV. The different curves correspond to vari-
ous approximations in the density expansion of the bind-
ing energy Eq. (10). For instance, the solid line in the
binding energy E/A corresponds to the 0-th order in the
density expansion Eq. (10) where only the quantity E∞
has been included, all other quantities being set to zero.
The dotted-line (E+K) takes into account the quantities
E∞and K∞, and the dashed line (E+K+Q) includes
the quantities E∞,K∞and Q∞. Similar approximations
have been performed in the case of the pressure Eq. (11)
and incompressibility Eq. (12). A good convergence is
found when successively including in the expressions for
the binding energy, the pressure and the incompressibil-
ity the quantities E∞,K∞and Q∞. These quantities
describe the density dependence of the equation of state
and are given in table II for a set of models considered
in this work.
It is clear from Table II that while the quantities E∞
and K∞are not varying by more than 20%, the values
5
-15
-10
-5
0
E/A (MeV)
E
E+K
E+K+Q
-0,5
0
0,5
P (MeV fm-3)
K
K+Q
00,05 0,1 0,15 0,2
ρ (fm-3)
-100
0
100
200
K (MeV)
0 0,2
0
200 K
K+Q
a)
b)
c)
FIG. 4. (color online) a) Binding energy E/A in MeV, b)
pressure in MeV fm−3and c) incompressibility K in MeV, as a
function of the density for various truncation in the expansion
Eq. (10). See text for more details.
ρ0E∞K∞Q∞
fm−3MeV MeV MeV
SLy5 0.160 -15.98 230 -363
SkM∗0.160 -15.79 217 -386
Sk255 0.157 -16.35 255 -350
Sk272 0.155 -16.29 272 -306
D1S 0.163 -16.02 210 -596
NL3 0.148 -16.24 271 189
DDME2 0.152 -16.14 251 478
FSUGold 0.148 -16.30 229 -537
TABLE II. Parameters appearing in the density expansion of
the binding energy E/A Eq. (10) for a set of models con-
sidered in this work: ρ0is the saturation density, E∞the
binding energy, K∞the incompressibility modulus, and Q∞
the skewness parameter.
for the skewness parameter Q∞is almost unconstrained
and can vary by more than 100% among the models. The
uncertainty in the determination of the skewness param-
eter Q∞gives, in a quantitative way, the main lack of
knowledge in the density dependence of the equation of
state. The uncertainty on Q∞also impacts the density
dependence of the pressure and more, interestingly here,
of the incompressibility.
-800 -600 -400 -200 0 200 400
Q (MeV)
210
220
230
240
250
260
270
280
K (MeV)
SkM*
SLy5
Sk255
Sk272
FSUGold
NL3
DDME2
8
8
D1S
BCPM
FIG. 5. K∞versus Q∞for several models. The solid line
corresponds to the fit on the Skyrme EDFs values.
B. Stability of Kc
Let us now provide an explanation for the stability of
Kcobserved in Table II. From Eq. (12), and assuming the
validity of a density expansion from ρ0to ρc, we obtain
Kc≃(1 + 3xc) [(1 + 9xc)K∞+ (1 + 6xc)xcQ∞],(13)
with xc= (ρc−ρ0)/(3ρ0).
In the case of Skyrme interaction, there is a good cor-
relation among the quantities K∞and Q∞, as shown
in Fig. 5. The parameters K∞and Q∞are mostly de-
termined by the same term, the term t3in ρα, in the
case of Skyrme interaction [10]. Due to their similar
density dependence (in ρα), the Skyrme EDFs exhibit
indeed a linear correlation among these two quantities
whereas the picture is blurred when considering at the
same time Skyrme and relativistic EDFs. The linear cor-
relation among the Skyrme EDFs can be described by,
K∞=a+bQ∞,(14)
with a= 338 ±9 MeV and b= 0.29 ±0.03. Injecting (14)
in Eq. (13), one gets
Kc≃(1 + 3xc) [(1 + 9xc)a+f(xc)Q∞],(15)
with f(x) = (6x2+ (9b+ 1)x+b). An EDF-almost-
independent value of Kcis therefore obtained for f(x)=0
since Q∞is the only EDF-dependent quantity in Eq.
