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\newcommand{\Zm}{Z$_{\text{max}}$ }
\newcommand{\zm}{Z$_{\text{max}}$}

\newcommand{\Sm}{$s_{\text{max}}$ }
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\newcommand{\Rat}{$R$ }
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\newcommand{\Xesn}{$^{129}$\text{Xe}+$^{nat}$\text{Sn} }
\newcommand{\xesn}{$^{129}$\text{Xe}+$^{nat}$\text{Sn}}

\newcommand{\etzt}{(Et$_{12}\times$Z$_{\text{tot}}$)}
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\begin{document}

\title{Nuclear multifragmentation time-scale and fluctuations of largest
fragment size}


\author{D. Gruyer}


\author{J.D. Frankland}


\thanks{Corresponding author}

\email{john.frankland@ganil.fr}


\affiliation{GANIL, CEA-DSM/CNRS-IN2P3, Bvd. Henri Becquerel, F-14076 Caen CEDEX,
France}


\author{R. Botet}


\affiliation{Laboratoire de Physique des Solides, CNRS UMR8502, Université de
Paris-Sud bât. 510, F-91405 Orsay CEDEX, France}


\author{M. P\l{}oszajczak}


\author{E. Bonnet}


\author{A. Chbihi}


\affiliation{GANIL, CEA-DSM/CNRS-IN2P3, Bvd. Henri Becquerel, F-14076 Caen CEDEX,
France}


\author{G. Ademard}


\affiliation{Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud
11, F-91406 Orsay CEDEX, France}


\author{M. Boisjoli}


\affiliation{GANIL, CEA-DSM/CNRS-IN2P3, Bvd. Henri Becquerel, F-14076 Caen CEDEX,
France}


\affiliation{Département de physique, de génie physique et d'optique, Université
Laval, Québec, G1V 0A6 Canada}


\author{B. Borderie}


\affiliation{Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud
11, F-91406 Orsay CEDEX, France}


\author{R. Bougault}


\affiliation{LPC, CNRS/IN2P3, Ensicaen, Université de Caen, F-14050 Caen CEDEX,
France}


\author{D. Guinet}


\author{P. Lautesse}


\affiliation{Institut de Physique Nucléaire, Université Claude Bernard Lyon 1,
CNRS/IN2P3, F-69622 Villeurbanne CEDEX, France }


\author{L. Manduci}


\affiliation{École des Applications Militaires de l'Énergie Atomique, B.P. 19,
F-50115 Cherbourg, France}


\author{N. Le Neindre}


\affiliation{LPC, CNRS/IN2P3, Ensicaen, Université de Caen, F-14050 Caen CEDEX,
France}


\author{P. Marini}


\affiliation{GANIL, CEA-DSM/CNRS-IN2P3, Bvd. Henri Becquerel, F-14076 Caen CEDEX,
France}


\author{P. Paw\l{}owski}


\affiliation{H. Niewodnicza\'{n}ski Institute of Nuclear Physics, PL-31342 Kraków,
Poland}


\author{M. F. Rivet}


\affiliation{Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud
11, F-91406 Orsay CEDEX, France}


\author{E. Rosato}


\author{G. Spadaccini}


\author{M. Vigilante}


\affiliation{Dipartimento di Scienze Fisiche e Sezione INFN, Universita di Napoli
\textquoteleft{}\textquoteleft{}Federico II,\textquoteright{}\textquoteright{}
I-80126 Napoli, Italy}


\author{J.P. Wieleczko}


\affiliation{GANIL, CEA-DSM/CNRS-IN2P3, Bvd. Henri Becquerel, F-14076 Caen CEDEX,
France}


\collaboration{INDRA collaboration}


\noaffiliation%{}
\begin{abstract}
We show that the biggest fragment charge distribution in central collisions
of \Xesn leading to multifragmentation is an admixture of two asymptotic
distributions observed for the lowest and highest bombarding energies.
The evolution of the relative weights of the two components with bombarding
energy is shown to be analogous to that observed as a function of
time for the largest cluster produced in irreversible fragmentation
for a finite system. This analogy allows us to speak of pseudo-critical
nuclear multifragmentation around $E_{beam}\approx30$ \mev. We infer
that the size distribution of the largest fragment in nuclear multifragmentation
is also characteristic of the time scale of the process, which is
largely determined by the onset of radial expansion in this energy
range.
\end{abstract}

