Higher Random Phase Approximation and Energy Spectra of Spherical Nuclei

A method which is an extension of the usual random phase approximation (RPA) is proposed in order to study the spectra of spherical nuclei. In contrast to the usual RPA method based on the linearized equations of motion of the one-nucleon excitation operators, in the higher random phase approximation (HRPA) the many-nucleon excitations are also included. Nuclear states are superpositions of the one- and many-nucleon excitations (i.e., particle-hole pairs). We have especially studied the modes consisting of one- and two-nucleon excitations. In this second RPA, one solves the secular problem obtained from the closed system of linearized equations of motion for the one particle-hole pair operators ("doubles") and the two-pair operators ("quadruples"). States of definite nuclear spin, parity, and isotopic spin are considered. The method of the second RPA is illustrated in the example of the 6.06-Mev, $J^{\pi }=0^{+}$, $T=0\mathrm{}$ state of ${\mathrm{O}}^{16}$. Satisfactory semiquantitative agreement with the experiment is obtained for this case. The second RPA is also discussed in connection with the "aligned coupling scheme." The importance of the method for the study of the vibrational $4^{+}$ states is indicated.