Temperature effects on the nuclear symmetry energy and symmetry free energy with an isospin and momentum dependent interaction

0.4 MB·English

by  Jun Xu

Temperature effects on the nuclear symmetry energy and symmetry free energy with
an isospin and momentum dependent interaction
Jun Xu,1 Lie-Wen Chen,1,2 Bao-An Li,3 and Hong-Ru Ma1
1Institute of Theoretical Physics, Shanghai Jiao Tong University, Shanghai 200240, China
2Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou, 730000, China
3Department of Physics, Texas A&M University-Commerce, Commerce,
TX 75429, and Department of Chemistry and Physics, P.O. Box 419,
Arkansas State University, State University, AR 72467-0419, USA
Within a self-consistent thermal model using an isospin and momentum dependent interaction
(MDI)constrained bytheisospin diffusiondatainheavy-ioncollisions, weinvestigatethetempera-
turedependenceof thesymmetryenergy Esym(ρ,T)and symmetry free energy Fsym(ρ,T)for hot,
isospin asymmetric nuclear matter. It is shown that the symmetry energy Esym(ρ,T) generally
7 decreaseswithincreasingtemperaturewhilethesymmetryfreeenergyFsym(ρ,T)exhibitsopposite
temperature dependence. The decrement of the symmetry energy with temperature is essentially
0
due to the decrement of the potential energy part of the symmetry energy with temperature. The
0
2 differencebetweenthesymmetryenergyandsymmetryfreeenergyisfoundtobequitesmallaround
thesaturation density of nuclear matter. While at very low densities, they differ significantly from
n
each other. In comparison with theexperimental dataof temperaturedependent symmetry energy
a
extracted from the isotopic scaling analysis of intermediate mass fragments (IMF’s) in heavy-ion
J
collisions, the resulting density and temperature dependent symmetry energy Esym(ρ,T) is then
5 used to estimate theaverage freeze-out density of theIMF’s.
2

PACSnumbers: 25.70.-z,21.65.+f,21.30.Fe,24.10.Pa
2
v
5
I. INTRODUCTION tryenergytodecreaseasthePauliblockingbecomesless
3
importantwhenthenucleonFermisurfacesbecomemore
0
diffused at increasingly higher temperatures [13, 14, 15].
9 Theequationofstate(EOS)ofisospinasymmetricnu-
0 clear matter, especially the nuclear symmetry energy, is Based on a simplified degenerate Fermi gas model at fi-
6 nite temperatures, two of present authors [15] have re-
essential in understanding not only many aspects of nu-
0 cently studied the temperature dependence of the sym-
clearphysics,butalsoanumberofimportantissuesinas-
/ metry energy and it was shown that the experimentally
h trophysics[1,2,3,4,5,6,7,8,9]. Informationaboutthe
-t symmetry energy at zero temperature is important for observed decrease of the nuclear symmetry energy with
l determining ground state properties of exotic nuclei and the increasing centrality or the excitation energy in iso-
c
topic scaling analyses of heavy-ion reactions can be well
u properties of cold neutron stars at β-equilibrium, while
understood analytically within the degenerate Fermi gas
n the symmetry energy or symmetry free energy of hot
: neutron-rich matter is important for understanding the model. In particular, it was argued that the symmetry
v
energyextractedfromisotopicscalinganalysesofheavy-
Xi liquid-gasphasetransitionofasymmetricnuclearmatter, ion reactions reflects the symmetry energy of bulk nu-
the dynamical evolution of massive stars and the super-
clear matter for the emission source. Furthermore, it
r novaexplosionmechanisms. Heavy-ionreactionsinduced
a wasfoundthatthe evolutionofthesymmetryenergyex-
byneutron-richnucleiprovideauniquemeanstoinvesti-
tractedfromtheisotopicscalinganalysisismainlydueto
gate the symmetry energy [1, 2, 7]. In particular, recent
the variation in the freeze-out density rather than tem-
analyses of the isospin diffusion data in heavy-ion reac-
peraturewhenthefragmentsareemittedinthereactions
tions [10, 11, 12] have already put a stringent constraint
carried out under different conditions.
on the symmetry energy of cold neutron-rich matter at
sub-normal densities. On the other hand, the temper- In the present work, within a self-consistent thermal
ature dependence of the symmetry energy or symmetry model using an isospin and momentum dependent inter-
free energy for hot neutron-rich matter has received so action(MDI)constrainedbytheisospindiffusiondatain
far little theoretical attention [13, 14, 15]. heavy-ioncollisions,westudysystematicallythetemper-
For finite nuclei at temperatures below about 3 MeV, ature dependence of the nuclear matter symmetry en-
the shell structure and pairing as well as vibrations of ergy E (ρ,T) and symmetry free energy F (ρ,T).
