Experimental study of the effective nucleon-nucleon interaction using the F21(d,p)F22 reaction

The single-neutron configurations of several low-lying states in $^{22}\mathrm{F}$ have been determined from neutron-adding cross sections of a ($d,p$) reaction using a $^{21}\mathrm{F}$ radioactive beam in inverse kinematics. Final states in $^{22}\mathrm{F}$ consisting of a proton in the $\ensuremath{\pi}0{d}_{5/2}$ orbital coupled to a neutron in either the $\ensuremath{\nu}0{d}_{5/2}$, $\ensuremath{\nu}1{s}_{1/2}$, or $\ensuremath{\nu}0{d}_{3/2}$ orbitals were observed up to an excitation energy of $\ensuremath{\sim}5$ MeV. Spectroscopic factors and strengths were determined from the angular distribution of the cross sections using a distorted wave Born approximation method. The distribution of the $\ensuremath{\nu}0{d}_{5/2}$ and $\ensuremath{\nu}1{s}_{1/2}$ strength was well described by all USD effective shell model interactions, while only the more recent USDA/USDB interactions showed a marked improvement in describing the observed $\ensuremath{\nu}0{d}_{3/2}$ strength. Diagonal two-body matrix elements of the ${(0{d}_{5/2})}_{J=0--5}^{2}$ effective nucleon-nucleon interaction were extracted from the data and compared with previous determinations of the same matrix elements from particle-particle and hole-hole spectra, as well as the calculated results of the USDA interactions. No significant discrepancies were observed. Inspection of the monopole energies showed that an improved agreement between the different empirical matrix elements is found if a mass dependence is included.