28
S. Mallik et al.
bution (hence the name bimodality). Different theoretical studies (from lattice gas
model and statistical models) [4–10] as well as experimental observations [11–13]
confirm this kind of signature of thermal phase transition in heavy-ion reactions.
On the other side, some other theoretical calculations [14–17] as well as experimental measurements [18–20] conclude that, a memory of the entrance channel is
clearly present and thermal equilibrium is not achieved. The signal was interpreted
in these studies as a dynamical bifurcation of reaction mechanism, induced by fluctuations of the collision rate, which leads to fluctuations of the collective momentum
distribution as expected in complex nonlinear dynamical systems.
Therefore, the origin of the experimentally observed bimodality is still not clear
completely. In the previous dynamical approaches used [14–17] to study the bimodality phenomenon, the collision final state was determined by the semiclassical onebody transport equation itself, considering simulations evolving until asymptotic
times. However, these approaches lack the necessary correlations to properly treat
fragment formation in the exit channel, even if they are known to very well describe
the entrance channel of heavy-ion reactions at intermediate energy. For this reason,
to have a quantitative reproduction of experimental data, the secondary decay of the
dynamically formed primary fragments is typically treated in two-step calculations,
coupling the transport dynamics to a statistical model (or “afterburner”). As the primary interest is phase transition in nuclear matter due to the nuclear force alone, most
theoretical models have considered symmetric nuclear matter where the Coulomb
force is switched off [21, 22]. Here, we follow the same practice.
In order to study the dynamical stage for phase transition, one needs to simulate
collisions between fairly large nuclei. In order to do that, a simplified yet accurate
method of transport model based on Boltzmann–Uehling–Uhlenbeck (BUU) equation is developed recently [23] which allows calculation of fluctuations in systems
much larger than what was considered feasible in a well-known and already existing model [27]. For studying the de-excitation phase, Canonical Thermodynamical
Model (CTM) is used.
From this theoretical study, it is observed that, depending on the incident energy
and impact parameter of the reaction, dynamical as well as statistical bimodality
mechanisms can appear, meaning that the different scenario proposed in the literature are both potentially observable in heavy-ion data. Specifically, fluctuations in
the stopping dynamics in central collisions lead to different reaction mechanisms
that can coexist in the sample characterized by a well-defined value of the impact
parameter. This gives rise to a bimodal behavior of the largest cluster probability
distribution that can survive to the secondary de-excitation if the deposited energy
is low enough, which happens at incident energies around the Fermi energy domain
(40 MeV/nucleon). At higher incident energies (100 MeV/nucleon), focusing on
binary mid-peripheral reactions, the fluctuations in the energy deposition leads to
an excitation energy distribution for the quasi-spectator source which is close to the
liquid–gas phase transition range. For these events, local equilibrium is achieved and
a thermal bimodality is observed in agreement with statistical expectations.
This article is structured as follows. The modifications in the fluctuation included
BUU model which is essential for studying liquid–gas phase transition is introduced
S. Mallik et al.
bution (hence the name bimodality). Different theoretical studies (from lattice gas
model and statistical models) [4–10] as well as experimental observations [11–13]
confirm this kind of signature of thermal phase transition in heavy-ion reactions.
On the other side, some other theoretical calculations [14–17] as well as experimental measurements [18–20] conclude that, a memory of the entrance channel is
clearly present and thermal equilibrium is not achieved. The signal was interpreted
in these studies as a dynamical bifurcation of reaction mechanism, induced by fluctuations of the collision rate, which leads to fluctuations of the collective momentum
distribution as expected in complex nonlinear dynamical systems.
Therefore, the origin of the experimentally observed bimodality is still not clear
completely. In the previous dynamical approaches used [14–17] to study the bimodality phenomenon, the collision final state was determined by the semiclassical onebody transport equation itself, considering simulations evolving until asymptotic
times. However, these approaches lack the necessary correlations to properly treat
fragment formation in the exit channel, even if they are known to very well describe
the entrance channel of heavy-ion reactions at intermediate energy. For this reason,
to have a quantitative reproduction of experimental data, the secondary decay of the
dynamically formed primary fragments is typically treated in two-step calculations,
coupling the transport dynamics to a statistical model (or “afterburner”). As the primary interest is phase transition in nuclear matter due to the nuclear force alone, most
theoretical models have considered symmetric nuclear matter where the Coulomb
force is switched off [21, 22]. Here, we follow the same practice.
In order to study the dynamical stage for phase transition, one needs to simulate
collisions between fairly large nuclei. In order to do that, a simplified yet accurate
method of transport model based on Boltzmann–Uehling–Uhlenbeck (BUU) equation is developed recently [23] which allows calculation of fluctuations in systems
much larger than what was considered feasible in a well-known and already existing model [27]. For studying the de-excitation phase, Canonical Thermodynamical
Model (CTM) is used.
From this theoretical study, it is observed that, depending on the incident energy
and impact parameter of the reaction, dynamical as well as statistical bimodality
mechanisms can appear, meaning that the different scenario proposed in the literature are both potentially observable in heavy-ion data. Specifically, fluctuations in
the stopping dynamics in central collisions lead to different reaction mechanisms
that can coexist in the sample characterized by a well-defined value of the impact
parameter. This gives rise to a bimodal behavior of the largest cluster probability
distribution that can survive to the secondary de-excitation if the deposited energy
is low enough, which happens at incident energies around the Fermi energy domain
(40 MeV/nucleon). At higher incident energies (100 MeV/nucleon), focusing on
binary mid-peripheral reactions, the fluctuations in the energy deposition leads to
an excitation energy distribution for the quasi-spectator source which is close to the
liquid–gas phase transition range. For these events, local equilibrium is achieved and
a thermal bimodality is observed in agreement with statistical expectations.
This article is structured as follows. The modifications in the fluctuation included
BUU model which is essential for studying liquid–gas phase transition is introduced