What Can be Learned Studying the Distribution of the Biggest Fragment ?
ABSTRACT In the canonical formalism of statistical physics, a signature of a first order phase transition for finite systems is the bimodal distribution of an order parameter. Previous thermodynamical studies of nuclear sources produced in heavyion collisions provide information which support the existence of a phase transition in those finite nuclear systems. Some results suggest that the observable Z1 (charge of the biggest fragment) can be considered as a reliable order parameter of the transition. This talk will show how from peripheral collisions studied with the INDRA detector at GSI we can obtain this bimodal behaviour of Z1. Getting rid of the entrance channel effects and under the constraint of an equiprobable distribution of excitation energy (E*), we use the canonical description of a phase transition to link this bimodal behaviour with the residual convexity of the entropy. Theoretical (with and without phase transition) and experimental Z1E* correlations are compared. This comparison allows us to rule out the case without transition. Moreover that quantitative comparison provides us with information about the coexistence region in the Z1E* plane which is in good agreement with that obtained with the signal of abnormal uctuations of configurational energy (microcanonical negative heat capacity).

Article: Inclusive selection of intermediatemassfragment formation modes in the spallation of 136Xe
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ABSTRACT: A correlation between the production and kinematic properties of the fragments issued from fission and multifragmentation is established in the study of the reaction 136Xe + hydrogen at 1 GeV per nucleon, measured in inverse kinematics at the FRagment Separator (GSI, Darmstadt). Such observables are analysed in a comprehensive study, selected as a function of the decay mode, and related to the isotopic properties of the fragments in the intermediatemass region. Valuable information can be deduced on the characteristics of the heaviest product in the reaction, which has been considered a fundamental observable for tagging the thermodynamic properties of finite nuclear systems.Journal of Physics G Nuclear and Particle Physics 10/2011; 38(11):115006. · 5.33 Impact Factor
Page 1
arXiv:0704.1396v1 [nuclex] 11 Apr 2007
WHAT CAN BE LEARNED STUDYING THE
DISTRIBUTION OF THE BIGGEST FRAGMENT ?
E. BONNET1,2, F. GULMINELLI3, B. BORDERIE1, N. LE NEINDRE1,
M.F. RIVET1
The INDRA and ALADIN Collaborations:
1Institut de Physique Nucl´ eaire, CNRS/IN2P3, Universit´ e ParisSud 11, F91406
OrsayCedex, France.
2GANIL, DSMCEA/IN2P3CNRS, B.P.5027, F14076 CaenCedex, France, France.
3LPC, IN2P3CNRS, ENSICAEN et Universit´ e de Caen, F14050 CaenCedex, France.
In the canonical formalism of statistical physics, a signature of a first order phase
transition for finite systems is the bimodal distribution of an order parameter.
Previous thermodynamical studies of nuclear sources produced in heavyion col
lisions provide information which support the existence of a phase transition in
those finite nuclear systems. Some results suggest that the observable Z1 (charge
of the biggest fragment) can be considered as a reliable order parameter of the
transition. This talk will show how from peripheral collisions studied with the
INDRA detector at GSI we can obtain this bimodal behaviour of Z1. Getting
rid of the entrance channel effects and under the constraint of an equiprobable
distribution of excitation energy (E∗), we use the canonical description of a phase
transition to link this bimodal behaviour with the residual convexity of the en
tropy. Theoretical (with and without phase transition) and experimental Z1− E∗
correlations are compared. This comparison allows us to rule out the case without
transition. Moreover that quantitative conparison provides us with information
about the coexistence region in the Z1−E∗plane which is in good agreement with
that obtained with the signal of abnormal fluctuations of configurational energy
(microcanonical negative heat capacity).
1Introduction
It is well known that a LiquidGas phase transition (PT) occurs in van der Waals fluids.
The similarity between intermolecular and nuclear interactions leads to a qualitatively
similar equation of state which defines the spinodal and coexistence zones of the phase
diagram. That is why we expect a “LiquidGas like” PT for nuclei. The order parameter
is a scalar (one dimension) and is, in this case, the density of the system (more precisely
the density difference between the ordered and disordered phase). The energy is also an
order parameter because the PT has a latent heat.
