Table Of ContentSymmetry energy from nuclear multifragmentation
Swagata Mallik and Gargi Chaudhuri
Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India
(Dated: January 24, 2013)
The ratio ofsymmetry energy coefficient to temperatureCsym/T is extracted from differentpre-
scriptionsusingtheisotopicaswellastheisobaricyielddistributionsobtainedindifferentprojectile
fragmentationreactions. Itisfoundthatthevaluesextractedfromourtheoreticalcalculationagree
with those extracted from the experimental data but they differ very much from the input value
of the symmetry energy used. The best possible way to deduce the value of the symmetry energy
coefficient is to use the fragment yield at the breakup stage of the reaction and it is better to use
the grand canonical model for the fragmentation analysis. This is because the formulas that are
used for the deduction of the symmetry energy coefficient are all derived in the framework of the
3 grand canonical ensemble which is valid only at the break-up(equilibrium) condition. The yield of
1 ”cold” fragments either from the theoretical models or from experiments when used for extraction
0 of thesymmetry energy coefficient using these prescriptions might lead to the wrong conclusion.
2
PACSnumbers: 25.70Mn,25.70Pq
n
a
J
Introduction:- Isospin-dependent phenomena in nu- mary (breakup) stage and hence extraction of Csym val-
3
clear physics has been an active area of research [1, 2] uesfromexperimentalyieldsthatcorrespondtothe’cold’
2
in recent years with the aim of enriching our knowl- secondaryfragmentsmightleadtothewrongconclusion.
] edge about the symmetry term of the nuclear equation This is primarily because the formulas that are used to
h of state. The study of this quantity at different regimes extract the values of C from the experimental yields
sym
t
- ofdensityandtemperatureisahottopic [3–5]inthe nu- areallderivedusingtheprescriptionofthegrandcanon-
cl clear physics community. This term plays an important ical ensemble, which is valid only at the break-up (pri-
u role in areas of astrophysical interest such as the study marystage)ofthemultifragmentingsystem. Theyieldof
n of supernova explosions and the properties of neutron thefragmentsatthebreakupstageofthereaction(before
[ stars [6]. This also has significant influence in deciding de-excitationbysecondarydecay)shouldonlybeusedto
1 the structure of neutron-rich and neutron-deficient nu- extract information about Csym using the existing pre-
v clei. The study of nuclear multifragmentation in heavy- scriptions. This is the main message of this paper and
5 ion reactions is an important tool for extracting infor- ithasbeenestablishedusingprojectilefragmentationre-
1 mation about the symmetry energy term, and this has actions. It has also been pointed out that theoretical
6
created much interest in the nuclear physics community simulations done by the grand canonical model are only
5
in recent years [7–17]. expected to reproduce the value of C used as input
. sym
1 In nuclear multifragmentation reactions, the neutron- to the model. Calculations by canonical or any other
0 protoncompositionofthebreak-upfragmentsisdictated ensemble that are otherwise better suited for describing
3
bytheasymmetrytermoftheequationofstateandhence intermediate energy nuclear reactions might lead to val-
1
the study of the multifragmentation process allows one ues widely different from the input value of C used.
: sym
v to obtain information about the symmetry term. Sta- Resultsfromcanonical[18]andgrandcanonicalensemble
Xi tistical models [18–20] that are simple and economic are [24] differ in generalfor finite nuclei [25]. This work also
verysuccessfulinpredictingtheoutcomeofnuclearmul- presentsacomparativeanalysisofthepredictivepowerof
r
a tifragmentation reactions. Different formulas [14, 21–23] the different existing formulas both atthe primarystage
have been proposed in the literature that connect the and also after evaporation and their relative agreement
measurablefragmentisotopicandisobaricobservablesof with experimental data.