\pacs{25.70.Pq}

\maketitle


Phase transitions occupy a central place in many fields of physics
such as condensed matter, collisions of atomic aggregates, and most
recently, in formation of quark-gluon plasma in ultra-relativistic
hadron collisions. They allow to investigate the equation of state
and phase diagram of the system under study. The case of nuclear multifragmentation,
as observed in intermediate energy heavy-ion collisions \citep{Borderie2008Nuclear},
provides a unique opportunity to study not only thermodynamical properties
of nuclear matter, but also phase transitions in \emph{finite} systems.

The universal character of order parameter fluctuations in finite
systems \citep{Botet2000Universal,UniversalFluctuationsBook}, provides
a good framework in which to address such questions. Using such a
model-independent analysis, it was shown that the size (atomic number)
of the largest fragment produced in multifragmentation events, \zm,
behaves like an order parameter, \emph{i.e.} the scaling properties
of its fluctuations change with increasing energy \citep{PhysRevLett.86.3514}.
In this sense nuclear multifragmentation belongs to a large class
of phase transitions or critical phenomena for which the size of the
largest cluster is the order parameter, for example site- or bond-percolation
on lattices, reversible or irreversible aggregation \citep{Botet2000Universal}.
Using the latter class of model we will show that we can extend this
analogy, by studying the evolution of the order parameter distribution
in the critical region.

In \citep{Frankland2005Modelindependent} it was shown that a distinct
asymptotic form of the largest fragment distribution in multifragmentation
can be associated with each scaling regime : a quasi-Gaussian symmetric
\Zm distribution at low bombarding energies, and an asymmetric Gumbel
distribution \citep{Gumbel1958Statistics} in the high-energy disordered
regime. In this letter we will study in more detail the transition
from one regime to the other, using new data on \Zm distributions
for \Xesn central collisions measured with INDRA \citep{Pouthas1995418}
at bombarding energies intermediate between the two asymptotic regimes.
We will show that, in fact, the \Zm distributions evolve continuously
from one asymptotic distribution to the other with increasing bombarding
energy, and, by analogy with the evolution of the order parameter
distribution in the irreversible aggregation model, deduce a link
between \Zm fluctuations and the time-scale of nuclear multifragmentation.

The irreversible aggregation model describes an out-of-equilibrium
clusterization process, whose order parameter is the average size
of the largest cluster at any given time, \sm. The evolution of the
droplet size population is given by the Smoluchowski equations \citep{PhysRevLett.54.1396},
\begin{equation}
\frac{dc_{s}}{dt}=\frac{1}{2}\sum_{i+j=s}K_{ij}c_{i}c_{j}-\sum_{i=1}^{N}K_{is}c_{i}c_{s}\label{eq:Smolu}
\end{equation}
where $c_{s}$ is the concentration of droplets of size $s$, $K_{\alpha\beta}=K_{0}\alpha\beta$
is the coalescence probability per unit of time between two droplets
of respective size $\alpha$ and $\beta$, $K_{0}$ is a constant
and $N$ is the size of the system. This model exhibits a second-order
phase transition after a critical time, $t_{c}$ \citep{PhysRevLett.54.1396}.

Precise limits of the critical domain for finite systems, where the
correlation length becomes comparable to the size of the system \citep{PhysRevLett.87.188901},
are not easily obtained. An estimation using the width of the largest
cluster size distribution, $\sigma$(\sm), has been proposed in \citep{Botet2011Tokyo}.
In the infinite system, this critical quantity must diverge at the
critical time. As shown in Fig. 1 \ref{fig:sigXx}, for a finite system
with $N=216$ particles, $\sigma$(\sm) reaches a sharp maximum at
the pseudo-critical time, $t_{c}^{*}=1.28t_{c}$. We define the lower
and upper limits of the critical domain, $t^{-}$ and $t^{+}$, as
the times at which $\sigma$(\sm) is above 10\% of its maximum: $t^{-}=0.5t_{c}$
and $t^{+}=2.6t_{c}$ (see Fig. \ref{fig:sigXx}). 
\begin{figure}[floatfix]
\centering \includegraphics[width=0.99\linewidth]{fig1} \caption{Standard deviation of the largest cluster size, $\sigma$(\sm), as
a function of the time $t$, for irreversible aggregation with $N=216$;
$t_{c}^{*}$, $t^{-}$ and $t^{+}$ are the pseudo-critical time and
the limits of the critical domain (see text). \label{fig:sigXx} }
\end{figure}


