Low density instabilities in asymmetric nuclear matter within QMC with $δ$-meson

0.22 MB·English

by  Alexandre M. Santos

Low density instabilities in asymmetric nuclear matter within QMC with δ-meson
Alexandre M. Santos
Centro de F´ısica Computacional - Dep. de F´ısica - Universidade de Coimbra - P-3004 - 516 - Coimbra - Portugal
Prafulla K. Panda
Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India and
Centro de F´ısica Computacional - Dep. de F´ısica - Universidade de Coimbra - P-3004 - 516 - Coimbra - Portugal
Constanc¸a Providˆencia
Centro de F´ısica Computacional - Departamento de F´ısica - Universidade de Coimbra - P-3004 - 516 - Coimbra - Portugal
In the present work we include the isovector-scalar δ-meson in the quark-meson coupling model
9 (QMC)andstudythepropertiesofasymmetricnuclearwithinQMCwithoutandwiththeδ-meson.
0 Recent constraints set by isospin diffusion on the slope parameter of the nuclear symmetry energy
0 at saturation density are used to adjust the model parameters. The thermodynamical spinodal
2 surfaces are obtained and the instability region at subsaturation densities within QMC and QMCδ
modelsare compared with mean-field relativistic models. Thedistillation effect in theQMC model
n
is discussed.
a
J

5 PACS number(s): 21.65.-f, 21.30.-x, 95.30.Tg ativistic interactions, where a balance between a scalar
2 (attractive)andavector(repulsive)potentialexists. The

δ-andρ-mesonsgiveriseto the correspondingattractive
]
h andrepulsivepotentialsintheisovectorchannel. Thein-
t troduction of the δ-meson will affect both the behaviour
- I. INTRODUCTION
l of the system at low and high densities. In the last case
c
due to Lorentz contraction, its contribution is reduced,
u
n The instabilitiespresentedbyasystemaredirectlyre- leading to a harder equation of state (EOS) at densities
[ latedwiththepossiblephasetransitionsitcanundertake. larger than 1.5ρ0 [6]. At low densities a reduction of
∼
At subsaturation densities a liquid-gas phase transition thesymmetryenergywilloccurwhichwillallowformore
2
in nuclear matter is predicted and it is normally tested asymmetric matter.
v
in nuclear reactions. The formation of highly excited
3
4 composed nuclei in equilibrium with a gas of evaporated In [7, 8] the instabilities in ANM have been investi-
2 particles can be interpreted in the framework of hydro- gated within relativistic mean field hadron models, both
3 dynamics as two coexisting phases of nuclear matter, a with constant and density-dependent couplings at zero
1. liquid and a gas phase. During these reactions, phase andfinite temperatures. Itwasshownthatthe maindif-
0 transitionsmayoccurdependingonthetemperatureand ferences occur at large isospin asymmetry and at finite
9 densities involved. The liquid-gas phase transition also temperature. In particular it has been shown that the
0 plays an importantrole in the description of the crustof predicteddensityattheinneredgeofthe crustofacom-
v: compact star matter at densities between 0.03 fm−3 and pactstar,fromthecrossingoftheβ-equilibriumequation
i saturation density ( 0.15 fm−3). It essentially consists of state (EOS), is model dependent [9].
X ∼
of neutron rich nuclei immersed in a gas of neutrons. It
r has been shown that this phase transition leads to an In the present work we investigate thermodynamical
a
isospin distillation phenomenon: the isospin content of instabilities within the QMC model with and without
each phase is different, most of the gas being composed theisovector-scalarδ-meson. Althoughinthis modelthe
ofneutronsandtheliquidbeingclosertosymmetricmat- isoscalar vector channel described by the ρ meson is in-
ter [1] cluded in a similar way to the non-linear Walecka model
Inthepresentpaper,weemploythe quark-mesoncou- (NLWM), the non linearities in the σ and δ-fields arise
pling model (QMC) [2, 3] to investigate the thermody- from the minimization of the bag energy. In particular,
namical instabilities of asymmetric nuclear matter. In the NLWMusedin[5]doesnotinclude non-linearitiesin
the QMCmodel,nuclearmatterisdescribedasasystem theδ-meson. Wemaythereforeexpectadifferentbehav-
ofnon-overlappingMIT bagswhich interactthroughthe ior of asymmetric matter.
exchange of scalar and vector mean fields. An earlier
study of ANM within this model has been focused on The paper is organized as follows: in section II an ex-
the effect of isospin asymmetry and temperature on the tensionofthe QMC model to include the δ-mesonis dis-
equation of state and on the coexistence surface [4]. We cussed, in section III we make a short review of the cal-
hereconsideranextensionofthemodelthatincludesthe culationof the spinodal surface,in section IV results are
scalar isovector virtual δ(a (980)) field [5]. Its presence presentedanddiscussedandsomeconclusionsaredrawn
0
introduces in the isovector channel the structure of rel- in the last section.