When an homogeneous system enters the spinodal region of the phase diagram, its en
tropy exhibits a convex intruder along the order parameter(s) direction(s). The system
becomes unstable and decomposes itself in two phases. For finite systems, due to surface
energy effects, we expect a residual convexity for the system entropy after the transition
leading directly to a bimodal distribution (accumulation of statistics for large and low
values) of the order parameter. The challenge is to select an observable connected to the
theoretical order parameter of the transition, and to explore sufficiently the phase dia
gram to populate the coexistence region and its neighbourhood. Quasiprojectile sources
produced in (semi)peripheral collisions cover a large range of dissipation and conse
quently permit this sufficient exploration.
Several theoretical1,2and experimental works3,4,5show that the biggest fragment has
a specific behaviour in the fragmentation process. In particular its size is correlated to
the excitation energy (E∗) of the sources. We can reasonably explore whether the Z1E∗
experimental plane shows a bimodal pattern. Other experimental signals obtained with
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multifragmentation data can be correlated with the presence of a phase transition in
hot nuclei. Indeed, abnormal fluctuations of configurational energy (AFCE)6,7can be
related to the negative heat capacity signal8, and the fossil signal of spinodal decom
position9can illustrate the density fluctuations occurring when the nuclei pass through
the spinodal zone10. These two signals are not direct ones and need some hypotheses
and/or high statistics. In this work we will present the study of the bimodality signal
which is expected to be more robust and direct. We will also show that its observation
reinforces the conclusions extracted from the two previous signals.
The idea is to show experimentally that the biggest fragment charge, Z1, can be
a reliable observable to the order parameter of the PT. After an introduction of the
canonical ensemble, we explain the procedure of renormalization which allows to get
rid of entrance channel and data sorting effects. Then, comparing experimental and
canonical (E,Z1) distributions, we will show that the observed signal of bimodality is
related to the abnormal convexity of the entropy of the system. At the end, we propose
a localisation of the coexistence zone deduced from a comparison between experimental
data and the canonical description of a PT.
2Canonical description of firstorder phase transition.
Let us consider an observable E, known on average, free to fluctuate. The least biased
distribution will be a BoltzmannGibbs distribution (def. 1)11. If this observable is an
order parameter of the system we have to distinguish two cases: with and without phase
transition.
Pcan
β (E) =
1
Zcan
β
e
S(E)−βEwith Zcan
β
=
?
dE e
S(E)−βE
(1)
S(E) ∼ S(Eβ) + (E − Eβ)
For a one phase system (PT is not present), the microcanonical entropy, S(E)=log W(E)
where W(E) is the number of microstates associated to the value of E, is concave every
where. We can perform on it a saddle point approximation (eq. 2) around the average
value of E, Eβ, meaning that the canonical distribution has a simple gaussian shape
(eq. 3).
d
dES??
Eβ+1
2(E − Eβ)2 d2
dE2S??
Eβ
(2)
P(s.g.)
β
(E) =
1
?2πσ2
√2πdetΣe−1
E
exp
?
−
1
2σ2
E
(E − Eβ)2
?
with σ2
E= −
?d2
dE2S??
σ2
ρ σEσZ
Eβ
?−1
(3)
P(s.g.)(E∗,Z1) =
1
2? xΣ−1? x, ? x =
?E∗− Eβ
Z1− Zβ
?
, Σ =
?
E
ρ σEσZ
σ2
Z
?
(4)
The parameters of this gaussian are directly linked to the characteristics of the entropy.
In the same way we can define the minimum biased two dimensional distribution for the
(E∗,Z1) observables leading to a 2D simple gaussian distribution12(def. 4). Parameters
of this function gathered in the variancecovariance matrix are also deduced from the
curvature matrix of the 2D microcanonical entropy12,13.
P(d.g.)(E∗,Z1) = Nliq× P(s.g.)
When a system passes through a phase transition and enters in the spinodal region, the
homogeneous system has a convex intruder in its microcanonical entropy along the order
liq
(E∗,Z1) + Ngaz× P(s.g.)
gaz(E∗,Z1)(5)
Bormio 2007: XLV International winter meeting on Nuclear Physics
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parameter(s) direction(s)14. Instabilities occur and, due to the finite size of the system,
the surface energy effects cause the nonadditivity of the entropy leading at the end of
the PT to a residual convex entropy for the twophase system even at equilibrium. We
cannot describe anymore the microcanonical entropy with a single saddle point approx
imation but we can introduce a double saddle point approximation. In this case the
canonical distribution of the (E∗,Z1) observables can be described as the sum of two 2D
simple gaussian distributions, one for each phase (def. 5)12,13.