The order parameter distribution for irreversible aggregation is not
known exactly, but it was proposed that the asymptotic \Sm distribution
for $t\ll t_{c}^{*}$ is that of Gumbel \citep{Botet2011Tokyo}. The
average size of the largest cluster evolves with time as more and
more coalescence of smaller clusters takes place. At long times, the
order parameter is then essentially of additive nature. From the central
limit theorem, this results in an asymptotic Gaussian distribution
of \Sm for $t\gg t_{c}^{*}$. In the critical domain, for $t^{-}<t<t^{+}$,
fluctuations are so large that similarly prepared systems can exhibit
one or the other behavior. This suggests that the effective \Sm distribution
for an ensemble of systems, at any given intermediate time, is an
admixture of a Gaussian distribution and a Gumbel distribution.

We performed numerical simulations of the Smoluchowski equations,
Eq.\eqref{eq:Smolu}, for finite systems with $N\sim50-200$ particles,
comparable with nuclear system sizes. The obtained \Sm distributions
were then fitted with the following admixture of Gaussian and Gumbel
distributions: 
\begin{eqnarray}
f(x)=\eta f_{Ga}(x)+(1-\eta)f_{Gu}(x),\label{fitEqua}
\end{eqnarray}
 where, 
\begin{align*}
f_{Ga}(x) & =\frac{1}{\sigma\sqrt{2\pi}}\exp{\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)},\\
f_{Gu}(x) & =\frac{1}{b_{m}}\exp{\left[-\frac{(x-a_{m})}{b_{m}}-\exp{\left(-\frac{(x-a_{m})}{b_{m}}\right)}\right]},
\end{align*}
and $x=$\sm. The five parameters ($\eta$, $\mu$, $\sigma$, $a_{m}$
and $b_{m}$) are left to vary freely, apart from the constraint that
$\eta$ should be between zero and one. Fits were performed using
a binned maximum-likelihood estimation method \citep{ROOFIT}. In
order to quantify the evolution of the relative weight of each distribution,
we introduce the ratio: 
\begin{equation}
R=\frac{I_{Ga}-I_{Gu}}{I_{Ga}+I_{Gu}}=2\eta-1\label{defREq}
\end{equation}
 where $I_{Ga}$ and $I_{Gu}$ are the integral of the components
defined in Eq.(\ref{fitEqua}). This ratio equals $+1$ for a pure
Gaussian distribution and $-1$ for a pure Gumbel distribution.

The results of the fitting procedure applied to the distribution of
the largest cluster size between $t^{-}$ and $t^{+}$ are presented
in Fig. \ref{fig:ResultSmolu}. At short times, $t^{-}<t<t_{c}^{*}$,
the distribution is almost purely ``Gumbellian'', with a small Gaussian
component (Fig. \ref{fig:ResultSmolu}(a)), and the ratio $R\approx-1$
(Fig. \ref{fig:ResultSmolu}(e)). As time increases, the Gaussian
component increases, such that between the infinite-system critical
time $t_{c}$ (Fig. \ref{fig:ResultSmolu}(b)) and the pseudo-critical
time $t_{c}^{*}$ (Fig. \ref{fig:ResultSmolu}(c)), both components
have comparable weights, and the ratio $R$ passes through zero (Fig.
\ref{fig:ResultSmolu}(e)). The very large fluctuations of \Sm observed
at these times in Fig.\ref{fig:sigXx} are due to the superposition
of the two distributions, whose combined width is far greater than
the intrinsic variance of each component. Finally, for the longest
time-scales, the Gumbel component disappears (Fig. \ref{fig:ResultSmolu}(d))
and $R$ tends towards +1 (Fig. \ref{fig:ResultSmolu}(e)).
\begin{figure}[th]
\begin{centering}
\subfigure{\includegraphics[width=0.99\linewidth]{fig2a}\label{FitSmolu}}
\par\end{centering}

\begin{centering}
\subfigure{\includegraphics[width=0.99\linewidth]{fig2b}\label{RvsTSmolu}}
\par\end{centering}

