Bimodality - a Sign of Critical Behavior in Nuclear Reactions

0.19 MB·English

by  A. Le Fevre

Bimodality - a Sign of Critical Behavior in Nuclear Reactions
A. Le F`evre1,2, J. Aichelin2
1 GSI, P.O. Box 110552, D-64220 Darmstadt, Germany
2
SUBATECH, Laboratoire de Physique Subatomique et des Technologies Associ´ees,
Universit´e de Nantes - IN2P3/CNRS - Ecole des Mines de Nantes
4 rue Alfred Kastler, F-44072 Nantes, Cedex 03, France
(Dated: February 1, 2008)
The recently discovered coexistence of multifragmentation and residue production for the same
totaltransverseenergyoflightchargedparticles,whichhasbeendubbedbimodalitylikeithasbeen
introduced in the framework of equilibrium thermodynamics, can be well reproduced in numerical
simulations of the heavy ion reactions. A detailed analysis shows that fluctuations (introduced
8 by elementary nucleon-nucleon collisions) determine which of the exit states is realized. Thus, we
0 can identify bifurcation in heavy ion reactions as a critical phenomenon. Also the scaling of the
0 coexistence region with beam energy is well reproduced in these results from the QMD simulation
2 program.

n
PACSnumbers: 24.10.Lx,24.60.Lz,25.70.Pq
a
J

8
Recently, the INDRA collaboration has discovered [1] In dynamical models, on the contrary, fragments are
1
that,incollisionsofheavyions–Xe+SnandAu+Aube- surviving initial state correlations which have not been

tween60and100A.MeV incidentenergy–,inasmallin- destroyedduring the reaction, and equilibrium is not es-
]
h tervalofthe totaltransverseenergyoflightchargedpar- tablished during the reaction. A detailed discussion of
-t ticles(Z ≤2),E⊥ 12 ,aquantitywhichisusuallyconsid- how the reaction proceeds in these models can be found
l eredasa goodmeasureforthe centralityofthe reaction, in [8].
c
u two distinct reactionscenariosexist. In this E⊥ 12 inter- Toquantifythebimodality,onemaydefineasinref.[1]
n val,inforwarddirection–i.e. quasi-projectile–,either a
[ heavy residue is formed which emits light charged parti- a2 =(Zmax−Zmax−1)/(Zmax+Zmax−1) (1)
clesonly,orthesystemfragmentsintoseveralintermedi-
2
v ate mass fragments. This phenomenon has been named where Zmax is the charge of the largest fragment, while
9 “bimodality”. In addition, as shown in [1], the mean Zmax−1isthechargeofthesecondlargestfragment,both
3 E⊥ 12 value of this transition interval scales with the observed in the same event in the forward hemisphere –
6 projectile energy in the center of mass of the system for at polar angles θcm < 90o – in the center of mass of the
3 Au+Au reactions, between 60A.MeV and 150A.MeV. system. If the system shows bimodality, we will observe
8. asuddentransitionfromsmalltolargea2 values. Inthis
This observation has created a lot of attention, be-
0 narrow transition region, we expect two types of events:
cause a couple of years before, the theory has predicted
7 Onewithalargea (onebigprojectileresiduewithsome
2
0 [2, 3] that in finite size systems, whose infinite counter- very light fragments), the other with a small a (two or
2
: parts show a first order transition, the system can - for
v more similarly sized fragments). Events with intermedi-
a given temperature - be in either of the two phases if
i ate values of a should be rare.
X this temperature is close to that of the phase transition. 2
In order to verify whether bimodality is a ’ smoking
r Assuming that E⊥ 12 is a measure for the excitation en- gun ’ signal for a first order phase transition in a finite
a ergy, and acts as the control parameter of the system, it
system, we have performed numerical simulations with
is tempting to identify the residue with the liquid phase
one of the dynamical models which has frequently been
ofnuclear matter, andthe creationofseveralmedium or
usedtointerpretthemultifragmentationobservables,the
smallsizefragmentswiththegasphase. Theexperimen-
Quantum Molecular Dynamics (QMD) approach [6, 8].
tal observation would then just be a realization of the
This approach simulates the entire heavy ion reaction,
theoretical prediction.