2
II. THE QUARK-MESON COUPLING MODEL The total energy density of the nuclear matter reads
1 1 1 1
InwhatfollowswepresentareviewoftheQMCmodel ε = m2σ2+ m2ω2+ m2ρ2 + m2δ2
2 σ 2 ω 0 2 ρ 03 2 δ 3
and its generalization to include the iso-vector-scalar δ-
meson. + 1 kN k2dk k2+M∗2(σ,δ) 1/2 (8)
In the QMC model, the nucleon in nuclear medium π2 N
N Z0
is assumed to be a static spherical MIT bag in which X (cid:2) (cid:3)
quarks interact with the scalar (σ, δ) and vector (ω, ρ) and the free energy density is given by
fields,andthosearetreatedasclassicalfieldsinthemean
field approximation (MFA) [2, 3]. The quark field, ψ , =ε µ ρ µ ρ ,
qi F − p p− n n
inside the bag then satisfies the equation of motion:
where the chemical potentials are given by
[i/∂ (m0 gqσ gqτ δ ) gq ωγ0
− q − σ − δ z 3 − ω g
+ 1gqτ ρ γ0 ψ (x)=0 , q =u,d (1) µp = kp2+Mp∗2+gωρ+ 2ρρ03,
2 ρ z 03 qi q
(cid:21)
gwqhdereenomte0q tihsethqeuacrukr-rmenetsoqnuacrokupmliansgs,caonndstagnσqt,sg.δqT,hgωeq naonrd- µn = kp2+Mn∗2+gωρ− g2ρρ03.
ρ q
malized ground state for a quark in the bag is given by
The vector mean field ω and ρ are determined
0 03
ψ (r,t) = exp( iǫ t/R ) through
qi Nqi − qi i
j0i(xqir/Ri) χq , (2) ω = gω(ρp+ρn), ρ = gρ(ρp−ρn), (9)
× (cid:18)iβqi~σ·rˆj1i(xqir/Ri)(cid:19)√4π 0 m2ω 03 m2ρ
where where g =3gq and g =gq. Finally, the mean fields σ
ω ω ρ ρ 0
and δ are fixed by
3
ǫ =Ω +R gq ω+ 1gqτ ρ ; β = Ωqi −Rim∗q ,
qi qi i(cid:18) ω 2 ρ z 03(cid:19) qi sΩqi +Rim∗q ∂ε =0, ∂ε =0. (10)
(3) ∂σ ∂δ3
with the normalization factor given by
In order to set the model parameters, we start by fix-
ing the free space bag properties. They are obtained by
−2 =2R3j2(x ) Ω (Ω 1)+R m∗/2 x2 , (4)
Nqi i 0 q q q− N q q fitting the nucleon mass and enforcing the stability con-
(cid:2) (cid:3). dition for the bag in free space. We consider two sets of
whereΩ x2 +(R m∗)2,m∗ =m0 gqσ gqτ δ , free space parameters, taking an equal proton and neu-
qi ≡ qi N q q q− σ − δ z 3 tron mass value in a first moment, and then proceeding
Ri is the bagqradius of nucleon i and χq is the quark byconsideringdifferentprotonandneutronmassesafter.
spinor. The bag eigenvalue for nucleon i, x , is deter-
qi In the first case, we consider the bare nucleon mass
mined by the boundary condition at the bag surface
M = 939 MeV and the bag radius, R = R = 0.6
p n
fm. The unknowns Z = Z = 3.986991 and B1/4 =
j0i(xqi)=βqij1i(xqi) . (5) 211.30305MeV arethepn obtaninedby setting the nuNcleon
Theenergyofastaticbagdescribingnucleoniconsisting bag energies to that (single) bare nucleon mass value.
