WHAT CAN BE LEARNED STUDYING THE
DISTRIBUTION OF THE BIGGEST FRAGMENT ?
E. BONNET
, F. GULMINELLI
, B. BORDERIE
, N. LE NEINDRE
M.F. RIVET
The INDRA and ALADIN Collaborations:
1Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud 11, F-91406
Orsay-Cedex, France.
2GANIL, DSM-CEA/IN2P3-CNRS, B.P.5027, F14076 Caen-Cedex, France, France.
3LPC, IN2P3-CNRS, ENSICAEN et Université de Caen, F-14050 Caen-Cedex, 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 heavy-ion 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).
1 Introduction
It is well known that a Liquid-Gas phase transition (PT) occurs in van der Waals fluids.
The similarity between inter-molecular 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 “Liquid-Gas 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. Quasi-projectile sources
produced in (semi-)peripheral collisions cover a large range of dissipation and conse-
quently permit this sufficient exploration.
Several theoretical 1,2 and experimental works 3,4,5 show 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 Z1-E
experimental plane shows a bimodal pattern. Other experimental signals obtained with
Bormio 2007: XLV International winter meeting on Nuclear Physics 1
http://arxiv.org/abs/0704.1396v1
multifragmentation data can be correlated with the presence of a phase transition in
hot nuclei. Indeed, abnormal fluctuations of configurational energy (AFCE) 6,7 can be
related to the negative heat capacity signal 8, and the fossil signal of spinodal decom-
position 9 can illustrate the density fluctuations occurring when the nuclei pass through
the spinodal zone 10. 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.
2 Canonical description of first-order phase transition.
Let us consider an observable E, known on average, free to fluctuate. The least biased
distribution will be a Boltzmann-Gibbs 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.
Pcanβ (E) =
e S(E)−βE with Zcanβ =
dE e S(E)−βE (1)
S (E) ∼ S (Eβ) + (E− 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).
(s.g.)
(E) =
2πσ2E
(E− Eβ)
with σ2E = −
P(s.g.)(E∗,Z1) =
2πdetΣ
~xΣ−1~x, ~x =
E∗ − Eβ
Z1 − Zβ
, Σ =
σ2E ρ σEσZ
ρ σEσZ σ
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 distribution 12 (def. 4). Parameters
of this function gathered in the variance-covariance matrix are also deduced from the
curvature matrix of the 2D microcanonical entropy 12,13.
P(d.g.)(E∗,Z1) = Nliq × P
(s.g.)
liq (E
∗,Z1) + Ngaz × P
(s.g.)
gaz (E
∗,Z1) (5)
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
Bormio 2007: XLV International winter meeting on Nuclear Physics 2
parameter(s) direction(s) 14. Instabilities occur and, due to the finite size of the system,
the surface energy effects cause the non-additivity of the entropy leading at the end of
the PT to a residual convex entropy for the two-phase 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 P canβ (E
∗) as well as the two-
dimensional distribution P canβ (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 distribution 12. Experimentally, this relation is not so clear: the weight
of the different states has no reason to be exponential and the measured distribution
(E∗) is modified by a factor gexp(E
∗) which is determined in a large part by en-
trance channel effects and data sorting : P exp(E∗, Z1) = e
S(E∗,Z1)gexp(E
∗). The relative
population of the different values of the E∗ distribution looses its thermostatistic meaning
(P exp(E∗) ∝ gexp(E
∗)P canβ (E
∗)eβE
). We cannot therefore directly compare experimen-
tal and canonical distributions and deduce entropy properties of the system.
Pexpω (E
∗,Z1) = ω(E)× P
exp(E∗,Z1) (6)
with ω(E∗) =
P(exp)(E∗,Z1) dZ1
2.1 Renormalization method.
In 12, 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 Z1 is not a one-to-one correspondence, it could reflect a residual convex intruder
of the entropy.
2.2 Spurious 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 models 12
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.
3 Data selection and first observation of Z1 distributions.
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 quasi-projectile 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 3
20 40 60 80
2000 Source
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 10
(E (exp)P
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 10
(E (exp)ω P
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 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 signal 13,15. We evaluate the
excitation energy using a standard calorimetry procedure 16,17. We compute the energy
balance event-by-event in the centre of mass of the QP sources calculated with fragments
only to minimize the effect of pre-equilibrium 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 Z1 and E
∗ observables
is shown, the latter covering a range between roughly 1 and 8 MeV/A (lower-right part).