(15): the solution of f(x)=0 shall therefore provide the
crossing point observed on Fig. 1. The function f(x)
has only one zero for positive densities, given by xc=
−0.095 ±0.002, which corresponds to ρ= (0.714 ±
0.005)ρ0=ρc.
In the Skyrme case there is therefore a density, ρc, for
which the incompressibility modulus K(ρc) is indepen-
dent from the quantities K∞and Q∞defined in Eq. (10),
6
and is
Kc≡K(ρc) = (1+ 3xc)(1 + 9xc)a=ρc
ρ0
(3 ρc
ρ0
−2)a. (16)
Taking the value for agiven by the linear correlation, one
finds Kc= 34±4 MeV, confirming the value of the cross-
ing point shown in Table I and on Fig. 1 for the Skyrme
EDFs. This approach confirms in a both quantitative
and qualitative way the existence of a crossing point, es-
pecially in the case of the Skyrme EDF. In the case of
the other EDFs, it is rather a crossing area that is ob-
tained (Fig. 1), due to their various density dependence,
as discussed above.
C. Relation between K∞and Mc
Fig. 6 displays the (Mc,K∞) correlation for four
Skyrme EDFs. Adding to this correlation graph EDFs
with other density dependences, such as the relativistic
one, drastically blurs this correlation. This shall orig-
inates from the large uncertainty on the value of the
skewness parameter Q∞among the EDFs discussed in
the previous section (Table II). Using Eqs. (7), (8) and
(12), the quantity Mccan be expressed as,
Mc≃3Kc+ (1 + 3xc)2h9K∞+ (1 + 12xc)Q∞i.(17)
The correlation between Mcand K∞depends on the den-
sity dependence of the binding energy reflected in the
skewness parameter Q∞, which can vary to a large ex-
tent, see Table II. More precisely, from Eq. (17), one can
deduce the value of the quantity K∞as,
K∞=1
9
Mc−3Kc
(1 + 3xc)2−1 + 12xc
9Q∞,(18)
where the second term of the r.h.s shows the theoretical
error on K∞induced by the uncertainty on Q∞, the un-
constrained density dependence of the EoS. Kcand xc
are fixed by the existence of a crossing point, and Mc
is extracted from the correlation analysis based on the
experimental EGMR. It is therefore clear that the un-
certainty on K∞is related to the lack of knowledge on
the density dependence of the equation of state, repre-
sented in the present analysis by the skewness parameter
Q∞. Taking a typical uncertainty for Q∞of ±400 MeV
(Table II), Eq. (18) provides a variation on K∞about
±40 MeV, compatible with the present uncertainty on
K∞.
In conclusion, the relation Eq. (18) clearly shows that
the uncertainty on the incompressibility modulus K∞is
mainly related to that on the quantities Q∞. The reduc-
tion of the error bar on K∞is therefore mostly related
to a better knowledge of the skewness parameter Q∞, for
which new experimental constraints shall be found. On
the contrary the density dependence around the crossing
point Mccan be more directly constrained from EGMR,
as it will be showed below.
1000 1100 1200 1300 1400 1500
Mc (MeV)
200
210
220
230
240
250
260
270
280
K (MeV)
SkM*
SLy5
Sk255
Sk272
NL3
DDME2
FSUGold
D1S
8
BCPM
FIG. 6. K∞as a function of Mcfor various Skyrme, Gogny
and relativistic functionals
IV. A SIMPLE EXPRESSION RELATING EGMR
AND Mc
To provide a complementary view to microscopic ap-
proaches [1–10], it may be useful to derive an analytic
relationship between the GMR centroid in nuclei and the
quantity Mc, in order to enlighten and confirm the results
obtained with a fully microscopic approach, see Sec. II
and Ref. [10]. In this section we aim to derive an analyt-
ical relationship between the centroid of the GMR and
the relevant quantity of the EoS, McEq. (7).