of three quarks in ground state is expressed as In the next step, we take the physical nucleon mass
values as M =938.272 MeV and M =939.56533MeV
p n
Ω Z 4 and the bag radius for protons as R =0.6 fm. The un-
Ebag = n qi i + πR3B , (6) p
i q q Ri − Ri 3 i N knowns Zp = 3.98865, Zn = 3.98471, BN1/4 = 211.26209
X MeV and the neutron radius R = 0.6002 are then ob-
n
where Zi is a parameter which accounts for zero-point tained. Note that for fixed proton bag radius Rp = 0.6
motionofnucleoni andBN isthe bagconstant. Theset we observe a decrease on Zn and on the bag parameter
of parameters used in the present work is given in Ref. B1/4 forthenucleons. Next,wefitthequark-mesoncou-
N
[10]. The effective mass of a nucleon bag at rest is taken pling constants gq, gq, g = 3gq and g = gq for the
σ δ ω ω ρ ρ
to be M∗ =Ebag. The equilibrium condition for the bag nucleons, so as to obtain the correct saturation proper-
i i
irsesopbetcatintoedthbeybmaginriamdiizuisng the effective mass, Mi∗ with tρie=s ρo0f n=uc0l.e1a5rfmm−at3t,ears,yEmN=≡33ǫ./7ρM−eVM.F=or−th1e5.c7ouMpelVingast,
we have gq = 5.981, g = 8.954. In our first case (when
dM∗ no effectivσe mass diffeωrence between p,n is considered),
i =0, i=p,n. (7)
dRi∗ gρ =8.615.
3
TABLEI:Nuclearmatterpropertiesofthemodelsusedinthepresentwork. Allquantitiesaretakenatsaturation,exceptthe
density ρs for which thepressure has a minimum and theincompressibility is zero.
Model B/A ρ0 K M∗/M Esym L Ksym Kasy ρs
(MeV) (fm−3) (MeV) (MeV) (MeV) (MeV) (MeV) (fm−3)
NL3 [16] 16.3 0.148 269 0.60 37.4 118.3 101 -608.8 0.096
NL3δ [18] 16.3 0.148 270 0.60 37.4 153.1 427.1 -491.5 0.096
TW [17] 16.3 0.153 240 0.56 32.0 55.3 -125 -456.8 0.096
QMC 15.7 0.150 291 0.77 33.7 93.5 -10 -570.8 0.098
QMCδ 15.7 0.150 291 0.77 34.2 102.1 34.8 -577.6 0.098
BHF [19] 14.7 0.182 176.5 0.79 33.2 63.4 6.04 -374.3 0.119
The properties of asymmetric nuclear matter have re- in the latter, as it defines the spinodal surface, which is
centlybeenrelatedtobothterrestrialdataandstarprop- determined by the values of T, ρ, and y for which the
p
ertiesfromVelapulsarglitches,whichsetsthesymmetry smallest eigenvalue of becomes negative. The asso-
ij
F
energy slope value to L = 88 25 MeV [11, 12]. We ciated eigenvector defines the instability direction of the
±
then consider the δ-meson and determine the values of system, in isospin space.
the couplingssoasto haveL=102.077MeV,whichsets Ithasrecentlybeenargued[14]thatinANMthespin-
gρN =12.599andgδq =12.6. Inthiscase,the(p,n)mass odal instabilities cannot be separately classified as me-
splittingmanifestsinthedifferentvaluesfortheeffective chanical or chemical instabilities. In fact, the two con-
masses: M∗ =727.718MeV and M∗ =729.007MeV, at ditions that give rise to the instability of the system are
p n
saturation. coupled so that the instability appears as an admixture
We take the standard values for the meson masses, ofnucleondensityandconcentrationfluctuations. Inthe
namely m = 550 MeV, m = 783 MeV, and m = 770 following we study the direction of instability and the
σ ω ρ
MeV. spinodal for the different models considered.
