Shell Theory and Collective Theory for the Pb Isotopes

The 0.570-Mev gamma ray in ${\mathrm{Pb}}^{207}$ and the 0.803-Mev gamma ray in ${\mathrm{Pb}}^{206}$ are electric quadrupole transitions and have transition rates which are much too fast for neutron transitions. The surface tension parameters necessary to give the observed transition rates are determined by assuming that the collective motion of the core is weakly coupled to the external nucleons. The calculated surface tension parameters are 1100 Mev for ${\mathrm{Pb}}^{207}$ and 520 Mev for ${\mathrm{Pb}}^{206}$. These two values are large compared to the hydrodynamic surface tension, as expected near closed shells, but the difference of a factor of 2 is unexpected. The quadrupole moment of ${\mathrm{Bi}}^{209}$ also indicates weak surface coupling. Also determined are the configurations which describe the low-lying energy levels of ${\mathrm{Pb}}^{204}$, assuming short-range two-body forces. Reasonable agreement with experiment indicates that the shell model can account for the main portion of the level splitting. Many levels are predicted which have not been observed as yet. The 0.375-Mev gamma ray in ${\mathrm{Pb}}^{204}$ is anomalously slow even with weak surface coupling. A possible explanation for this slowdown is indicated by the configurations describing the initial and final states. In connection with the ${\mathrm{Pb}}^{204}$ problem, formulas are given for antisymmetrical wave functions in the $j-j$ coupling scheme for three and four particles.