from the initial separation of projectile and target up to
Ifthiswerethecase,thelongstandingproblemtoiden- the final state, composed of fragments and single nucle-
tify the reaction mechanism which leads to multifrag- ons. Here, nucleons interact by mutual density depen-
mentation would be solved. This problem arrived be- dent two body interactions and by collisions. The two
cause many observables could be equally well described body interaction is a parametrization of the Bru¨ckner
inthermodynamicalorstatisticaltheories[4, 5]asindy- G-Matrix supplemented by an effective Coulomb inter-
namical models [6, 7], although the underlying reaction action. For this work, we have used a soft equation of
mechanism was quite different. The statistical models state. Theinitialpositionsandmomentaofthe nucleons
assume that at freeze out,when the system is well below are randomly chosen, and respect the measured rms ra-
normalnuclearmatterdensity,thefragmentdistribution dius of the nuclei. Collisions take place if two nucleons
is determined by phase space. come closer than r = σ/π, where σ is the energy de-
p
2
pendent cross section for the corresponding channel (pp function of E⊥ 12 /(Ec.m./A) in the quasi-target for dif-
orpn). Thescatteringangleischosenrandomly,respect- ferentbeamenergies,whereE /Aisthecenterofmass
c.m.
ing the experimentally measured dσ/dΩ. Collisions are energy per nucleon of the colliding system. Accordingly
Pauli blocked. For details we refer to ref.[6, 7]. For the with[1],thetwoobservableshavebeendeterminedindis-
later discussion it is of importance that, even for a given tinctangularrangesinordertominimizepossiblecorrela-
impact parameter, two simulations are not identical, be- tions between the total transversekinetic energy of light
cause the initial positions and momenta of the nucleons particles and the size of the two biggest fragments in-
as well as the scattering angles are randomly chosen. sidethesamespectator(quasi-projectileorquasi-target).
Fig. 1 shows the INDRA experimental results (left We observe that in the experiment as in the calculation,
the sudden transition between large and small a val-
2
ues scales with EQT /(E /A). Even the numerical
⊥ 12 c.m.
value of this transition agrees between experiment and
theory. In order to see whether this phenomenon sur-
vives at higher incident energies, the simulations have
beenextendedupto150A.MeV bombardingenergy. We
observe that it is the case.
Thebottomrowshowsthetransitionintervalindetail.
Here, we display the differential reaction cross-sectionof
aQPas a function of EQT for Au+Au at 100A.MeV.
2 ⊥ 12
From the experimental data, we observe that there is
no smooth transition between the two event classes. In
the simulations as well as in the experiment we see two
maxima for aQP, separated by a minimum of the dis-
2
tribution. QMD simulations reproduce the experimental
findings qualitatively and quantitatively.
In QMD simulations, the system does not even come
to a local thermal equilibrium. It is therefore necessary
toexplorethe originofthe observeddependence ofa as
2
a function of E⊥ 12 . The first step toward this goal is
to identify when, in the course of the reaction, the frag-
mentpatternis determined. Thisis allbuttrivial. Frag-
mentscaneasilybeidentifiedattheendoftheheavyion
reaction,when they are clearly separated in coordinate
space, by a minimum spanning tree procedure. At ear-
liertimes,however,theyoverlapincoordinatespaceand,
consequently,anothermethodhastobeemployed. Ithas
beenproposedbyDorsoandRandrup[9],andlaterbeen
FIG. 1: (Color online) Top: most probablevalueof a2 in the verified in QMD simulations [10], that an early identifi-
quasi-projectile angular range θcm<90o (aQ2P) as afunction cationoffragments is possibleif one usesin additionthe
ofE⊥QT12,thetotaltransverseenergyE⊥ 12inthequasi-target momentum space information: If one identifies at each
angular range θcm ≥90o, scaled by Ecm, the energy per nu- time step during the simulation the most bound con-
cleonofthesysteminthecenterofmass. WedisplayINDRA figuration, one can establish that the fragment pattern
exprimentalresults(leftpanels)extractedfrom[1]andQMD changes only little during the time, and that the early
