The Statistical Multifragmentation Model with Skyrme Effective Interactions

0.39 MB·English

by  S. R. Souza

The Statistical Multifragmentation Model with Skyrme Effective Interactions
S.R. Souza1,2, B.V. Carlson3, R. Donangelo1,4, W.G. Lynch5, A.W. Steiner5, and M.B. Tsang5
1Instituto de F´ısica, Universidade Federal do Rio de Janeiro Cidade Universit´aria,
CP 68528, 21941-972, Rio de Janeiro, Brazil
2Instituto de F´ısica, Universidade Federal do Rio Grande do Sul
Av. Bento Gonc¸alves 9500, CP 15051, 91501-970, Porto Alegre, Brazil
3Departamento de F´ısica,
Instituto Tecnol´ogico de Aeron´autica - CTA, 12228-900
S˜ao Jos´e dos Campos, Brazil
4Instituto de F´ısica, Facultad de Ingenier´ıa, Universidad de la Repu´blica,
Julio Herrera y Reissig 565, 11.300 Montevideo, Uruguay and
5 Joint Institute for Nuclear Astrophysics, National Superconducting Cyclotron Laboratory,
and the Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
9 (Dated: January 20, 2009)
0
0 The Statistical Multifragmentation Model is modified to incorporate the Helmholtz free energies
2 calculated in the finite temperature Thomas-Fermi approximation using Skyrme effective interac-
tions. In this formulation, the density of the fragments at the freeze-out configuration corresponds
n
to the equilibrium value obtained in the Thomas-Fermi approximation at the given temperature.
a
J Thebehavior of thenuclear caloric curveat constant volume is investigated in the micro-canonical
ensemble and a plateau is observed for excitation energies between 8 and 10 MeV per nucleon. A
0
kink in the caloric curve is found at the onset of this gas transition, indicating the existence of
2
a small excitation energy region with negative heat capacity. In contrast to previous statistical

calculations, this situation takes place even in this case in which the system is constrained to fixed
]
h volume. The observed phase transition takes place at approximately constant entropy. The charge
t distribution and other observables also turn out to be sensitive to the treatment employed in the
-
l calculation of the free energies and the fragments’ volumes at finite temperature, specially at high
c
excitationenergies. Theisotopicdistributionisalsoaffectedbythistreatment,whichsuggeststhat
u
thisprescription may help to obtain information on thenuclear equation of state.
n
[
PACSnumbers: 25.70.Pq,24.60.-k,31.15.bt

1
v
5 I. INTRODUCTION nuclearMultifragmentationprocess[32,33,34],therehas
8 notbeenmuchefforttoincorporateinformationbasedon
9 the EOS in the main ingredients of statistical multifrag-
2 Understandingthebehaviorofnuclearmatterfarfrom
mentation models. Yet, these models have recently been
. equilibrium, besides its intrinsic relevance to theoretical
1 applied to investigate, for instance, the Isospin depen-
nuclear physics, is a subject of great interest to nuclear
0 dence of the nuclear energy at densities below the sat-
9 astrophysics,wherethe fateofsupernovaeorthe proper- uration value [35, 36, 37]. These calculations have sug-
0 tiesofneutronstarsareappreciablyinfluencedbythenu-
gested an appreciable reduction of the symmetry energy
: clearequationofstate(EOS)[1,2,3]. Thus,thisareahas
v coefficient at low densities but other statistical calcula-
been intensively investigatedin different contexts during
i tions [38, 39, 40] indicate that surface corrections to the
X the last decades [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Nuclear
symmetryenergymayalsoexplainthebehaviorobserved
r collisions, at energies starting at a few tens of MeV per
a inthose studies. Therefore,statisticaltreatments,which
nucleon,providea suitablemeansto study hotandcom-
consistently include density effects, are most advisable
pressed nuclear matter [9, 10, 11, 12, 13, 14, 15, 16, 17].
for these studies.
The determination of the nuclear caloric curve is of par-
ticular interest as it allows one to infer on the existence
In this work, we modify the Statistical Multifragmen-
ofaliquid-gasphasetransitioninnuclearmatter. Never-
tation Model (SMM) [41, 42, 43] and calculate some of
theless,owingtoexperimentaldifficulties, conflictingob-
its key ingredients from the finite temperature Thomas-
servationshavebeenmadeindifferentexperimentalanal-
Fermi approximation[44, 45, 46, 47] using Skyrme effec-
yses [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
tiveinteractions. Thisversionofthe modelishenceforth
Althoughtherehavebeenattemptstoreconcilethesere-
labeled SMM-TF. The internal Helmholtz free energies
sults [31], this issue has not been settled.
of the fragments are calculated in a mean field approxi-
The properties of the disassembling system in central mation,whichisfairlysensitivetotheSkyrmeforceused
collisions, as well as the outcome of the reactions, have [48]. Thismakespossibletoinvestigatewhethersuchsta-
been found to be fairly sensitive to the EOS employed tisticaltreatmentsmayprovideinformationonthe EOS.
in the many theoretical studies using dynamical models Furthermore, this approach allows to consistently take
thathavebeenperformed[9,10,11,12,13]. However,in into account contributions to the free energy due to ex-
spite of their success in describing many features of the citations in the continuum, in contrastto the traditional
2
SMM [49]. For consistency with the mean field treat- charge, and energy conservation and thus the following
ment, the equilibrium density of the fragments at the constraints are imposed for each partition:
freeze-out stage is also provided by the Thomas-Fermi
calculations. Thus, in contrast with former SMM calcu-
lations, fragments are allowed to be formed at densities A0 = NA,ZA, (2)
below their saturation value. For a fixed freeze-out vol- A,Z
X
ume, this leads to a systematic reduction of the free vol-
ume, which directly affects the entropy of the fragment-
Z = N Z , (3)
ing system, the fragment’s kinetic energies, and, also, 0 A,Z
the system’s pressure. As a consequence, other proper- XA,Z
ties,suchasthecaloriccurveandthemultiplicitiesofthe
and
different fragment species produced, are also affected.
