Constraining simultaneously nuclear symmetry energy and neutron-proton effective mass splitting with nucleus giant resonances from a dynamical approach

0.34 MB·

by  Hai-Yun Kong

Constraining simultaneously nuclear symmetry energy and neutron-proton effective
mass splitting with nucleus giant resonances from a dynamical approach
Hai-Yun Kong,1,2 Jun Xu∗,1 Lie-Wen Chen,3,4 Bao-An Li,5,6 and Yu-Gang Ma1,7
1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3Department of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai 200240, China
4Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
5Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429-3011, USA
6Department of Applied Physics, Xi’an Jiao Tong University, Xi’an 710049, China
7Shanghai Tech University, Shanghai 200031, China
7 (Dated: March 14, 2017)
1
0 Withanewlyimprovedisospin- andmomentum-dependentinteractionandanisospin-dependent
2 Boltzmann-Uehling-Uhlenbeck transport model, we have investigated the effects of the slope pa-
rameter L of the nuclear symmetry energy and the isospin splitting of the nucleon effective mass
ar m∗n−p = (m∗n −m∗p)/m on the centroid energy of the isovector giant dipole resonance and the
M electric dipole polarizability in 208Pb. With the isoscalar nucleon effective mass m∗s = 0.7m con-
strained by the empirical optical potential, we obtain a constraint of L = 64.29 ±11.84(MeV)
1 and m∗n−p = (−0.019±0.090)δ, with δ being the isospin asymmetry of nuclear medium. With
1 the isoscalar nucleon effective mass m∗s = 0.84m extracted from the excitation energy of the
isoscalargiantquadrupleresonancein208Pb,weobtainaconstraintofL=53.85±10.29(MeV) and
] m∗n−p =(0.216±0.114)δ.
h
t
- PACSnumbers: 24.30.Cz,21.65.+f,21.30.Fe,24.10.Lx
l
c
u
n I. INTRODUCTION ramificationsinthe thermodynamicpropertiesofisospin
[ asymmetric nuclear matter as well [20, 21]. Moreover,

the neutron-proton effective mass splitting is actually
2 One of the main tasks of nuclear physics is to under- inter-relatedtothenuclearsymmetryenergythroughthe
v
stand the in-medium nuclear interactions and the equa- Hugenholtz-Van Hove theorem [10, 22]. For a recent re-
2
0 tion of state (EoS) of nuclear matter. The uncertainties viewontheisospinsplittingofthenucleoneffectivemass,
5 of the isospin-dependent part of the EoS, i.e., the nu- we refer the reader to Ref. [23].
4 clear symmetry energy (Esym), has hampered our accu-
0 rate understandingof nuclearmatter properties,while it Giant resonances of nuclei serve as a useful probe of
. hasimportantramificationsinheavy-ionreactions,astro- nuclear interactions and the EoS of nuclear matter at
1
0 physics,andnuclearstructures[1–8]. Thankstothegreat subsaturationdensities. The studies on giantresonances
7 efforts made by nuclear physicists in the past decade, a mainly follow two methods, i.e., the random-phase ap-
1 morestringentconstraintonE atsubsaturationden- proximation and the transport model calculations. As a
sym
: sities has been obtained from various analysis, with the breathingoscillationmodeintheradialdirectionofanu-
v
i slope parameter of the nuclear symmetry energy so far cleus,theisoscalargiantmonopoleresonance(GMR)isa
X constrained within L = 60± 20 MeV [9–11], although goodprobeoftheincompressibilityofnuclearmatter[24–
r further verifications are still needed. On the other hand, 28], while the isoscalar giant quadruple resonance (IS-
a
the in-medium isospin splitting of the nucleon effective GQR), an oscillation mode with quadruple deformation
mass m∗ = (m∗ −m∗)/m has become a hot topic re- of a nucleus, has been found to be much affected by the
n−p n p ∗
cently. Comparedtothebarenucleonmassinfreespace, isoscalarnucleoneffectivemassm [29–34]. Ontheother
s
the in-medium nucleon effective mass comes from the hand,the isovectorgiantdipole resonance(IVGDR) and
momentum-dependent potential in non-relativistic mod- the pygmy dipole resonance (PDR), with the former an
els(see,e.g.,Chapter3ofRef.[7]). Theisospinsplitting oscillationmode betweenthe centersofmassofneutrons
ofthein-mediumnucleoneffectivemassisthusrelatedto and protons and the latter that between the neutron
themomentumdependenceofthesymmetrypotentialin skin and the nucleus core, are valuable probes of the nu-
non-relativistic models [12]. It has been found that the clearsymmetryenergyatsubsaturationdensities[35–49].
isospin splitting of the nucleon effective mass is as im- Since the nuclear symmetry energy acts as a restoring
portantasthenuclearsymmetryenergyinunderstanding force for the IVGDR, the main frequency of the IVGDR
theisospindynamicsinnuclearreactions[13–19],andhas oscillation, i.e., the centroid energy E−1, is related to
E atsubsaturationdensities orits slope parameterL
sym
atthe saturationdensity [36, 45,49]. The electric dipole
polarizability α has a strong correlation with the neu-
D
∗correspondingauthor: [email protected] tron skin thickness ∆rnp [38, 49], and αD times Esym at
2
the saturation density shows a good linear dependence be expressed as [21, 56]
onL [44,47,48]. Itwasarguedthatthe accurateknowl-
edgeofαD and∆rnp canhelpconstrainsignificantlythe V(ρ,δ) = Auρnρp + Al (ρ2 +ρ2)+ B ρσ+1
nuclearsymmetry energyat subsaturationdensities [40]. ρ 2ρ n p σ+1 ρσ
0 0 0
The effect of the neutron-proton effective mass splitting 1
onthe IVGDRpropertieswasrealizedonlyrecently[34]. × (1−xδ2)+ Cτ,τ′
ρ
It is also worth mentioning that the dynamical isovec- 0 Xτ,τ′
′
tor dipole collective motion in fusion reactions could be × d3pd3p′fτ(~r,p~)fτ′(~r,~p). (1)
a probe of the nuclear symmetry energy as well from 1+(p~−~p′)2/Λ2
Z Z
transport model studies [50–55].
