Effects of Configuration Mixing on the $3^{-}$ State of ${\mathrm{Pb}}^{208}$ and the ($p,p^{\prime }$) Reaction in the Distorted-Wave Approximation

In this paper it is shown that the experimental mean life of the first excited $3^{-}$ state of ${\mathrm{Pb}}^{208}$ can be approximately accounted for if configuration mixing is introduced instead of considering this state as a pure single-particle shell-model state. For this purpose an octupole-octupole force is assumed (an extension of the quadrupole-quadrupole force of Elliott). Further, the direct interaction theory of inelastic scattering in the distorted-wave Born approximation is used to calculate the cross section of 23-MeV protons for excitation of the 0.57-MeV and 0.90-MeV levels of ${\mathrm{Pb}}^{207}$ [corresponding to the transitions ${\left(p_{\frac{1}{2}}\right)}^{-1\mathrm{}}\to {\left(f_{\frac{5}{2}}\right)}^{-1\mathrm{}}$ and ${\left(p_{\frac{1}{2}}\right)}^{-1\mathrm{}}\to {\left(p_{\frac{3}{2}}\right)}^{-1\mathrm{}}$, respectively] and for excitation of the anomalous peak at 2.6 MeV ($3^{-}$ state of ${\mathrm{Pb}}^{208}$) using the configuration-mixed nuclear wave function. It is suggested that the anomalous peak at 2.6 MeV for ${\mathrm{Pb}}^{206}$ and ${\mathrm{Pb}}^{207}$ is also due to superposition of many single-particle transitions as in ${\mathrm{Pb}}^{208}$. As a result of the configuration mixing, reasonable agreement is obtained between the experimental and theoretical cross sections for the above-mentioned cases. Finally, various suggestions which can lead to better understanding of this process are made.