III. STABILITY CONDITIONS
IV. RESULTS AND DISCUSSIONS
The stability conditions for asymmetric nuclear mat-
Inthepresentsectionwecomparethemodelproperties
ter, keeping constant volume and temperature are ob-
of QMC and QMCδ, respectively with and without the
tainedfromthefreeenergydensity ,imposingthatthis
F δ-meson, the non-linear Walecka model (NLWM) NL3
function is a convex function of the densities ρ and ρ ,
p n [16], with and without the δ-meson, and the density de-
i.e. the symmetric matrix with elements
pendent relativistic hadronmodel TW [17]. We will also
∂2 refer to the nuclear matter properties obtained within a
= F , (11) microscopic Brueckner-Hartree-Fock (BHF) approxima-
ij
F ∂ρ ∂ρ
(cid:18) i j(cid:19)T tionusingtherealisticArgonneV18nucleon-nucleonpo-
tential plus a three-body force of Urbana type [19].
is positive [13, 14, 15]. This is equivalent to imposing
∂µ ∂(µ ,µ )
p p n
>0, >0, (12)
∂ρ ∂(ρ ,ρ ) A. Model properties
p p n
where we have used µ = ∂F . Wewillfirstcomparetheequilibriumpropertiesofnu-
i ∂ρi T,ρj6=i clear matter described by the different models consid-
The two eigenvalues of the(cid:12)stability matrix are given
(cid:12) ered. The parameters of these models have been fitted
by [15] (cid:12)
to similar binding energy and saturation density values
1 as seen in Table I. At saturation, the effective mass in
λ± = Tr( ) Tr( )2 4Det( ) , (13) QMC is much largerthan the correspondingmass in the
2 F ± F − F
(cid:16) p (cid:17) other models, which is a characteristic of the model [3].
and the eigenvectors δρ± by In Fig. 1, it is seen that the QMC mass decreases much
slower with density. Even the hadronic models we study
±
δρi = λ±−Fjj, i,j =p,n. show quite different behaviour among themselves. NL3
δρ± has an almost linear decrease on the mass whereas TW
j Fji
has much faster drop at low densities, and shows a less
The largest eigenvalue is always positive whereas the dramatic fall as density increases, crossing the curve for
other can take on negative values. We are interested NL3 at ρ 0.18 fm−3. Incompressibility is one of the
∼
4
1000 60
QMCδ
900 50 QMC
NL3
800 NL3δ
) V) 40 TW
V e
e 700 M
M ( 30
*M ( 600 ym
s
ε 20
500 QMC
NL3 (a)
400 TW 10
300 0
0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25
ρ (fm − 3 ) ρ (fm− 3 )
250
FIG. 1: (Color online) Effective mass for QMC (brown, dot-
dashed), NL3 (green, dashed) and TW (red) in symmetric
200
matter.
)
V 150
e
bulk properties that distinguishes the different models, M
butitisontheisovectorchannelthatliesthelargestdis- L ( 100
tinctions among the different models we use. Although
havingidentical(orbarelydifferent)valuesfortheirbulk
isoscalarproperties,similarmodelsdifferconsiderablyon 50 (b)
the isovector parameters we discuss next.
We now compare the symmetry energy and its slope 0
0 0.05 0.1 0.15 0.2 0.25
and compressibility for all models in this work (Fig. 2
ρ (fm−3 )
and Table I). The symmetry energy in our relativistic
mean field models is given by 600
400
k 2 ρ g2 g2 M∗ 2
Esym = 6ǫFF2 + 2"4mρ2ρ − mδ2δ (cid:18) ǫF0(cid:19) #, (14) V) 200
e 0
M
owTnhhsee,raNenǫLdF3M=m0∗opdisPeltFs2heh+iarMveeff0∗te2hcietsivltaheremgFeasestrsmvianileusneyemragtmyseoatftruitchraemtniaoutntceloer-f. K (sym −−420000
the models considered, 37.4 MeV. −600 (c)
The symmetry energy slope L(ρ) is defined by L = −800
3ρ ∂ /∂ρ. The curvature parameter of the symme-
torfy0ineEntseeyrrmegsyt Kbescyamus=e i9tρ20d∂is2tEisnygmu/is∂hρe2s (bFeitgw.ee2nc))diiffseraelnsot −1000 0 0.05 0.ρ1 (fm−3 0).15 0.2 0.25
parametrizations. In particular, the quantity K =
asy
Ksym 6Lcanbe directly extractedfrommeasurements FIG.2: (Coloronline)Symmetryenergy(a)anditsslopepa-
of the−isotopic dependence of the giant monopole reso- rameter L=3ρ0Es′ym (b), and Ksym (c) in the QMCδ (blue,
nance (GMR) [20]. Recent measurements of the GMR dotted), QMC (brown, dot-dot-dashed), NL3 (dark green,
on even-A Sn isotopes give a quite stringent value of dashed),NL3δ (light green, dot-dashed) and TW (red) mod-
K = 550 100 MeV. According to this value, the els, for symmetric matter.