The single-particle potential of a nucleon with isospin τ
Recently we have improved our isospin- and andmomentump~intheasymmetricnuclearmatterwith
momentum-dependent interaction (MDI) [56, 57], which the isospin asymmetry δ and nucleon number density ρ
was previously extensively used in the studies of ther- can be expressed as
modynamic properties of nuclear matter, dynamics of
ρ−τ ρτ
nuclear reactions, and properties of compact stars (see U (ρ,δ,p~) = A +A
τ u l
ρ ρ
Ref. [58] for a review). In the improved isospin- and 0 0
mmoenmteunmtudmep-deenpdeenndceenotfitnhteermacetaionn-fi(eIlmdMpoDteI)nt[2ia1l],htahsebmeeon- + B ρρ σ(1−xδ2)−4τxσB+1ρρσ−σ1δρ−τ
(cid:18) 0(cid:19) 0
fitted to that extracted from proton-nucleus scatterings 2C f (~r,p~′)
up to the nucleon kinetic energy of about 1 GeV, and + ρ l d3p′1+(p~τ−p~′)2/Λ2
moreisovectorparametersarefurtherintroducedsothat 0 Z
′
the density dependence of the symmetry energy and the + 2Cu d3p′ f−τ(~r,p~) . (2)
momentum dependence of the symmetry potential, or ρ0 Z 1+(p~−p~′)2/Λ2
equivalently, the isospin splitting of the nucleon effec-
In the above, ρ and ρ are the number density of neu-
tive mass, can be mimicked separately. In the present n p
trons and protons, ρ is the saturation density, and
study,wearegoingtoinvestigatetheeffectofthenuclear 0
δ = (ρ − ρ )/ρ is the isospin asymmetry. f (~r,~p) is
symmetry energy and the neutron-proton effective mass n p τ
the phase-space distribution function with τ = 1(−1)
splitting on the centroid energy E−1 of IVGDR as well being the isospin label of neutrons (protons). The po-
astheelectricdipolepolarizabilityα ,byemployingthe
D tential energy density of Eq. (1) can be derived based
ImMDI interaction together with the isospin-dependent
on the Hartree-Fock calculation from an effective inter-
Boltzmann-Uehling-Uhlenbeck(IBUU) transportmodel.
action with a zero-range density-dependent term and a
With the experimental data of E−1 and αD from the finite-range Yukawa-type term [60].
IVGDRin208Pbavailable[39,47,59],weareabletocon-
In the ImMDI interaction [21], the isovector parame-
strain both the slope parameter of the symmetry energy
ters x, y, and z are introduced to vary respectively the
and the isospin splitting of the nucleon effective mass,
slopeparameterofthe symmetryenergy,the momentum
once the isoscalar nucleon effective mass is well deter-
dependence of the symmetry potential or the neutron-
mined. SectionIIgivesabriefintroductiontotheImMDI
protoneffectivemasssplitting, andthevalueofthe sym-
interactionandthenecessaryformalismsforIVGDR.De-
∗ metry energy at the saturationdensity, via the following
tailedstudies onthe extractionof m and the constraint
∗ s relations
on L and m from IVGDR are presented in Sec. III.
n−p
We conclude in Sec. IV. 2B
A (x,y) = A +y+x ,
l 0
σ+1
2B
A (x,y) = A −y−x ,
u 0
σ+1
p2
f
C (y,z) = C −2(y−2z) ,
l l0 Λ2ln[(4p2 +Λ2)/Λ2]
f
p2
II. THEORY C (y,z) = C +2(y−2z) f ,
u u0 Λ2ln[(4p2 +Λ2)/Λ2]
f
(3)
A. An improved isospin- and
momentum-dependent interaction where p = ~(3π2ρ /2)1/3 is the nucleon Fermi momen-
f 0
tum in symmetric nuclear matter at the saturation den-
sity. Wesetz =0allthroughthemanuscript. Thevalues
The potential energy density functional in the asym- of the parameters A , B, C , C , σ, Λ, x, and y can
0 u0 l0
metric nuclearmatter with isospinasymmetryδ and nu- be fitted or solved from the saturation density ρ , the
0
cleon number density ρ from the ImMDI interaction can binding energy E at ρ , the incompressibility K , the
0 0 0
3
∞
mean-field potential U at infinitely large nucleon mo- Since both the strength function S(E) and the photoab-
0
∗
mentumatρ ,theisoscalarnucleoneffectivemassm at sorption cross section σ are related to the energy spec-
0 s f
ρ , the symmetry energy E and its slope parameter trum of the excited states in the nucleus, the centroid
0 sym
∗
L at ρ , and the isovector nucleon effective mass m at energy and the electric dipole polarizability can be ex-
0 v
ρ , as detailed in APPENDIX A. pressedintermsofthemomentsm ofthestrengthfunc-
0 k
tion as
B. Isovector giant dipole resonance E−1 = m1/m−1 (9)
p
and
Theisovectorgiantdipoleresonance(IVGDR)isacol-
lective vibration of protons against neutrons, with the αD =2e2m−1. (10)
isovector dipole operator defined as
We note thatthe relationbetweenαD andm−1 is differ-
NZ
Dˆ = Xˆ, (4) ent from that in Ref. [34], because we only consider one-
A dimensional oscillation, and the definition of the isovec-
tor giant dipole operator is different from that used in
whereN,Z,andA=N+Z arethenumberofneutrons,
Ref. [34] based on a random-phase approximation ap-
protons, and the mass number of the nucleus, respec-
tively,andXˆ is thedistance betweenthe centersofmass proach.