The energy centroid of the GMR is used to define the
incompressibility in nuclei KA[1]:
EGMR =s~2KA
mhr2i.(19)
In order to derive an analytical relationship, hr2ican be
approximated by 3R2/5 [32] where R≃1.2A1/3is the
nuclear radius, yielding:
EGMR ≃~
Rr5KA
3m(20)
We shall derive an analytic relation between KAand
Mcusing the LDA, in order to check, in a complemen-
tary way to microscopic approaches, the role of Mcin
determining the centroid of the GMR.
The following step consists in dividing KAinto a nu-
clear and a Coulomb contributions, as
KA=KNucl +KCoul ·Z2A−4/3(21)
where, in the liquid drop approach, KNucl is defined as
KNucl =K∞+Ksurf A−1/3+KτN−Z
A2
,
7
as in the Bethe Weiss¨acker formula for the binding en-
ergy [1]. The accuracy of this approach can be enhanced
with the inclusion of higher order terms [33]. The quan-
tities K∞,Ksurf and Kτare however poorly constrained
by the relative small data [1, 34]. We prefer instead to ex-
tract KNucl from the LDA which has the advantage that
i) it was proven to be a good approximation of the mi-
croscopic calculation [12], and ii) the consistency between
the value obtained for KAand the Skyrme functional is
guarantied.
A. The local density approximation (LDA)
The nuclear contribution KNucl is related to the density
dependence of the incompressibility K(ρ) as [12],
KNucl =ρ2
0
AZd3rK(ρ(r))
ρ(r)(22)
Eq. (22) allows to perform the LDA by consider-
ing the density profile of nuclei, ρA(r), in Eq. (8),
where ρ=ρA(r). The LDA give accurate estimation
of KNucl [12]. It should be noted that in Eq. (22), the
value of K(ρ) at saturation density (i.e. K∞) doesn’t
have any specific impact on the KAvalue and therefore
nor on the prediction of EGMR. Further, due to the ex-
istence of the crossing area (ρc,Kc)≃(0.1 fm−3,40 MeV),
which takes into account both the Skyrme EDFs cross-
ing (Table I) and the relativistic one (Fig 1), K(ρ) can
be approximated to the first order around the crossing
point by:
K(ρ) = Mcρ
3ρc
−Mc
3+Kc(23)
where Mcis related to the first derivative of the incom-
pressibility, Eq. (7).
This first order approximation is relevant as observed
on the (EGMR,Mc) correlation of Figs. 2 and 3. Of course
taking the density dependence of the incompressibility
as its first derivative around the crossing point remains
an approximation, which explains the non exactly linear
(EGMR,Mc) correlation on Fig. 2 considering Skyrme
and relativistic EDFs. But still, the correlation among
the EDF families (Skyrme, Relativistic) are ordered.
The integral in Eq. (22) is taken between ρ0/2 and
ρ0, which is adapted to the linear regime around ρcand
corresponds to the typical dispersion of the density val-
ues around the mean density in nuclei [10]. Injecting
expression (23) in (22) and assuming a Fermi shape of
the nuclear density, with diffusivity a ≃0.5 fm [32], yield
the analytical relation between the centroid of the GMR
and Mc, using Eq. (20) and (21):
EGMR =~
R(20π
3mA Zρ0
ρ0/2aln ρ0
ρ−1+R2Mcρ
3ρc
−Mc
3+Kca
1−ρ/ρ0
ρ2
0
ρ2dρ+5KCoul
3mZ2A−4/3)1/2
(24)
The integral in Eq. (24) denotes the nuclear contribu-
tion whereas the second part comes from the Coulomb
effects. The Coulomb contribution is evaluated using
KCoul=-5.2 MeV [1, 35]. This value is obtained from
the liquid drop expansion of the incompressibility and
applied to several Skyrme interactions [35]. It should be
noted that the Fermi shape is a good approximation of
the density and we have checked that the diffusivity of
the density obtained from microscopic Hartree-Fock cal-
culations (0.47 fm) is very close to 0.5 fm. The use of
the Fermi density is legitimized by the aim of tracing
the analytical impact of the quantity Mcon the GMR
centroid. Equation (24) also underlines the important
role of the quantity Mcon the GMR centroid. On the
contrary, the incompressibility at saturation density K∞
doesn’t play any specific role in Eq. (24). It is therefore
rather the quantity Mcwhich is the relevant quantity to
be constrained by the GMR measurements.