We have organized the remainder of this work as fol-
lows. InSect.IIwediscussthemodificationstotheSMM Eg.s. +E∗ =E (T)+ N −B +ǫ∗
source trans A,Z A,Z A,Z
andpresenttheresultsobtainedwiththismodifiedtreat-
A,Z
mentinSect.III. ConcludingremarksaredrawninSect. X (cid:2) (cid:3)
C Z2 C Z2
IV. In Appendix A we provide a brief description of the + Coul 0 − Coul N . (4)
Thomas-Fermi calculations employed in this work. (1+χ)1/3A1/3 (1+χ)1/3 A,ZA1/3
0 A,Z
X
Intheaboveequations,Eg.s. isthegroundstateenergy
source
II. THEORETICAL FRAMEWORK ofthesource,N denotesthemultiplicityoffragments
A,Z
with mass and atomic numbers A and Z, respectively,
In the SMM [41, 42, 43], it is assumed that a source, BA,Z correspondsto the binding energyofthe fragment,
∗
made up of Z0 protons and A0−Z0 neutrons, is formed and ǫA,Z(T) represents its excitation energy at temper-
at the late stages of the reaction, with total excitation ature T. The Coulomb repulsion among the fragments
energy E∗. This excited source then undergoes a si- is taken into account by the last two terms of the above
multaneous statistical breakup. As the system expands, equation which, together with the self energy contribu-
there is a fast exchange of particles among the different tionincludedinBA,Z,givetheWigner-Seitz[58]approx-
fragments until a freeze-out configuration is reached, at imation discussed in Ref. [41]. The coefficient CCoul is
whichtimeparticleexchangeceasesandthecomposition given in Ref. [59]. As discussed in Ref. [49], the frag-
of the primary fragments is well defined. One then as- ment’s binding energy BA,Z is either taken from experi-
sumes that thermal equilibrium has been reached and mental values [60] or is obtained from a careful extrapo-
calculates the properties of the possible fragmentation lation if empirical information is not available. The spin
modes through the laws of equilibrium statistical me- degeneracy factors, which enter in the calculation of the
chanics. Apossiblescenarioconsistsinconjecturingthat translational energy Etrans, are also taken from experi-
the freeze-out configuration is always attained when the mental data for A≤4. In the case of heavier fragments,
system reaches a fixed pressure, i.e. the nuclear multi- thisfactorisneglected,i.e.,itissettounityforallnuclei.
fragmentation is an isobaric process. In this case, differ- One should notice that the freeze-out temperature
entstatisticalcalculationspredictaplateauinthecaloric varies from one fragmentation mode f = {NA,Z} to an-
curve [42, 50, 51, 52, 53, 54, 55]. The situation is quali- other,sinceitisdeterminedfromtheenergyconservation
tatively different if one assumes that, for a given source, constraint of Eq. (4). Therefore, the average tempera-
thefreeze-outconfigurationisreachedatafixedbreakup ture is calculated, as any other observable O, through
volume V . As found in many different calculations, a the usual statistical averages:
χ
monotonic increase of the temperature with excitation
energy takes place in this case [55, 56, 57]. In what fol-
O exp(S )
lows we demonstrate that this is a consequence of the hOi= f f f , (5)
exp(S )
properties assumedfor the fragments formed, and notof P f f
the fixed volume assumption.
where S denotes the entPropy associated with the mode
In this work we keep the breakup volume fixed for all f
f. This entropy is calculatedthroughthe standardther-
fragmentationmodes,andparametrizeitthroughtheex-
modynamical relation
pression:
dF
V =(1+χ)V , (1) S =− , (6)
χ 0 dT
where V0 denotes the volume of the system at normal where
density and χ≥0 is an input parameter.
In the micro-canonical version of SMM, the sam-
pled fragmentation modes [43] are consistent with mass, F =E−TS (7)
3
is the Helmholtz free energy. In the following, we write Thevaluesoftheparametersintheaboveexpressionare
this quantity as ǫ = 16.0 MeV, T = 18.0 MeV and β = 18.0 MeV
0 c 0
[49]. This expression is used for all nuclei with A ≥
5. Lighter fragments are assumed to behave as point
F = NA,Z −BA,Z +fA∗,Z(T)+fAtr,aZns(T) +FCoul particles, except for the alpha particle, for which one
A,Z retains the bulk contribution to the free energy in order
X (cid:2) (cid:3)
(8) to take its excited states into account.
where the contributions from the fragment’s internal ex- In Ref. [49], the calculation of f∗ has been modified
citation (fA∗,Z) and translational motion (fAtr,aZns) are ex- to include empirical information oAn,Zthe excited states of
plicitly separated. The latter reads: lightnuclei. We labelthis versionofthe modelas ISMM
and it is used throughout this work.
g V A3/2 log(N !)