simulations (right panels) for the Au+Au collisions at three identified fragments are the prefragments of the finally
different bombarding energies. Bottom: differential reaction observed fragments. The most bound configuration in a
cross-section (linear color scale, arbitrary unit) of aQPas a
2 simulation with N fragments is that in which
function of EQT , in the transition region, for Au+Au at
⊥ 12
100A.MeV bombardingenergy. WeshowtheINDRAexperi- N
mentaldataandthefilteredQMDsimulations(leftandright Ebind =X Ei
panels respectively). As in [1], for calculating a2,in both ex- i=1
perimentaland QMD results, it isrequired thatat least 80%
ofthetotalchargeoftheprojectileisdetectedbytheINDRA is minimal. Ei is the binding energy of the fragment i
set-up in the forward hemisphere. which contains m(i) nucleons and is given by
m(i) m(i)
panels) in comparison with those of QMD calculations 1
Ei = X(pk−<pi >)2+ XVkl
(right panels). The calculations have been acceptance 2m
k=1 k<l
corrected. Qualitatively, we see the same behavior also
in the unfiltered simulations. In the top row, we present where < p > is the average momentum of the nucleons
i
the mostprobablevalue ofa inthe quasi-projectileasa entrained in the fragment i. Please note that E does
2 bind
3
notcontainthe interactionamongfragments. Therefore, 60 A.MeV 100 A.MeV 150 A.MeV
P
its numerical value can vary, although the total energy Q
a2 1
is conserved in the simulations. The most bound config- n
uration has to be determined by a simulated annealing a
e
procedure[10]. Withthisprocedure,thefragmentmulti- m0.75
plicity of a given event can already be determined when final aQP≥0.4
the systemhaspassedthehighestdensityandisstarting 2
0.5
to expand. Later, the prefragments may still emit some
final aQP<0.4
nucleons, but the nucleons which are entrained at the 2
0.25
end in the fragment are part of this prefragment. These
methods allow us to trace back at which time point in
the reaction it is determined whether the event has a 0
0
large or a small final a value. Fig. 2 shows QMD re- 3
2 r/
sults for Au+Au at 80, 100 and 150A.MeV bombarding r
energy. We see that, whatever the incident energy, the
two event classes are already formed shortly after the 2
system has passed the highest compression stage. There
we observethe highest rate ofhard NN collisions. These
collisionstransportnucleonsinunoccupiedregionsofmo-
1
mentum space leaving behind holes in momentum space
which create in time (due to the different trajectories)
holes in coordinate space. These holes weaken the bind-
ingofprojectileandtargetmatter,leadingtoafragment 00 100 0 100 0 100
t (fm/c)
formation still at rather high density. Within statisti-
cal models [4, 13], droplets are created with a spherical
shapeandhaveanormaldensity. Thisisonlypossibleat
a density of the system of ρ ≤ ρ /3. The observation of FIG.2: (Coloronline)ResultsofQMDsimulationsofAu+Au
0
an early creation of cluster partitions, above the critical at80,100and150A.MeVincidentenergies(left,middleand
density,asbeenfoundtooin[11]withlattice-gascalcula- right panels respectively). Top: aQ2Pas a function of time for
tionswheredroppletsarealsodefinedaccordingtoenergy the two different event classes: final (at 200 fm/c) aQ2P<0.4
considerations. The same conclusion has been obtained corresponding to multifragmentation events and final aQ2P≥
in [12] from classical molecular dynamics (CMD) calcu- 0.4 corresponding to a projectile residue. Bottom: central
density of thesystem as a function of time.
lations where particles interact through Lennard-Jones
plus Coulomb potentials.
Consequently, bimodality in QMD has nothing to
isknownfromnonlineartheory[14],andis called“bifur-
do with the final state interaction, or with how the
cation”. Weseehereinasystemwithaverylimitedpar-
neckbetweenprojectileandtargetresiduefinallybreaks.
ticle number, a nonlinear behavior. Can we understand
Whetherwefindamultifragmentationoraheavyresidue
where it comes from? In order to answer this question,
event is determined when projectile and target nucleons
we go back to the search of the most bound configura-
still overlap almost completely in coordinate space [8].