\centering{}\caption{(color online) Analysis of Smoluchowski calculations for $N=216$
particles: (a-d) distribution of the size of the largest cluster $P(\text{\sm})=(1/N)dN/d\text{{\sm}}$
for different times in the critical domain; (Black solid curve) best
fit to the data using Eq.(2); (Red dashed curve) Gumbel component;
(Blue dotted curve) Gaussian component; (e) evolution of the ratio
$R$ between the two components (Eq.(3)) in the critical domain (see
Fig. \ref{fig:sigXx}), as a function of the time $t$ in unit of
$t_{c}$ (see text); vertical lines are the same as in Fig. \ref{fig:sigXx}.\label{fig:ResultSmolu} }
\end{figure}
 

The analysis of Smoluchowski calculations shows that the decomposition
of the \Sm distributions into an admixture of two asymptotic distributions
seems to be relevant. Similar results were obtained for all simulated
system sizes, $N$: as $N$ increases the transition becomes more
abrupt and the value $R=0$ occurs closer and closer to $t_{c}$.
It should be noted that we have obtained similar results with a three-dimensional
percolation model \citep{StaufferPerco}, which describes the at-equilibrium
formation of clusters on a geometric lattice as a function of bond
probability, $p$, with a second-order transition at the critical
probability, $p_{c}$. Such models have often been used to study criticality
in nuclear multifragmentation \citep{Bauer1988Percolation,CampiEurPhysJD2000}.
In all cases the value $R=0$ occurs in the critical region of the
underlying phase transition, between the infinite-system critical
point and the $N$-dependent maximum of \Sm fluctuations. We will
now apply the same analysis to the largest fragment distributions
measured for nuclear multifragmentation in central heavy-ion collisions.
\begin{figure}[ht]
\centering \includegraphics[width=0.99\linewidth]{fig3} \caption{(color online) (Points) Log-log correlation between the first two
cumulant moments ($\left\langle \text{{\zm}}\right\rangle ^{2}$and
$\sigma^{2}$) of the order parameter's distribution; (Lines) linear
fits performed in the range 50-32 \Mev (resp. 32-25 \mev) which
correspond to a slope $\Delta\sim1$ (resp. $\Delta\sim0.5$) (see
text).\label{fig:deltaScaling} }
\end{figure}


Collisions of \Xesn were measured using the INDRA $4\pi$ array at
the GANIL accelerator facility. This charged product detector, composed
of 336 detection cells arranged according to 17 rings centered on
the beam axis, covers 90\% of the solid angle. More experimental details
can be found in \citep{Pouthas1995418,Frankland2005Modelindependent,Bonnet2009Fragment}.
Data were taken during two separate campaigns. Beam energies of 27,
29, and 35 \Mev have been performed specifically in order to probe
in more detail the energy range around the change of scaling regime
observed in \citep{PhysRevLett.86.3514}: those data are presented
here for the first time.

As in previous works \citep{PhysRevLett.86.3514,Frankland2005Modelindependent},
we study multifragmentation events occurring in central collisions,
requiring the geometrical overlap between projectile and target to
be as close as possible to total. In addition, the correct measurement
of the largest fragment distribution requires the detection of nearly
all fragments event by event. For this purpose we will use two global
variables, \Et and \zt. \Et is the total transverse energy of light
charged particles ($Z=1,2$), which increases with collision centrality
\citep{Plagnol1999Onset}, while \Zt is the sum of the atomic numbers
of all detected charged products in each event. In \citep{Frankland2005Modelindependent}
separate energy-dependent cuts were made in the \Et and \Zt distributions.
In this new analysis, in order to ensure equivalent selections for
all beam energies we retain events which maximize the quantity \etzt.
For each beam energy, a cut was defined corresponding to the last
percentile of the \Etzt distribution measured with the online trigger
condition (charged product multiplicity $M\geq4$).