sym sym
nuclear surfaces are important and the symmetry en- It is shown that the nuclear matter symmetry energy
ergywaspredictedtoincreaseslightlywiththeincreasing E (ρ,T) generally decreases with increasing tempera-
sym
temperature [16, 17]. Interestingly, an increase by only ture while the symmetry free energy F (ρ,T) exhibits
sym
about8%inthesymmetryenergyintherangeofT from oppositetemperaturedependence. Thedecrementofthe
0 to 1 MeV was found to affect appreciably the physics symmetry energy with temperature is essentially due to
ofstellar collapse,especially the neutralizationprocesses thedecrementofthepotentialenergypartofthesymme-
[16]. At higher temperatures, one expects the symme- tryenergywithtemperature. Furthermore,thedifference
2
betweenthenuclearmattersymmetryenergyEsym(ρ,T) Au(x),Al(x),B,Cτ,τ,Cτ,−τ and Λ have been assumed
andsymmetryfreeenergyF (ρ,T)isfoundtobequite to be temperature independent and are obtained by fit-
sym
small around nuclear saturation density. Using the re- ting the momentum-dependence of U(ρ,T =0,δ,~p,τ) to
sultingdensityandtemperaturedependentsymmetryen- that predicted by the Gogny Hartree-Fock and/or the
ergyE (ρ,T),weestimate the averagefreeze-outden- Brueckner-Hartree-Fock calculations, the zero tempera-
sym
sity of the fragment emission source based on the mea- ture saturation properties of symmetric nuclear matter
suredtemperaturedependentsymmetryenergyfromthe andthesymmetryenergyof31.6MeVatnormalnuclear
isotopic scaling analysis in heavy-ioncollisions. matter density ρ =0.16 fm−3 [18]. The incompressibil-
0
Thepaperisorganizedasfollows. InSectionII,wein- ity K of cold symmetric nuclear matter at saturation
0
troduce the isospin and momentum dependent MDI in- density ρ is set to be 211 MeV. The parameters A (x)
0 u
teraction and the detailed numerical method to obtain and A (x) depend on the x parameter according to
l
the EOS of the symmetric nuclear matter and pure neu-
tron matter at finite temperatures. Results on the tem- 2B 2B
A (x)=−95.98−x , A (x)=−120.57+x .
perature dependence of the symmetry energy and sym- u σ+1 l σ+1
metry free energy are presented in Section III. In Sec- (3)
tion IV, we discuss the experimentaldata of the isotopic The different x values in the MDI interaction are in-
scaling in heavy-ion collisions by means of the obtained troduced to vary the density dependence of the nuclear
densityandtemperaturedependentsymmetryenergy. A symmetry energy while keeping other properties of the
summary is given in Section V. nuclear equation of state fixed [11] and they can be ad-
justedtomimicpredictionsonthedensitydependenceof
nuclear matter symmetry energy by microscopic and/or
II. HOT NUCLEAR MATTER EOS IN phenomenological many-body theories. The last two
MOMENTUM DEPENDENT INTERACTION terms of Eq. (2) contain the momentum-dependence of
thesingle-particlepotential. Themomentumdependence
Our study is based on a self-consistent thermal model of the symmetry potential stems from the different in-
usingamodifiedfinite-rangeGognyeffectiveinteraction, teraction strength parameters Cτ,−τ and Cτ,τ for a nu-
i.e., the isospin- and momentum-dependent MDI inter- cleon of isospin τ interacting, respectively, with unlike
action[18]. In the MDI interaction,the potential energy and like nucleons in the background fields. More specif-
density V(ρ,T,δ) of a thermal equilibrium asymmetric ically, we use Cτ,−τ = −103.4 MeV and Cτ,τ = −11.7
nuclear matter at total density ρ, temperature T and MeV. We note that the MDI interactionhas been exten-
isospin asymmetry δ is expressed as follows [11, 18], sivelyusedinthetransportmodelforstudyingisospinef-
fects in intermediate energy heavy-ion collisions induced
V(ρ,T,δ) = Auρnρp + Al (ρ2 +ρ2) by neutron-rich nuclei [11, 12, 19, 20, 21, 22, 23, 24]. In
ρ0 2ρ0 n p particular, the isospin diffusion data from NSCL/MSU
B ρσ+1 have constrained the value of x to be between 0 and −1