In the following calculations, we fit the time evolution
of protons and neutrons in the nucleus. The strength
of the isovector giant dipole moment with the function
function of IVGDR can be calculated from the isovector
dipole moment via D(t)=asin(bt)e−ct, (11)
−Im D˜(ω) where a, b, and c are fitting constants characterizing
S(E)= , (5) the amplitude, the frequency, and the damping time of
πhη i
IVGDR, respectively. With the help of Eq. (11), the
where D˜(ω) = tmaxD(t)eiωtdt is the fourier transfor- strength function S(E), the moments of the strength
mation of the isto0vector dipole moment with E = ~ω. function m−1 and m1, the centroid energy E−1 of
In order to initiRalize the isovector dipole oscillation, the IVGDR, and the electric dipole polarizability αD can be
expressed analytically in terms of the fitting constants
initialmomentaofprotonsandneutronsaregivenaper-
respectively as
turbation in the opposite direction according to [61]
−ac 1 1
pi →(cid:26)ppii+−ηηNANA ((nperuottroonnss)), (6) S(E) = 2πη "c2+ b+ E~ 2 − c2+ b− E~ 2#,
−ab
with pi being the momentum of the ith nucleon along m−1 = 2η(b2+c2)(cid:0), (cid:1) (cid:0) (cid:1)
Xˆ, and the perturbation parameter η = 25 MeV/c used
−ab
in the present study. One sees that a larger η leads m = ,
1
to a larger amplitude of the oscillation D(t), while the 2η
strength function S(E) is independent of η as the am- E−1 = m1/m−1 = b2+c2,
plitude is cancelled from the denominator. Once the
strength function of IVGDR is obtained, the moments αD = 2pe2m−1 = 2η−(pb22e2+abc2). (12)
of the strength function can be calculated from
∞
mk = dEEkS(E). (7) III. RESULTS AND DISCUSSIONS
Z0
Boththecentroidenergyandthe electricdipole polar- In the followingstudy, we employ the IBUU transport
izability can be measured from photoabsorption experi- model together with the ImMDI interaction to investi-
ments. The centroid energy, corresponding to the peak gate the giant resonances of nuclei. The positions of the
energy in the photoabsorption energy spectrum, is the projectile and target in the IBUU transport model are
main frequency of IVGDR. The electric dipole polariz- fixed, i.e., with zero beam energy. The initial density
ability, characterizing the response of the nucleus to the distributionissampledaccordingto thatgeneratedfrom
external electric field, is related to the photoabsorption Skyrme-Hartree-Fockcalculationswith the same physics
cross section σ via [39] quantities used in the ImMDI interaction, such as L,
f
∗
m , etc., listed in Table I. The initial nucleon momen-
v
~c σ tum is sampled accordingly to the local density from
f
α = dω. (8)
D 2π2 ω2 the Thomas-Fermi approximation. We generate events
Z
4
from 40 runs with each run 200 test particles. Since operator written as
the oscillation generally lasts for hunderds of fm/c, in
order to improve the stability in the calculation with A A 5
the momentum-dependent mean-field potential, we use Qˆ = ri2Y20(rˆi)= 16π 3zi2−ri2 , (13)
the nuclear matter approximation in the real calcula- i=1 i=1r
X X (cid:0) (cid:1)
tion by taking the phase-space distribution function as
wherer andz arerespectivelytheradialandz-direction
f (~r,p~)= 2 Θ(p −p)andusingtheanalyticalexpres- i i
τ h3 fτ coordinate of the ith nucleon, and Y is the spherical
sion(Eqs. (2) and(A.23)) for the momentum-dependent 20
harmonic function. Noticing that the following scaling
mean-field potential. This is similar to the spirit of the
relation in the ISGQR is observed (see, e.g., Ref. [64])
Thomas-Fermiapproximationinthe casethatthe vibra-
tion compared to the stable distribution is small. With
x→x/λ p →λp
this treatment, we found that up to t = 500 fm/c only x x
about 17% of the total nucleons become free particles. y →y/λ  py →λpy , (14)
z →λ2z pz →pz/λ2
we choose λ =1.1 in oursimulation to initialize the os-
cillation. The value of λ is close to 1 corresponding to a
A. Extract the isoscalar nucleon effective mass
small vibration with respect to the equilibrium distribu-
tion. Again,wefoundthatthevalueofλonlyaffectsthe
We first extract the isoscalar nucleon effective mass amplitude but has almost no effect on the frequency of
from the opticalpotential. The single-particle potentials ISGQR.Thetimeevolutionoftheisoscalargiantquadru-
in symmetric nuclear nuclear matter at the saturation plemoment,withdifferentvaluesoftheisoscalarnucleon
density, with different values of the isoscalar nucleon ef- effective mass but same other isoscalar properties of nu-
∗
fective mass m but same other isoscalar properties of clear interaction, is displayed in the left panel of Fig. 2.
s
the nuclear interaction, are displayed in Fig. 1. We can It is seen that a smaller isoscalar nucleon effective mass
seethatonlytheparametersetthatleadstom∗ =0.7m, m∗ leads to a larger frequency of the oscillation. From
s s
with m being the nucleon mass in free space, can de- the FouriertransformationofQ(t), the linearcorrelation
scribe reasonably well the real part of the optical poten- between the isoscalar nucleon effective mass m∗ and the
s
tial extracted from proton-nucleus scatterings by Hama excitationenergyofISGQRisobservedintherightpanel
et al. [62, 63]. of Fig. 2. It is found that the result from the isoscalar
∗
nucleon effective mass m = 0.84m reproduces best the
s
excitationenergyE =10.9±0.1MeVofISGQRin208Pb
x
extracted from α-nucleus scattering experiments [65].
V) 60
e
M
ntial ( 40 4 (a) (b) 13
e 20
ot ISGQR
p
art of optical --42000 mmm*s*s*s===000...788mm4m 2 Q (fm) 02 12xVeM( E
al p m*s=0.9m * )
Re -60 Hama et al. ms*=0.7m 11
-2 ms=0.8m
-80 *
ms=0.84m
0 200 400 600 800 1000 *
ms=0.9m
E-m (MeV)
-4 10
FIG.1: (Coloronline)Single-particlepotentialsinsymmetric 0 200 400 0.6 0.7 0*.8 0.9 1.0
nuclear nuclear matter at the saturation density as a func- t (f m/c) ms( m)
tion of the nucleon energy subtracted by the nucleon rest
FIG. 2: (Color online) The time evolution of the isoscalar
massfromtheImMDI∗interactionwithdifferentisoscalar nu- giant quadruple moment in 208Pb (a) and the correlation
cleon effective mass m . The real part of the optical poten-
tial extracted from prsoton-nucleus scatterings by Hama et between the corresponding excit∗ation energy Ex and the
isoscalar nucleon effective mass m (b).
al. [62, 63]is shown byscatters for comparison. s
∗
The different values of the isoscalar effective mass m
s
∗
Next, we extract the value of m from the isoscalar extractedfromtheopticalpotentialandfromtheexcita-
s
giant quadruple resonance (ISGQR) in 208Pb, with the tion energy of ISGQR in 208Pb represent the theoretical
5
TABLE I: The physicsquantities and thecorresponding parameters for the ImMDI interaction as well as theresults from the
IVGDRin 208Pb.
