Muon Capture and Supermultiplet Symmetry Breaking inO16

The validity of assuming the equality of the unretarded-dipole muon-capture matrix elements, ${\left({M_V}^2\right)}_{\mathrm{U},\mathrm{D},}$, ${\left({M_P}^2\right)}_{\mathrm{U},\mathrm{D},}$, and ${\left({M_A}^2\right)}_{\mathrm{U},\mathrm{D}}$, is investigated for muon capture in ${\mathrm{O}}^{16}$. Earlier investigation has employed supermultiplet theory to obtain ${\left({M_V}^2\right)}_{\mathrm{U},\mathrm{D},}={\left({M_P}^2\right)}_{\mathrm{U},\mathrm{D},}={\left({M_A}^2\right)}_{\mathrm{U},\mathrm{D},}$. Therefore, we also consider the correctness of a supermultiplet grouping of the important muon-capture nuclear final states when realistic supermultiplet-symmetry-breaking interactions are included in the nuclear Hamiltonian. Basis wave functions are taken to be simple harmonic-oscillator shell-model $\mathrm{LS}$-coupled states belonging to the ${\mathrm{}\left(1p\right)\mathrm{}}^{-1\mathrm{}}2s$ and ${\mathrm{}\left(1p\right)\mathrm{}}^{-1\mathrm{}}1d$ configurations of ${\mathrm{O}}^{16}$. It is found that all particle-hole states of interest for muon capture belong to vector supermultiplets. These states will not be mixed with the other supermultiplet type (scalar) of the configuration if isotopic spin is a good quantum number. Energy eigenvalues and eigenfunctions of various nuclear Hamiltonians are obtained by diagonalizing the residual interaction matrices that mix the simple basis states. The results indicate that the spin-independent interparticle forces isolate the unretarded-dipole muoncapture strength in the supermultiplet lying highest in energy. The energy difference between this supermultiplet and lower lying supermultiplets has a tendency to suppress the large amount of supermultiplet mixing one might expect in the giant-resonance region when tensor and spin-orbit forces are added to the Hamiltonian. Therefore, it is found that the muon-capture strength remains concentrated in the giant-dipole region, and the muon-capture supermultiplet remains substantially pure. In all cases, ${\left({M_V}^2\right)}_{\mathrm{U},\mathrm{D},}={\left({M_P}^2\right)}_{\mathrm{U},\mathrm{D},}={\left({M_A}^2\right)}_{\mathrm{U},\mathrm{D},}$ remains a valid assumption to within 12%.