In the canonical ensemble, the energy distribution Pcan
dimensional distribution Pcan
β
(E∗,Z1) are conditioned by the number of available states
expS with a Boltzmann factor ponderation. The convex intruder in S leads to a bi
modality in the distribution12. Experimentally, this relation is not so clear: the weight
of the different states has no reason to be exponential and the measured distribution
Pexp
β
(E∗) is modified by a factor gexp(E∗) which is determined in a large part by en
trance channel effects and data sorting : Pexp(E∗,Z1) = eS(E∗,Z1)gexp(E∗). The relative
population of the different values of the E∗distribution looses its thermostatistic meaning
(Pexp(E∗) ∝ gexp(E∗)Pcan
tal and canonical distributions and deduce entropy properties of the system.
β
(E∗) as well as the two
β
(E∗)eβE∗). We cannot therefore directly compare experimen
Pexp
ω (E∗,Z1) = ω(E) × Pexp(E∗,Z1)
??
(6)
with ω(E∗) =
P(exp)(E∗,Z1) dZ1
?−1
2.1Renormalization method.
In12, a method was proposed to get rid of the experimental effects. Assuming that
the experimental bias gexp(E∗) affects the Z1 distribution only through its correlation
with the deposited energy E∗(phase space dominance), a renormalization of the (E∗,Z1)
distribution under the constraint of an equiprobable distribution of E∗(eq. 6) allows to
be E∗shape independent. If the system passes through a PT and the correlation between
E∗and Z1is not a onetoone correspondence, it could reflect a residual convex intruder
of the entropy.
2.2Spurious bimodality
In principle one can ask whether the renormalization procedure given by eq. 6 can create
spurious bimodality. This does not seem to be the case for different schematic models12
but cannot be excluded a priori. Another ambiguity arises from the fact that a physical
bimodality can be hidden by the renormalization procedure if the correlation between Z1
and E∗is too strong. Bimodality can be also difficult to spot if the energy interval is not
wide enough. For these reasons in the following we will compare the two canonical cases
(with and without transition) with the experimental distribution, to check the validity
of the obtained signal.
3Data selection and first observation of Z1distributions.
Data used in this present work are 80 MeV/A Au+Au reactions performed at the GSI
facility and detected with the INDRA 4π multidetector. We focus on peripheral and semi
peripheral collisions to study quasiprojectile sources (forward part of each event). To
perform thermostatistical analyses, we select a set of events with a dynamically compact
Bormio 2007: XLV International winter meeting on Nuclear Physics
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1
Z
20 40 60 80
dN / dZ
1
0
500
1000
1500
2000
Source
1
10
2
10
3
10
(MeV/A)
*
E
123456789 10
1
Z
10
20
30
40
50
60
70
80
90
)
1
,Z
*
(E
(exp)
P
3
10
2
10
1
10
(MeV/A)
*
E
123456789 10
1
Z
10
20
30
40
50
60
70
80
90
)
1
,Z
*
(E
(exp)
ω
P
(MeV/A)
*
E
12345678 9 10
1
)
*
(E
ω
=
*
dN /dE
1
10
1
10
2
10
3
10
4
10
5
10
Figure 1. Upper part : left : experimental distribution of the argest size fragment (Z1) of source events;
right : experimental correlation between Z1 and the excitation energy (E∗). Lower part : left : ex
perimental reweighted correlation between Z1 and the excitation energy; right : excitation energy (E∗)
experimental distribution of source events in black squares; the open red circles show this distribution
after the renormalization process. For this, we keep only E∗bins with a statistics greater than 100.
configuration for fragments, to reject dynamical events which are always present in heavy
ion reactions at intermediate energies. We require in addition a constant size of the
sources to avoid size evolution effects in the bimodality signal13,15. We evaluate the
excitation energy using a standard calorimetry procedure16,17. We compute the energy
balance eventbyevent in the centre of mass of the QP sources calculated with fragments
only to minimize the effect of preequilibrium particles. Afterwards we keep only particles
emitted in the forward part of the QP sources and double their contribution, assuming
an isotropic emission. In figure 1 information on the experimental Z1and E∗observables
is shown, the latter covering a range between roughly 1 and 8 MeV/A (lowerright part).
