Reﬁned Thomas-Fermi description of hot nuclei J. N. De and N. Rudra* Variable Energy Cyclotron Centre, 1/AF, Bidhannagar, Calcutta-700 064, India Subrata Pal and S. K. Samaddar Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Calcutta-700 064, India Received 26 April 1995 Self-consistent density proﬁles of two-component hot nuclei in equilibrium with an external gas are calcu- lated in the semi-classical Thomas-Fermi model with a new prescription. The energy functional is calculated with a momentum and density dependent ﬁnite range two-body effective interaction. The evolution of equi- librium nuclear masses as a function of temperature and densities of the external neutron and proton gas is investigated in this description. Limiting temperatures of nuclei, their lifetimes against neutron evaporation, temperature dependence of incompressibilites of ﬁnite nuclei and a few other observables are also studied in the present model. PACS numbers: 21.60.-n, 21.10.-k, 21.30.Fe, 21.65.+f I. INTRODUCTION The properties of hot nuclear systems created in the labo- ratory in intermediate energy heavy ion collisions can now be investigated experimentally 1. They give us information about the temperature dependence of the surface tension of ﬁnite nuclei, their incompressibility, level density parameter, etc., which are extremely useful in answering some impor- tant questions of astrophysical interest. The hot nuclei so created are not thermodynamically stable; they deexcite by emission of nucleons and light particles. The theoretical modeling of such an evaporating nucleus poses some prob- lems. The continuum states of a nucleus at nonzero tempera- ture are occupied with probability given by a Fermi factor 2 as a result of which the particle density does not vanish at large distances. The extracted observables then depend on the size of the box in which the calculation is performed. For not too high temperatures, the evaporation times are quite long; the nucleus can then be considered to be in a meta- stable state, very much like a superheated liquid drop 3.A free variation of the density proﬁles in the Thomas-Fermi TF description can then lead to a solution of the density in a sphere of radius R which is also given variationally with zero pressure outside, but with externally given boundary conditions; i.e., the derivatives of the density at the center and at the surface should be zero 4. In earlier thermal Hartree-Fock HF5 or semiclassical calculations 6,7, this problem was circumvented either by imposing artiﬁcial conditions on the size of the basis states or by demanding an exponentially decreasing density at large distances. In later calculations, this problem was addressed in the HF 8 and TF 9 approaches in a so-called subtraction procedure, where the nucleus was studied by means of a thermodynamic potential calculated as the difference between the thermody- namic potentials for the nucleus in equilibrium with a sur- rounding gas and that for the gas alone. This has the desir- able feature that asymptotically the nuclear plus gas density falls off smoothly to the gas density and the result is inde- pendent of the size of the box. The solution does not exactly correspond to that of an isolated nucleus; the subtracted grand potential contains a residual contribution due to the coupling of the liquid and gas parts. This contribution is, however, expected to be small. To compensate for the tendency of the nucleons to leave the hot nucleus, a suitable external pressure has to be im- posed 7,10 on the system to maintain thermodynamic equi- librium. This constraint appears somewhat artiﬁcial for the description of an isolated hot nucleus in the laboratory, but is more relevant in the astrophysical context where a nucleus embedded in a hot nucleon gas is a possible scenario. In this paper, we look for a solution of the density proﬁle of this hot nucleus immersed in a nucleon gas at the same temperature, which supplies the necessary external pressure. Mechanical equilibrium is maintained from the equality of the pressure in the condensed phase the nucleus with the external pressure exerted by the gas phase, and chemical equilibrium is main- tained from the equality of the average number of particles leaving the hot nucleus with the average number of particles entering it; i.e., the pressure and the chemical potential in the two phases are equal. Since each phase is a two-component phase of neutrons and protons, the chemical potentials of neutrons and protons in both the liquid and gas phase are equal separately. With the above-mentioned thermodynamic equilibrium conditions, the density proﬁles of nuclei at different tempera- tures are calculated in a reﬁned TF approximation scheme in this paper. Beyond a certain temperature, the equilibrium conditions, however, cannot be maintained. The nucleus is unstable, and can no longer exist as a bound system. This temperature is called the limiting temperature T lim . We ex- plore this limit for different nuclear systems. For a particular temperature, density, and composition neutron-proton ratio of the nucleon gas, we ﬁnd that only one nuclear isotope can coexist in equilibrium. The evolution of this equilibrium nuclear mass as a function of temperature, density, and com- position of the nucleon gas is then calculated. From the cal- * Permanent address: Department of Physics, University of Kaly- ani, West Bengal 741235, India. PHYSICAL REVIEW C FEBRUARY 1996 VOLUME 53, NUMBER 2 53 0556-2813/96/532/78010/$06.00 780 © 1996 The American Physical Society