+ σ+1 ρσ (1−xδ2) fornuclearmatterdensitieslessthanabout1.2ρ0[11,12],
0 we will thus in the present work consider the two values
′
+ ρ10 Xτ,τ′Cτ,τ′Z Z d3pd3p′1fτ+(~r(,p~~p−)fτp~′′()~r2,/p~Λ(1)2). ofAxt=ze0roantdemxp=er−at1u.re, fτ(~r,p~) = h23Θ(pf(τ)−p) and
the integralinEqs.(1) and(2) canbe calculatedanalyt-
Inmeanfieldapproximation,Eq. (1)leadsto thefollow- ically[18]. For anasymmetricnuclearmatter atthermal
ingsingleparticlepotentialforanucleonwithmomentum equilibriumwitha finite temperatureT,the phasespace
p~ and isospin τ in the thermal equilibrium asymmetric distribution function becomes the Fermi distribution
nuclear matter, i.e., [11, 18]
2 1
f (~r,p~)= (4)
U(ρ,T,δ,~p,τ) = A (x)ρ−τ +A(x)ρτ τ h3exp(ǫ(ρ,T,δ,Tp~,τ)−µτ)+1
u l
ρ ρ
0 0
+ B(ρρ )σ(1−xδ2)−8τxσB+1ρρσ−σ1δρ−τ twhheetroetaµlτsiinsgtlheepachrteimcleicaenleprogtyenfotriaal nauncdleǫo(nρ,wTi,tδh,~pis,oτs)piins
0 0 τ and momentum ~p, which includes the kinetic energy
′
2C f (~r,p~)
+ τ,τ d3p′ τ and the single particle potential U(ρ,T,δ,~p,τ), i.e.,
ρ 1+(p~−~p′)2/Λ2
0 Z
′
+ 2Cρτ0,−τ Z d3p′1+f(−p~τ−(~r~p,′~p)2)/Λ2. (2) ǫ(ρ,T,δ,~p,τ) = 2pm2τ +Auρρ−0τ +Alρρτ0
Iσn=th4e/a3b;ofvτe(~rτ,~p)=is1/th2e(−ph1a/s2e)sfpoarceneduitsrtornibsut(ipornotfounnsc)-; + B(ρρ0)σ(1−xδ2)−8τxσB+1ρρσ−σ01δρ−τ
tion at coordinate ~r and momentum ~p. The parameters + Rτ,τ(ρ,p~)+Rτ,−τ(ρ,~p) (5)
3
with Furthermore, we can obtain the entropy per nucleon
Rτ1,τ2(ρ,p~)= 2Cρτ1,τ2 d3p′1+f(p~τ2−(~r,p~p~′)′2)/Λ2 (6) Smτa(tρt,eTr,aδs)ofthethermalequilibriumasymmetricnuclear
0 Z ∞
8π
where τ1 and τ2 can be chosen as the same or different S (ρ,T,δ)=− p2[n lnn +(1−n )ln(1−n )]dp
to mimic the last two terms of Eq. (2). τ ρh3 τ τ τ τ
Z0
The chemical potential µ is therefore independent of (10)
τ
the nucleon momentum p~ and can be determined from with the occupation probability
8π ∞ p2dp 1
ρ = . (7) n = . (11)
τ h3Z0 exp(ǫ(ρ,T,δ,Tp~,τ)−µτ)+1 τ exp(ǫ(ρ,T,δ,Tp~,τ)−µτ)+1
From Eq. (7), we can see that, at finite temperature,
Finally,thefreeenergypernucleonF(ρ,T,δ)ofthether-
to obtain the chemical potential µ requires knowing
τ mal equilibrium asymmetric nuclear matter can be ob-
ǫ ρ,T,δ,~p,τ) (and thus R (ρ,~p)) for all p~, while
( τ1,τ2 tained from the thermodynamic relation
from Eq. (6) knowing R (ρ,~p) needs further to know
τ1,τ2
f (~r,p~) which again depends on the chemical potential
τ
µτ from Eq. (4). Therefore. Eqs. (4), (5), (6), and (7) F(ρ,T,δ)=E(ρ,T,δ)−T Sτ(ρ,T,δ). (12)
constitute closedsets of equations whose solutioncanbe
τ
X
obtained by a self-consistency iteration, just as in the
Hartree-Fock theory.
Following the recipe used in Ref. [25], the self-

consistency problem of Eqs. (4), (5), (6), and (7) can
20 20
be solved by the following iterative scheme. Firstly, we
Symmetric Nuclear Matter
make an initial guess for Rτ1,τ2(ρ,~p) from the zero tem- 10 MDI interaction 10
perature condition, i.e.,
Rτ01,τ2(ρ,p~)= 2Cρτ01,τ2 Z d3p′h123+Θ((p~pf−(τp~2′))2−/Λp′2) (8) MeV) -100 -010 F (M
wherepf(τ)=h¯(3π2ρτ)1/3istheFermimomentum. The E ( eV
right hand side of Eq. (8) is not related to the chemical )
-20 T (MeV) -20
potential µτ and thus the initial form of the single nu- 0
cleon energy ǫ0(ρ,T,δ,~p,τ) can be obtained. Secondly, 5
-30 10 -30
substituteǫ0(ρ,T,δ,~p,τ)intoEq.(7)toobtaintheinitial 15
chemical potential µ0 for protons and neutrons. Then,
τ -40 -40
useǫ0(ρ,T,δ,~p,τ)andµ0τ toobtainnewRτ1,τ2(ρ,p~)func- 0 1 2 0 1 2 3
tion, namely, Rτ11,τ2(ρ,p~) from Eq. (4) and (6). This in / 0
turn gives the new single nucleon energy ǫ1(ρ,T,δ,~p,τ)
FIG. 1: (Color online) Density dependenceof theenergy per
fromEq. (5)andthenthenewchemicalpotentialµ1τ can nucleon E (left panel) and free energy per nucleon F (right
be obtained from Eq. (7). The cycle is repeated and a panel) for symmetric nuclear matter at T =0 MeV, 5 MeV,
fewiterationsaresufficienttoachieveconvergenceforthe 10 MeV and 15 MeV with the MDI interaction.