multifragmentation reactions to the symmetry energy of
Model description:- For the study of the nuclear sym-
excited nuclei and these have been applied to the anal-
metry energy coefficient from projectile fragmentation
ysis of heavy-ion collision data. These formulas have all
reactions we use a model [26–28] that consists of three
beendeducedusingthegrandcanonicalversionofthenu-
stages: (i) abrasion, (ii) multifragmentation, and (iii)
clearmultifragmentationmodelassuminganequilibrium
evaporation. In heavy-ion collisions, if the beam energy
scenario for the break-upstage of the disintegrating sys-
is high enough then at a particular impact parameter
tem. They have been used to analyze experimental data
three different regions are assumed to be formed: (i) the
from different projectile fragmentationas well as central
participant,(ii)theprojectilelikefragment(PLF)orpro-
collision reactions and the extracted values for the sym-
jectilespectator,and(iii)thetargetlikefragment(TLF)
metry energy coefficient C ranges between 15 and 30
sym or target spectator. Here we are interested in the frag-
MeV [7–11]. It has been observed from model calcula-
mentation of the PLF.
tions that the fragment distributions of the secondary
In the abrasion part (first stage), we have calculated
fragments after evaporation vary from those at the pri-
the PLFvolumeV (b)atdifferentimpactparameters(b)
s
2
using straightline geometry. By knowing V (b), we can motion, which is the difference between the freeze-out
s
determine the average number of neutrons and protons volume and normal nuclear volume. For projectile frag-
present in the projectile spectator [26] by assuming the mentation calculation the freeze-out volume is assumed
same neutron-to-protonratio of projectile spectator and to be three times the normal nuclear volume.
projectile. The impact parameter dependence of tem- We now list the properties of the compos-
perature is considered as T(b) = 7.5 −4.5[(As(b)/A0)] ites used in this work. The proton and the
whereAs(b)andA0 aremassnumbersofprojectilespec- neutron are fundamental building blocks, thus
tator and original projectile respectively. This tempera- z1,0(int) = z0,1(int) = 2, where 2 takes care of the
ture profile successfully reproduces [27, 28] the differen- spin degeneracy. For deuteron, triton, 3He and 4He we
tial charge distribution, total charge and mass distribu- use z (int) = (2s + 1)exp[−βE (gr)] where
N,Z N,Z N,Z
tion, and isotopic distribution obtained in different ex- β = 1/T,E (gr) is the ground-state energy of the
N,Z
periments [29–32]. composite and (2s + 1) is the experimental spin
N,Z
In the statistical models [18–20] of nuclear multifrag- degeneracy of the ground state. Excited states for these
mentation, it is assumedthat the hot nuclear systemex- very-low-massnuclei are not included. For mass number
pands and then, due to density fluctuations, it breaks A ≥ 5 we use the liquid-drop formula. For nuclei in
up (multifragments) into composites of different sizes. isolation, this reads
Thermodynamic equilibrium is assumed to be reached
at this expanded freeze-out configuration, where the in-
t(eerxaccetpitonfobretthweeelnontgh-eracnogmepCoosuitleosmbbecfoormcee).unTimhepodritsainnt- zN,Z(int) = expT1[W0A−σ(T)A2/3−a∗cAZ1/23
tegration of hot nuclear system into various composites (N −Z)2 T2A
−C + ] (4)
canbecalculatedusingdifferentstatisticalensembles. In sym
A ǫ0
our model, in general canonical ensemble is used as it is
more appropriate than the grand canonicalfor finite nu- Theexpressionincludes thevolumeenergy[W0 =15.8
clei calculations. Results from these ensembles differ in MeV],thetemperaturedependentsurfaceenergy[σ(T)=
generalthough they are found to convergeunder certain σ0{(Tc2 −T2)/(Tc2 +T2)}5/4 with σ0 = 18.0 MeV and
conditions [25] for finite nuclei. The multifragmentation T = 18.0 MeV], the Coulomb energy [a∗ = 0.31a with
c c c
calculation of each PLF (second stage) formed at differ- a = 0.72 MeV and Wigner-Seitz correction factor 0.31
c
ent impact parameters is done separately by using the [19]] and the symmetry energy (C =23.5 MeV). The
sym