Let us first examine the scaling properties of \Zm fluctuations including
the three new data points at 27, 29 and 35 \mev. Figure \ref{fig:deltaScaling}
shows the correlations between the first two cumulant moments of the
\Zm distribution ($\left\langle \text{{\zm}}\right\rangle ^{2}$and
$\sigma^{2}$), for each beam energy. As in Fig. 1(c) of \citep{PhysRevLett.86.3514},
the data fall on two branches, $\sigma^{2}\sim\left\langle \text{{\zm}}\right\rangle ^{2\Delta}$,
with different values for the scaling parameter, $\Delta$ \citep{Botet2000Universal,UniversalFluctuationsBook}:
$\Delta\sim1$ above 32 \mev, and $\Delta\sim1/2$ below 32 \mev.
The new data points present a consistent behavior which follows the
systematic scaling trend. It should also be noted that the scaling
properties of \Zm distributions in central collisions are quite independent
of the detailed selection procedure, as different methods were used
to select events in \citep{PhysRevLett.86.3514,Frankland2005Modelindependent}.
\begin{figure}[h]
\begin{centering}
\subfigure{\label{fitXeSn50}\includegraphics[width=0.99\linewidth]{fig4a}}
\par\end{centering}

\centering{}\subfigure{\label{RvsE}\includegraphics[width=0.99\linewidth]{fig4b}}\caption{(color online) Analysis of \Xesn reactions: (a-d) distribution of
the charge of the largest detected fragment $P(\text{\zm})=(1/N)dN/d\text{{\zm}}$
for different beam energies, (Black solid curve) best fit to the data
using Eq.(2), (Red dashed curve) Gumbel component, (Blue dotted curve)
Gaussian component; and (e) evolution of the ratio R between the two
components (Eq.(3)), as a function of the beam energy.\label{fig:fitXeSn} }
\end{figure}


The results of the fits performed on the \Zm distribution using Eq.(\ref{fitEqua})
are shown in Fig.\ref{fig:fitXeSn}. It can be seen that, for all
analyzed energies, a good fit to the experimental distribution is
achieved: reduced $\chi^{2}$ values all lie between approximately
4 and 10, and can further be reduced if \Zm distributions are smoothed
to remove odd-even staggering of the yields \citep{Yang1999Oddeven}:
in that case all $\chi^{2}$ values are close to 2, except for data
at 32 \Mev for which a significantly larger value $(\chi^{2}\sim4)$
is obtained. At the highest energy considered here (50 \mev), the
result of the fit is an almost pure Gumbel distribution (Fig. \ref{fig:fitXeSn}(d)).
For the other bombarding energies, both contributions are present,
and the relative importance of the Gaussian component increases with
decreasing energy. The quantity \rat, defined in Eq.(\ref{defREq}),
is evaluated for each bombarding energy, and its evolution is presented
in Fig. \ref{fig:fitXeSn}(e). The vertical bars show the uncertainty
coming from the fitting procedure: $\sigma_{R}=2\sigma_{\eta}$, calculated
from Eq.(\ref{defREq}). The value $R=0$ is reached between 29 and
32 \mev, in the same bombarding energy range as the change of $\Delta$-scaling
(Fig. \ref{fig:deltaScaling}).

A strong similarity can be seen between the results of the analysis
for irreversible aggregation (Fig. \ref{fig:ResultSmolu}) and central
collisions of \Xesn (Fig. \ref{fig:fitXeSn}). In the aggregation
model the order parameter distribution depends on the length of time
during which clusters can form. In central heavy-ion collisions, a
determining factor for the time-scale of fragment formation is the
radial expansion of the multifragmenting system, which increases with
bombarding energy \citep{Borderie2008Nuclear}. Fragment sizes can
evolve as long as exchanges of nucleons can take place between them,
\emph{i.e.} until freeze-out at which the short-range nuclear interaction
between fragments ceases and they begin to move apart. It has been
shown that for central \Xesn reactions the onset of significant radial
expansion occurs for beam energies above $25$ \mev \citep{Bonnet2009Fragment}.
The similarity between Figs. \ref{fig:ResultSmolu} and \ref{fig:fitXeSn}
can therefore be understood in terms of fragment size distributions
being determined on shorter and shorter time-scales due to increasingly
rapid expansion.

