Phase transition in hot $\mathrm{\Lambda }$ hypernuclei within the relativistic Thomas-Fermi approximation

A self-consistent description for hot $\mathrm{\Lambda }$ hypernuclei in hypothetical big boxes is developed within the relativistic Thomas-Fermi approximation in order to investigate directly the liquid-gas phase coexistence in strangeness finite nuclear systems. We use the relativistic mean-field model for nuclear interactions. The temperature dependence of $\mathrm{\Lambda }$ hyperon density, $\mathrm{\Lambda }$ hyperon radius, excitation energies, specific heat, and the binding energies of $\mathrm{\Lambda }$ hypernuclei from ${}_{\mathrm{\Lambda }}{}^{16}\mathrm{O}$ to ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ in phase transition region are calculated by using the subtraction procedure in order to separate the hypernucleus from the surrounding baryon gas. The $\mathrm{\Lambda }$ central density is very sensitive to the temperature. The radii of $\mathrm{\Lambda }$ hyperon at high temperature become very large. In the relativistic Thomas-Fermi approximation with the subtraction procedure, the properties of hypernuclei are independent of the size of the box in which the calculation is performed. The level density parameters of hypernuclei in the present work are confirmed to be almost constant at low temperature. It is also found that the single-$\mathrm{\Lambda }$ binding energies of $\mathrm{\Lambda }$ hypernuclei are largely reduced with increasing temperature.

Figure 1

The density distributions of the $\mathrm{\Lambda }$ hyperon for ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ at $T=8$ MeV obtained with different box sizes $R=16,18$, and 20 fm. The density distributions from the gas phase ($G$) and the liquid-plus-gas phase ($NG$) are shown in the top and bottom panels, respectively.

Figure 2

The density distributions of $\mathrm{\Lambda }$ hyperon (left panels), neutron (middle panels), and proton (right panels) for ${}_{\mathrm{\Lambda }}{}^{40}\mathrm{Ca}$ at $T=0,4,$ and 8 MeV obtained using the TM1 parametrization. The density distributions from liquid-plus-gas ($NG$), gas phase ($G$), and subtracted liquid phase ($L$) are shown in the top, middle, and bottom panels, respectively.

Figure 3

Same quantities as Fig. 2, but for ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$.

Figure 4

The rms radii of neutrons, protons, and $\mathrm{\Lambda }$ hyperon as a function of temperature $T$ for ${}_{\mathrm{\Lambda }}{}^{40}\mathrm{Ca}$ and ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$.

Figure 5

The scalar and vector $\mathrm{\Lambda }N$ potentials as a function of hypernuclei radius at $T=0,4$, and 8 MeV. The results of ${}_{\mathrm{\Lambda }}{}^{40}\mathrm{Ca}$ and ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ are shown in the left and right panels, respectively.

Figure 6

The caloric curves, i.e., the temperature $T$ as functions of the excitation energy per particle $E^{*}/A$ for ${}^{40}\mathrm{Ca}$, ${}_{\mathrm{\Lambda }}{}^{40}\mathrm{Ca}$, ${}^{208}\mathrm{Pb}$, and ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$.

Figure 7

The specific heat $C_v$ as a function of temperature for ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ calculated with different box sizes $R=15$ and 20 fm (left panel) and those for ${}_{\mathrm{\Lambda }}{}^{40}\mathrm{Ca}$ and ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ with $R=20$ fm (right panel).

Figure 8

The single-$\mathrm{\Lambda }$ binding energies from ${}_{\mathrm{\Lambda }}{}^{16}\mathrm{O}$ to ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$ at different temperatures $T=0,2,4,6$, and 8 MeV and compared with the experimental data for $1s$ states at zero temperature [31].