cbaanseodnicoanletqhuerilmibordiuymnasmtaictaisltimcaoldmele(cChaTnMic)s[a1n8d] winhviochlveiss term Tǫ20A (ǫ0 =16.0 MeV) represents contribution from
excitedstatessincethecompositesareatanon-zerotem-
the calculation of partition functions. In this model the
perature. Different equations have been deduced using
average number of composites with N neutrons and Z
the grand canonical model which relates this symmetry
protons formed from a PLF N ,Z is given by
s s energy coefficient C to temperature as well as the
sym
hnN,Zic = ωN,ZQNs−N,Zs−Z (1) sthouernceexatnsdecftriaognm. ent compositions. This is dealt with in
Q
Ns,Zs The excited fragments produced after multifragmen-
where Q is the total partition function, which can tation decay to their stable ground states. They can
Ns,Zs
be easily calculated using the recursion relations [33]. γ-decay to shed energy but may also decay by light
Thegrandcanonicalensemblecanalsobeusedinstead particle emission to lower mass nuclei. We include emis-
of the canonical, and the average number of composites sions of n,p,d,t,3He and 4He. Particle-decay widths are
in this ensemble is given by [24], obtained using Weisskopf’s evaporation theory. Fission
is also included as a de-excitation channel though for
hn i = eβµnN+βµpZω (2) the nuclei of A<100 its role will be quite insignificant.
N,Z gc N,Z
The details of the evaporation stage (last stage) are
Theneutronandprotonchemicalpotentialsµn andµp described in Ref. [34].
are determined by the baryon and charge conservation
conditions which amounts to solvingthe equations Ns = Symmetryenergy from different formulas:-Thereexist
NeβµnN+βµpZωN,Z and Zs = ZeβµnN+βµpZωN,Z. different formulas in the literature from which the sym-
PIn both the models, the partitPion function ωN,Z of a metry energy coefficient has been extracted. Here we
compositehavingN neutronsandZ protonsisaproduct
make a short review of these existing formulas.
of two parts and is given by
The formula that connects the symmetry energy co-
efficient with the isospin asymmetry of the source was
V
ωN,Z = h3(2πmT)3/2A3/2×zN,Z(int) (3) first proposed in the framework of the Expanding evap-
orating source (EES) model [21]. This is based on the
Thefirstpartisduetothetranslationalmotionandthe grandcanonicalensemble,andassumingthermodynamic
secondpartz (int)istheintrinsicpartitionfunctionof equilibrium at the time of fragmentation of two systems
N,Z
thecomposite. V isthevolumeavailablefortranslational having same charge Z0 but different masses A01, A02
3
(A01 < A02) at the same temperature T, Csym is given Results:- The isotopic and isobaric yield distributions
by are related to the nuclear symmetry energy coefficient
through different prescriptions, which have already been
α(Z)T
describedbriefly in the previous section. These relations
C (Z)= . (5)
sym 2 2 are derived using the yields of the fragments obtained
4 Z0 − Z0
(cid:20)(cid:16)A01(cid:17) (cid:16)A02(cid:17) (cid:21) in the grand canonical ensemble. We first compare the
symmetry energy coefficient obtained using the different
where α(Z) is the isoscaling parameter [21, 35, 36] of prescriptionsinboththe canonicalandthegrandcanon-
fragments having charge Z which can be measured from ical ensemble. In this calculation we have used the sta-
theratiooftheisotopicyields[37]. Thesuffix0indicates tistical models (canonical or grand canonical) in order
the fragmenting source. This formula, which is referred to obtain the yields of the composites formed after mul-
toastheisoscaling(source)formula,hasbeenextensively tifragmentation (second stage) of the hot single source.
used on experimental data [7–11]. The sourcesused for the first two methods [Fig 1(a)and
The isoscaling (fragment) formula also derived [22] 1(b)] are A01 =55 and A02 =60 and Z0 =25, while for
from the grand canonical ensemble assumes equilibrium the later two methods [Fig 1(c) and 1(d)] where a sin-
at the breakupstage of two fragmenting sources in iden- gle source is required the source A0 =55 and Z0 =25 is
tical thermodynamical states that differ in their isospin used. Thetemperatureusedforthecalculationis5MeV.
