Progress of Theoretical Physics, Vol. 76, No.2, August 1986 Thermal Properties of Finite Nuclear System in the Constrained Hartree-Fock Approximation Hiroyuki SAGA W A and Hiroshi TOKI* Cyclotron Laboratory, Michigan State University East Lansing, MI 48824-1321 and Department of Physics, University of Tokyo, Tokyo 113 * Cyclotron Laboratory, Michigan State University East Lansing, MI 48824-1321 and Department of Physics, Tokyo Metropolitan University, Tokyo 158 (Received March 1, 1986) 433 In order to study dilute thennal nuclei, we have perfonned constrained Hartree·Fock calculations at finite temperatures for 90Zr and 208Pb. We find that the nuclear compression modulus at T=6 MeV is less than half of its value at T=O MeV. This decrease implies a very low excitation energy for the giant monopole state at temperature above T=6 MeV. The equation of state is also discussed in relation to the critical temperature of the liquid· gas phase transition. Thermal properties of nuclei have been of great interest recently in the context of the interpretation of data from intermediate-energy heavy-ion experiments. l ),2) The satura- tion properties of nuclear systems at finite temperatures are particularly interesting in relation to the giant compression mode 3 ) and the liquid-gas phase transition. 1 ),2),4) So far, these problems have been discussed in terms of nuclear matter calculation and the Thomas-Fermi approximation. l )-4) It is well known that the saturation properties of nuclei can be described very successfully over a wide range of mass regions at the zero temperature by the Hartree· Fock (H·F) approximation. S ) H-F calculations at finite-temperature have been perfor- med using two types of interactions, namely, the Skyrme-type interaction 6 ),7) and the Bruckner G-matrix obtained from the Reid soft-core potentiaP) In order to study the nuclear compression modulus at finite-temperature and also the equation of state of the finite nucleus, we must perform a constrained H-F calculation with various densities. Until now, such constrained H-F calculations have been done only at zero temperature. 9 ),lO) In this work, we discuss results of constrained H-F calculations of dilute system at several finite-temperatures. We have chosen medium and heavy nuclei 90Zr and 208Pb for this study. The Hartree-Fock theory at finite-temperature can be formulated by considering the variation of the thermodynamical potential with respect to the grand canonical.partition function of the system. 6 ),ll) The expectation value for any operator can be expressed in terms of the occupation probability function of the single particle state fi( T). The entropy 5 is given byll) 5 = - {f;( T)ln(fi( T)) + (1- f;( T) )In(l- f;( T))} . i (1) The variation of the thermodynamical potential with respect to fi( T) gives the Downloaded from https://academic.oup.com/ptp/article-abstract/76/2/433/1848586 by guest on 21 May 2020