Fragment isospin distributions and the phase diagram of excited nuclear systems

Fragment average isospin distributions are investigated within a microcanonical multifragmentation model in different regions of the phase diagram. The results indicate that in the liquid phase $\langle N/Z\rangle$ versus Z is monotonically increasing, in the phase coexistence region it has a rise-and-fall shape, and in the gas phase it is constant. Deviations from this behavior may manifest at low fragment multiplicity as a consequence of mass-charge conservation. Characterization of the “free” and “bound” phase functions of the fragment charge reconfirms the neutron enrichment of the free phase with respect to the bound one irrespectively of the localization of the multifragmentation event in the phase diagram.

Figure 1

Phase diagram of a (200, 82) nuclear system without Coulomb and hard-core interactions in (a) temperature-excitation energy (b) pressure-excitation energy, and (c) pressure-temperature representations. The solid curves correspond to the considered iso-$\beta P$ trajectories; the dashed curves indicate the borders of the phase-coexistence region; and the dotted curves indicate the borders of the spinodal region. The $\beta P$ values of the iso-$\beta P$ curves are labeled in units of inverse cubic femtometers.

Figure 2

(Top) fragment charge distributions and (bottom) charge distributions of the largest fragment for the (200, 82) nuclear system without Coulomb interaction at different excitation energies. For all considered situations the system in constrained by $\beta P=3\times {10}^{-3}$ fm${}^{-3}$. The key legend is as follows: open circles, $E=3$ MeV/nucleon; filled circles, $E=4$ MeV/nucleon; open squares, $E=5$ MeV/nucleon; open triangles, $E=6$ MeV/nucleon; open diamonds, $E=8$ MeV/nucleon; crosses, $E=10$ MeV/nucleon, stars, $E=12$ MeV/nucleon.

Figure 3

Fragment average isospin distributions as functions of fragment charge for the (200, 82) nuclear system without Coulomb interaction at different excitation energies. For all considered situations the system in constrained by $\beta P=3\times {10}^{-3}$ fm${}^{-3}$. The horizontal line indicates the source's isospin, $I_s=1.44$. The thick solid curve corresponds to the most probable isospin value obtained from the liquid-drop formula of the binding energy, Eq. (2).

Figure 4

Phase diagram of a (50, 23) nuclear system with Coulomb interaction and free-volume parametrization in (a) temperature-excitation energy, (b) pressure-excitation energy, and (c) pressure-temperature representations. The solid curves correspond to the considered iso-$\beta P$ trajectories; the dashed curves indicate the borders of the phase-coexistence region; and the dotted curves indicate the borders of the spinodal region. The $\beta P$ values of the iso-$\beta P$ curves are labeled in units of inverse cubic femtometers.

Figure 5

(Top) Fragment charge distributions and (bottom) charge distributions of the largest fragment for the (50, 23) nuclear system with Coulomb interaction at different excitation energies. For all considered situations the system in constrained by $\beta P=1\times {10}^{-3}$ fm${}^{-3}$.

Figure 6

Fragment average isospin distributions as functions of fragment charge for the (50, 23) nuclear system with Coulomb interaction in the liquid ($E=2.5$ MeV/nucleon), liquid-gas coexistence ($E=5$ MeV/nucleon) and gas phase ($E=8$ MeV/nucleon) regions. For all considered situations the system in constrained by $\beta P=1\times {10}^{-3}$ fm${}^{-3}$. The horizontal line indicates the source's isospin, $I_s=1.174$. The thick solid curves corresponds to the most probable isospin value obtained from the liquid-drop formula of the binding energy, Eq. (2).

Figure 7

Phase diagram of (200, 82) nuclear system with Coulomb interaction and free-volume parametrization in (a) temperature-excitation energy (b) pressure-excitation energy, and (c) pressure-temperature representations. The solid curves correspond to the considered iso-$\beta P$ trajectories; the dashed curves indicate the borders of the phase-coexistence region; the dotted curves indicate the borders of the spinodal region. The $\beta P$ values of the iso-$\beta P$ curves are labeled in units of inverse cubic femtometers.

Figure 8

Fragment average isospin distributions as functions of fragment charge for the (200, 82) nuclear system with Coulomb interaction. Upper-left-hand panel, liquid + undersaturated vapor phase for $\beta P=1\times {10}^{-4}$ and $3\times {10}^{-4}$ fm${}^{-3},E=2$, 3 MeV/nucleon; upper-right-hand panel, supercritical fluid phase along $\beta P=2\times {10}^{-3}$ fm${}^{-3}$; lower panel, freeze-out case (i) and different freeze-out volume approximations as indicated in the figure. The horizontal line indicates the source's isospin, $I_s=1.44$. The thick solid curve corresponds to predictions of Eq. (2).

Figure 9

(Top) Fragment charge distributions and (bottom) charge distributions of the largest fragment for the (200, 82) nuclear system with Coulomb interaction at different excitation energies. For all considered situations the system in constrained by $\beta P=1\times {10}^{-4}$ fm${}^{-3}$.

Figure 10

Fragment average isospin distributions corresponding to three sources characterized by $Z=82$ and $A=182$, 192, and 200 with excitation energy 6 MeV/nucleon and $\beta P=2\times {10}^{-3}$ fm${}^{-3}$. For all cases Coulomb and hard-core interactions are considered.

Figure 11

Fragment average isospin distributions in the breakup (filled symbols) and asymptotic (open symbols) stages of the reaction. The left-hand panel corresponds to the case of (200, 82) nucleus, $\beta P=1\times {10}^{-4}$ fm${}^{-3}$ with $E=2$, 6, and 9 MeV/nucleon; the right-hand panel corresponds to the case of (50, 23) nucleus, $\beta P=1\times {10}^{-3}$ fm${}^{-3}$ with $E=5$, 8 MeV/nucleon. In all cases the Coulomb interaction is present. Freeze-out case (ii) was used.

Figure 12

Left-hand panel, relative neutron enrichment of the “free” phase with respect to the “bound” phase for the (200, 82) nuclear system with and without the Coulomb interaction when the associated phase spaces are explored following the $\beta P$ constant paths $2\times {10}^{-3}$ and $1\times {10}^{-4}$ fm${}^{-3}$ (with the Coulomb interaction) and $3\times {10}^{-3}$ fm${}^{-3}$ (without the Coulomb interaction); right panel, Relative neutron enrichment of the “free” phase with respect to the “bound” phase as a function of the source's isospin in the case of a system with $Z=82$, with Coulomb interaction at 2 and 6 MeV/nucleon excitation energy and $\beta P=2\times {10}^{-3}$ fm${}^{-3}$.