Incorporating Radial Flow in the Lattice Gas Model for Nuclear Disassembly

We consider extensions of the lattice gas model to incorporate radial flow. Experimental data are used to set the magnitude of radial flow. This flow is then included in the Lattice Gas Model in a microcanonical formalism. For magnitudes of flow seen in experiments, the main effect of the flow on observables is a shift along the E∗/A axis. 25.70.Pq, 24.10.Pa, 64.60.My Typeset using REVTEX 1 It is expected that when nuclei disintegrate after heavy ion collisions, there will be a radial flow in the disintegrating system in addition to chaotic motion which is usually described by thermal motion. This was first proposed for collisions in the Bevalac [1] but is expected and seen in collisions at lower energies as well [2–4]. The amount of radial flow is larger for central collisions. In this paper we address the issue of incorporating radial flow in statistical models for nuclear disassembly. This is automatically taken into account in models based on transport equations such as BUU (Boltzmann-Uehling-Uhlenbeck) [5]. But in many statistical models such as the SMM (statistical multifragmentation model [6]), thermodynamic model [7] or the microcanonical model [8] the flow can only be included a posteriori. The idea here would be that the energy which is lost in radial flow is lost for thermalisation, thus essentially less energy is available for thermal disassembly. While this idea is certainly quite attractive, radial flow may do more than just take away energy. As far as we know this was first pointed out in [9]. It is this aspect that we study quantitatively here. Some additional insight can be gained by incorporating a radial flow in the Lattice Gas Model (LGM) which is being applied more and more to fit experimental data [10,11]. In the usual formulation of the LGM, equilibrium statistical mechanics is done before composites are calculated [12]. We combine statistical mechanics with radial flow and then examine how it affects the composite production. The important issue here is the relative kinetic energy of two nearest neighbours. If this is less than the attractive bond, the two nearest neighbours will be part of the same cluster. Now, if one has a radial flow which diverges outward from the centre, the flow will affect the relative kinetic energy and hence the composite production. The argument of merely some energy being unavailable for thermalisation is strictly valid when the collective velocity is the same for every nucleon. We are not claiming that radial flow arises in LGM in a fundamental way. But it can be included with a reasonable prescription. Inclusion of radial flow in a model similar to LGM was considered by Elattari et al [13]. Here we base our calculations on experimental data. The data used here is from a work by Williams et al. [4]. In that paper experimental 2 data of average radial flow is plotted as a function of excitation energy per nucleon (E∗/A). This is converted here to Ef/A against E ∗/A (see Fig. 1) where Ef/A is the flow energy per nucleon. Our calculation, by construct, will reproduce this curve. The model we use is this. From Fig. 1 we can also construct a Ef/A against Estat/A where Estat/A = E ∗/A− Ef/A. The part Estat/A is generated by the LGM. In LGM we generate events which pertain to a Estat/A. If there were no flow then from these events we would generate clusters and compare with experiments. But since experiments dictate that there is also energy tied up with flow we impose a flow energy on each event. The amount of flow energy is taken from the experimental Ef/A against Estat/A curve. We take the flow velocity to be proportional to the distance from the center of mass of the exploding nucleus. Since in an event the position of each nucleon is known, this can be done uniquely. This is the principle of this calculation; below we provide some more details. The most straightforward approach would be to use a microcanonical Lattice gas model. This is as simple as a canonical Lattice gas model calculation (see ref. [14]). Our simulations are done for A = 84, N = 48, Z = 36. We use a 6 lattice. The neutron-proton bond is -5.33 MeV; like particle bonds are set at 0. Coulomb interaction between protons is taken into account as in [11]. We use Metropolis Monte-Carlo simulations to obtain microcanonical samplings. We start from a suitable initial lattice configuration which gives an interaction energy Epot. The statistical energy Estat is fixed. The statistical kinetic energy for this configuration is then Estat + Eground − Epot = Ekin. The phase space Ωk(Ekin) available for this kinetic energy is well-known: Ωk(Ekin) = ∫ dp1d p2.....d pAδ(Ekin − p 2 i 2m ) = (2πm) 3A/2 Γ(3A/2) (Ekin) 3A/2−1 We now try to switch to a different configuration in the lattice. As a result, the potential energy in this new configuration would change to a new value E ′ pot. In this configuration, since we are doing a microcanonical simulation, the statistical kinetic energy would have to adapt to a new value:E ′ kin = Estat + Eground − E ′ pot. Correspondingly, the new phase-space will be Ωk(E ′ kin). If this is bigger than Ωk(Ekin), the switch is made. Otherwise the switch