Isospin dependence of the nuclear equation of state near the critical point

We discuss experimental evidence for a nuclear phase transition driven by the different concentrations of neutrons to protons. Different ratios of the neutron to proton concentrations lead to different critical points for the phase transition. This is analogous to the phase transitions occurring in ${}^4\mathrm{He}$-${}^3\mathrm{He}$ liquid mixtures. We present experimental results that reveal the $N/A$ (or $Z/A$) dependence of the phase transition and discuss possible implications of these observations in terms of the Landau free energy description of critical phenomena.

Figure 1

Typical $Z_{\mathrm{Real}}$ spectra for He and Li isotopes. Accidental events are generated only for $p$, $d$, and $t+\alpha$ and are shown separately by shaded histograms as indicated.

Figure 2

Mass distribution for the ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ system at 40 MeV/nucleon for $I=0$. The lines are power-law fits with exponents $2.3\pm 0.02$ (odd-odd nuclei, dashed line) and $3.4\pm 0.06$ (even-even nuclei, solid line), respectively.

Figure 3

Ratio versus fragments ground-state binding energy (top panel) and symmetry energy (bottom panel) for the ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ case at 40 MeV/nucleon. $a_{\mathrm{sym}}=25$ MeV is used. The $I<0$ and $I>0$ ($I=0$) isotopes are indicated by the open and solid circles, respectively (solid squares). The dashed lines (bottom panel) are fits using a ground-state symmetry energy, Eq. (1), and a “temperature” of 6 MeV. Notice that the given experimental ${}^8\mathrm{He}$ yield is the upper limit.

Figure 4

Ratio versus symmetry energy $+$ Coulomb energy for the ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ case at 40 MeV/nucleon. The panels from top to bottom are for different combinations of the symmetry and Coulomb energies. The $I<0$ and $I>0$ ($I=0$) isotopes are indicated by the open and solid circles, respectively (solid squares).

Figure 5

Free energy versus $m$ for the case ${}^{64}\mathrm{Ni}+{}^{232}\mathrm{Th}$. The solid line is a free fit based on Landau $O(m^6$) free energy. The dashed-dotted-dotted-dotted line is obtained by imposing in the fit $b=-\sqrt{16/3\mathit{ac}}$ and it is located on a line of first-order phase transitions. The short-dashed line corresponds to $b=-\sqrt{4\mathit{ac}}$, that is, superheating. The $O(m^2)$ case, $F/T=a(m-m_s){}^2$, that is, $b=c=0$, $m_s=0.1$, is given by the long-dashed line.

Figure 6

$H/T$ versus $(I/A)$ of the compound nucleus obtained from the data fit to the Landau free energy, Eq. (3). The solid circles are for ${}^{64}\mathrm{Ni}$, the solid triangles are for ${}^{70}\mathrm{Zn}$, and the squares for ${}^{64}\mathrm{Zn}$ projectiles impinging on various targets (see text).

Figure 7

The parameters $a$, $b$, and $c$ versus $(I/A)$ of the compound nucleus obtained from the data fit to the Landau free energy, Eq. (3). The symbols are like in Fig. 6.

Figure 8

$F/T$ $(H/T=0)$ versus $m$ of the fragments obtained from the $a$, $b$, and $c$ parameter fits to the Landau free energy, Eq. (3). Results for all experimentally investigated reactions are displayed.

Figure 9

$F/T$ $(H/T=0)$ versus $m$ of the fragments obtained from the $a$, $b$, and $c$ parameter fits to the Landau free energy, Eq. (3), for ${}^{64}\mathrm{Ni}+{}^{232}\mathrm{Th}$ (error bars are of the same size as the data points). The five curves correspond to vapor (solid line), superheating (short-dashed line), line of first-order transition (dashed-dotted-dotted-dotted line), the experimental data fit line (dashed-dotted line), and first-order phase transition (long-dashed line) (see text).

Figure 10

Order parameter versus reduced temperature for all studied systems. The dashed line is given by Eq. (10); the vertical line indicates the critical temperature $T_3$. To the right of this line the system is in a superheated state. Supercooling occurs on the left of the vertical line and $M=0$ [2].

Figure 11

Scaling form for magnetization $M$ vs external field for nickel [2], open symbols. The corresponding quantities for nuclei normalized to the metal case are given by the solid symbols.

Figure 12

The parameters $x$, $z$, and $h/T$ versus $(I/A)$ of the compound nucleus obtained from the data fit to the Landau free energy, Eq. (3). The symbols are like in Fig. 6.

Figure 13

Free energy versus mass for $m=0$ isotopes for the ${}^{70}\mathrm{Zn}+{}^{124}\mathrm{Sn}$ system (top panel), the dashed line is a fit using Coulomb and pairing contributions. Free energy times $\delta$ (see text) (bottom panel) versus mass for $m=0$ isotopes. The lines are separate fits suggested by the Coulomb (dashed line) and pairing (solid line) energy mass number dependence.

Figure 14

Free energy vs time in AMD calculations (see text) for ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ system at 40 MeV/$A$ and central collisions, that is, impact parameter less than 3 fm. Different panels correspond to $T=200$, 300, 500, 1000, 1500, and 2000 fm/$c$, respectively.

Figure 15

Fit parameters $a$, $b$, and $c$ vs time (see text) for the same system of Fig. 14. Solid circles refer to AMD calculations while the open symbol in the $M$ vs $x$ plot is the experimental value for this system.

Figure 16

Reduced pressure versus $m$ of the fragments obtained from the $a$, $b$, and $c$ parameter fits to the Landau free energy, Eq. (3), for the ${}^{64}\mathrm{Ni}+{}^{232}\mathrm{Th}$. The curves correspond to vapor (open circles), superheating (open squares), first order (three critical line-solid stars) (see text). The solid circle is for $N=Z$ nuclei at the critical point.

Figure 17

Comparison to the analytical result, Eq. (20) (solid circles), for a first-order phase transition.

Figure 18

Reduced compressibility versus $m$ of the fragments as in Fig. 16.