Set I(a) Set I(b) Set I(c) Set II(a) Set II(b) Set II(c)
A0 (MeV) -66.963 -66.963 -66.963 -96.799 -96.799 -96.799
B (MeV) 141.963 141.963 141.963 171.799 171.799 171.799
Cu0 (MeV) -99.70 -99.70 -99.70 -90.19 -90.19 -90.19
Cl0 (MeV) -60.49 -60.49 -60.49 -50.03 -50.03 -50.03
σ 1.2652 1.2652 1.2652 1.2704 1.2704 1.2704
Λ (p ) 2.424 2.424 2.424 3.984 3.984 3.984
f
x 0 1 1 0 1 1
y (MeV) -115 -115 115 -115 -115 115
ρ0 (fm−3) 0.16 0.16 0.16 0.16 0.16 0.16
E0(ρ0) (MeV) -16 -16 -16 -16 -16 -16
K0 (MeV) 230 230 230 230 230 230
U0∞ (MeV) 75 75 75 75 75 75
m∗ (m) 0.7 0.7 0.7 0.84 0.84 0.84
s
Esym(ρ0) (MeV) 32.5 32.5 32.5 32.5 32.5 32.5
L (MeV) 58.57 8.70 60.00 72.63 11.24 36.03
∗
m (m) 0.537 0.537 0.853 0.712 0.712 0.928
v
a (fm) -28.13±0.33 -27.16±0.25 -19.55±0.14 -24.30±0.21 -22.69±0.21 -19.34±0.16
b (fm−1) 0.0782±0.0001 0.0883±0.0001 0.0614±0.0001 0.0662±0.0001 0.0760±0.0001 0.0640±0.0001
c(fm−1) 0.0075±0.0001 0.0099±0.0001 0.0040±0.0001 0.0052±0.0001 0.0075±0.0001 0.0054±0.0001
E−1 (MeV) 15.50±0.0001 17.53±0.0001 12.13±0.0001 13.10±0.0001 15.06±0.0001 12.68±0.0001
αD (fm3) 20.5±0.2 17.5±0.2 18.3±0.1 21.0±0.2 17.0±0.2 17.3±0.1
uncertaintiesinthepresentstudy. Inthefollowingstudy, in the transport model calculation, a damping factor of
we will thus employ the parameter Set I and Set II that exp(−γt) withthe widthγ =2 MeVis multiplied to the
∗ 2~
leadrespectively to m =0.7m and0.84mtogether with isovectordipolemomentD(t)incalculatingthe strength
s
different combinations of the isovectorparametersx and function S(E) from Eq. (5) as in Ref. [61]. This slightly
y to study the isovector giant dipole resonances of nu- affectsthe dampingcoefficientc inEq.(11) buthasvery
clei. Thevaluesoftheparametersandthecorresponding small effects on the final results as can be seen from the
physics quantities are detailed in Table I. analytical formulaes of Eq. (12). The resulting strength
functions from the numerical Fourier transformation is
displayedintherightpanelofFig.3. Itisseenthatwith
asoftersymmetryenergy(Set I(b)), the mainfrequency,
B. Constrain the symmetry energy and the
neutron-proton effective mass splitting i.e., the centroid energy in the IVGDR, is larger, due
tothelargersymmetryenergyatsubsaturationdensities
∗ acting as a stronger restoring force. On the other hand,
We first employ the parameter Set I with m =0.7m,
s the centroid energy is sensitive to the isovector nucleon
whichisabletoreproducetheempiricalopticalpotential
effective mass as well, with a larger isovector effective
asshowninFig.1,tostudythepropertiesoftheIVGDR
mass (Set I(c)) leading to a smaller centroid energy of
in 208Pb. Three different parameter sets, with different
IVGDR. This could be understood since the oscillation
combinations of the slope parameter L of the symmetry
∗ frequency is smaller with a larger reduced mass of the
energyandtheisovectoreffectivemassm detailedasSet
v system, with the latter attributed to the largerisovector
I(a),SetI(b),andSetI(c)inTableI,areemployedinthe
effective mass once the isoscalar effective mass is fixed.
study. The initial momenta of neutrons and protons are
modified with the perturbation parameter η as detailed The constants a, b, and c can be obtained from fitting
inSec.IIB.Theoscillationamplitudeisproportionaltoη the isovector dipole moment D(t) according to Eq. (11),
whileitsfrequencyisfoundtobeinsensitivetothechoice andtheirdetailedvaluesforthe threeparametersetsare
ofη. The resulting time evolutionofthe isovectordipole listed in Table I. The resulting centroid energy E−1 and
moment is displayedin the left panel of Fig. 3. One sees theelectricdipolepolarizabilityα calculatedanalytical
D
that the time evolution of D(t) follows a good damping accordingto Eq. (12) are also listed in Table I. It is seen
∗
oscillationmode as inEq.(11). Inorderto avoidoscilla- thatforagivenisovectoreffectivemassm ,alargerslope
v
tion in the Fourier transformation due to the finite tmax parameter L leads to a smaller centroid energy E−1 and
6
be

30 (a) (b) 16 −0.041L−10.480m∗v+23.512=E−1, (15)
∗
IVGDR ( 0.061L−7.405mv+20.951=αD,
20 *
ms=0.7m 12 with L in MeV, m∗v in m, E−1 in MeV, and αD in fm3.