B. Results and comparison with the microscopic
method
The stability of the results obtained with Eq. (24) has
been studied with respect to the diffusivity value a, the
LDA prescription (Eq. (22)), the crossing point (ρc,Kc)
values and the integration range. A sound stability is
obtained against these quantities: the predicted GMR
centroid doesn’t change by more than 10 % by making
all these variations in relevant physical ranges.
We first study the behavior of the nuclear contribution
(KCoul=0 in Eq. (24)). Fig. 7 displays the correlation
between the centroid of the GMR and the Mcvalue us-
ing Eq. (24), for nuclei with A=208 and A=120. A good
qualitative agreement is obtained with the fully micro-
scopic results (see Fig. 2 and 3) in view of the approxima-
tions performed to derive Eq. (24). The A-dependence
is also well described. These results confirm the validity
of the present approach, and emphasize Mcas a relevant
EoS quantity to be constrained by the GMR measure-
ments. It also qualitatively agrees with the microscopic
results.
In order to perform a more quantitative study, the
8
1000 1200 1400 1600 1800 2000
Mc (MeV)
14
15
16
17
18
19
20
21
EGMR (MeV)
A=208
A=120
FIG. 7. Centroid of the GMR in A=208 (solid line) and
A=120 (dashed line) nuclei calculated with the local density
approximation for the nuclear incompressibility and using its
first derivative at the crossing point (Eq. (24) without the
Coulomb term).
1000 1100 1200 1300 1400 1500
Mc (MeV)
13
14
15
16
17
18
EGMR (MeV)
208Pb
120Sn
LDA
SkM*
SLy5
Sk255 Sk272
SkM*
SLy5 Sk255
Sk272
DDME2
D1S
FSUGold NL3
D1S FSUGold
NL3
DDME2
BCPM
FIG. 8. Centroid of the GMR in 208Pb and 120 Sn calculated
with the LDA and including the Coulomb effects (solid lines,
see text) as a function of Mc. The values for various EDF
obtained microscopically (squares and dots) are also displayed
for comparison.
curves on Fig. 8 displays the centroid of the GMR in
208Pb and 120 Sn nuclei predicted using the full Eq. (24)
with both the nuclear and the Coulomb contributions.
The comparison with the microscopic results using vari-
ous EDFs is also shown. A good agreement is obtained
in both cases, strengthening again the present analytical
LDA approach, and emphasising the role of Mc. Com-
paring Figs. 7 and 8, the Coulomb effect on the GMR
centroid can be evaluated to be about 1 MeV in heavy
nuclei.
The almost linear correlation between EGMR and Mc
observed on Fig. 8 can be further investigated. Eq. (24)
can be rewritten as:
EGMR =α(A, ρ0)Mc+β(A, Z, ρ0)1/2(25)
with
α(A, ρ0)≡20π~2
9mAR2Zρ0
ρmin aln ρ0
ρ−1+R2
×
ρ
ρc
−1a
1−ρ/ρ0
ρ2
0
ρ2dρ(26)
β(A, Z, ρ0)≡5~2KCoul
3mR2Z2A−4/3+
20π~2
3mAR2Zρ0
ρmin aln ρ0
ρ−1+R2aKc
1−ρ/ρ0
ρ2
0
ρ2dρ
(27)
Fixing ρcand Kc, the coefficients αand βonly depend
on the nucleus’ mass and charge (A,Z), and on the sat-
uration density ρ0, which is constrained by the charge
radii. Typical values are α=0.12 MeV and β=42 MeV2
in the case of 208Pb and α=0.16 MeV and β=75 MeV2
in the case of 120Sn.