content (different masses A01, A02 but same charge Z0). These are the results for the primary fragments and no
This is given by secondary decay is used for the de-excitation of the ex-
cited primary fragments. There are some differences in
α(Z)T
C (Z)= (6) the results from the canonical and the grand canonical
sym 2 2
4 Z − Z ensemble,andthesedifferencesarealmostsameinallthe
(cid:20)(cid:16)<A1>(cid:17) (cid:16)<A2>(cid:17) (cid:21) methods used. In the canonical ensemble, the extracted
symmetry energy coefficient changes with the fragment
Inthisexpressionitisassumedthattheisotopicdistri-
mass or charge and this variation is more or less same
butions are essentially Gaussian and that the free ener-
forallthefourmethods usedforextractionofthis coeffi-
giescontainonlybulkterms. Here,α(Z)istheisoscaling
cient. Thevalueofextractedsymmetryenergycoefficient
parameter of fragments having Z protons, <A > is the
i variesbetween25and50MeVwhiletheinputsymmetry
averagemassnumberofafragmentofchargeZ produced
by source i=1(less neutron rich), 2 (more neutron rich),
and T is the temperature of the decaying systems. This
60 60
formula, which is similar in structure to the isoscaling
(source) formula, connects the symmetry energy coeffi-
cient to the isotopic composition of fragments instead of 40 40
m m
theAinsoatloteprincactoemepxposrietsisoinono(fflsuocutrucaestioans ifnorEmqu.la()5)h.asbeen C sy20 20C sy
derived in Ref. [23, 38] connecting the symmetry energy
of a cluster of size A to the width of its isotopic distri- (a) (b)
0 0
bution. Indeed, aGaussianapproximationonthe grand- 5 10 5 10 15
canonical expression for cluster yields gives Z Z
60 60
AT
C (A)= , (7)
sym 2σ2(A)
I 40 40
m m
where σI2(A) indicates the width of the isobaricdistribu- C sy C sy
tion of a cluster of size A and I =A−2Z. 20 20
According to the isobaric yield ratio method [14–
(c) (d)
17,39],obtainedfromthe grandcanonicalexpressionfor
0 0
cluster yields, Csym can be expressed as 10 20 10 20 30
A A
AT
C (A)=− [lnR(3,1,A)−lnR(1,−1,A)] (8)
sym 8 FIG.1: (Coloronline)Symmetryenergycoefficientobtained
from canonical (red solid lines) and grand canonical model
where, R(I + 2,I,A) is the yield ratio between two (blackdottedlines)atT =5MeVandinputCsym=23.5MeV.
isobars differing by 2 units in isospin content as (a) and (b) represents the variation of extracted Csym with
R(I + 2,I,A) = Y(I + 2,A)/Y(I,A). This equa- proton number Z by using Eq. (5) and Eq. (6) respectively
tion assumes the Coulomb terms in lnR(3,1,A) and from two sources having same Z0 = 25 but different A0 =
lnR(1,−1,A) are same and in both ratios mixing 60 and 55 where as (c) and (d) indicates the variation of
entropy terms are neglected [39]. extractedCsymwithmassnumberAcalculatedbyusingEqs.
(7) and (8) for Z0 =25 and A0 =55.
4
energy coefficient used is fixed at 23.5 MeV. formula [Eq. (6)], C /T depends on the isoscaling pa-
sym
In contrast, in the results from the grand canonical rameter and on the Z/<A> values of the fragments. In
ensemble, C is independent of the fragment mass or the fluctuation formula[Eq. (7)], the coefficient depends
sym
charge. The value of C extracted from the grand onthewidthoftheisobaricdistributionofthefragments
sym
canonical ensemble lies between 23 and 24 MeV and andtheir mass while in the isobaricyield ratio[Eq. (8)],
hence matches almost exactly with the input value used it depends on the ratio of the fragment yields and their
for the calculation. This difference in results between mass. In the break-up stage of the multifragmentation
two ensembles is mainly because these formulas are all reaction,theyieldsoftheprimaryfragmentscanbeused
derivedusing the prescriptionof the grandcanonicalen- to deduce the values of C /T from all the four for-
sym
semble and hence when this ensemble is used to extract mulas. The projectile fragmentation reactions involved
the value, the results agree with the input value. The are 58Ni on 9Be and 64Ni on 9Be at 140 MeV/n [30, 31].