chemicalpotentialµ withenoughaccuracy. Itshouldbe
τ
mentioned that the neutron and proton chemical poten- Using the MDI interaction, we can now calculate the
tials are coupled with each other in asymmetric nuclear energypernucleonE(ρ,T,δ)andfreeenergypernucleon
matter and thus the convergence condition must be sat- F(ρ,T,δ) of nuclear matter at finite temperature from
isfied simultaneously for neutrons and protons. Eq. (9) and (12). Shown in Fig. 1 is the density depen-
With the self-consistency iteration, we can finally ob- dence of E(ρ,T,δ) and F(ρ,T,δ) for symmetric nuclear
tain the chemical potential µ and the single nucleon matteratT =0MeV,5MeV,10MeVand15MeVusing
τ
energy ǫ(ρ,T,δ,~p,τ) for an asymmetric nuclear matter the MDI interaction with x =0 and −1. For symmetric
atthermalequilibrium with a finite temperatureT. The nuclear matter (δ = 0), the parameter x = 0 would give
potentialenergydensityV(ρ,T,δ)ofthethermalequilib- the same results as the parameter x = −1 as we have
rium asymmetric nuclear matter then can be calculated discussedabove,andthusthe curvesshowninFig. 1are
fromEq.(1)andtheenergypernucleonE(ρ,T,δ)isthen the same for x = 0 and −1. From Fig. 1, one can see
obtained as that the energy per nucleon E(ρ,T,δ) increases with in-
creasingtemperatureT whilethefreeenergypernucleon
F(ρ,T,δ) decreases with increment of T. The increment
1 p2
E(ρ,T,δ)= V(ρ,T,δ)+ d3p f (~r,p~) . oftheenergypernucleonE(ρ,T,δ)withthetemperature
τ
ρ" τ Z 2mτ # reflects the thermal excitation of the nuclear matter due
X
(9) to the change of the phase-space distribution function
4
f (~r,p~). With the increment of the temperature, more
τ
nucleonsmovetohighermomentumstatesandthus lead 60
tolargerinternalenergypernucleon. Ontheotherhand, 50 (a) T=0 MeV (b) T=5 MeV
the decrement of the free energy per nucleon F(ρ,T,δ) ) =0.5
V40
with T is mainly due to the increment of the entropy e =1.5
per nucleon with increasing temperature. This feature (M 30 =2.5
also implies that the increment of TS(ρ,T) with T is 0)
=20
larger than the increment of E(ρ,T) with T. Further-
more, the temperature effects are seen to be stronger at T,10
,
lowerdensitieswhiletheybecomemuchweakerathigher E( 0
idsesnmsiatilelesr. aAntdlotwhuersdteemnspiteireast,utrheeeFffeercmtsiomnotmheenetnuemrgpyfp(τer) ) 50 (c) T=10 MeV (d) T=15 MeV
nucleon E(ρ,T,δ) are expected to be stronger. Mean- T,40
,
while, the entropy per nucleon becomes larger at lower E(30
densities where the particles become more free in phase
spaceandthusleadstoasmallerfreeenergypernucleon. 20
10
0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
2
60 60
Pure Neutron Matter
MDI interaction FIG.3: (Coloronline)E(ρ,T,δ)−E(ρ,T,δ=0)asafunction
x= 0 ofδ2 at temperatureT =0MeV (a),5MeV (b),10 MeV(c)
40 40
x= 1 and 15 MeV (d) for three different baryon number densities
ρ = 0.5ρ0,1.5ρ0 and 2.5ρ0 using the MDI interaction with
V) F x=0.
e 20 20 (
M M
( e
E V
0 T15 (MeV) T (MeV) 0 ) equation of state of hot neutron-rich matter at density
10 0 ρ, temperature T, and an isospin asymmetry δ can also
5 5 be written as a parabolic function of δ, i.e.,
0 10
-20 15 -20
E(ρ,T,δ)=E(ρ,T,δ =0)+E (ρ,T)δ2+O(δ4). (13)
sym
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0
/ 0 The temperature and density dependent symmetry en-
ergy E (ρ,T) for hot neutron-rich matter can thus be
sym
FIG. 2: (Color online) Same as Fig. 1 but for pure neutron extractedfromE (ρ,T)≈E(ρ,T,δ =1)−E(ρ,T,δ =
sym
matterusingtheMDIinteractionwithx=0(solidlines)and
0). The symmetry energy E (ρ,T) is the energy cost
−1 (dashed lines). sym
to convert all protons in symmetry matter to neutrons
at the fixed temperature T and density ρ. In order to
Similarly, shown in Fig. 2 are the density dependence
check the empirical parabolic law Eq. (13) for the MDI
of the E(ρ,T,δ) and F(ρ,T,δ) for pure neutron matter interaction, we show in Fig. 3 E(ρ,T,δ)−E(ρ,T,δ =0)
at T = 0 MeV, 5 MeV, 10 MeV and 15 MeV using the as a function of δ2 at temperature T = 0 MeV, 5 MeV,
MDI interaction with x = 0 and −1. The temperature
10 MeV and 15 MeV for three different baryon number
dependence of the E(ρ,T,δ) andF(ρ,T,δ) for pure neu-
densities ρ = 0.5ρ ,1.5ρ and 2.5ρ using the MDI in-
0 0 0
tronmatterisseentobesimilartothatofthesymmetric
teraction with x = 0. The clear linear relation between
nuclear matter as shown in Fig. 1. However,the param- E(ρ,T,δ)−E(ρ,T,δ = 0) and δ2 shown in Fig. 3 indi-
eters x=0 and −1 display different density dependence
catesthe validity ofthe empiricalparaboliclawEq. (13)
for the energyper nucleon E(ρ,T,δ)and free energyper
for the hot neutron-rich matter. We note that the em-
nucleon F(ρ,T,δ), which just reflects that the parame-
piricalparaboliclawEq. (13)isalsowellsatisfiedforthe
ters x = 0 and −1 give different density dependence of parameter x=−1.