D (fm)100 8 S (fm/Me 2 TeisxhpEee−rci1emn=etnrot1ai3dl.l4ye6nferMorgmeyVto[hf5e9t]hp,ehwoIhtVoilaGebDtshoRerpietnlieocn2t0r8miPcebdasiopubortelaeminpeenodt-
V larizability is α = 19.6±0.6 fm3 measured from pho-
) D
-10 toabsorption cross section by Tamii et al. [39] and with
Set I(a) 4 further correction by subtracting quasideuteron excita-
-20 Set I(b) tions [47]. With the above experimental data available,
Set I(c) the constraintson the slope parameterL and the isovec-
-30 0 tor effective mass can be solved from Eq. (15) as
0 200 400 0 10 20 30
L = 64.29±11.84 (MeV), (16)
t (fm/c) E (MeV)
∗
m /m = 0.710±0.046, (17)
FIG.3: (Coloronline)Thetimeevolutionoftheisovectorgi- v
ant dipole moment (a) and the corresponding strength func-
wheretheerrorbars,whichoriginatefromthefittingand
tion (b) in 208Pb with m∗ =0.7m.
s the statistical error of a, b, and c listed in Table I, are
calculatedfromtheerrortransfer. WefoundthattheIm-
∗
MDIparameterizationwiththemeanvaluesofLandm
v
in Eq. (16) gives very close results of E−1 and αD com-
1.0 pared with the experimental data, justifying the linear
relation of Eq. (15). The corresponding isospin splitting
0.8 -0.041L-10.480mv*+23.512=13.46 (mL*v==604.7.2190 mM)e V, of the nuc(mleo∗nn−effme∗pc)t/ivme m=a(s−s0d.e0d1u9c±ed0.f0ro9m0)δE.q. (A.35()18is)
)
* (mm0v.6 m* v+20.951=19.6 Towbhittehaiinnabeadonvbeisyocsoapnnisnatl-rydazeiinpntegnidsn,eunchtloeowopnet-vinceuar,cllmeduioffsdeesrclean[7ttt0e]fr.rionmg dthaatat
0.4 0.061L-7.405 ISmeMt ID I beNme∗sxt=,w0e.8c4hmoowsehitlheekeiseopsicnaglaorthneurcplehoynsiecffseqcutiavnetimtiaesssthtoe
0.2 * same in the ImMDI parameterization, and the resulting
m =0.7m
s parameter sets are listed as Set II(a), Set II(b), and Set
II(c) in Table I with different combinations of the slope
0.0
parameter L of the symmetry energy and the isovector
0 20 40 60 80 100 120 ∗
nucleon effective mass m . With the same calculation
v
L (MeV) method, the time evolution of the isovector dipole mo-
ment in 208Pb and the corresponding strength function
FIG. 4: (Color online) The linear relations between thesym-
are displayed respectively in the left and right panel of
metry energy slope parameter L as well as the isovector nu-
cleon effective mass m∗v and the centroid energy E−1 as well Fig. 5. With a, b, and c fitted by Eq. (11), and the ana-
as the electric dipole polarizability αD from the IVGDR in lyticalresultsofthecentroidenergyE−1 andtheelectric
208Pb, respectively, with the isoscalar nucleon effective mass dipole polarizability αD obtained according to Eq. (12),
m∗ =0.7m. we can get the similar linear relationfrom transportcal-
s
culations as
∗
a larger electric dipole polarizability αD. Analogously, −0.032L−7.346mv+20.651=E−1, (19)
foragivenslopeparameterL,alargerisovectoreffective ( 0.065L−6.368m∗v+20.845=αD.
∗
mass mv leads to a smaller centroid energy E−1 and a
smaller electric dipole polarizability αD. The rigorous With the available experimental data of E−1 and αD,
tool to treat such a problem of multiple experimental the slope parameter L of the symmetry energy and the
results with multiple parameters is the Bayesian frame- isovector nucleon effective mass from the constraint of
work [66–69], which is beyond the scope of the present the ISQGR and the IVGDR in 208Pb are
study. Ontheotherhand,wefoundbothE−1andαDare
∗ L = 53.85±10.29 (MeV), (20)
linearlycorrelatedwithLandm ,andthecorresponding
v
∗
relationsbasedonourtransportcalculationsturnsoutto m /m = 0.744±0.045. (21)
v
7
∗
Again,wefoundthatthemeanvaluesofLandm repro- IV. CONCLUSIONS
v
duce very well the experimental results of E−1 and αD
withinthe statisticalerrorbasedonourtransportmodel
Based on an improved isospin- and momentum-
calculations, justifying the linear relation of Eq. (19).
dependent interaction (ImMDI) and an isospin-
The corresponding neutron-proton effective mass split-
dependent Boltzmann-Uehling-Uhlenbeck (IBUU)
ting deduced from Eq. (A.35) is thus
transport model, we have studied the effect of the
slope parameter L of the nuclear symmetry energy
(m∗ −m∗)/m=(0.216±0.114)δ. (22) and the isovector nucleon effective mass m∗ on the
n p v
centroid energy E−1 of the isovector giant dipole res-
The constraint from both ISGQR and IVGDR on the onance (IVGDR) and the electric dipole polarizability
neutron-protoneffective masssplitting is consistentwith αD in 208Pb. We found that both E−1 and αD are
∗
that obtained in Ref. [70]. almost linearly correlated with L and m . With a
v
given isoscalar nucleon effective mass, we are able to
∗
constrain the values of L and m with the available
v
experimental data of E−1 and αD. From the isoscalar
∗
nucleon effective mass m = 0.7m constrained by the
30 18 s
(a) (b) empirical optical potential, we obtain a constraint of
∗
IVGDR L = 64.29± 11.84(MeV) and m /m = 0.710± 0.046,
20 * 15 v
ms=0.84m resulting in the isospin splitting of the nucleon effective
∗ ∗
D (fm)100 912S (fm/Me 2 iifmnsroogasmsctsahtwlehaiertishoenixsnupcc(iiltmneaonatnis−oynemmffepemn)ce/etritmvgreyy=omoff(a−ntshus0ec.0mlie1sa∗so9rs±=cma0l.e0a0d.r98iu0g4)mimδa,.nwteFxirqttouhrmaaδdcttbrheuede--
V
) ple resonance (ISGQR) in 208Pb, we obtain a constraint
-10 6 ∗
of L = 53.85±10.29(MeV) and m /m = 0.744±0.045,
Set II(a) v
resultingintheisospinsplittingofnucleoneffectivemass
-20 Set II(b) 3 within (m∗ −m∗)/m=(0.216±0.114)δ. The constraint
Set II(c) n p
on the neutron-protoneffective mass splitting from both
-30 0 ISGQR and IVGDR in 208Pb is consistent with that
0 200 400 0 10 20 30 from optical model analyses of nucleon-nucleus elastic
t (fm /c) E (M eV) scatterings. The uncertainty of the isoscalar nucleon
FIG. 5: (Color online) Same as Fig. 3 but with the isoscalar effective mass has hampered our accurate constraint on
nucleon effective mass m∗ =0.84m. the neutron-proton effective mass splitting.
