Bimodality emerges from transport model calculations of heavy ion collisions at intermediate energy

This work is a continuation of our effort [S. Mallik, S. Das Gupta, and G. Chaudhuri, Phys. Rev. C 91, 034616 (2015)] to examine if signatures of a phase transition can be extracted from transport model calculations of heavy ion collisions at intermediate energy. A signature of first-order phase transition is the appearance of a bimodal distribution in $P_m\left(k\right)$ in finite systems. Here $P_m\left(k\right)$ is the probability that the maximum of the multiplicity distribution occurs at mass number $k$. Using a well-known model for event generation [Botzmann-Uehling-Uhlenbeck (BUU) plus fluctuation], we study two cases of central collision: mass 40 on mass 40 and mass 120 on mass 120. Bimodality is seen in both the cases. The results are quite similar to those obtained in statistical model calculations. An intriguing feature is seen. We observe that at the energy where bimodality occurs, other phase-transition-like signatures appear. There are breaks in certain first-order derivatives. We then examine if such breaks appear in standard BUU calculations without fluctuations. They do. The implication is interesting. If first-order phase transition occurs, it may be possible to recognize that from ordinary BUU calculations. Probably the reason this has not been seen already is because this aspect was not investigated before.

Figure 1

Largest cluster probability distribution for $A_p=40$ on $A_t=40$ reaction at beam energies (a) 20, (b) 42.5, and (c) 100 MeV/nucleon. The average value of 2 mass units are shown. At each energy 1000 events are chosen. The results shown in this figure are calculated at $t=300\text{fm}/c$.

Figure 2

Largest cluster probability distribution for $A_p=120$ on $A_t=120$ reaction at beam energies (a) 20, (b) 60.125, and (c) 100 MeV/nucleon. The average value of 5 mass units are shown. At each energy 1000 events are chosen. The results shown here are calculated at $t=600\text{fm}/c$.

Figure 3

Largest cluster probability distribution for $A_p=40$ on $A_t=40$ reaction at beam energies (a) 41.5, (b) 42.5, (c)43.5, and (d) 44.5 MeV/nucleon. The average value of 2 mass units are shown. At each energy 1000 events are chosen. The results shown in this figure are calculated at $t=300\text{fm}/c$

Figure 4

Upper panel: Dependence of kinetic energy per nucleon (red), potential energy per nucleon (blue), and total energy per nucleon (black) for the$A_p=40$ on $A_t=40$ reaction on the projectile beam energy per nucleon. Lower panel: Dependence of first-order derivatives of kinetic energy and potential energy with respect to total energy on total energy per nucleon for the $A_p=40$ on $A_t=40$ reaction.

Figure 5

Same as Fig. 4 but here the nuclear reaction is $A_p=120$ on $A_t=120$.

Figure 6

Same as Fig. 5 but here the calculation is done by the standard BUU method (i.e., without fluctuation).