the nuclear symmetry energy and symmetry free energy
Similarly, we can define the symmetry free energy
as will be discussed in the following.
F (ρ,T) by the following parabolic approximation to
sym
the free energy per nucleon
III. NUCLEAR SYMMETRY ENERGY AND F(ρ,T,δ)=F(ρ,T,δ =0)+F (ρ,T)δ2+O(δ4). (14)
sym
SYMMETRY FREE ENERGY
The temperature and density dependent symmetry free
As in the case of zero temperature, phenomenologi- energy F (ρ,T) for hot neutron-rich matter can thus
sym
cal and microscopic studies [13, 14] indicate that the be extracted from F (ρ,T) ≈ F(ρ,T,δ = 1) −
sym
5

60
50 (a) T=0 MeV (b) T=5 MeV 50 50
MDI interaction
=0.5
) x= 0
eV 40 =1.5 40 x= 1 T (MeV) 40
=0) (M 2300 =2.5 (MeV) 30 11 5050 30 sym F(
,T, 10 (,T) 20 20 ,T) (M
) F( 500 (c) T=10 M eV (d) T=15 MeV Esym 10 T05 (MeV) 10 eV)
T, 40 10
, 15
F( 30 0 0
0.0 0.5 1.0 0.0 0.5 1.0 1.5
20
/ 0
10
FIG. 5: (Color online) Density dependence of the symmetry
0 energy Esym(ρ,T) (left panel) and thesymmetry free energy
0.0 0.2 0.4 0.6 0.8 0.20 0.2 0.4 0.6 0.8 1.0 Fsym(ρ,T) (right panel) at T =0MeV, 5MeV, 10 MeV and
15 MeV using theMDI interaction with x=0 and −1.
FIG.4: (Color online)SameasFig. 3butforthefreeenergy
per nucleon F(ρ,T,δ).
namely, with increasing temperature T, E (ρ,T) de-
sym
creaseswhile F (ρ,T)increases. This also means that
sym
F (ρ,T) always has larger values than E (ρ,T) at
sym sym
F(ρ,T,δ = 0) which is just the free energy cost to
fixeddensity andtemperature since they are identicalat
convert all protons in symmetry matter to neutrons at
zero temperature. At higher temperatures, one expects
the fixed temperature T and density ρ. In order to
thesymmetryenergyE (ρ,T)todecreaseasthePauli
sym
check if the empirical parabolic law is also valid for the
blocking(apurequantumeffect)becomeslessimportant
free energy per nucleon of hot neutron-rich matter, we
when the nucleon Fermi surfaces become more diffused
show in Fig. 4 F(ρ,T,δ) − F(ρ,T,δ = 0) as a func-
at increasingly higher temperatures [13, 14, 15]. On the
tion of δ2 at temperature T = 0 MeV, 5 MeV, 10 MeV
other hand, the symmetry free energy F (ρ,T) is re-
sym
and 15 MeV for three different baryon number densities
lated to the entropy per nucleon of the asymmetric nu-
ρ = 0.5ρ ,1.5ρ and 2.5ρ using the MDI interaction
0 0 0 clear matter, which is not a pure quantum effect, and
with x = 0. One can see from Fig. 4 that the parabolic
its increment with increasingtemperature can be under-
law Eq. (14) is also approximately satisfied though at
stood by the following expression
low densities and high temperatures, the linear relation
between F(ρ,T,δ) − F(ρ,T,δ = 0) and δ2 is violated F (ρ,T) = E (ρ,T)
sym sym
slightly. Forthe parameterx=−1,wealsoobtainedthe
+T [S (ρ,T,δ =0)+S (ρ,T,δ =0)]
n p
similar conclusion.