ftrans =−T log A,Z f − A,Z . (9)
A,Z λ3 N
(cid:20) (cid:18) T (cid:19) A,Z (cid:21) B. The SMM-TF
In the above expression, λ = 2π~2 is the thermal
T mnAT The Thomas-Fermi approximation, briefly outlined in
wavelength, mn is the nucleon qmass, gA,Z is the spin Appendix A, allows one to calculate the internal free
degeneracy factor, and Vf denotes the free volume, i.e., energy of the fragments f∗ from Skyrme effective in-
A,Z
it is the difference between Vχ and the volume occupied teractions. Equations (A19,A26) clearly show that f∗
A,Z
by all the fragments at freeze-out. The quantity FCoul contains, besides those from the nuclear interaction tra-
corresponds to the last two terms in Eq. (4). ditionally used in SMM, contributions associated with
Before we presentthe changesin the model associated the Coulombenergyinadditionto the onesappearingin
withtheThomas-Fermicalculations,webrieflyrecallbe- [Eq. (13)]. The additional Coulomb contribution arises,
low the calculation of Helmholtz free energy F in the in the present case, because the equilibrium density of
SMM. the nucleus at temperature T does not correspond, in
general, to its ground state value. This is illustrated in
Fig.1,whichshowstheratiobetweentheaveragedensity
A. The standard SMM
hρi at a temperature T, and the corresponding ground
state value hρ i, for severalselected light and intermedi-
0
Initsoriginalformulation[41],theSMMassumesthat atemassnuclei. Wedefinethesharpcutoffdensityhρias
the diluted nuclear system undergoes a prompt breakup thatwhichgivesthesamerootmeansquareradiusasthe
and that the resulting pieces of matter collapse to nor- actualnucleardensityobtainedintheThomas-Fermical-
mal nuclear density, although being at temperature T. culation. Oneobservesthathρidecreasesasoneincreases
Therefore, the volume occupied by the fragments corre- thetemperatureofthenucleusandthatitquicklygoesto
sponds to V0, so that zero as T approaches its limiting temperature, since the
nuclear matter tends to move to the external border of
the box due to Coulomb instabilities [47, 48, 61]. In our
V =χV . (10)
f 0 SMM-TF calculations presented below, we only accept
a fragmentation mode at temperature T if it is smaller
Theenergyandentropyassociatedwiththetranslational
thanthe limiting temperature ofallthe fragmentsofthe
motion of the fragment are respectively given by:
partition. If this is not the case, the entire partition is
discardedasnotbeingphysicallypossibleandwesample
3 another one.
ǫtrans =ftrans+Tstrans = T (11)
A,Z A,Z A,Z 2 Thus,afragment’svolumeattemperatureT isdefined
as:
and
strans = − d ftrans VA,Z = hρA0,Zi , (14)
A,Z dT A,Z V0 hρA,Zi
A,Z
3 g V A3/2 log(N !)
A,Z f A,Z
= +log − .(12) where V0 represents the ground state value. Since it
2 λ3 N A,Z
(cid:20) T (cid:21) A,Z is useful to have analytical formulae to use in practical
The internal free energy f∗ has contributions from SMMcalculations,weperformedafitofhρA,Ziusingthe
A,Z
bulk and surface terms: following expression
fA∗,Z =−Tǫ02A+β0A2/3"(cid:18)TTcc22+−TT22(cid:19)5/4−1# . (13) hhρρA0A,,ZZii =1+TaAn,Zni=−01aAi ,ZTi , (15)
X
4
500
10..08 10B 31P * 150Nd E(Mtra-n1s) 3/2 T
E / A = 8 MeV
400 ∆E
0.6
M = 29
0.4 TF
ρ >0 0.2 IfSitMM eV)300
ρ < > / < 010...008 20Ne 40Ca Energy (M200
0.6
100
0.4
0.2
0
0.0
0 2 4 6 8 0 2 4 6 8 10
0 1 2 3 4 5 6
T (MeV) T (MeV)
FIG.1: (Coloronline)Ratiobetweentheaverageequilibrium FIG. 2: (Color online) Kinetic energy (full line) of a partic-
densityofthenucleusattemperatureT andthegroundstate ular partition of the 150Nd nucleus into M = 29 fragments,
∗
value as a function of the temperature. For details, see the for E /A=8 MeV and Vχ/V0 =3, as a function of the tem-
text. perature. Forcomparison,thestandardaveragetranslational
energy, (M −1)3T, is also displayed (dashed line). The dif-
2
ference between the left and the right hand sides of Eq. (4),
where{aA,Z}arethefitparameters. Thisexpressionhas
i ∆E,is also shown (dashed-dottedline).
proventobe accurateenoughfornumericalapplications,
as is illustrated in Fig. 1, which shows a comparison be-
tween Eq. (15) (full lines) and the results obtained with
which, for a given temperature, is also lower than the
the Thomas-Fermi calculation (circles). The fit was car-
corresponding SMM value. As a matter of fact, if the
ried out using n = 6. The dashed lines emphasize the
secondfactordominatesthe firstone,forT >T , where
facItnsttheaatdhρoAf,bZeiin=ghgρiA0ve,ZnibiynEthqe. s(t1a0n),dathrde fSrMeeMv.olume of ǫtAr,aZns(TK) = 0, it can even become negative. KWe also
discardallpartitionsforwhichthereisnosolutionofEq.
a fragmentation mode now reads
(4)satisfyingT <T . ThisaspectisillustratedinFig.2,
K
whichshowsthe totalkinetic energyE as a function
trans
n−1 −1 of T for the 150Nd nucleus, with E∗/A = 8 MeV, for
Vf =(1+χ)V0− NA,ZVA0,Z 1+TaAn,Z aAi ,ZTi . a partition containing M = 29 fragments. The full line
" # representsE ,whereasthedashedlinecorrespondsto
A,Z i=0 trans
X X
(16) the standard SMM formula. The factor, M−1 is due to
For the values of χ usually adopted in statistical calcu- the fact that the center of mass motion is consistently
lations (0≤χ≤5), this expressionshows that, for some removed in all the kinetic formulae, although it is not
partitions, there may be a temperature T for which explicitly stated above.