results from the canonicalensemble deviate from that of FromEqs. (5)and(6),the variationofthereducedsym-
the grandcanonicalensemble in generalfor finite nuclei. metryenergycoefficientC /T withfragmentchargeZ
sym
Hence extraction of C leads to values that can differ isshowninFig. 2(a). Inthesamefiguretheresultsfrom
sym
widely from the input value used. Eq. (7) and (8) are shown as functions of the fragment
mass A. The first two methods [Eq. (5) and (6)] depend
8 16 A 24 18 27 A 36 45 ontheisoscalingparameterwhichiscalculatedusingtwo
25 25 projectiles Ni58 and Ni64 having same value of Z. The
fluctuation method [Eq. (7)] and the isobaric yield ratio
20 20
method [Eq. (8)] depend on one source for the calcula-
15 15T tion of the reduced symmetry energy coefficient and the
/TC sym10 10C/ sym fprroomjectthileepNriim58arhyasfrbaegemneunstesda.rIetcislosseeentotheaatchthoethreesruflotsr
allfourformulaandC /T increaseswiththefragment
5 5 sym
mass or charge. Since these values are for the break-up
(a) (b)
stage before the final de-excitation, no comparison has
0 0
4 8 12 8 12 16 20 been made to the experimental data.
Z Z
We have repeated the same calculation [Fig. 2(b)] for
FIG. 2: (Color online) Comparison of Csym/T calculated different projectiles (Xe124 and Xe136). The projectile
by using (i) Eq. (5) (black dotted lines) (ii) Eq. (6) (red fragmentation reactions involved were 124Xe on 208Pb
dashed lines) (iii) Eq. (7) (green solid lines) and (iv) Eq. (8) and 136Xe on 208Pb at 1 GeV/n [32]. The beam energy
(blue dash dotted lines) for primary fragments in projectile is1GeV/nwhichismuchhigherthanthepreviousvalue
fragmentation reactions Nion Be (left panel) and Xe on Pb
of 140 MeV/n [30, 31]. The trend of the results remains
s(eritgshotfpraeancetl)io.nFso5r8(Ni)ioannd9B(iei)a,nCdsy6m4N/Tioins9cBalecu(llaetftedpafrnoeml)atwndo almost the same irrespective of the beam energy. The
124Xeon208Pband136Xeon208Pb(rightpanel)andplotted results from all four formulas are close to each other for
theprimaryfragments(atthebreak-upstage)forawide
againstZ whereasfor(iii)and(iv),Csym/T iscalculatedfor
only neutron less reactions and plotted against A. range of beam energies from 140 MeV/n to 1 GeV/n.