−TS (ρ,T,δ =1). (15)
n
InFig. 5,weshowthedensity dependence ofthe sym-
metry energy Esym(ρ,T) and the symmetry free energy The first term of the right hand side in Eq. (15) is the
Fsym(ρ,T) at T =0 MeV, 5 MeV, 10 MeV and 15 MeV symmetry energy Esym(ρ,T), which decreases with in-
using the MDI interaction with x = 0 and −1. For dif- creasing temperature as discussed above. However, the
ferentchoice of the parameterx=0 and−1,Esym(ρ,T) total entropy per nucleon of the symmetric nuclear mat-
andFsym(ρ,T)displaydifferentdensitydependencewith ter is larger than that of the pure neutron matter and
x = 0 (−1) giving larger (smaller) values for the sym- their difference becomes larger with increasing tempera-
metry energy and the symmetry free energy at lower ture, which leads to a positive value for the difference
densities while smaller (larger) ones at higher densities between the last two terms of the right hand side in
for a fixed temperature. Similar to the E(ρ,T,δ) and Eq. (15). Therefore, F (ρ,T) has larger values than
sym
F(ρ,T,δ) as shown in Figs. 1 and 2, the temperature E (ρ,T) at fixed density and temperature. Further-
sym
effects onthe symmetryenergyEsym(ρ,T)andthe sym- more, the increment of TS(ρ,T)with T is stronger than
metry freeenergyFsym(ρ,T)arefoundto be strongerat theincrementofE(ρ,T)withT asmentionedabove,and
lowerdensitieswhiletheybecomemuchweakerathigher the combinational effects thus cause the symmetry free
densities. energy F (ρ,T) increase with increasing temperature.
sym
Interestingly, we can see from Fig. 5 that the sym- Within the present self-consistent thermal model, be-
metry energy E (ρ,T) and the symmetry free energy cause the single particle potential is momentum depen-
sym
F (ρ,T) exhibit opposite temperature dependence, dent with the MDI interaction, the potential part of the
sym
6
symmetry energy is expected to be temperature depen-

dent. Itisthusinterestingtostudyhowthepotentialand
kinetic parts of the symmetry energy Esym(ρ,T) may 30
MDI interaction with x = 0
vary respectively with temperature. However, for the (a) = 0
symmetry free energy F (ρ,T), one cannot separate 25 Total
sym Potential
its potential and kinetic parts since F (ρ,T) depends
sym 20 Kinetic
on the entropy that is determined by the phase space
distribution function. Fig. 6 displays the temperature 15
dependence of the symmetry energy Esym(ρ,T) as well V) 10
as its potential and kinetic energy parts using the MDI e
M
tinhteerpaacrtaiomnetweirthxx==−01,atthρe=samρ0e, 0co.5nρc0l,usainodn0is.1oρb0t.aWineitdh. T) ( 18 (b) =0.5 0
, 15
ItisseenthatboththesymmetryenergyE (ρ,T)and
sym (
itspotentialenergypartdecreasewithincreasingtemper- m12
y
s
ature at all three densities considered. While the kinetic E
9
energy part of the E (ρ,T) increases slightly with in-
sym
creasing temperature for ρ=ρ and 0.5ρ and decreases
for ρ = 0.1ρ . These features0are uniqu0ely determined 5 (c) = 0
0
by the momentum dependence in the MDI interaction 4
within the present self-consistent thermal model. The 3
decrementofthekineticenergypartofthesymmetryen-
2
ergywith temperature atvery low densities is consistent
with predictions of the Fermi gas model at high temper- 1
atures and/orvery lowdensities [15, 26]. In the study of 0 5 10 15
T (MeV)
Ref.[15]usingthesimplifieddegenerateFermigasmodel
FIG. 6: (Color online) Temperature dependence of the sym-
the potential part of the symmetry energy was assumed
to be temperature independent for simplicity. The de- metry energy Esym(ρ,T) as well as its potential energy part
and kinetic energy part using MDI interaction with x=0 at
crease of the symmetry energy observed there is thus
ρ=ρ0 (a), 0.5ρ0 (b), and 0.1ρ0 (c).
completely due to the decrease in the kinetic contribu-
tion. However,wenotethatthetemperaturedependence
of the total symmetry energy Esym(ρ,T) is consistent where△[(Z/A)2]≡(Z /A )2−(Z /A )2 isthedifference
with each other for the two models. This is due to the 1 1 2 2
betweenthe(Z/A)2valuesofthetwofragmentingsources
fact that the phase space distribution function will vary
created in the two reactions.
self-consistently accordingto if the singleparticle poten-
AsmentionedinRef. [15],however,becauseofthedif-
tial is or not momentum dependent. From the present
ferent assumptions used in the various derivations, the
self-consistentthermalmodelwithmomentumdependent
validity of Eq. (16) is still disputable as to whether
MDI interaction,ourresults indicate thatthe decreasing
and when the C is actually the symmetry energy or
symmetry energy with increasing temperature is essen- sym
the symmetry free energy. Moreover, the physical inter-
tially due to the decrement of its potential contribution.
