Nuclear liquid-gas phase transition studied with antisymmetrized molecular dynamics

The nuclear liquid-gas phase transition of the system in ideal thermal equilibrium is studied with antisymmetrized molecular dynamics. The time evolution of a many-nucleon system confined in a container is solved for a long time to get a microcanonical ensemble of a given energy and volume. The temperature and the pressure are extracted from this ensemble and the caloric curves are constructed. The present work is the first time that a microscopic dynamical model which describes nuclear multifragmentation reactions well is directly applied to get the nuclear caloric curve. The obtained constant pressure caloric curves clearly show the characteristic feature of the liquid-gas phase transition, namely negative heat capacity (backbending), which is expected for the phase transition in finite systems.

Figure 1

The EOS of symmetric nuclear matter. Each full line shows the pressure as a function of the density for the given temperature (0, 5, 10, 15, and 20 MeV). This EOS is obtained by adopting the Gogny force and assuming uniform density in the Hartree-Fock framework. Below the dotted line, the liquid and gas phases coexist if nonuniform density is allowed. The dashed lines are the EOS in this coexistence region obtained by the Maxwell construction.

Figure 2

The caloric curves of infinite nuclear matter obtained by applying the Maxwell construction to the EOS (Fig. 1). The line of $E/A=(3/2)T$ is drawn for the comparison. (a) The constant volume caloric curves drawn for the average densities $\rho =0.40{\rho }_0,0.30{\rho }_0,0.23{\rho }_0,0.18{\rho }_0$, $0.15{\rho }_0,0.10{\rho }_0,0.07{\rho }_0,0.04{\rho }_0,0.02{\rho }_0$, and $0.015{\rho }_0$, where ${\rho }_0=0.17$ ${\text{fm}}^{-3}$ is the normal nuclear matter density. (b) The constant pressure caloric curves for the pressures $P=0.02$, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.25, 0.30, and $0.40\mathrm{MeV}/{\mathrm{fm}}^3$. The meaning of the diamond marks is explained in Sec. 5c.

Figure 3

The constant volume caloric curves for the $A=36$ system obtained by AMD. The lines correspond to the container size $r_{\mathrm{wall}}=5$, 5.5, 6, 6.5, 7, 8, 9, and 11 fm from the top. Statistical uncertainty is shown by error bars. The line of $E/A=(E^{*}+E_{\mathrm{g}.\mathrm{s}.})/A=(3/2)T$ is drawn for comparison.

Figure 4

The constant pressure lines drawn on $E\text{−}V$ plain. The circles indicate the points where the microcanonical ensembles are constructed. The lines correspond to the constant pressure lines with $P=0.02$, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.25, 0.30, and 0.40 MeV/${\text{fm}}^3$, which are obtained by performing the transformation between volume $V$ and pressure $P$ with the interpolation at each energy $E$. Statistical uncertainty is shown by error bars.

Figure 5

The constant pressure caloric curves for the $A=36$ system obtained by AMD. The lines correspond to the pressure $P=0.02$, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.25, 0.30 and 0.40 MeV/${\text{fm}}^3$ from the bottom. Statistical uncertainty is shown by error bars. The curves of $E^{*}/A=T^2/(8\mathrm{MeV})$ and $E^{*}/A=T^2/(13\mathrm{MeV})$, and the line of $E/A=(3/2)T$ are drawn for comparison.

Figure 6

The fragment mass distributions along the $P=0.05$ MeV/${\text{fm}}^3$ line. Full lines are the distributions obtained with $r_{\mathrm{clust}}=2.5$ fm. Dashed lines are the distributions obtained with $r_{\mathrm{clust}}=3.0$ fm.

Figure 7

The temperatures obtained with different parameter sets ($r_{\mathrm{G}}$ and ${\rho }_{\mathrm{G}}$) for choosing G-subsystem. The results of the systems with $r_{\mathrm{wall}}=5,9,15$ fm and $E^{*}/A=6,14,22$ MeV are shown. The abscissa $\langle A_{\mathrm{G}}\rangle$ is the average number of nucleons in G-subsystem and the parameters are chosen so that $\langle A_{\mathrm{G}}\rangle$ becomes about half and about a quarter of the original choice [$r_{\mathrm{G}}=3$ fm and ${\rho }_{\mathrm{G}}=(1/200){\rho }_0$]. The open points show the dependence of $r_{\mathrm{G}}$ and the solid points show the dependence of ${\rho }_{\mathrm{G}}$, which are connected by lines to guide eyes. Statistical uncertainty is similar to or smaller than the size of points.

Figure 8

The AMD results of the constant pressure caloric curves with $P=0.02$, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.25, 0.30, and 0.40 MeV/${\mathrm{fm}}^3$ from the bottom when the coherence time is varied: (a) ${\tau }_0=250$ fm/$c$, (b) ${\tau }_0=1000$ fm/$c$, (c) the same procedure as in Refs. [30, 33].