Continuous phase transition and negative specific heat in finite nuclei

The liquid-gas phase transition in finite nuclei is studied in a heated liquid-drop model where the nuclear drop is assumed to be in thermodynamic equilibrium with its own evaporated nucleonic vapor, conserving the total baryon number and isospin of the system. It is found that in the liquid-vapor coexistence region the pressure is not a constant on an isotherm, indicating that the transition is continuous. At constant pressure, the caloric curve shows some anomalies; namely, the systems studied exhibit negative heat capacity in a small temperature domain. The dependence of this specific feature on the mass and isospin of the nucleus, Coulomb interaction, and the chosen pressure is studied. The effects of the presence of clusters in the vapor phase on specific heat have also been explored.

Figure 1

The isotherms for the system ${}^{150}\mathrm{Re}$ at $T=$ 7.0 MeV with and without the Coulomb interaction (top panel) and at $T=$ 7.0 and 7.2 MeV with Coulomb off in a narrow density interval (bottom panel). For the horizontal line $a\text{−}b$, see text.

Figure 2

The isotherms for the system ${}^{150}\mathrm{Nd}$ with and without the Coulomb interaction. The inset magnifies the loop in a narrow density region.

Figure 3

The isotherms for the nuclei ${}^{40}\mathrm{Ca}$ and ${}^{50}\mathrm{Ca}$ at T = 7 MeV with the Coulomb interaction.

Figure 4

The proton fraction $Y^l$ of the liquid drop as a function of its depleting mass number $A_l$ (upper panels). The neutron and proton chemical potentials in the liquid and gas phase as a function of the proton fraction in the respective phases (lower panels). The liquid and gas phase results are well separated and are marked in the figure. The vertical arrows on the abscissa show the proton fraction of the total system. The results correspond to constant pressure P = 0.0014 MeV fm${}^{-3}$.

Figure 5

The caloric curve for the system ${}^{40}\mathrm{Ca}$ at $P=0.0014$ MeV fm${}^{-3}$ with (filled circles) and without (open circles) the Coulomb interaction (top panel). The middle panel displays the same at $P=0.028$ MeV fm${}^{-3}$. The bottom panel is the same as in the top panel, but for the system ${}^{150}\mathrm{Re}$.

Figure 6

The caloric curves for the system ${}^{150}\mathrm{Nd}$ at P = 0.0014 MeV fm${}^{-3}$. The dashed line is obtained after switching off the Coulomb (C) and the surface (S) effects in the single phase (sp), the full line refers to that with the coexistence phase (cp), the open circles correspond to the one after switching on the surface effect, and the filled circles refer to caloric curve with both Coulomb and surface on.

Figure 7

Heat capacity at constant pressure for the system ${}^{40}\mathrm{Ca}$ with (bottom panel) and without (top panel) the Coulomb interaction. The arrows indicate the points of discontinuity.

Figure 8

Same as Fig. 7 for the system ${}^{150}\mathrm{Nd}$.

Figure 9

Entropy per particle, $S/A$, as a function of temperature for the system ${}^{150}\mathrm{Nd}$ at P = 0.0014 MeV fm${}^{-3}$. The notation is the same as in Fig. 6.

Figure 10

Entropy per particle as a function of excitation energy per particle, $E^{*}/A$, for the same case as in Fig. 9. The notation is the same as in Fig. 6.

Figure 11

Caloric curve with monomers (filled circles) and clusters (filled triangles) in the vapor phase for the systems ${}^{150}\mathrm{Re}$ (top panel) and ${}^{150}\mathrm{Nd}$ (bottom panel).

Figure 12

Heat capacity at constant pressure for the system ${}^{150}\mathrm{Re}$ (top panel) and ${}^{150}\mathrm{Nd}$ (bottom panel). The arrows indicate the points of discontinuity.

Figure 13

Entropy as a function of temperature (top panel) and excitation energy (bottom panel) for the system ${}^{150}\mathrm{Nd}$ at P = 0.0014 MeV fm${}^{-3}$.