pretation of the C (ρ,T) is also not clear, sometimes
sym
even contradictory, in the literature. The main issue is
whether the C measures the symmetry energy of the
sym
fragmenting source or that of the fragments formed at
IV. ISOTOPIC SCALING IN HEAVY-ION freeze-out. This ambiguity is also due to the fact that
COLLISIONS the derivation of Eq. (16) is not unique. In particular,
within the grand canonical statistical model for multi-
It has been observed experimentally and theoretically fragmentation [34, 35] the Csym refers to the symmetry
in many types of reactions that the ratio R (N,Z) of energy of primary fragments. While within the sequen-
21
yields of a fragment with proton number Z and neu- tial Weisskopf model in the grand canonical limit [34]
tron number N from two reactions reaching about the it refers to the symmetry energy of the emission source.
sametemperatureT satisfiesanexponentialrelationship Following the arguments in Ref. [15], we assume in the
R21(N,Z)∝ exp(αN) [27, 28, 29, 30, 31, 32, 33, 34, 35, presentworkthattheCsym reflectsthesymmetryenergy
36, 37, 38]. Particularly, in several statistical and dy- of bulk nuclear matter for the emission source.
namical models under some assumptions [34, 35, 36], it In Fig. 7, we show the symmetry energy Esym(ρ,T)
has been shown that the scaling coefficient α is related and symmetry free energy Fsym(ρ,T) as a function of
to the symmetry energy C (ρ,T) via temperature using MDI interaction with x = 0 and −1
sym
at different densities from 0.1ρ to ρ . The temperature
0 0
α= 4Csym(ρ,T) △[(Z/A)2], (16) dbeepveenrdyesnicmeiolafrthtoetshyamtminetRryefe.n[1er5g]ywEhesyrme(aρs,iTm)pilsifiseedendteo-
T
7
squares) [31, 32]. From Fig. 7, it is seen clearly that
the experimentally observed evolution of the symmetry
35
V) MDI interaction with x = 0 MDI interaction with x = 1 energyismainlyduetothechangeindensityratherthan
Me 30 / 0=1 / 0=1 temperature, as shown in Ref. [15]. Meanwhile, we can
T) ( 25 00..89 INDRA@GSI 0.9 INDRA@GSI efrsatgimmaetnetfermomissFioign.s7outrhceeafvroermagtehefrmeeezaes-uouretddteenmsiptyeroaftuthree
, 0.7 0.8
(m 0.6 dependentsymmetryenergybasedontheisotopicscaling
and Fsy1250 00..45 TAMU 000...567 TAMU wasynimtahlmyxseits=riyn0,ehnweeearvgfiyyn-diEontsyhmceo(alρlvi,seTiroa)ngsfe.rofIrmneepzthea-erotuMictuDdlaIerni,nsuittesyirnaogcftttiohhnee
) 0.3
T 0.4 fragmentemissionsourceρ isbetweenabout0.41ρ and
, 10 f 0
(m 0.2 0.3 0.52ρ0 for TAMU data while about 0.42ρ0 and 0.75ρ0
Esy 5 0.1 0.2 for INDRA-ALADIN collaboration data. On the other
Esym Fsym 0.1 hand, using the symmetry energy Esym(ρ,T) from the
0 MDI interaction with x = −1, the ρf is found to be
0 2 4 6 8 0 2 4 6 8 10 between about 0.57ρ0 and 0.68ρ0 for TAMU data while
T (MeV) about0.58ρ0 and0.84ρ0 forINDRA-ALADIN collabora-
tiondata. Itisinterestingtoseethattheextractedvalues
FIG. 7: (Color online) Temperature dependence of the sym-
ofρ fromtheMDIinteractionwithx=0isverysimilar
metry energy (solid lines) and symmetry free energy (dashed f
lines) using MDI interaction with x = 0 (left panel) and to those extracted in Ref. [30] using different models.
−1 (right panel) at different densities from 0.1ρ0 to ρ0. Furthermore, if the symmetry free energy Fsym(ρ,T)
The experimental data from Texas A&M University (solid from the MDI interactionwith x=0 is used to estimate
squares) and the INDRA-ALADIN collaboration at GSI theρ ,wefindtheρ isbetweenabout0.36ρ and0.49ρ
f f 0 0
(open squares) are included for comparison. forTAMUdataandabout0.33ρ and0.72ρ forINDRA-
0 0
ALADINcollaborationdata. While ifthe symmetryfree
energyF (ρ,T)fromtheMDIinteractionwithx=−1
sym
is used, the ρ is between about 0.52ρ and 0.66ρ for
generateFermigasmodelatfinitetemperatureshasbeen f 0 0
TAMU data and about 0.51ρ and 0.83ρ for INDRA-
used. The symmetry energy does not change much with 0 0
ALADINcollaborationdata. Therefore,theextractedρ
thetemperatureatagivendensity,especiallyaroundthe f
values are not sensitive to if the measured C (ρ,T) is
saturation density ρ . Furthermore, it is seen that while sym
0
thesymmetryenergyorthesymmetryfreeenergy. How-
the symmetry energy E (ρ,T) deceases slightly with
sym
ever, the extracted ρ values are indeed sensitive to the
the increasing temperature at a given density, the sym- f
x parameter used in the MDI interaction, namely, the
metry free energy F (ρ,T) increases instead. Around
sym
density dependence of the symmetry energy. We note
the saturation density ρ , it is found that the difference
0
thatthe zero-temperaturesymmetryenergyforthe MDI
between the symmetry energy E (ρ,T) and the sym-
sym interaction with x = 0 and −1 can be parameterized,
metryfreeenergyF (ρ,T)isquitesmall,i.e.,onlysev-
sym
respectively, as 31.6(ρ/ρ )0.69 MeV and 31.6(ρ/ρ )1.05
eral percents, though at higher temperature, compared 0 0
MeV [11]. Therefore, the isotopic scaling in heavy-ion
withtheirvaluesatT =0MeV.Thisfeatureconfirmsthe
collisionsprovidesapotentialgoodprobe forthe density
assumption on identifying C (ρ,T) to E (ρ,T) at
sym sym
dependence ofthe nuclear matter symmetry energyonce
lower temperatures and not so low densities [28, 29, 30].