V
V ≤ 0. Therefore, if Eq. (4) leads to T ≥ T , the The observed drop of the kinetic energy may lead to
f V
partition is discarded as it is not a physically acceptable nontrivialconsequences. Inthecaseofthe150Ndnucleus
solution. and for E∗/A . 7.0 MeV, the fragment multiplicity is
From Eq. (9), the entropy associated with the kinetic relatively low. Therefore, in this lower excitation energy
motion of the fragment (A,Z) becomes rangethe behaviorof the kinetic energydoes notleadto
any qualitative changes arising from the energy conser-
vation constraint. However, for higher excitation ener-
3 g V A3/2 log(N !) T dV
strans = +log A,Z f − A,Z + f . gies,andconsequentlylargerfragmentmultiplicities, the
A,Z 2 λ3 N V dT
(cid:20) T (cid:21) A,Z f kinetic energy is comparable to the total energy of the
(17)
system E . In this case, for a given value of E ,
One should notice that, besides the smaller free volume, total total
there may be two values of T which are acceptable solu-
the last term in the above expression does not appear
tions to Eq. (4). This is also illustrated in Fig. 2, which
in the earlier version of the SMM. Since dV /dT ≤ 0,
f shows the difference ∆E between the left and the right
the expression above gives a smaller contribution to the
hand sides of this expression. Since all the micro-states
total entropy than Eq. (12). Owing to this change in
corresponding to the same total energy E should be
the entropy, the average kinetic energy of the fragment total
included,bothsolutions,inthiscaseassociatedwithtem-
becomes:
peratures T ≈ 5.3 MeV and T ≈ 6.2 MeV must be con-
sidered. They contribute, however, with different statis-
ǫtrans = 3T 1+ 2 T ∂Vf , (18) tical weights, due to the different number of states asso-
A,Z 2 3V ∂T ciated with each of these two solutions.
(cid:18) f (cid:19)
5
1.5 0 0
150Nd -20 10B 31P -50
-100
-40
-150
-60 TF
nsity1.0 V) -80 IfSitMM --225000
e e
ability d *f (MA,Z-500 20Ne 40Ca --015000
b -150
o0.5 -100
Pr -200
-150 -250
-300
-200 -350
0 2 4 6 8 0 2 4 6 8 10
0.0 T (MeV)
4 5 6 7 8
T (MeV)
FIG. 4: (Color online) Internal free energy of selected nuclei
as a function of the temperature. For details, see thetext.
FIG. 3: (Color online) Temperature distribution for the
breakup of the the 150Nd nucleus at E∗/A = 8 MeV and
for Vχ/V0 = 3. The full line corresponds to all events while
those in which there are two temperatures associated with inthe caseofthe lighter nuclei. Particularly,manymore
Etotal are depicted by thedashed line. statesaresuppressedintheISMMthanintheSMM-TF,
whichsuggeststhat the latter shouldpredictlargerfrag-
ment multiplicities than the former. This is due to the
Basedonthisscenario,thedeterminationofthefreeze- empirical information on excited states which are taken
outtemperaturefromisotopicratios[62],where onetac- into account in the ISMM [49]. In the case of heavier
∗
itly assumes that T is univocally determined from E , nuclei,thedifferencesaremoreimportantathighertem-
should be carefully reexamined. To give a quantitative peratures where the ISMM has more contributions from
estimate of these effects, we show, in Fig. 3, the tem- statesinthecontinuumthantheSMM-TF.However,the
perature distribution for the fragmentation of the 150Nd determinationof the free energyat hightemperatures in
∗
nucleus, at E /A = 8 MeV and V = 3V . The full line the ISMM is not as reliable as in the Thomas-Fermi ap-
χ 0
in this picture shows the results when all the partitions proximationinthesensethatthenumericalvaluesofthe
are considered whereas the dashed line represents only parameters ǫ , T , and β , used in actual calculations,
0 c 0
thosewhichleadtotwodifferenttemperatures. Thecases are not obtained from a fundamental theory. They cor-
wheretherearetwotemperaturesolutionscorrespondto respond to average values [41, 42] which, sometimes, are
43%oftheeventsandaccountfor76%ofthetotalstatis- slightly changed by different authors [42, 49, 63].
tical weight. These numbers are drastically changed at From the above parametrization to f∗ , the en-
A,Z
lower excitation energies where, for instance, one finds, tropyandexcitationenergyassociatedwiththefragment
∗
at E /A = 6 MeV, 0.03% and 0.09%, respectively. In (A,Z) read:
spite of the great importance of these solutions at high
excitation energy, the temperature distribution does not
m m
exhibittwocleardominantpeaks,separatedbyagap,as s∗ =2T bA,ZTi+T2 ibA,ZTi−1 (20)
itcouldbe expectedfromFig.2. This isbecause the nu- A,Z i i
i=0 i=1
mericalvalueofthetwosolutionsvaryfromonepartition X X
to the other and the expected signature is thus blurred. and
We have also fitted the internal free energies of the
nuclei through a simple analytical formula:
m m
ǫ∗ =T2 bA,ZTi+T3 ibA,ZTi−1 . (21)
A,Z i i
m i=0 i=1
f∗ =−T2 bA,ZTi , (19) X X
A,Z i
The free energies and equilibrium volumes are calcu-
i=0
X lated using the aboveexpressionsfor alpha particles and
where {bA,Z} are the fit coefficients. The results are de- all nuclei with A≥5.
