Exact form of the random phase approximation equation at finite temperature including the entropy effect

Abstract

The thermal random phase approximation (TRPA) equation is derived from the variational principle applied to the grand potential with an entropy term. This form of the TRPA equation is exact within the framework of the random phase approximation at finite temperature, whose matrix representation coincides with the stability matrix for the solution to the thermal Hartree-Fock-Bogoliubov equation. It is, however, shown that the $\ensuremath{\gamma}$-ray energy-dependence of the response function for a giant resonance built on a heated nucleus is not altered within a perturbation treatment of the entropy effect based on a simple microscopic model. Thus, an application of the TRPA formalism neglecting the entropy effect is justifiable as for giant resonance shape. The calculation employing a simple microscopic model shows that the increase of the Landau splitting of giant resonance levels with temperature is mainly attributed to the contributions of $\mathrm{pp}$ and $\mathrm{hh}$ configurations.