In the next stage we investigated the results for the
Inthenextpart,weshowtheresultsofthecalculations secondary fragments i.e, after the evaporation of the ex-
fromtheprojectilefragmentationmodelfortwodifferent cited primary fragments. In eachfigure,the results from
reactions. Since for finite nuclei, the canonical ensemble theprimaryfragmentsarealsoshownforthesakeofcon-
is physically more acceptable than the grand canonical venience in comparison. In Fig. 3(a) we have shown the
ensemble,inourmodel(asdescribedpreviously)wehave result for the isoscaling (source) formula [Eq. (5)]. Here
usedthe canonicalensemble forthe fragmentationofthe C /T decreases after evaporation because the isoscal-
sym
excitedPLF.Inthis casewe haveusedthe reducedsym- ing parameter α decreases after evaporation at the tem-
metry energy coefficient which is nothing but the ratio perature range used here while the denominator of the
of the symmetry energy coefficient to temperature i.e, right-hand side of Eq. (5), which depends only on the
C /T. Recent studies on this parameter using differ- source sizes remain unchanged. In both results from the
sym
ent methods canbe found in ref. [13–16]. This is mainly modelandfromtheexperimentaldata,thereducedsym-
because it is difficult to estimate temperature both from metry energy coefficient does not depend very much on
experimental yields and from the model results. Hence thefragmentsize. Thenumbersobtainedfromtheexper-
it is better to express the results in the form of C /T imentaldataarelessthanthoseobtainedfromourmodel
sym
instead of using only C since unambiguous extrac- for the projectile fragmentation. In Fig. 3(b) we have
sym
tion of temperature is very difficult. In the isoscaling plotted the results for the isoscaling (fragment) formula
(source)formula[Eq. (5)],symmetryenergyisrelatedto [Eq. (6)]. Inthis case, C /T increasesafter secondary
sym
the isoscalingparameterα and the Z/A value of the two decay and the results more or less agree with those ex-
sources. The other three formula depend solely on the tracted from the experimental data. The isoscaling
properties of the fragments. In the isoscaling (fragment) (fragment)formuladependsontheZ/<A>valuesofthe
5
25 25 isobaric yield ratio method in Antisymmetrized molecu-
20 20 lar dynamics (AMD) model+GEMINI calculations is re-
portedin[14]. InthiscaseasseenfromEq. (8), C /T
C/T sym1105 1105/T sym dreedpuecneddssoynmtmheetrryateionserogfyycioeledffiscoiefndtiffinecrreenatseissoabfaterrss.yemvTahpe-
C
orationbecausethewidthoftheisobaricdistributionde-
5 5
creasesdue tosecondarydecay,whichresultsindecrease
(a) (b)
0 0 of the yield Y(−1,A) and Y(3,A) in Eq. (8) while the
5 10 5 10 15
yield Y(1,A) remains almost unchanged. In the results
Z Z
25 25 from Eq. (6)-(8), Csym/T increases after evaporation,
as compared to the results obtained from the primary
20 20
fragments.
T15 15T We have repeated this calculation for another projec-
C/ sym10 10C/ sym tainledftrhagismiesnsthaotiwonn rineaFctiigo.n4(.XeTohne Pobbs)eravta1tioGnesVa/nnd[3re2-]
5 5 sultsaresimilartothoseofthepreviousreactioninspite
(c) (d) of the vast difference in the projectile energy and widely
0 0
10 15 20 25 10 15 20 25 30 different target-projectile combination.
A A All the four formulas are derived from the yields of
the grand canonical ensemble and they hold good for
FIG. 3: (Color online) Variation of Csym/T with atomic the breakup stage of the reaction. Hence any attempt
numberZ calculatedbyusing(a)Eq. (5)and(b)Eq. (6)for
to deduce the value of the symmetry energy coefficient
58Ni on 9Be and 64Ni on 9Be reactions. Panels (c) and (d)
depictthevariation ofCsym/T with massnumberAfor58Ni from the yields obtained after evaporationmight lead to
on9BereactioncalculatedfromEqs. (7)and(8)respectively. the wrong conclusion. Neither the experimental yields
(which are the values after evaporation from the excited
Experimentaldata(blacksquares)comparedwiththeoretical
results: primary fragments (bluedotted lines) and secondary fragments) nor the yields of the secondary fragments
fragments (red solid lines). from the model should be used to deduce the values of
the symmetry energy coefficient. It might be possible
to deduce the value of C from the break-up stage
sym
of the reaction, i.e., from the hot primary fragments,
fragments[see Eq. (6)]. For the less neutronrichsource,
but since it is difficult to access this stage from an
this quantity does not change very much after evapora-
experimental point of view, no attempts are made to do
tion but for the more neutron-rich nuclei, this quantity
such calculations.
