Dynamical and statistical bimodality in nuclear fragmentation

The origin of bimodal behavior in the residue distribution experimentally measured in heavy ion reactions is reexamined using Boltzmann-Uehling-Uhlenbeck simulations. We suggest that, depending on the incident energy and impact parameter of the reaction, both entrance channel and exit channel effects can be at the origin of the observed behavior. Specifically, fluctuations in the reaction mechanism induced by fluctuations in the collision rate, as well as thermal bimodality directly linked to the nuclear liquid-gas phase transition, are observed in our simulations. Both phenomenologies were previously proposed in the literature but presented as incompatible and contradictory interpretations of the experimental measurements. These results indicate that heavy ion collisions at intermediate energies can be viewed as a powerful tool to study both bifurcations induced by out-of-equilibrium critical phenomena, as well as finite-size precursors of thermal phase transitions.

Figure 1

Variation of average mass of largest cluster $A_{\text{max}}$ (red solid lines) and second-largest cluster $A_2$ (blue dashed lines) with time as calculated from BUU model for (a) $b=0$ fm, (b) $b=3$ fm, (c) $b=6$ fm, and (d) $b=9$ fm at projectile beam energy 100 MeV/nucleon.

Figure 2

Variation of isotropy of momentum distribution ($I$) of the largest cluster with time calculated from BUU model for (a) $b=0$ fm, (b) $b=3$ fm, (c) $b=6$ fm, and (d) $b=9$ fm at projectile beam energy 100 MeV/nucleon.

Figure 3

Probability distribution of largest cluster $P\left(A_{\text{max}}\right)$ (red solid lines) and normalized mass asymmetry of two largest masses $P\left(a_2\right)$ (black dashed lines) at constant projectile beam energy 100 MeV/nucleon but four different impact parameters (a) $b=0$ fm, (b) $b=3$ fm, (c) $b=6$ fm, and (d) $b=9$ fm calculated from BUU model at freeze-out time $t$ = 175 $\text{fm}/\mathrm{c}$. At each impact parameter 2000 events are simulated.

Figure 4

Largest cluster scattering angle (left panel) and momentum (right panel) probability distribution for $A_{\text{max}}\ge 37$ (red lines) and $A_{\text{max}}<37$ (blue lines) for central collisions ($b=0$ fm) at projectile beam energy 100 MeV/nucleon calculated from BUU model at freeze-out time $t$ = 175 $\text{fm}/\mathrm{c}$. To study this, 2000 events are simulated. The average value of 10 degrees and 10 MeV are shown for angle and momentum, respectively.

Figure 5

Excitation ($E^{*}$) (upper panels) and temperature ($T$) (lower panels) probability distribution for largest (red dash dotted lines) and second-largest cluster (blue dotted lines) at constant projectile beam energy 100 MeV/nucleon but two different impact parameters $b=0$ fm (left panels), $b=3$ fm (middle panels), and $b=6$ fm (right panels). At each impact parameter 2000 events are simulated. The average value of 1 MeV/nucleon and 1 MeV are shown for excitation and temperature, respectively.

Figure 6

Final largest cluster probability distribution (upper panels) and mass distributions (lower panels) studied after CTM calculation at constant projectile beam energy 100 MeV/nucleon but three different impact parameters $b=0$ fm (left panels), $b=3$ fm (middle panels), and $b=6$ fm (right panels). At each impact parameter 2000 events are simulated.

Figure 7

Largest cluster probability distribution for crossed events (blue lines) and stopped events (red lines) studied after BUU model calculation (left panel) and CTM calculation (right panel) for central collisions ($b=0$ fm) at projectile beam energy 40 MeV/nucleon. To study this 500 events are simulated and the BUU calculation is stopped at $t=400\text{fm}/\mathrm{c}$.

Figure 8

Probability distribution of normalized mass asymmetry of two largest masses $P\left(a_2\right)$ studied after BUU model calculation (black dashed line) and CTM calculation (red solid line) for central collisions ($b=0$ fm) at projectile beam energy 40 MeV/nucleon. To study this 500 events are simulated and the BUU calculation is stopped at $t=400\text{fm}/\mathrm{c}$.