Spinodal zone limits obtained with the AFCE signal are around 2.5 and 5.8 MeV/A
for this set of data 13. The shape of the distribution P exp(E∗, Z1) (upper right part)
shows the dominance of low dissipation-large Z1 events and reflects the cross-section
distribution and data selection. If we look at the corresponding Z1 distribution (upper
left part) we do not see any clear signal of bimodality: a large part of statistics is around
65-70, and only a shoulder is visible around 30-40. This particular shape could reflect
Bormio 2007: XLV International winter meeting on Nuclear Physics 4
the lack of statistics for the ”gas-like” events. If we apply the renormalization procedure
(eq. 6) we obtain (lower left part) a P expω (E,Z1) distribution which has a double humped
shape, tending to prove that this procedure can reveal bimodality.
10 20 30 40 50 60 70 80 90
  (exp)ωP
  (s.g.)ωP
  (d.g.)ωP
> 1 <Z
20 30 40 50 60 70
10 20 30 40 50 60 70 80 90
[1.25 , 3.00[
[3.00 , 6.25[
[6.25 , 9.75[
10 20 30 40 50 60 70
 [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 squares-left Y axis) and skewness (colored
triangles-right Y axis) are plotted as a function of the mean value (Z1-lower 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 Canonical-Experimental comparisons.
To confirm that the two-hump distribution of Z1 signals 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 5
Parameters [10,79] [25,55] Errors (%)
Ē∗liq 2,10 1,67 23
2,09 1,66 23
60,2 62,2 3
12,9 9,85 4
Ē∗gaz 7,11 6,81 4
3,07 2,97 3
21,1 23,8 12
σZgaz
15,2 18,8 2
ρ -0,906 -0,860 4
Nliq/Ngaz 1,12 0,66 52
Ndof 605 387 -
χ2 1488 646 -
χ2/Ndof 2,459 1,669 -
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; Ē∗,Z̄1,σE∗ ,σZ1 stand 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.
the analytic expectation for a system exhibiting or not a first order PT. We apply the
same renormalization to P
(s.g.)
(E) and P
(d.g.)
(E) and try to reproduce the data. We
focus on the projection on the Z1 axis 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 no-transition case can not curve itself in the Z1=40-50 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 Z1 distribution 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
30-40 for < Z1 >. This maximum of fluctuations signs the core of the spinodal zone
which corresponds to the hole in P expω (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 distribution 2
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
Bormio 2007: XLV International winter meeting on Nuclear Physics 6
select the region E∈[3.00,6.25[ to compare the two reweighted distributions P expω (E,Z1)
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
Z1 where fits are performed ([10,79] and [25,55]). These two best solutions are shown for
the projection on the Z1 axis.
 (MeV/A) *E
2 4 6 8 10
2 / Tk
2σ> s<A
2 10×)  s / Z max (Z
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 Z1 range 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 Ē∗, Z̄1, 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 events 13,18. We can further explore the coherence between the two signals by
looking at the fluctuations associated to Z1 and to the Freeze-Out configurational kinetic
energy 8: 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 7
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 Z1 observable 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 in 12 and analyses on data
are in progress.
References
1. R. Botet et al., Phys. Rev. E 62 (2000) 1825.
2. F. Gulminelli et al., Phys. Rev. C 71 (2005) 054607.
3. J. D. Frankland et al. (INDRA and ALADIN collaborations), Phys. Rev. C 71
(2005) 034607.
4. J. D. Frankland et al. (INDRA and ALADIN collaborations), Nucl. Phys. A 749
(2005) 102.
5. M. Pichon et al. (INDRA and ALADIN collaborations), Nucl. Phys. A 279 (2006)
6. N. Le Neindre, thèse de doctorat, Université de Caen (1999),
http://tel.ccsd.cnrs.fr/tel-00003741.