Spinodal zone limits obtained with the AFCE signal are around 2.5 and 5.8 MeV/A
for this set of data13. The shape of the distribution Pexp(E∗,Z1) (upper right part)
shows the dominance of low dissipationlarge Z1 events and reflects the crosssection
distribution and data selection. If we look at the corresponding Z1distribution (upper
left part) we do not see any clear signal of bimodality: a large part of statistics is around
6570, and only a shoulder is visible around 3040. This particular shape could reflect
Bormio 2007: XLV International winter meeting on Nuclear Physics
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the lack of statistics for the ”gaslike” events. If we apply the renormalization procedure
(eq. 6) we obtain (lower left part) a Pexp
ω
(E,Z1) distribution which has a double humped
shape, tending to prove that this procedure can reveal bimodality.
1
Z
10 20 30 40 50 60 70 80 90
1
/ dZ
ω
dN
0
0.2
0.4
0.6
0.8
1
(exp)
ω
P
(s.g.)
ω
P
(d.g.)
ω
P
>
1
<Z
2030 405060 70
)
(Z
σ
1
0
2
4
6
8
10
1.3751.375 3.1253.1254.625 4.6256.3756.375
)
skw(Z
1
1
0.5
0
0.5
1
*
E
1
Z
10 20 30 40 50 60 70 80 90
1
/dZ
ω
dN
0
0.2
0.4
0.6
0.8
1
[1.25 , 3.00[
[3.00 , 6.25[
[6.25 , 9.75[
1
Z
10203040506070
/dZ
ω
dN
1
0.1
0.2
0.3
0.4
0.5
[3.00,6.25[
∈
*
E
Figure 2. Upper part: left: Largest size fragment (Z1) experimental reweighted distribution (black
squares with error bars); the blue dashed curve corresponds to the best solution obtained by comparing
data and a single gaussian function (concave entropy, no PT), the red continuous curve corresponds
to the best solution obtained by comparing data and a double gaussian function (convex entropy, PT);
right: microcanonical sampling (fixed E∗) of the mean, the RMS and the skewness of the Z1 distri
butions. For each bin of E∗(upper X axis), RMS (colored squaresleft Y axis) and skewness (colored
trianglesright Y axis) are plotted as a function of the mean value (Z1lower X axis); the two vertical
dashed lines delimit the evaluated experimental spinodal zone where a quantitative comparison between
data and PT case is performed. Lower part: left: same reweighted distribution of Z1 as above (black
curve); the three other distributions correspond to the three regions delimited by the vertical dashed lines
(from left to right E∈[1.25,3.00[,[3.00,6.25[ and [6.25,9.75[); right: best solutions obtained after the 2D
comparison between data and canonical PT case; results are plotted for the Z1 axis projection; the two
solutions correspond to two different ranges of Z1 where fits have been performed Z1∈[25,55] (dashed
curve) and Z1∈[10,79] (continuous curve). The corresponding parameter values are listed in table 1.
4 CanonicalExperimental comparisons.
To confirm that the twohump distribution of Z1signals a convex intruder in the under
lying entropy, in this section we compare the experimental reweighted distributions with
Bormio 2007: XLV International winter meeting on Nuclear Physics
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Parameters
¯ E∗liq
σE∗
¯ Z1
σZliq
¯ E∗gaz
σE∗
gaz
¯ Z1
σZgaz
1
ρ
Nliq/Ngaz
Ndof
χ2
χ2/Ndof
[10,79]
2,10
2,09
[25,55]
1,67
1,66
Errors (%)
23
23
liq
liq
60,2
12,9
7,11
3,07
21,1
15,2
0,906
1,12
605
1488
2,459
62,2
9,85
6,81
2,97
23,8
18,8
0,860
0,66
387
646
1,669
3
4
4
3
1
gaz
12
2
4
52



Table 1. Parameters values for the two best reproductions of data by the double gaussian function P(d.g.)
for two ranges in Z1 [10,79] (first column) et [25,55] (second column); the third column gives relative
errors computed with the previous values; ¯
E∗,¯
Z1,σE∗,σZ1stand respectively for centroids and RMS in
the two directions (E,Z1) of each phase (liquid and gas). The ρ parameter is the correlation factor
]1,1[ between Z1 and E∗and the ratio Nliq/Ngaz indicates the repartition of statistics between the two
phases. The three last lines give the number of degrees of freedom and the absolute and normalized χ2
estimator values.
ω
the analytic expectation for a system exhibiting or not a first order PT. We apply the
same renormalization to P(s.g.)
β
(E) and P(d.g.)