s
Acknowledgments

WethankChenZhongformaintainingthehigh-quality
1.0 performance of the computer facility, and acknowledge
helpful communications with Zhen Zhang. JX acknowl-
(L=53.85 MeV, edges support from the Major State Basic Research De-
0.8 *
mv=0.744m) velopmentProgram(973 Program)of China under Con-
* (m)mv0.6 m* v+20.845=19.6 -0.032L-7.346mv*+20.651=13.46 tNGTraaralatceintnottnNaNPol.olaN.n2a”0t11uo51rfC4a7lSB5hS82a5c4ni63eg9nh0aca4neidaIFnnoNdsutoniN.tduoat.te1i1o2o4n0f211Ao45fCp0pBC5l,8ihe4itdn5h4aeP01hu”,yn1sdt0ihec0ers-
0.4 0.065L-6.368 ISmeMt IDI I ufrnodmerthGeraCnhtinNeose. AYc2a9d0e0m6y10o1f1SacniedncNeso,. thYe5S26h0a1n1g0h1a1i
0.2 * Key Laboratory of Particle Physics and Cosmology un-
ms=0.84m der Grant No. 15DZ2272100, and the ”Shanghai Pu-
jiangProgram”underGrantNo. 13PJ1410600. LWCac-
0.0 knowledgestheMajorStateBasicResearchDevelopment
0 20 40 60 80 100 120 Program (973 Program) in China under Contract No.
L (M eV) 2013CB834405 and No. 2015CB856904, the National
Natural Science Foundation of China under Grant No.
FIG. 6: (Color online) Same as Fig. 4 but with the isoscalar
11275125 and No. 11135011, the ”Shu Guang” project
∗
nucleon effective mass m =0.84m.
s supported by Shanghai Municipal Education Commis-
sion and Shanghai Education Development Foundation,
8
theProgramforProfessorofSpecialAppointment(East- with the kinetic energy per nucleon calculated from
ernScholar)atShanghaiInstitutionsofHigherLearning,
and the Science and Technology Commission of Shang- 1 p2 p2
haiMunicipality(11DZ2260700). BALacknowledgesthe Ek(ρ,δ) = d3p fn(~r,~p)+ fp(~r,~p)
ρ 2m 2m
National Natural Science Foundation of China under Z (cid:20) (cid:21)
4π
GrantNo. 11320101004,theU.S.DepartmentofEnergy, = p5 +p5 , (A.26)
Office of Science, under Award Number de-sc0013702, 5mh3ρ fn fp
and the CUSTIPEN (China-U.S. Theory Institute for (cid:0) (cid:1)
Physics with Exotic Nuclei) under the US Department where p =~ 3π2ρ 13 is the Fermi momentum of
fn(p) n(p)
of Energy Grant No. DEFG02- 13ER42025. YGM ac-
neutrons (protons), and m is the nucleon mass.
knowledges the Major State Basic Research Develop- (cid:0) (cid:1)
By setting ρ = ρ = ρ and p = p = p , we can
ment Program (973 Program) of China under Contract n p 2 fn fp f
express the binding energy per nucleon for symmetric
No. 2014CB845401 and the National Natural Science
nuclear matter as [71]
Foundation of China under ContractNos. 11421505and
11220101005.
E (ρ)
0
σ
8π ρ B ρ
APPENDIX A. EXPRESSIONS FOR PHYSICS = 5mh3ρp5f + 4ρ (Al+Au)+ σ+1 ρ
QUANTITIES FROM THE IMMDI 0 (cid:18) 0(cid:19)
INTERACTION 1 4π 2
+ (C +C ) Λ2
3ρ ρ l u h3
0 (cid:18) (cid:19)
At zero temperature, the phase-space distribution
2p
function can be written as f (~r,~p) = 2 Θ(p −p), × p2(6p2 −Λ2)−8Λp3arctan f
τ h3 fτ f f f Λ
with p = ~(3π2ρ )1/3 being the Fermi momentum of (cid:20) (cid:18) (cid:19)
fτ τ
nucleons with the isospin label τ, and the momentum- + 1(Λ4+12Λ2p2)ln 4p2f +Λ2 . (A.27)
dependent part of the single-particle potential as well as 4 f Λ2
!#
that in the potential energy density can be integrated
analytically as [71]
The saturation density is determined by the zero point
f (~r,p~′) ofthe first-orderderivativeofthe bindingenergyper nu-
d3p′ τ
1+(p~−p~′)2/Λ2 cleon, with the latter expressed as [71]
Z
2 p2 +Λ2−p2 (p+p )2+Λ2
= πΛ3 fτ ln fτ dE0(ρ)
h3 2pΛ (p−p )2+Λ2
( (cid:20) fτ (cid:21) dρ
2pfτ p+pfτ p−pfτ 16π 1 Bσ ρσ−1
+ −2 arctan −arctan = p5 + (A +A )+
Λ Λ Λ 15mh3ρ2 f 4ρ l u σ+1 ρσ
(cid:18) (cid:19)(cid:27) 0 0
(A.23) 2
1 4π
+ (C +C ) Λ2 (A.28)
and 3ρ ρ2 l u h3
0 (cid:18) (cid:19)
d3pd3p′1fτ+(~r(,p~p~−)fτ~p′′()~r2,/~pΛ′)2 × 2p4f +Λ2p2f − 41(Λ4+4Λ2p2f)ln 4p2fΛ+2Λ2 .
Z Z " !#
1 4π 2
= 6 h3 Λ2{pf(τ)pf(τ′)[3(p2fτ +p2fτ′)−Λ2] The incompressibilityofsymmetric nuclearmatter isde-
(cid:18) (cid:19)
+ 4Λ (p3fτ −p3fτ′)arctan pfτ −Λpfτ′ fidnereidvaatsivKe0of=t9hρe20(bdin2Edi0n/gdρe2n)eρr=gρy0,pwerithnuthcleeosnecoenxpdr-oesrsdeedr
(cid:20) (cid:18) (cid:19) as [71]
− (p3fτ +p3fτ′)arctan pfτ +Λpfτ′
(cid:18) (cid:19)(cid:21) d2E (ρ)
0
1
+ 4[Λ4+6Λ2(p2fτ +p2fτ′)−3(p2fτ −p2fτ′)2] dρ2
16π Bσ(σ−1)ρσ−2
× ln (pfτ +pfτ′)2+Λ2 }. (A.24) = −45mh3ρ3p5f + σ+1 ρσ
(cid:20)(pfτ −pfτ′)2+Λ2(cid:21) 2 0
2 4π
The binding energy per nucleon for asymmetric nuclear + (C +C ) Λ2 (A.29)
3ρ ρ3 l u h3
matter can be expressed as 0 (cid:18) (cid:19)
2 Λ2 2 4p2 +Λ2
V (ρ,T =0,δ) × − p4 −Λ2p2 +Λ2 + p2 ln f .
