Microscopic Description of the Inelastic Proton Decay of Analog Resonances

A microscopic description of analog resonance is presented in the framework of the shell-model theory. The separation of the total Hamiltonian into an independent-particle Hamiltonian $H_0$ and the residual interaction $V$ is discussed. A choice of $H_0$ is made that minimizes the residual interaction—in particular, the matrix elements between the continuum eigenstates of $H_0$. The diagonalization of the two-body residual interaction on an appropriately chosen set of $2p-1h$ configurations is carried out, and the numerical results are presented for ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$. A group of ${\frac{9}{2}}^{+}$ states of analog spin $W_{<}$ is predicted to be about 11 MeV below the $g_{\frac{9}{2}}$ analog resonance in ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$. Coupling of the various continua to the $W_{>}$ state gives rise to the isobaric analog resonance in the corresponding channels. The inelastic proton decay of the $g_{\frac{9}{2}}$ analog resonance in ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$ to the particle-hole states of ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ is calculated, and the results are compared with experimental data. An expression for the energy-average $S$-matrix elements is derived starting from the multilevel expression describing the coupling of both $T_{>}$ and $T_{<}$ states to the various continua. There expressions differ from those given by Weidenmüller and by Mekjian and MacDonald and are in agreement with Robson's results in the one-channel case. Our expressions also agree with recent results obtained by Tamura.