β
focus on the projection on the Z1axis to perform the fit. The results are shown in the
upper left part of fig. 2. The scatter points with errors bars correspond to the data; the
continuous (respectively dashed) curve corresponds to the best solution obtained for the
double (respectively simple) reweighted gaussian. We can clearly distinguish the two be
haviours, the notransition case can not curve itself in the Z1=4050 region and can only
reproduce one phase. The fact that data are reproduced with the functional describing
a first order transition allows us to associate the experimental bimodality signal to a
genuine convexity of the system entropy. This confirms also that the Z1 observable is
linked to the order parameter of the transition. To obtain more quantitative information
we have to better localize the spinodal region. To do this, we look at the second and
third moments of the Z1distribution for each bin of E∗. Their evolution is plotted on
the upper right part of fig. 2 as a function of the mean value of Z1(lower X axis) and E∗
(upper X axis). The squares (left Y axis) stand for the sigma (σ) of the distribution and
the triangles (right Y axis) for the skewness (skw). σ shows a maximum in the range
3040 for < Z1 >. This maximum of fluctuations signs the core of the spinodal zone
which corresponds to the hole in Pexp
ω
(E,Z1) distribution. All values of Z1, for a given
E∗, are more or less equivalent. In the same region the skewness changes sign, illustrat
ing the change in the distribution of asymmetry, with a value close to zero when the
distribution approaches a normal one. The two vertical dashed lines on the plot delimit
three regions (E∈[1.25,3.00[,[3.00,6.25[ and [6.25,9.75[) and the three corresponding Z1
reweighted distributions are plotted in the lower left part of the same figure. The middle
one, flat and broad, is very close to the behavior expected for a critical distribution2
and illustrates the effect of an energy constraint on the order parameter distribution. If
we had made a thinner range, we would have approached the microcanonical case. We
(E) and try to reproduce the data. We
Bormio 2007: XLV International winter meeting on Nuclear Physics
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Page 7
select the region E∈[3.00,6.25[ to compare the two reweighted distributions Pexp
and P(d.g.)
ω
(E,Z1) (eq. 5). The best solutions obtained after this 2D fit procedure are
shown in the lower right part of figure 2 and table 1. They correspond to two ranges of
Z1where fits are performed ([10,79] and [25,55]). These two best solutions are shown for
the projection on the Z1axis.
ω
(E,Z1)
(MeV/A)
*
E
2468 10
(u.a.)
0
0.5
1
1.5
2
2
/ T
k
σ
>
s
2
<A
2
10
×
)
s
/ Z
max
(Z
2
σ
Figure 3. Microcanonical sample (fixed E∗) of the fluctuations of normalized FO kinetic energy (open
circles) and largest fragment charge (full squares). T, As and Zs stand respectively for the temperature,
the mass and charge of the source.
Using two different ranges for the Z1range allows us to estimate the sensitivity of the
different parameters. The description of the two phases, given by a set of four parameters
for each phase, can be summarized as follows: the average characteristics of the phases,
given by¯E∗,¯ Z1, are well defined. The ratio between the populations of the liquid and
gas phase strongly depends on the interval used to perform the fit. In the two cases
the normalized estimator, χ2, is good. Concerning the E∗axis, the values obtained for
the liquid and gas phase centroids reflect the location of the coexistence zone, and are
consistent with the location of the spinodal zone obtained with the AFCE signal with the
same set of events13,18. We can further explore the coherence between the two signals by
looking at the fluctuations associated to Z1and to the FreezeOut configurational kinetic
energy8: we observe in fig. 3 that their evolution with excitation energy has a similar
behaviour and exhibits a maximum for E∼5MeV/A. This observation shows that we can
consistently characterize the core of the spinodal zone with the maximum fluctuations of
different observables connected to the order parameter of the phase transition.
Bormio 2007: XLV International winter meeting on Nuclear Physics
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5 Conclusion and outlook.
In this contribution we have shown that, taking into account the dynamics of the entrance
channel and sorting effects with a renormalization procedure, the distribution of the
largest size fragment (Z1) of each event shows a bimodal pattern. The comparison with
an analytical estimation assuming the presence (the absence) of a phase transition, shows
that the experimental signal can be unambiguously associated to the case where the
system has a residual convex intruder in its entropy. This link makes the Z1observable a
reliable order parameter for the PT in hot nuclei. A bijective relation between the order
of the transition and the bimodality signal has been proposed in12and analyses on data
are in progress.
References
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1
Z
10 20 30 40 50 60 70 80 90
/ dZ
ω
dN
1
0
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0.4
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0.8
1
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