Electron Scattering and Muon Capture byC12andO16in a Continuum Model

The giant resonance region of ${\mathrm{C}}^{12}$ and ${\mathrm{O}}^{16}$ is investigated using a continuum model, which is based on configuration mixing of negative-parity particle-hole states, with the particle in the continuum. Using this model $T=1\mathrm{}$, $J^{\pi }=1^{-}, \mathrm{and} 2^{-}$ electron-scattering spectra are calculated as well as the $0^{-}$, $1^{-}$, and $2^{-}$ nuclear-excitation spectra (or equivalently the neutrino spectra) in $\mu$ capture. The $\mathrm{SU}\left(4\right)\mathrm{}$ relationship ${M_P}^2={M_V}^2={M_A}^2$ is tested and found to be satisfied in both nuclei to better than 10%. In addition, the relationship $\left({M_V}^2\right)={\left({M_V}^2\right)}_{\mathrm{UD}}{\left|F_{\mathrm{el}}\right({{\nu }_{\mathrm{res}}}^2\left)\right|}^2$ is tested and found to hold to within 3%. Both of these results justify the previous work of Foldy and Walecka on calculations of total-muon-capture rates. The detailed behavior of the electron scattering spectra is examined as a function of momentum transfer with emphasis on the magnetic contribution to the $1^{-}$ transverse electric spectra and the giant magnetic quadrupole resonances. The momentum dependence of integrated form factors for certain regions of the excitation spectra is also investigated.