It should be noted that the LDA approximation (25)
of EGMR can be obtained because of the existence of the
crossing point. In Eq. (25), the energy of the GMR de-
pends on the functional mostly through the parameter
Mc. In conclusion, the LDA allows to obtain expres-
sion (25) relating EGMR with Mcin a simple and accurate
form.
Introducing (M0,E0) as the reference point, where
M0≡1200 MeV and E0is the corresponding GMR en-
ergy, one can go one step further and linearise Eq. (25)
with respect to Mc−M0, as
EGMR ≃α
2E0
Mc+E0−αM0
2E0(28)
This is justified for the typical values of Mc, ranging be-
tween 1000 MeV and 1500 MeV as shown on Fig. 8.
The almost linear correlation between EGMR and Mcob-
served on Fig. 8 is therefore understood by the present
approach (Eq. (28)). It clearly shows that the mea-
surement of the GMR position constrains Mc, which is a
first information on the density dependence of the incom-
pressibility. It should be recalled that such a quantitative
description is not possible with K∞because there is no
crossing point of the incompressibility at saturation den-
sity: Eq. (24) is not applicable in that case.
V. CONCLUSIONS
The relationship between the isoscalar GMR and the
equation of state raises the question of which EoS quan-
tity is constrained by GMR centroid measurements. The
9
incompressibility modulus K∞alone may not the rel-
evant one nor the most direct because the more gen-
eral density dependence of the incompressibility should
be considered. A crossing area is observed on K(ρ) at
ρc≃0.1 fm−3among various functionnals. Using a mi-
croscopic approach, such as constrained-HFB, the slope
Mcof K(ρ) at the crossing density can be directly con-
strained by GMR measurements. This shall assess the
change of the method in extracting EoS quantities from
GMR: Mcis first constrained, and an approximate value
of K∞can be deduced in a second step [10].
The stability of Kchas been demonstrated in the case
of Skyrme EDFs. A general relationship between Mcand
K∞is obtained, showing the contribution of the uncer-
tainty in the density dependence of the EoS which has
been casted into the quantities Q∞. The K∞value can
be determined in a second step from the knowledge of
the Mcvalue, requiring a better constraint on the skew-
ness parameter Q∞, being the main uncertainty for the
density dependence of the incompressibility between the
crossing density and the saturation density. One should
recall that the K∞value remains 230 ±40 MeV (17% un-
certainty), whereas the quantity Mcis better constrained
to be Mc=1100 ±70 MeV (6% uncertainty) [10]. A bet-
ter knowledge of higher order density dependent terms of
E/A(ρ), e.g. the skewness parameter Q∞, shall help to
more accurately relate the parameter Mcto the incom-
pressibility modulus K∞.
Using the LDA approach and an analytical approxima-
tion of the density, the microscopic results have been con-
firmed: the measurement of the centroid of the isoscalar
giant monopole resonance constrains the first derivative
Mcof the incompressibility around the crossing point
ρc≃0.1 fm−3. A analytical relation between the cen-
troid of the GMR and the quantity Mcis derived and
the predicted GMR centroid are found in good agree-
ment with the microscopic method.
ACKNOWLEDGEMENT
The authors thank I. Vida˜na for fruitful discussions.
This work has been partly supported by the ANR SN2NS
contract, the Institut Universitaire de France, and by
CompStar, a Research Networking Programme of the Eu-
ropean Science Foundation.
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