One may conjecture that, due to the random character tion. For E⊥ 12 values below the transition (and hence
large a values), the most bound configurationis a large
of the scattering angle, events with the same E⊥ 12 de- 2
residue. Abovethetransition(wherea issmall),several
celeratedifferently,and,therefore,adifferentbehaviorof 2
small fragments give a more bound configuration. The
the average momenta may be at the origin of the differ-
sum of the internal kinetic energies of the clusters
ent a values. For this purpose, we study with Au+Au
2
at 150A.MeV incident energy, at 60 fm/c, when a is
2
decided, the dependence with the final a of the average N m(i)
2 1
momentum of all target nucleons which are at the end Emult = XX (pk−<pi >)2
2m
entrained in A>4 fragments. In fig. 3, we display their i=1 k=1
longitudinal and transverse momentum as a function of
a2. Clearly, both average momentum are almost inde- is there smaller than
pendentofthefinala . Thefluctuationsofthemomenta
2
around the mean values are by far larger than the dif- m(i)+m(2)+..+m(N)
1
ference between the mean values for different a2 values. Eres = X (pk−<p>)2,
Thisexcludesmeandecelerationofthesimulationevents 2m
k=1
as reason for the different reaction scenarios.
Obviously fluctuations around the mean values are at the internal kinetic energy for a residue configuration,
theoriginofthedifferenteventclasses. Thisphenomenon and compensates the increase of the attractive potential
4
c) c) ments. (Inorderto countthe micro-states,it is assumed
V/ V/ that the fragments are in one if their eigenstates, some-
P (Gez 0 P (Ge⊥ 0.3 teinmeregsypararanmgee,tebriimzeoddbaylitaylaepvepleadresnsaistyafogrlmobualal.p)rIonpetrhtiys
of the systems which is dependent on the total energy
-0.2 0.2 of all nucleons present in the reaction. In QMD events,
the essential quantity is the total binding energy of the
0.1 nucleons bound in medium size or large clusters. As ex-
-0.4 plained above, in the transition region, this energy is al-
mostidenticalforamultifragmentandfora residuecon-
0 0.5 1 00 0.5 1 figuration. Therefore, both configurations appear, and
a a
2 QP final 2 QP final we see bimodality. The fragments are not in the ground
state, their nucleons are not isotropic neither in coordi-
nate space nor in momentum space. Thus, bimodality
FIG. 3: (Color online) QMD simulations of Au+Au at is a local quantity in QMD simulations, depending only
150A.MeV incident energy, at 60 fm/c: differential cross- on the total binding energy of a subset of the nucleons.
section (colored contour levels, linear scaled) of the longitu- Therefore, in QMD, bimodality makes no reference to a
dinal (left) and transverse momentum (right) of all nucleons statistical or thermal equilibrium, neither of the system
which are finally entrained in a fragment of size A > 4 as a nor of the population of the excited states of the frag-
function of final (at 200 fm/c) aQ2P. The symbols represent ments.
themean values of momentum.
In summary, we have shown that the experimentally
observed bimodality, the sudden transition between a
energy residueandamultifragmentexitstates,andtheexistence
of a small interval in E⊥ 12 in which both channels are
m(1)+m(2)+..+m(N) N m(i) coexistent,isinquantitativeagreementwiththeresultof
1
Vres−Vmult = ( X Vkl−X X Vkl). QMDsimulations. Eventhe scaling ofthis transitionre-
2
k,l=1 i=1k,l=1 gionwith the center of mass energyof the system is well
reproduced. Fromadetailedinvestigationofthereaction
In the transition region, we see that E + V ≈
mult mult mechanism in QMD, we have seen that bimodality has
E +V . In some events, both configurations differ
res res properties observed in nonlinear systems: The system
by 100keV only. Therefore it may happen that the scat-
shows bifurcation as a function of the control parame-
teringangleofonesinglenucleon-nucleoncollision,which
ter E⊥ 12 . Fluctuations around the mean value in the
is –see above– randomlychosen,determines the type of
longitudinalandtransversemomentumdecidewhichexit
the most bound configuration.
channel the simulation will take.
It is interesting to see the differences and similarities
Beingreproducedinstatisticalaswellasindynamical
of the originof bifurcation in a statistical model as com-
models, bimodality reflects the same ambiguity already
pared to the analysis of QMD events. In both cases, the
observed for other observables [15].
energy is the essentialquantity. In the statistical model,
there is, for a given number of nucleons in a given vol- AcknowledgementWewouldliketothankthemembers
ume,asmallrangeoftotalenergiesforwhichthenumber oftheINDRAcollaborationformanydiscussionsandfor
of micro-states with one residue is of the same order of giving us access to their data. We would like as well to
magnitudeasthenumberofmicro-stateswithmanyfrag- thank Dr. W. Trautmann for fruitful discussions.
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