7. M. D’Agostino et al., Phys. Lett. B 473 (2000) 219.
8. P. Chomaz et al., Nucl. Phys. A 647 (1999) 153.
9. G. Tăbăcaru et al., Eur. Phys. J. A 18 (2003) 103.
10. P. Chomaz et al., Phys. Rep. 389 (2004) 263 and references therein.
11. R. Balian, Cours de physique statistique de l’École Polytechnique, Ellipses (1983).
12. F. Gulminelli (2007), Nucl. Phys. A, in press.
13. E. Bonnet, thèse de doctorat, Université Paris-XI Orsay (2006),
http://tel.ccsd.cnrs.fr/tel-00121736.
14. D. H. E. Gross, Microcanonical Thermodynamics-Phase Transitions in “Small” Sys-
tems, Singapore-World Scientific, 2001.
15. E. Bonnet et al. (INDRA and ALADIN collaborations) (2007) in preparation.
16. D. Cussol et al., Nucl. Phys. A 561 (1993) 298.
17. E. Vient et al. (INDRA Collaboration), Nucl. Phys. A 700 (2002) 555.
18. N. Le Neindre et al. (INDRA and ALADIN collaborations), Nucl. Phys. A (2007)
submitted.
Bormio 2007: XLV International winter meeting on Nuclear Physics 8
10 20 30 40 50 60 70 80 90
Source

  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 heavy-ion 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 Z1-E* 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
Z1-E* plane which is in good agreement with that obtained with the signal of
abnormal uctuations of configurational energy (microcanonical negative heat
capacity).

Introduction
It is well known that a Liquid-Gas phase transition (PT) occurs in van der Waals fluids.
The similarity between inter-molecular 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 “Liquid-Gas 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. Quasi-projectile sources
produced in (semi-)peripheral collisions cover a large range of dissipation and conse-
quently permit this sufficient exploration.
Several theoretical 1,2 and experimental works 3,4,5 show 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 Z1-E
experimental plane shows a bimodal pattern. Other experimental signals obtained with
Bormio 2007: XLV International winter meeting on Nuclear Physics 1
http://arxiv.org/abs/0704.1396v1
multifragmentation data can be correlated with the presence of a phase transition in
hot nuclei. Indeed, abnormal fluctuations of configurational energy (AFCE) 6,7 can be
related to the negative heat capacity signal 8, and the fossil signal of spinodal decom-
position 9 can illustrate the density fluctuations occurring when the nuclei pass through
the spinodal zone 10. 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.
2 Canonical description of first-order phase transition.
Let us consider an observable E, known on average, free to fluctuate. The least biased
distribution will be a Boltzmann-Gibbs 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.
Pcanβ (E) =
e S(E)−βE with Zcanβ =
dE e S(E)−βE (1)
S (E) ∼ S (Eβ) + (E− 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).
(s.g.)
(E) =
2πσ2E
(E− Eβ)
with σ2E = −
P(s.g.)(E∗,Z1) =
2πdetΣ
~xΣ−1~x, ~x =
E∗ − Eβ
Z1 − Zβ
, Σ =
σ2E ρ σEσZ
ρ σEσZ σ
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 distribution 12 (def. 4). Parameters
of this function gathered in the variance-covariance matrix are also deduced from the
curvature matrix of the 2D microcanonical entropy 12,13.
P(d.g.)(E∗,Z1) = Nliq × P
(s.g.)
liq (E
∗,Z1) + Ngaz × P
(s.g.)
gaz (E
∗,Z1) (5)
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
Bormio 2007: XLV International winter meeting on Nuclear Physics 2
parameter(s) direction(s) 14. Instabilities occur and, due to the finite size of the system,
the surface energy effects cause the non-additivity of the entropy leading at the end of
the PT to a residual convex entropy for the two-phase 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 P canβ (E
∗) as well as the two-
dimensional distribution P canβ (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 distribution 12. Experimentally, this relation is not so clear: the weight
of the different states has no reason to be exponential and the measured distribution
(E∗) is modified by a factor gexp(E
∗) which is determined in a large part by en-
trance channel effects and data sorting : P exp(E∗, Z1) = e
S(E∗,Z1)gexp(E
∗). The relative
population of the different values of the E∗ distribution looses its thermostatistic meaning
(P exp(E∗) ∝ gexp(E
∗)P canβ (E
∗)eβE
). We cannot therefore directly compare experimen-
tal and canonical distributions and deduce entropy properties of the system.