i
picted in Fig. 4 by the full lines, whereas the Thomas-
Fermi calculations are represented by the circles. As
in the previous case, an excellent agreement is obtained III. RESULTS AND DISCUSSION
with a small number of parameters (m = 5). The free
energies used in the ISMM are also shown in this pic- The SMM-TF modeldescribed in the previous section
ture and are represented by the dashed lines. One sees is now applied to study the breakup of the 150Nd nu-
thattherearenoticeabledifferencesatlowtemperatures, cleus at fixed freeze-out density. We use V /V = 3
χ 0
6
10 60
150Nd 50 SMM-TF 150Nd
40 ISMM
8 Nlp30 Ntotal
20
V) 6 10
Me 0
T ( 4 10
SMM-TF 8
2 IES*M / AM = 3/2 T Nα 6 NIMF
E* / A = T2/a 4
TF 2
0 0
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
E* / A (MeV) E* / A (MeV) E* / A (MeV)
FIG. 5: (Color online) Caloric curve associated with the FIG.6: (Color online) Averagemultiplicityof light particles,
breakupof the150Nd nucleus. TheISMM calculation of Ref. alphas, IMF’s and the total fragment multiplicity, as a func-
[49](triangles)andtheSMM-TFcalculationpresentedinthis tion of theexcitation energy. Fordetails, see thetext.
∗
work(circles) areseentodifferforE /A>8MeV.Forrefer-
encetheexcitationenergyofthecompoundnucleuscalculated
withintheFermigasmodel(fullline),theclassicalgasmodel 20 MeV, this strongly increases the free energy at low
(dashed line) as well as the Thomas-Fermi approach (dotted
temperatures within the ISMM calculation, in contrast
line) have also been presented. For further details, see the
to the Thomas-Fermi model calculations. In the case of
text.
the other multiplicities, the agreement between the two
model calculations is fairly good for excitation energies
∗
up to E /A ≈ 8 MeV. The small discrepancy between
in all calculations below. The caloric curve of the sys- N in the two calculations can be attributed to the
total
tem is displayed in Fig. 5. Besides the SMM-TF (cir- differences in the alpha multiplicities. All multiplicities
cles) and the ISMM (triangles) results, the Thomas- risesmoothlyuptoapproximatelythisexcitationenergy.
Fermi calculations for the 150Nd nucleus are also shown Then, at E∗/A ≈ 8 MeV, in the SMM-TF calculations,
(dotted line), as well as the Fermi gas (full line) and N andN reachamaximumanddecreasefromthere
α IMF
the Maxwell-Boltzmann (dashed line) expressions. For on. ThisbehaviorisnotobservedinthecaseoftheISMM
E∗/A . 8.0 MeV, both SMM calculations agree fairly because it takes place beyond the energy range consid-
wellonthepredictionofthebreakuptemperatures. How- ered in the figure. Another feature also observed in this
ever, a kink in the caloric curve is observed at this pictureisthesuddenchangeintheslopeoftheN and
total
point, in the case of the SMM-TF, indicating that the N multiplicities calculated using the SMM-TF model,
lp
heat capacity of the system is negative within a small whichalsotakesplaceattheexcitationenergymentioned
excitation energy range around this value. Negative above, and which is not seen in the ISMM results.
heatcapacitieshavebeenpredictedbymanycalculations Althoughthe Helmholtz free energiesof the fragments
and have been strongly debated in the recent literature are somewhat different in both calculations, the differ-
[42, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68]. However, this ences are not large enough to quantitatively explain this
feature is normally observed at the onset of the multi- peculiar behavior, as illustrated in Fig. 4. Therefore,
fragmentemission,i.e. atthebeginningofthe liquid-gas the differences in the multiplicities calculated within the
phase transition [42, 53], whereas it appears much later ISMMandSMM-TFmodelsmustbeassociatedwiththe
in the present calculation. behavior of the kinetic terms, due to changes in the free
In order to understand the qualitative differences be- volume in the SMM-TF calculations.
tween the two SMM approaches, we show, in Fig. 6, To examine this aspect more closely, we show, in Fig.
