Weak-Coupling Model forPb212andPb204

The weak-coupling model is applied to the study of the low-lying states of ${}_{}{}^{212}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$. We first obtain the phonon states of ${}_{}{}^{210}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$ by shell-model calculations. The states of ${}_{}{}^{212}{}_{}{}^{}\mathrm{Pb}$ (${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$) are then obtained by solving the eigenvalue problem in a truncated Hilbert space consisting of a small number of basis vectors, each of which is a product of two phonon states of ${}_{}{}^{210}{}_{}{}^{}\mathrm{Pb}$ (${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$). A canonical transformation is used to orthonormalize these two-phonon basis vectors. The energies as well as the wave functions of many low-lying states of ${}_{}{}^{212}{}_{}{}^{}\mathrm{Pb}$ (${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$) given by the exact shell-model calculations are reproduced surprisingly well by the weakcoupling model where only three or four low-lying ${}_{}{}^{210}{}_{}{}^{}\mathrm{Pb}$ (${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$) phonons are included.