Pexpω (E
∗,Z1) = ω(E)× P
exp(E∗,Z1) (6)
with ω(E∗) =
P(exp)(E∗,Z1) dZ1
2.1 Renormalization method.
In 12, 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 Z1 is not a one-to-one correspondence, it could reflect a residual convex intruder
of the entropy.
2.2 Spurious 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 models 12
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.
3 Data selection and first observation of Z1 distributions.
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 quasi-projectile 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 3
20 40 60 80
2000 Source
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 10
(E (exp)P
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 10
(E (exp)ω P
 (MeV/A) *E
1 2 3 4 5 6 7 8 9 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 signal 13,15. We evaluate the
excitation energy using a standard calorimetry procedure 16,17. We compute the energy
balance event-by-event in the centre of mass of the QP sources calculated with fragments
only to minimize the effect of pre-equilibrium 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 Z1 and E
∗ observables
is shown, the latter covering a range between roughly 1 and 8 MeV/A (lower-right part).
Spinodal zone limits obtained with the AFCE signal are around 2.5 and 5.8 MeV/A
for this set of data 13. The shape of the distribution P exp(E∗, Z1) (upper right part)
shows the dominance of low dissipation-large Z1 events and reflects the cross-section
distribution and data selection. If we look at the corresponding Z1 distribution (upper
left part) we do not see any clear signal of bimodality: a large part of statistics is around
65-70, and only a shoulder is visible around 30-40. This particular shape could reflect
Bormio 2007: XLV International winter meeting on Nuclear Physics 4
the lack of statistics for the ”gas-like” events. If we apply the renormalization procedure
(eq. 6) we obtain (lower left part) a P expω (E,Z1) distribution which has a double humped
shape, tending to prove that this procedure can reveal bimodality.
10 20 30 40 50 60 70 80 90
  (exp)ωP
  (s.g.)ωP
  (d.g.)ωP
> 1 <Z
20 30 40 50 60 70
10 20 30 40 50 60 70 80 90
[1.25 , 3.00[
[3.00 , 6.25[
[6.25 , 9.75[
10 20 30 40 50 60 70
 [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 squares-left Y axis) and skewness (colored
triangles-right Y axis) are plotted as a function of the mean value (Z1-lower 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 Canonical-Experimental comparisons.
To confirm that the two-hump distribution of Z1 signals 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 5
Parameters [10,79] [25,55] Errors (%)
Ē∗liq 2,10 1,67 23
2,09 1,66 23
60,2 62,2 3
12,9 9,85 4
Ē∗gaz 7,11 6,81 4
3,07 2,97 3
21,1 23,8 12
σZgaz
15,2 18,8 2
ρ -0,906 -0,860 4
Nliq/Ngaz 1,12 0,66 52
Ndof 605 387 -
χ2 1488 646 -
χ2/Ndof 2,459 1,669 -
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; Ē∗,Z̄1,σE∗ ,σZ1 stand 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.
the analytic expectation for a system exhibiting or not a first order PT. We apply the
same renormalization to P
(s.g.)
(E) and P
(d.g.)
(E) and try to reproduce the data. We
focus on the projection on the Z1 axis 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 no-transition case can not curve itself in the Z1=40-50 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 Z1 distribution 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
30-40 for < Z1 >. This maximum of fluctuations signs the core of the spinodal zone
which corresponds to the hole in P expω (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 distribution 2
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
Bormio 2007: XLV International winter meeting on Nuclear Physics 6
select the region E∈[3.00,6.25[ to compare the two reweighted distributions P expω (E,Z1)
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
Z1 where fits are performed ([10,79] and [25,55]). These two best solutions are shown for
the projection on the Z1 axis.
 (MeV/A) *E
2 4 6 8 10
2 / Tk
2σ> s<A
2 10×)  s / Z max (Z
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 Z1 range 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 Ē∗, Z̄1, 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 events 13,18. We can further explore the coherence between the two signals by
looking at the fluctuations associated to Z1 and to the Freeze-Out configurational kinetic
energy 8: 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 7
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 Z1 observable 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 in 12 and analyses on data
are in progress.
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