the multiplicity of light particles Nlp (all particles with 7, the energy dependence of hVfi. It confirms the expec-
∗
A≤4, except for alpha particles), the alpha particle Nα tation that hVfi should decrease as E increases, ow-
andthe IntermediateMassFragment(IMF, 3≤Z ≤15) ing to the expansion of the fragments’ volumes at fi-
N multiplicities, as well as the total fragment mul- nite temperature. However, it reaches a minimum at
IMF
∗
tiplicity N as a function of the excitation energy. E /A ≈ 8.0 MeV and rises from this point on. The
total
It is important to notice that neutrons are included in logarithmic volume term of the entropy [Eq. (17)] dis-
N and N . One sees that there is a clear disagree- favors partitions with small free volumes. Furthermore,
lp total
ment between the two SMM calculations in the predic- the last term in Eq. (17) also gets larger as T increases
tion of the alpha particle multiplicity. This is due to the since,besides being explicitly proportionalto T,the fac-
construction of the internal free energies in the ISMM tor | dVf | grows faster at high temperatures, as it can
dT
[49],whichconsidersempiricallowenergydiscretestates. be inferred from the behavior of the densities shown in
Sincethefirstexcitedstateofthealphaparticleisaround Fig. 1. Therefore,the system favorsthe emissionof very
7
light particles, Nlp, which cannot become excited in our 101 E* = 5 MeV E* = 7 MeV
treatment,inordertominimizethereductionofV . Nev-
f 10-1
ertheless, this preference is closely related to the energy
conservationconstraintgivenby Eq.(4). Itis only when 10-3
theexcitationenergybecomessufficientlyhighthatthere
is enough energy for the system to produce a significant ds10-5 SISMMMM-TF 150Nd
nthuemebnetrroopfyvpereyrnliugchlteopnarptriceldeisc.teTdhbeyinthseettiwnoFSiMg.M7 sthreoawts- Yiel 100 E* = 6 MeV E* = 8 MeV
ments. It reveals that, while in the ISMM case it rises
10-2
steadily,theentropysaturates,andevendecreasesinthe
SMM-TF model for 8.0 . E∗/A . 11.0 MeV. The large 10-4
emission of particles which have no internal degrees of
freedom prevents the entropy from falling off from this 10-60 10 20 30 40 50 0 10 20 30 40 50 60
point on, since they lead to larger dVf (smaller absolute Atomic number
dT
values) by increasing V , as they do not expand. One
f FIG. 8: (Color online) Charge distribution in the breakup of
shouldnoticethatthereductionofthecomplexfragment the150Ndnucleus at four different excitation energies.
multiplicities does not mean that the limiting tempera-
ture ofthe fragments in the different partitions has been
reached. In fact, the breakup temperatures obtained in sible fragmentation modes. In particular, the SMM-TF
thepresentcalculationsaremuchlowerthanthelimiting model systematically gives much lighter fragments than
temperatures of most nuclei, except for the very asym- the ISMM, for the reasons just discussed.
metric ones, as may be seen in the examples given in Even though the fragments are not directly affected
Fig. 1 andin Refs. [47, 48]. This effect onthe fragments
by their limiting temperatures at the excitation energies
produced should appear at much higher excitation ener- weconsider,thereductionoftheentropyassociatedwith
gies, as those fragments have excitation energies much
thevolumeaffectsthefragmentspeciesindifferentways.
smaller than the original nucleus, since an appreciable Indeed, since the proton rich nuclei tend to be more un-
amount of energy is used in the breakup of the system.
stable, they are hindered due to these dilatation effects
Therefore, the back bending of the caloric curve and the more strongly than the other isotopes. Owing to their
small plateau observed in Fig. 5 are strongly ruled by
largervolumesat a giventemperature T,partitions con-
the changes in the free volume. As a consequence of this tainingprotonrichfragmentshavesmallerentropiesthan
fact, the phase transitionathighexcitationenergytakes
the others. Therefore,oneshouldexpectto observeare-
place at approximately constant entropy.
ductionintheyieldsofthesefragments. Thisqualitative
This observation is also corroborated by the charge reasoningis confirmedby the results presentedin Fig. 9,
distributions shownin Fig. 8 for four different excitation which displays the isotopic distribution of some selected
energies: E∗/A = 5, 6, 7 and 8 MeV. It shows that the light fragments, produced at E∗/A=6.0 MeV. One sees
multiplicityofheavyfragmentsisstronglyreducedinthe that, even though both SMM models make similar pre-
SMM-TFcalculationsasthe excitationenergyincreases, dictions for many observables at this excitation energy,
although they are not completely ruled out of the pos- theroleplayedbythefreevolumeeffectsjustdiscussedis
non-negligible. Since the limiting temperatures, as well
as the equilibrium density at temperature T, is sensitive
to the effective interaction used [47, 48], these findings
0.8
SMM-TF 150Nd suggestthatcarefulcomparisonswith experimentaldata
ISMM may provide valuable information on the EOS.
0.6
χ IV. CONCLUDING REMARKS
V
> / 0.4 2.5
V f 2.0 SISMMMM-TF We have modified the SMM to incorporate the
< A1.5
S / 1.0 Helmholtz free energies and equilibrium densities of nu-
0.2 0.5 clei at finite temperature from the results obtained with
0.0 theThomas-FermiapproximationusingSkyrmeeffective
0 2 4 6 8 10
E* / A (MeV) interactions. Owing to the reduction of the fragments’
0.0 translationalenergyatfinitetemperature,themodelpre-
0 2 4 6 8 10
E* / A (MeV) dicts the existence of two temperatures associated with
the sametotalenergy. This featureis directly associated
FIG. 7: (Color online) Average free volume and entropy with the reduction of the free volume due to the expan-
per nucleon as a function of the excitation energy calculated sion of the fragments’ volumes. If this statistical treat-
within theISMM and SMM-TF models. ment proves to be more appropriate to describe the nu-
8
100 was supported in part by the National Science Founda-
10-1 Li C tion under Grant Nos. PHY-0606007 and INT-0228058.
10-2 AWS is supported by Joint Institute for Nuclear Astro-
-3
10 physics at MSU under NSF-PFC grant PHY 02-16783.
s
d
el
yi10-1 Be N
ve 10-2 APPENDIX A: THE FINITE TEMPERATURE
elati10-3 THOMAS-FERMI APPROXIMATION
R
B O
-1
10 The Thomas-Fermi approximation to nuclear systems
-2
10 is thoroughly discussed in Refs. [44, 45, 46, 47]. Thus,
-3 SMM-TF
10 ISMM we review its essential features below in order to give a
-4
10 -4 -2 0 2 4 6 8 -2 0 2 4 6 8 10 full account of all calculations presented in this work.