We can use this interpretation to understand the systematic evolution
of $\Delta$-scaling with beam energy and system mass presented in
\citep{Frankland2005Modelindependent}: for $^{58}$Ni+$^{58}$Ni
collisions, measured from 32 \Mev to 90 \mev, a change of $\Delta$-scaling
and of the form of the \Zm distribution were observed, as for \Xesn
but at a higher bombarding energy of 52 \mev; for the lighter system
$^{36}$Ar+KCl, studied from 32 \Mev to 74 \mev, only the $\Delta=1/2$
regime was observed, with quasi-Gaussian \Zm distributions; on the
other hand, for the much heavier $^{197}$Au+$^{197}$Au system, at
bombarding energies between 40 \Mev and 80 \mev, only the $\Delta=1$
regime occurs, with Gumbel \Zm distributions. 

Radial expansion in central heavy-ion collisions occurs after significant
compression of the incoming nuclear fluid, and as such depends not
only on static nuclear matter properties such as incompressibility,
but also on transport properties such as the degree of stopping achieved
in the collision. The latter increases with the mass of the colliding
nuclei, as shown in \citep{Lehaut2010Study}. Thus for light systems,
such as $^{36}$Ar+KCl or $^{58}$Ni+$^{58}$Ni, the bombarding energy
required to achieve sufficient initial compression for there to be
significant radial expansion is higher than for the heavier systems
like \Xesn and $^{197}$Au+$^{197}$Au. This explains why the $\Delta$-scaling
transition occurs at higher energy for $^{58}$Ni+$^{58}$Ni than
for \Xesn. For the very light $^{36}$Ar+KCl system the threshold
is higher still than for $^{58}$Ni+$^{58}$Ni, outside the range
of measured bombarding energies. On the other hand, for $^{197}$Au+$^{197}$Au
both the greater initial compression and the far larger Coulomb contribution
may come into play in order to reduce the fragment formation time-scale
even at the lowest bombarding energy.

We have shown that, for finite systems, the largest cluster size distribution
in critical aggregation models is an admixture of the two asymptotic
distributions observed far below and above the critical region. This
result holds true for both equilibrium (percolation) and out-of-equilibrium
(irreversible aggregation) models. A similar decomposition has been
shown for the experimentally-observed charge distribution of the largest
fragment per event produced in nuclear multifragmentation, indicating
that the critical domain lies around $E_{beam}\approx30$\Mev for
the \Xesn system. By analogy with the irreversible aggregation model,
where the form of the order parameter distribution depends on the
time-scale of the process, we interpret such criticality along with
the corresponding change of $\Delta$-scaling as the onset of an ``explosive''
multifragmentation regime in which initially compressed heated nuclear
liquid clusterizes in the presence of significant radial expansion
\citep{Bauer1993ExplosiveFlow}. The mass-dependence of the energy
at which the onset occurs for different (symmetric) systems is related
to nuclear stopping and hence to transport properties of hot nuclear
matter. Such an overall picture is both consistent with, and provides
a link between, recent results on the role of radial expansion in
nuclear multifragmentation \citep{Bonnet2009Fragment,Bonnet2010New}
and the systematic study of nuclear stopping around the Fermi energy
\citep{Lehaut2010Study}.
\begin{acknowledgments}
The authors would like to thank the staff of the GANIL Accelerator
facility for their continued support during the experiments. D. G.
gratefully acknowledges the financial support of the Commissariat
à l'Énergie Atomique and the Conseil Régional de Basse-Normandie.
This work was also supported by the Centre National de la Recherche
Scientifique and by the Ministère de l'Éducation Nationale.
\end{acknowledgments}
\bibliographystyle{apsrev}
\bibliography{paper}

\end{document}
