Variational principle model for the nuclear caloric curve

Following the lead of a recent work, I perform a variational principle model calculation for the nuclear caloric curve. A Skyrme-type interaction with and without momentum dependence is used. The calculation is done for a large nucleus, i.e., in the nuclear matter limit. Thus I address the issue of volume fragmentation only. Nonetheless, the results are similar to the previous, largely phenomenological calculation for a finite nucleus. I find that the onset of fragmentation can be sudden as a function of temperature or excitation energy.

Figure 1

Free energy per particle as function of density at different temperatures. This is plotted as a difference from the minimum value of F in the frame. At a temperature of 12 MeV there is no minimum, and the system will roll down to lower density. Already for this temperature at lower values of ρ in the figure the derivative $\partial p/\partial \rho$ is negative; i.e., the system has entered a region of mechanical instability.

Figure 2

Temperature versus $e^{*}$ in the finite-temperature mean-field model with momentum dependence and in the Fermi-gas model. As in Ref. 6, the curve with momentum dependence lies above the Fermi-gas curve till they meet (around 11.5 MeV). Beyond, the mean-field curve flattens. Just below 12 MeV the curve stops. The nucleus will break up into many pieces at higher temperatures.

Figure 3

Isotherms at temperatures 6, 8, 10, 12, 14, and 15.64 MeV in the mean-field model with density dependence but no momentum dependence. The dashed curve is the coexistence line. Also shown is the $p=0$ line. The intersections of this line with the isotherms give the minima (on the high-density side) and maxima (on the low density side) of free energy.

Figure 4

Caloric curve according to the finite-temperature mean-field theory (solid curve), and according to the finite-excitation mean-field theory and $e^{*}={\mathit{aT}}^2$ Fermi-gas approximation (dashed curve). The mean field is density dependent but without any momentum dependence. The finite-temperature and finite-excitation answers are the same up to $e^{*}\approx 13.6$ MeV. The finite-temperature model stops here with zero slope for T. The finite-excitation model continues beyond but with an unphysical negative slope for T. This part belongs to the negative compressibility zone of Fig. 3.

Figure 5

Pressure versus ρ curves for fixed excitation energies, which in this case are, starting from the lowest, for $e^{*}=6$, 8, 10, 12, 14, 15 MeV. The curves terminate on the low-density side when the energy cost of expansion equals $e^{*}$. The curves intersect the $p=0$ line at only one point, unlike the case for isotherms (Fig. 3).