increases after evaporation since <A> decreases as the
peak of the isotopic distribution shifts to the left (lower
valuesofA)afterevaporation. Hencethedenominatorof
the right-hand side of Eq. (6) decreases. The isoscaling 25 25
parameter α in the numerator also decreases after evap-
20 20
oration but the denominator decreases much more and
fhreonmcethCesyFmig/.T3(inbc)r.eTasheissaisftienrcsoenctornadstartyodtheecaryesausltissfsreoemn C/T sym1105 1105/T sym
C
the isoscaling(source)formulawhere the denominatoris
independent of the property of the fragments. In Fig. 5 5
(a) (b)
3(c)we haveplotted the resultsfromthe fluctuationfor- 0 0
mula[Eq. (7)]. Herealsothe trendofthe resultsis same 8 12 16 8 12 16 20
Z Z
as inFig. 3(b). Csym/T increasesafter secondarydecay, 25 25
and this is due to the fact that σ2 which is a measure
of the width of the isobaric distribution, decreases after 20 20
secondarydecayandonecanseefromEq. (7)thatifthis
T15 15T
dwehcarteahsaepsptehnesnasCwsyems/eTefrwoimllFinicgr.e3a(sce).anTdhetheixspiesriemxaencttlayl C/ sym10 10C/ sym
values obtained in this case also are quite close to those
5 5
obtained from the model. In Fig. 3(d) we have plotted (c) (d)
0 0
the results from the isobaric yield ratio method. In this
20 24 28 32 20 24 28 32 36
case also the result is similar to that of the previous two A A
cases and the reduced symmetry energy coefficient in-
creases after evaporation. The numbers extracted from FIG. 4: (Color online) Same as Fig.-3 except that here
the experimental data are close to those from the theo- theprojectilefragmentation reactionsinvolvedare124Xeand
retical calculation. A similar trend of results from the 136Xe on 208Pb instead of 58Niand 64Nion 9Be.
6
Summary and conclusion:- In this work we have com- canonical ensemble is used to calculate the yields of the
paredthevaluesofC obtainedfromtheprimaryfrag- hot primary fragments, the values of C /T obtained
sym sym
ments for the canonical and the grand canonical ensem- from the secondary fragments after evaporation are
bles for a single source at a fixed temperature using the close to those obtained from the experimental yields
four different prescriptions. The results from the canon- but they differ from those obtained from the primary
ical ensemble calculations show that the values obtained fragments and from the input value of C used in the
sym
fromthefourformulasagreewitheachother. Theresults model. The messagewe wishto conveyisthatto deduce
also display that for the canonical model, C varies the value of the symmetry energy coefficient using the
sym
with the fragment size and differs from the input C prescriptions described earlier [Eqs. (5) to (8)], it is
sym
value which is equal to 23.5 MeV. In the grand canoni- advisabletousethe grandcanonicalmodeltoobtainthe
cal model, this value is independent of the fragment size yield of the fragments and one should use the yields at
and is almost equal to the input value used. This is be- the break-up stage (primary fragments) of the reaction.
cause all the four formulas are deduced using the grand The experimental yields (which are from the ”cold”
canonical ensemble at the break-up stage of multifrag- fragments) should not be used to deduce the value of
mentation. Hence it is possible to get back the value the symmetry energy coefficient since the formulas used
of the input symmetry energy coefficient using only the for the deduction are all valid at the equilibrium stage
grand canonical model for calculating the yields of pri- or the breakup stage of the reaction and secondary
mary fragments at the break-up stage. The results of decay disturbs the equilibrium scenario of the breakup
canonical and grand canonical ensembles differ in gen- stage. Attempts to deduce the value of C from the
sym
eral for finite nuclei[25] and hence in this case also the secondary fragments or from the experimental yields
results from the two ensembles are different. might lead to values that are very different from the
We have also extracted the value of C /T from actual value.
sym
the yields of projectile fragmentation reactions using
our model for projectile fragmentation. The results Acknowledgement:- The authors thank M. Mocko,
from the primary fragments are close to each other for Kelic-HeilAleksandraandKarl-HeinzSchmidtforaccess
all the four formulas used. In this model, where the to experimental data.
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