asy
− ±
hadronicandQMCmodels weuse here(see TableI)sat-
isfy the above constraint, whereas the BHF results lie
slightly below. the nucleon effective mass with density. Although still
The symmetry energy within QMC and QMCδ shows quitehardabovesaturationdensity,theQMCsymmetry
an extremely linear behaviour with density (Fig. 2a)), energy is softer than NL3, but harder than TW. At sub-
in comparison with all hadron models shown. This is saturationdensitiesandconsideringonlymodelswithout
quite visible from the symmetry energy curves, but un- theδ-meson,theQMCsymmetryenergytakesthesmall-
doubtedly clear from the slope parameter L (Fig. 2b). est values. The introduction of the δ has the expected
At larger densities the symmetry energy in QMC is es- effect: atsubsaturationdensitiesthe symmetryenergyis
sentially defined by the second term of Eq. (14), pro- softerbutabovesaturationvalues itbecomes harderdue
portional to the density, due to the small variation of to the saturation of the δ-meson field [5]. In Fig. 2 we
5
showboththeNL3δ andQMCδ symmetryenergies. The 16
effect of the δ-meson on the QMC at subsaturation den- 14 QMCδ
NL3δ (x0.1)
sities is quite small, much smaller than the effect seen 12
)
in NL3. It is above the saturation density that the δ- V
e 10
meson has a largereffect in QMC. From the slope of the M
symmetryenergyitisseenthatwhileforQMCthe slope * (n 8
decreases slightly with density, for the QMCδ model it M 6
−
increases slightly, with a value close to 100 MeV. In the *Mp 4 yp=0.1
bottomfigurewealsoplotKsym. Theδhasaverystrong 2 (a)
effectintheNL3model. TheQMCmodelislessaffected
0
butinbothcasesthepresenceoftheδ increasesthesym-
−2
metryincompressibilityK ,becomingslightlypositive
sym 0 0.05 0.1 0.15 0.2 0.25
for QMCδ while it was slightly negative for QMC. The ρ (fm− 3 )
model TW is presenting the smallest values.
16
Fig. 3 shows the proton and neutron radii for proton
14
fractionsy =0.5and0.0. Forsymmetricmatter,theneu-
p
tron bag is larger in this model due to the mass proton- ) 12
V
neutron difference. This result has also been reported e 10 yp=0.3
M
in [3]. Decreasing the proton fraction increases the pro- ( 8
ton radius and the neutron and proton radii cross at a *M n 6
certain value of the density, isospin dependent: density: −
0.175fm−3fory =0.3, 0.11fm−3fory =0.1and 0.1 *Mp 4
p p
f∼m−3 for neutron matter∼. The bag radius is sensitiv∼e to 2 (b)
isospin content by a small amount. It is, however, quite 0
clear from the neutron matter results (yp=0.0) that the −2
isospin contents of the nucleons leads to higher radii dif- 0 0.05 0.1 0.15 0.2 0.25
ferences at higher densities. Moreover, proton bags be- ρ (fm− 3 )
come larger than neutrons in medium, as matter goes
denser. The neutron radius does not change much with FIG.4: (Coloronline)EffectivemassdifferencesMp−Mn,for
isospin and for symmetric matter and neutron matter QMCδ (blue,dotted)andNL3δ(lightgreen,dot-dashed),for
QMC and QMCδ neutron radii almost overlap. differentprotonfractions: yp=0.1(a)andyp=0.3(b). Notice
In Fig. 4 we show the effective mass difference M∗ that thevalues for NL3δ havebeen scaled by a 0.1 factor.
M∗ for both QMCδ and NL3δ, for y =0.3 and 0.1. pFo−r
n p
NL3, we show the mass difference multiplied by a fac-
tor of 0.1 in order to compare with QMCδ. The most p,n mass splitting in NL3 and QMC. In both models,
striking result is the factor of ten difference between the this effect increases with baryon density, but it is worth
remarking that the effective mass is larger for neutrons
than for protons at lower densities, which is represented
by the negative values in the figures. As referred before,
1
this occurs because the proton and neutron masses were
considered different at zero density. The effect of the δ-
0.998
mesonis to increase the M∗ M∗ difference as it occurs
p − n
0.996 in NL3δ and other relativistic mean field models and in
0 contrast to Brueckner-Hartree-Fock calculations [21, 22]
R
0.994 In addition, for the same proton fraction,the crossingof
/
R i thep,neffectivemasscurves(equaleffectivep,nmasses)
0.992 { 0.5 does not occur for the same densities as the proton and
R /R ,y
n 0 pp 0.0 neutron radius, Rp and Rn (Fig. 3).