N - Z
The equilibrium configuration of a nucleus at temper-
atureT isfoundbyminimizingthethermodynamicalpo-
FIG. 9: (Color online) Isotopic distribution of selected nu-
clearspeciesproducedinthebreakupofthe150Ndnucleusat tentialΩwithrespecttothenumberdensityρα (α=p,n
∗ for protons or neutrons):
E /A=6.0 MeV.
Ω=F[ρ]− d3~r µ ρ (A1)
clear multifragmentation process than its standard ver- α α
sion, the determination of the isotopic temperatures, at Xα Z
highexcitationenergies,shouldbe carefullyreexamined,
where the Helmholtz free energy is given by
sinceonetacitlyassumesaunivocalrelationshipbetween
the temperature and the excitationenergyin the deriva-
tion that leads to the corresponding formulae [62].
F[ρ]= H +H −T S d3~r . (A2)
The thermal dilatation of the fragments’ volumes also nucl Coul α
" #
hasimportantconsequencesonthefragmentationmodes. Z Xα
Forexcitationenergieslargerthanapproximately8MeV In the above expression, S denotes the entropy density
α
per nucleon, it favors enhanced emission of particles associated with the species α, µ is the corresponding
α
whichhavenointernaldegreesoffreedom(verylightnu- chemicalpotential,H isthe nuclearenergydensityof
nucl
clei, protons and neutrons), leading to the onset of a gas the system, and the Coulomb term reads:
transition at excitation energies around this value. The
existenceofasmallkink inthe caloriccurve,aswellasa
plateau, for a system at constant volume is qualitatively H = e2ρ (~r) ρp(r~′) d3r~′
different from the results obtained in previous SMM cal- Coul 2 p |~r−r~′ |
Z
culations where these features were observed only at (or
3 3 1/3
at least at nearly) constant pressure [55]. − e2 ρ4/3(~r). (A3)
Since many-particle multiplicities, such as those asso- 4 π p
(cid:18) (cid:19)
ciated with the IMF’s and the light particles, are very
Thesecondtermabovecorrespondstoanapproximation
different in both statistical treatments for excitation en-
tothe exchangecontributionto the Coulombenergy[69,
ergies larger than 8 MeV per nucleon, we believe that
70].
careful comparisons with experimental data may help to
The expression for H given in Ref. [44] may be
establish which treatment is better suited for describing nucl
rewritten as
the multifragment emission. Furthermore, since the iso-
topic distribution turns out to be sensitive to the treat-
mentevenatlowerexcitationenergies,thissuggeststhat
H =H +H +H +H (A4)
nucl 0 τ grad J
one may obtain important information on the EOS by
usingdifferentSkyrmeeffectiveinteractionsintheSMM- where
TF calculations. Particularly,this modified SMM model
is appropriate to investigate the density dependence of
t x 1
the symmetry energy recently discussed [35, 36, 37, 40]. H = 0 (1+ 0)ρ2−(x + )(ρ2 +ρ2) (A5)
0 2 2 0 2 n p
(cid:20) (cid:21)
t x 1
+ 3ρσ (1+ 3)ρ2−(x + )(ρ2 +ρ2) ,
Acknowledgments 12 2 3 2 n p
(cid:20) (cid:21)
We would like to acknowledge CNPq, FAPERJ,
~2 ~2
and the PRONEX program under contract No E-
H = τ + τ , (A6)
26/171.528/2006,forpartialfinancialsupport. Thiswork τ 2m∗ p 2m∗ n
p n
9
where
1 4 2
H = 9t −5t (1+ x ) ▽~ρ (A7)
grad 1 2 2
64 5 1
1 (cid:20) (cid:21)(cid:16) (cid:17) 2 µ(αk) = N d3~r Bα(k)(~r)ρ(αk)(~r), (A15)
− [3t (1+2x )+t (1+2x )] ▽~ρ −▽~ρ , α Z
1 1 2 2 n p
64
(cid:16) (cid:17)
N =Z, N =A−Z, and
p n
1
H = W J~·▽~ρ+J~ ·▽~ρ +J~ ·▽~ρ , (A8)
J 0 n n p p
2
δF
h i B(k) = . (A16)
the total density is denoted by ρ = ρn + ρp and J~ = α δρ(αk)
J~ +J~ is the spin-orbitdensity. The kinetic factorτ is
p n α
given by The parameter λ is chosen to be small enough in order
toensurethatthefirstorderapproximationgivenbyEq.
(A14) remains valid.
∗ 5/2
1 2m In our numerical implementation, we have assumed
τ = α T5/2I (y ) (A9)
α 2π2 ~2 3/2 α spherical symmetry, and discretized the space using a
(cid:18) (cid:19)
mesh spacing ∆R = 0.1 fm, which suffices for our pur-
where
poses. As suggested in Refs. [46, 47], the second term of
Eq.(A7)isneglectedsinceitissmallandmayleadtonu-
~2 ∂ merical instabilities. Similarly to the treatment adopted
2m∗ = ∂τ Hnucl in Ref. [46], the gradient density terms are calculated at
α α the mesh point r =(i+1/2)∆R, using [72]
~2 1 i+1/2
= + [t (1−x )+3t (1+x )]ρ
1 1 2 2 α
2m 8
+ 14 t1(1+ x21)+t2(1+ x22) ρα′ (A10) ∂∂rρ(ri+1/2)= ρ(r2i)(∆−Rρ/(r2i)−1) +O[(∆R)2], (A17)
h i
and ρα′ =ρp(ρn) if α=n(p). The Fermi-Dirac integral which turned out to be numerically stable.