the average density of the emission source has been de-
At low densities, on the other hand, the symmetry free
terminedintheisotopicscalingmeasurement,aspointed
energy F (ρ,T) exhibits a stronger temperature de-
sym
out in Ref. [15].
pendenceanditissignificantlylargerthanthesymmetry
energy E (ρ,T) at moderate and high temperatures.
sym
This is due to the fact that the entropy contribution to
the symmetry free energy F (ρ,T) becomes stronger V. SUMMARY
sym
at low densities as mentioned in Sec. II and Sec. III. It
shouldbenotedthat,atlowdensitiestheentropymaybe Within a self-consistent thermal model using the
affected strongly by the clustering effects [33, 39] which isospin and momentum dependent MDI interaction with
are not included in the present work. x = 0 and −1 constrained by the isospin diffusion data
Experimentally, the temperature T and scaling coeffi- in heavy-ioncollisions, we have investigated the temper-
cient α (thus the C ) of the fragment emission source ature dependence of the nuclear matter symmetry en-
sym
can be directly measured while the determination of the ergy E (ρ,T) and symmetry free energy F (ρ,T).
sym sym
density of emission source usually depends on the model It is shown that the nuclear matter symmetry energy
used. AlsoincludedinFig.7aretheexperimentaldataof E (ρ,T) generally decreases with increasing tempera-
sym
the measured temperature dependent symmetry energy ture while the symmetry free energy F (ρ,T) exhibits
sym
fromTexasA&MUniversity(TAMU)(solidsquares)[30] oppositetemperaturedependence. Thedecrementofthe
and the INDRA-ALADIN collaboration at GSI (open symmetry energy with temperature is essentially due to
8
thedecrementofthepotentialenergypartofthesymme- ergy F (ρ,T) in the temperature and density ranges
sym
tryenergywithtemperature. Thetemperatureeffectson reached in the TAMU and INDRA/GSI experiments.
the nuclear matter symmetry energy and symmetry free They are, however, sensitive to the x parameter used
energy are found to be stronger at lower densities while in the MDI interaction, namely, the density dependence
become much weaker at higher densities. Furthermore, of the symmetry energy. Therefore the isotopic scaling
the difference between the nuclear matter symmetry en- in heavy-ioncollisions provides a potentially good probe
ergy E (ρ,T)and symmetry free energy F (ρ,T) is forthedensitydependenceofthenuclearmattersymme-
sym sym
found to be quite small around nuclear saturation den- try energy provided the average density of the emission
sity, although significantly large at very low densities. source can be determined simultaneously in the isotopic
Comparing the theoretical density and tempera- scaling measurements.
ture dependent symmetry energy E (ρ,T) with the
sym
C (ρ,T) parameter extracted from the isotopic scal-
sym
ing data from TAMU and the INDRA-ALADIN collab-
Acknowledgments
oration at GSI, we found that the experimentally ob-
served evolution of the symmetry energy is mainly due
to the change in density rather than temperature, as This work was supported in part by the National
shown in the previous work [15]. Meanwhile, we have Natural Science Foundation of China under Grant Nos.
estimated the averagefreeze-out density of the fragment 10334020, 10575071, and 10675082, MOE of China un-
emission source formed in these reactions by comparing der project NCET-05-0392, Shanghai Rising-Star Pro-
the calculated E (ρ,T) or F (ρ,T) with the mea- gram under Grant No. 06QA14024,the SRF for ROCS,
sym sym
suredC (ρ,T). Ourresultsindicatethattheextracted SEM of China, the US National Science Foundation un-
sym
average freeze-out densities are not sensitive to whether der Grant Nos. PHY-0652548 and PHY-0456890, and
theexperimentallymeasuredC (ρ,T)parameteristhe the NASA-Arkansas Space Grants Consortium Award
sym
symmetry energy E (ρ,T) or the symmetry free en- ASU15154.
sym
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