0.99
{ 0.5
0.988 R /R ,y
p 0 p 0.0 B. Instabilities
0.986
0 0.05 0.1 0.15 0.2 0.25
ρ −3 Inthe presentsubsectionwediscussthe resultsforthe
(fm )
instability region at subsaturation densities with QMC,
QMCδ and the hadron models we have considered. In
FIG. 3: (Color online) Radii ratios for neutron and proton, Fig. 5 we plot the spinodal curves for np matter. As
for symmetric matter and for yp=0.0 in the QMCδ model. referredbefore,theyaredefinedbythepoints,foragiven
Here we havetaken R=0.6 fm−3. temperature, density and isospin asymmetry, that make
6
The nuclear liquid-gas coexistence phase is character-
QMCδ
0.08 ized by different isospin contents for eachphase, i.e., the
QMC
clusterized regions are more isospin symmetric than the
NL3
surroundingnucleargas,the so-calledisospindistillation
NL3δ [11, 24]. The extension of the distillation effect is model
3 ) 0.06 TW dependent and it has been shown that NL3 and other
−m NLWMparametrizationsleadtolargerdistillationeffects
thanthe densitydependenthadronmodels [6,8,23]. On
f
(p 0.04 the other hand, the distillation effect was also studied
ρ within a BHF calculation [19] and a smaller distillation
effect was generally obtained.
0.02
1
QMCδ
QMC
0 0.9 NL3 yp=0.05
0 0.02 0.04 0.06 0.08 NL3δ
ρ −3 TW
(fm ) −n 0.8
n ρ
δ
−/p
ρ 0.7
FIG.5: (Coloronline)Spinodal(thermodynamicalinstability δ
border) for QMC, QMCδ, NL3, NL3δ and TW.
0.6
(a)
the curvature matrix of the free energy vanish. 0.5
0 0.02 0.04 0.06 0.08 0.1
Both QMC and QMCδ present larger instability re-
ρ (fm−3 )
gions than NL3, NL3δ and TW on the isoscalar direc-
tion, ρ = ρ . This is possibly due to the σ contribu- 1
p n
tion inthe Lagrangianfor QMC (both with andwithout
δ): the fields here attain magnitudes so as to minimize 0.9 yp=0.1
the bag energy, whereas in hadron models their mean-
field values are determined by solving the relevant set of
−n 0.8
equationswherenonlinearitiesshowexplicitlyorthrough ρ
δ
density dependent couplings. The extension of the spin- −/p
odal for ρp = ρn defines the density ρs, corresponding δρ 0.7
to the density value for which the pressure of symmetric
(b)
nuclearmatter hasaminimumandthe incompressibility 0.6
is zero [23]. We have included the values of this density
for the different models in Table I. Recently [19] it was
0.5
shown that within a Brueckner-Hartree-Fockcalculation 0 0.02 0.04 0.06 0.08 0.1
the isoscalar extension of the spinodal was much larger ρ (fm−3 )
than NLWM and Skyrme interaction predictions. 1
Forlargeisospinasymmetriesthepresenceofδreduces
the instability region both in QMC and NL3. NL3 has 0.9
a higher symmetry energy than NL3δ for densities rang-
ing from 0 up to ρ , and the same occurs for QMC
although the diffe∼rence0s are smaller. This means that ρ−n 0.8
δ
/
highlyasymmetricmatterislessboundinNL3andQMC −p
thaninNL3δ andQMCδ. Amereinspectionofthe sym- δρ 0.7 yp=0.3
metryenergyisnotenoughtoaccountforthedifferences
intheinstabilityregionifdifferentmodelsareconsidered 0.6 (c)
[23]. With the introduction of the δ-meson the scalar
channel is not affected, and therefore we are essentially 0.5
0 0.02 0.04 0.06 0.08 0.1
changing the isovector channel. We also notice that at
ρ (fm−3 )
ρ = ρ the curvature of the QMC spinodal is inter-
p n
mediate between NL3 and TW. As shown in [23] this
curvatureisdefinedbythe symmetryenergyandits first FIG.6: (Coloronline)Directionofinstability(eigenvectorfor
and second derivatives. negative eigenvalue λ−) for yp=0.05 (a), 0.1 (b) and 0.3 (c).