E(ρ,δ)= ρ +Ek(ρ,δ) (A.25) " 3 f f (cid:18) 4 3 f(cid:19) Λ2 !#
9
Thesymmetryenergybydefinitioncanbewrittenas[71] the mean-field potential U in symmetric nuclear matter
0
via
E (ρ)
sym
1 ∂2E ∗ mdU0 −1
= 2 ∂δ2 ms =m 1+ p dp . (A.32)
(cid:18) (cid:19)δ=0 (cid:18) (cid:19)p=pf
σ
8π ρ Bx ρ
= 9mh3ρp5f + 4ρ (Al−Au)− σ+1 ρ In asymmetric nuclear matter, the isovector nucleon ef-
0 (cid:18) 0(cid:19) fective mass can be calculated through the following re-
C 4π 2 4p2 +Λ2 lation
+ l Λ2 4p4 −Λ2p2ln f (A.30)
9ρ ρ h3 f f Λ2
0 (cid:18) (cid:19) " !# ~2 2ρ ~2 2ρ ~2
n(p) n(p)
+ Cu 4π 2Λ2 4p4 −p2 4p2 +Λ2 ln 4p2f +Λ2 . 2m∗n(p) = ρ0 2m∗s2 +(cid:18)1− ρ0 (cid:19)2m∗v,(A.33)
9ρ ρ h3 f f f Λ2
0 (cid:18) (cid:19) " !#
(cid:0) (cid:1) with the neutron (proton) effective mass defined as
The slope parameter of the symmetry energy at the sat-
uration density is defined as L =3ρ0[dEsym(ρ)/dρ]ρ=ρ0, ∗ mdUn(p) −1
with the first-order derivative of the symmetry energy m =m 1+ . (A.34)
n(p) p dp
expressed as [71] (cid:18) (cid:19)p=pf
dE (ρ) Keeping the first-order term of the isospin asymmetry
sym
δ, the neutron-proton effective mass splitting is related
dρ
to the isoscalar and isovector effective mass through the
16π 1
= p5 + (A −A ) following relation
27mh3ρ2 f 4ρ l u
0
− Bxσ ρσ−1 + Cl+Cu 4π 2Λ2 (A.31) m∗ −m∗ ≈ 2m∗s(m∗−m∗)δ. (A.35)
σ+1 ρσ 27ρ ρ2 h3 n p m∗ s v
0 0 (cid:18) (cid:19) v
Finally, the mean-field potential of a nucleon with in-
4p2 +Λ2 8Λ2p4
× 4p4 +Λ2p2ln f − f finitely large momentum in symmetric nuclear matter at
" f f Λ2 ! 4p2f +Λ2# the saturation density can be expressed as
4C 4π 2 4p2 +Λ2 8p2
− u Λ2p4 ln f + f . ∞ Al+Au
27ρ0ρ2 (cid:18)h3(cid:19) f" Λ2 ! 4p2f +Λ2# U0 = 2 +B. (A.36)
∗
The isoscalar nucleon effective mass m is defined as The values of the parameters A , B, C , C , σ, Λ, x,
s 0 u0 l0
the nucleon effective mass in symmetric nuclear matter andy canbeobtainedfromEqs.(A.27-A.36),withgiven
∞ ∗ ∗
at the saturation density, and it can be calculated from ρ , E (ρ ), K , U , m , E (ρ ), L, and m .
0 0 0 0 0 s sym 0 v
[1] B.A. Li, C.M. Ko, and W. Bauer, Int. J. Mod. Phys. E [11] M. Oertel, M. Hempel, T. Kla¨hn, and S. Typel,
7, 147 (1998). arXiv:1610.03361 [astro-ph.HE].
[2] Isospin Physics in Heavy-Ion Collisions at Intermediate [12] B.A. Li, Phys.Rev. C 69, 064602 (2004).
Energies, Eds. B.A. Li and W. Uuo Schr¨oder (Nova Sci- [13] J.Rizzo,M.Colonna,andM.DiToro,Phys.Rev.C72,
encePublishers, Inc,New York,2001). 064609 (2005).
[3] P.Danielewicz, R.Lacey,andW.G.Lynch,Science298, [14] V. Giordano et al.,Phys. Rev.C 81, 044611 (2010).
1592 (2002). [15] Z.Q. Feng, Phys. Rev.C 84, 024610 (2011).
[4] V.Baran, M.Colonna,V.Greco, andM.DiToro,Phys. [16] Z.Q. Feng, Nucl. Phys.A 878, 3 (2012).
Rep.410, 335 (2005). [17] Y.X.Zhang,M.B.Tsang,Z.X.Li,andH.Liu,Phys.Lett.
[5] A.W.Steiner,M.Prakash,J.M.Lattimer,andP.J.Ellis, B 732, 186 (2014).
Phys.Rep.411, 325 (2005). [18] W.J.XieandF.S.Zhang,Phys.Lett.B735,250(2014).
[6] J.M. Lattimer and M. Prakash, Phys. Rep. 442, 109 [19] H.Y.Kong,Y.Xia,J.Xu,L.W.Chen,B.A.Li,andY.G.
(2007). Ma, Phys. Rev.C 91, 047601 (2015).
[7] B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep.464, 113 [20] L. Ou et al.,Phys.Lett. B 697, 246 (2011).
(2008). [21] J.Xu,L.W.Chen,andB.A.Li,Phys.Rev.C91,014611
[8] B.A. Li, A`. Ramos, G. Verde, and I. Vidanˇa, Topical (2015).
issue on nuclear symmetry energy, Eur. Phys. J. A 50, [22] C.Xu,B.A.Li,andL.W.Chen,Phys.Rev.C82,054607
No.2 (2014). (2010).