Due to the important contributions associated with
∞ xn/2 unbound states at high temperatures, the above treat-
I (y)= dx (A11)
n/2 1+exp(x−y) ment is not accurate for T & 4 MeV, as pointed out
Z0 by Bonche, Levit, and Vautherin [61]. Therefore, these
is efficiently calculated using the formulae given in Ref. authors have proposed a method to extend the Hartree-
[71], where one also finds approximations to the inverse Fock calculations to higher temperatures. As they have
functiony(I ). Thelatterisdeterminedfromthenum- noticed,therearetwosolutionsoftheHartree-Fockequa-
n/2
ber density tions for a given chemical potential. One of them cor-
responds to a nucleus in equilibrium with its evaporated
particleswhereastheotherisassociatedwiththenucleon
1 2m∗ 3/2
ρ = α T3/2I (y ). (A12) gas. Thus, in their formalism, the properties of the hot
α 2π2 ~2 1/2 α nucleus is obtained by subtracting the thermodynamical
(cid:18) (cid:19)
potential associated with an introduced nucleon gas Ω
The entropy density S can then be easily calculated G
α from that corresponding to the nucleus in equilibrium
with its evaporated gas Ω . Except for the Coulomb
NG
5 ~2 τ energy, there is no interaction between the gas and the
α
S = −ρ y . (A13)
α 32m∗ T α α nucleus-gas system.
α
This approach has been successfully applied by these
The parameter set {xi,ti,σ,W0}, i = 0,1,2,3, for the authors [48, 61] and has been adapted to the finite tem-
Skyrme interaction used in this work, SKM, is listed in perature Thomas-Fermi approximation by Suraud [47].
Ref. [44]. Since we stay in the zero-th order approxima- More precisely, the thermodynamical potential associ-
tion in ~, J~ = 0 and then H does not contribute to ated with the nucleus is given by
α J
H [44].
nucl
Following Suraud and Vautherin [46, 47], the equilib-
rium configuration is found by iterating the densities at Ω =Ω −Ω +E . (A18)
N NG G Coul
the k-th step according to
Oneshouldnoticethat,byconstruction,Ω andΩ do
NG G
notcontainanyCoulombcontribution. Morespecifically,
ρα(k+1) =ρ(αk) 1−λ Bα(k)−µ(αk) (A14) one defines the subtracted free energy
h (cid:16) (cid:17)i
10
since ρNG and ρG are constrained by
Fsub = HNG −HG −T SNG−SG d3~r
nucl nucl α α
" #
Z α
X(cid:0) (cid:1)
+ HCsuobul d3~r (A19) Nα = d3~r ρα(k,NG)(~r)−ρ(αk,G)(~r) . (A24)
Z Z h i
wherethesubtractedCoulombenergydensity,inthelast
term of this expression, reads
One then starts with a reasonable guess for ρNG and
α
ρG, which can be a Woods-Saxon density for the for-
α
e2 ρ (r~′) mer and a small constant value for the latter (subject
Hsub ≡ ρ (~r) d3r~′ p (A20)
Coul 2 p |~r−r~′ | to the condition ρα > 0), obeying the constraint given
Z by the above expression, and apply the iteration scheme
− 3e2 3 1/3 ρNG 4/3− ρG 4/3 , just described. Ideally, convergence is reached when
4 π p p B(k,γ)(~r)−µ(k) vanishes, so that ρ(k,γ) becomes station-
(cid:18) (cid:19) h(cid:0) (cid:1) (cid:0) (cid:1) i arαy. In practiαce, one can monitor tαhe quantity [47]
and the subtracted density ρ :
p
ρp(~r)=ρNpG(~r)−ρGp(~r) (A21) ∆E2 = d3~r B(k,NG)(~r)−µ(k) 2ρ(k,NG)
α α α α
is the quantity that enters in the direct part of the Z n (cid:16) (cid:17)
2
Coulomb energy. + B(k,G)(~r)−µ(k) ρ(k,G) (A25)
α α α
The iteration scheme given by Eq. (A14) remains un-
(cid:16) (cid:17) o
changed if one rewrites B(k) as
α
and stop the iteration when the established tolerance is
δFsub reached. The Helmholtz free energy of the nucleus can
B(k,γ) =± , (A22)
α (k,γ) then be easily calculated throughEq. (A19), so that the
δρ
α
internal free energy of the nucleus is
wherethe super-index(k,γ)denotesthequantityassoci-
ated with the gas (γ = G) or the nucleus-gas (γ = NG)
at the k-th stage of the iteration. The positive sign is
associated with the NG solution whereas the negative f∗ (T)=Fsub(T)−Fsub(T =0). (A26)
sign is used in the other case. The proton and neutron A,Z
chemicalpotentials are givenby an expressionsimilar to
Eq. (A15)
We haveusedthe approximationjust describedinthis
∗
Appendixtocalculatef forallthefragmentsentering
1 A,Z
µ(k) = d3~r B(k,NG)(~r)ρ(k,NG)(~r) in the SMM, with A≥5 (and alpha particles) from T =
α N α α
α Z n 0MeVuptothelimitingtemperature[47,48,61]insteps
− B(k,G)(~r)ρ(k,G)(~r) , (A23) of 0.1 MeV.
α α
o
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