7
In Fig. 6 we show the ratio of the proton versus the saturation density is larger, 0.182 fm−3. The shape of
neutron density fluctuations correspondingto the unsta- the spinodal itself depends on the density dependence of
ble mode. This ratio defines the direction of instability thesymmetryenergyanditsderivatives. Itwouldbeim-
of the system. We show the results for different proton portant to identify the properties that define the shape
fractions (including rather small values), for the sake of of the spinodal for the more asymmetric matter.
studying the effectiveness of the models in restoring the
symmetry in the liquid phase. Wehavestudiedsubsaturationnuclearinstabilities for
We first compare the three models not including the both symmetric and asymmetric matte within the QMC
δ-meson: QMC, NL3 and TW. We see that QMC has a model,withandwithouttheinclusionoftheδ-meson. In
behaviour which is intermediate between NL3 and TW. thismodelthenucleonsaredescribedasnon-overlapping
The distillation effect for densities above 0.02 fm−3 is bags. We propose a parametrization for QMCδ with a
largerthanthe predictionofTW, butforthe largerden- the symmetry energy slope value L = 102 MeV within
sities it also shows a tendency to decrease, contrary to the interval L = 88 25 MeV proposed in [11, 12] and
±
NL3. The presence of δ-meson makes the distillation ef- which was determined from nuclear laboratory data. It
fect more efficient. was interesting to notice that for QMC and QMCδ the
In Fig. 7 we plot the proton-neutron density fluctu- quantity Kasy defined in [11], and which can be directly
ation ratio as a function of the isospin asymmetry for extractedfrommeasurementsofthe isotopicdependence
a fixed nuclear density, ρ=0.06 fm−3. We compare all ofthegiantmonopoleresonances,fallsinsidetheinterval
models under study,NL3, NL3δ,TW,QMC andQMCδ, predictedbyexperiments. The BHFcalculationpredicts
and include also the results derived from the BHF ap- a value of Kasy which is not very far, though lower than
proach [19] referred above. All the relativistic models theintervalobtainedfromGMR,Kasy =550 100MeV.
±
predict larger fluctuation ratios than the corresponding
Acomparisonwasdonewiththeresultsobtainedfrom
valueofρ /ρ ,dottedline. Thebehaviourgivesrisetoa
p n
a NLWM parametrization(NL3), one density dependent
distillation effect, which, as referred before, is larger for
relativistic model (TW), along with BHF with the Ar-
NL3andsmallerforTW,withQMCpresentinginterme-
gonne V18 potential calculation. It was shown that the
diate values. The δ-meson stresses the distillation effect,
restoration of isospin symmetry, obtained by a distilla-
clearly seen both in QMCδ and NL3δ.
tion effect, was more efficient in QMC with respect to
However,exceptfortheveryasymmetricmatter(y <
p
TW but less efficient when compared with NL3. The
0.02), both QMC and the other relativistic models pre-
spinodal surface within QMC is closer to TW although
dict fluctuations with larger proton fractions than BHF.
with a largercurvature at ρ =ρ and a slightly smaller
The instability properties within the BFH calculation of p n
instability extension atlargerasymmetries,while the in-
[19] show also differences for the spinodal surface: the
clusion of the δ-meson (QMCδ) shrinks the asymmetric
unstable region is larger, extending to larger densities
partsoftheinstabilityenvelope. TheBHFresultsof[19],
both in the isoscalaran isovectordirections and the cur-
althoughwithsimilargeneralproperties,differinthe ex-
vature of the spinodal at ρ = ρ is much larger than
p n
tension and shape of the spinodal and in the amount of
theoneofalltherelativisticmodelsconsidered. Alarger
distillation predicted.
extension of the unstable region is justified because the
A study of the QMC instability properties at finite
1 temperatures is now under preparation. It is also im-
portant to identify how the density dependence of the
symmetry energy determines the shape of the spinodal
0.8
for large isospin asymmetries.