[9] L.W. Chen, arXiv:1212.0284 [nucl-th]. [23] B.A.LiandL.W.Chen,Mod.Phys.Lett.A30,1530010
[10] B.A. Li and X. Han,Phys. Lett. B 727, 276 (2013). (2015); B.A. Li, B.J. Cai, L.W. Chen, and X.H. Li, Nu-
10
clear Science and Techniques, 27, 141 (2016). [48] Z. Zhang and L.W. Chen, Phys. Rev. C 93, 031301(R)
[24] J. Blaizot et al.,Nucl. Phys.A 591, 435 (1995). (2015).
[25] D.H.Youngblood,H.L.Clark,andY.W.Lui,Phys.Rev. [49] H.Zheng,S.Burrello, M.Colonna, andV.Baran, Phys.
Lett.82, 691 (1999). Rev. C 94, 014313 (2016).
[26] B. K. Agrawal, S. Shlomo, and V. Kim Au, Phys. Rev. [50] V. Baran, D.M. Brink, M. Colonna, and M. Di Toro,
C 68, 031304 (2003). Phys. Rev.Lett. 87, 182501 (2001).
[27] G. Col`o et al., Phys.Rev.C 70, 024307 (2004). [51] V.Baran, M.Cabibbo, M.Colonna, M.DiToro, andN.
[28] B.G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. Tsoneva, Nucl. Phys.A 679, 373 (2001).
95, 122501 (2005). [52] M. Papa et al.,Phys. Rev.C 72, 064608 (2005).
[29] A. Bohr and B.R. Mottelson, Nuclear Stucture, Vols. I [53] V. Baran, C. Rizzo, M. Colonna, M. Di Toro, and D.
and II (W. A.Benjamin Inc., Reading, MA, 1975). Pierroutsakou, Phys. Rev.C 79, 021603(R) (2009).
[30] O.Bohigas,A.M.Lane,andJ.Martorell,Phys.Rep.51, [54] H.L. Wu et al., Phys.Rev.C 81, 047602 (2010).
267 (1979). [55] S.Q.Yeetal.,Phys.Rev.C88,047602(2013);S.Q.Yeet
[31] J.-P. Blaizot, Phys.Rep.64, 171 (1980). al., Nuclear Scienceand Techniques,25, 030501 (2014).
[32] P. Klu¨pfel, P.-G. Reinhard, T.J. Bu¨rvenich, and J.A. [56] C.B. Das, S. Das Gupta, C. Gale, and B.A. Li, Phys.
Maruhn,Phys. Rev.C 79, 034310 (2009). Rev. C 67, 034611 (2003).
[33] X.Roca-Maza et al., Phys.Rev.C 87, 034301 (2013). [57] L.W. Chen, C.M. Ko, and B.A. Li, Phys.Rev. Lett.94,
[34] Z. Zhang and L.W. Chen, Phys. Rev. C 93, 034335 032701 (2005).
(2016). [58] L.W. Chen, C.M. Ko, B.A. Li, C. Xu, and J. Xu, Eur.
[35] A. Klimkiewicz et al., Phys. Rev. C 76, 051603(R) Phys. J. A 50, 29 (2014).
(2007). [59] S.S.DietrichandB.L.Berman,At.DataNucl.DataTa-
[36] L. Trippa, G. Col`o, and E. Vigezzi, Phys. Rev. C 77, bles 38, 199 (1988).
061304(R) (2008). [60] J. Xu and C.M. Ko, Phys. Rev.C 82, 044311 (2010).
[37] A.Carbone et al.,Phys. Rev.C 81, 041301(R) (2010). [61] M. Urban,Phys. Rev.C 85, 034322 (2012).
[38] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 81, [62] S. Hama et al.,Phys. Rev.C 41, 2737 (1990).
051303(R) (2010). [63] E.D. Cooper et al.,Phys. Rev.C 47, 297 (1993).
[39] A.Tamii et al.,Phys. Rev.Lett. 107, 062502 (2011). [64] S.K. Patra et al.,Nucl. Phys.A 703, 240 (2002).
[40] J. Piekarewicz, B.K. Agrawal, G. Col`o, W. Nazarewicz, [65] M.Buenerd,J.Phys.Colloques45,C4-115(1984);D.H.
N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vrete- Youngblood, P. Bogucki, J.D. Bronson, U. Garg, Y. W.
nar, Phys.Rev.C 85, 041302(R) (2012). Lui, and C.M. Rozsa, Phys. Rev. C 23, 1997 (1981);
[41] X.Roca-Maza, G. Pozzi, M. Brenna, K.Mizuyama, and S. Brandenburg, Ph.D. thesis, University of Groningen,
G. Col`o, Phys. Rev.C 85, 024601 (2012). 1985; D.H.Youngblood,Y.-W.Lui,H.L.Clark,B.John,
[42] D.Vretenar,Y.F.Niu,N.Paar,andJ.Meng,Phys.Rev. Y. Tokimoto, and X. Chen, Phys. Rev. C 69, 034315
C 85, 044317 (2012). (2004).
[43] V.Baran,B.Frecus,M.Colonna,andM.DiToro,Phys. [66] S. Habib et al., Phys.Rev.D 76, 083503 (2007).
Rev.C 85, 051601(R) (2012). [67] F.A. G´omez et al., The Astrophysical Journal 760, 112
[44] X. Roca-Maza, M. Brenna, G. Col`o, M. Centelles, X. (2012).
Vin˜as, B.K. Agrawal, N. Paar, D. Vretenar, and J. [68] S. Pratt et al., Phys.Rev.Lett. 114, 202301 (2015).
Piekarewicz, Phys. Rev.C 88, 024316 (2013). [69] E. Sangaline and S. Pratt, Phys. Rev. C 93, 024908
[45] C.Taoet al.,Phys.Rev.C87,014621 (2013); C.Taoet (2016).
al.,Nuclear Science and Techniques,24, 030502 (2013). [70] X.H. Li et al.,Phys. Lett. B 743, 408 (2015).
[46] G. Col`o, U. Garg, and H. Sagawa, Eur. Phys. J. A 50, [71] J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, The Astro-
26 (2014). physical Journal 697, 1549 (2009); J. Xu, Ph.D. thesis,
[47] X. Roca-Maza, X. Vin˜as, M. Centelles, B.K. Agrawal, Shanghai Jiao Tong University.
G.Col`o,N.Paar,J.Piekarewicz,andD.Vretenar,Phys.
Rev.C 92, 064304 (2015).