−n 0.6
ρ
δ
/
−p
δρ 0.4 QMCδ
QMC
NL3
0.2 NL3δ
ρ /ρ n ρ=0.06 fm−3 TW V. ACKNOWLEDGMENTS
p BHF
0
0 0.1 0.2 0.3 0.4 0.5
y
p This work was partially supported by
FEDER and FCT (Portugal) under the projects
FIG.7: (Coloronline)Proton-neutrondensityfluctuationra- POCI/FP/81923/2007, CERN/FP/83505/2008 and
tio versus the isospin asymmetry for a fixed nuclear density, SFRH/BPD/29057/2006. The authors acknowledge
ρ=0.06fm−3. TheBHFresultswereobtainedbyVidan˜aand I. Vidan˜a for kindly providing us with some data for
Polls [19].
comparison in this work.
8
[1] H.S. Xu,et al., Phys.Rev.Lett. 85, 716 (2000). C 77, 014302 (2008).
[2] P.A. M. Guichon, Phys. Lett. B200 235 (1988). [12] J. Xu, L. W. Chen, B. A. Li and H. R. Ma,
[3] K.SaitoandA.W.Thomas,Phys.Lett.B327,9(1994); arXiv:0807.4477v1 [nucl-th].
335, 17 (1994); 363, 157 (1995); Phys. Rev. C 52, 2789 [13] V.Baran, M. Colonna, M. DiToro, andA.B. Larionov,
(1995); P.A.M. Guichon, K. Saito, E. Rodionov, and Nucl. Phys. A632, 287 (1998).
A.W. Thomas, Nucl. Phys. A601 349 (1996); K. Saito, [14] J. Margueron and P. Chomaz, Phys. Rev. C 67,
K.Tsushima,andA.W.Thomas,Nucl.Phys.A609,339 041602(R) (2003); P. Chomaz, M. Colonna and J. Ran-
(1996);Phys.Rev.C55,2637(1997);Phys.Lett.B406, drup,Phys. Rep.389, 263 (2004)
287 (1997); P. K. Panda, A. Mishra, J. M. Eisenberg, [15] H.Mu¨llerandB.D.Serot,Phys.Rev.C52,2072(1995).
W. Greiner, Phys. Rev. C 56, 3134 (1997); P.K. Panda, [16] G.A.Lalazissis, J.K¨onigandP.Ring,Phys.Rev.C55,
D.P. Menezes, C. Providˆencia. Phys. Rev. C 69, 025207 540 (1997).
(2004). [17] S. Typel and H. H. Wolter, Nucl. Phys. A656, 331
[4] P.K.Panda,G.Krein,D.P.MenezesandC.Providˆencia, (1999).
Phys.Rev C 68, 015201 (2003). [18] A. Santos, C. Providˆencia and L. Brito, unpublished.
[5] B.Liu,V.Greco,V.Baran,M.ColonnaandM.DiToro, [19] I. Vidan˜a and A. Polls, Phys. Lett. B 666 232 (2008).
Phys.Rev.C 65, 045201 (2002). [20] Bao-AnLi,Lie-WenChenandCheMingKo,Phys.Rep.
[6] Constan¸ca Providˆencia, Int. J. Mod Phys. E 16, 2680 464, 113 (2008).
(2007). [21] C.FuchsandH.H.Wolter,Eur.Phys.J.A30,5(2006).
[7] S.S. Avancini, L.Brito, D. P. Menezes and C. [22] W. Zuo, A. Lejeune, U. Lombardo, J.F. Mathiot, Nucl.
Providˆencia, Phys. Rev. C 70, 015203 (2004). Phys. A 706, 418 (2002);T. Frick, Kh. Gad, H. Muther,
[8] S.S.Avancini,L.Brito, Ph.Chomaz, D.P.Menezes, C. P. Czerski, Phys. Rev.C 65, 034321 (2002).
Providˆencia, Phys. Rev. C74 (2006) 024317. [23] C. Ducoin, C. Providˆencia, A. M. Santos, L. Brito and
[9] S.S. Avanciniet al., Phys. Rev.C 78, 015802 (2008). Ph. Chomaz, Phys. Rev.C 78, 055801 (2008).
[10] P.K. Panda, D.P. Menezes, C. Providˆencia. Phys. Rev. [24] Ph. Chomaz, Nucl. Phys.A 681, 199c (2001).
C69, 025207 (2004).
[11] J. Xu, L. W. Chen, B. A. Li and H. R. Ma, Phys. Rev.