Experimental study of the effective nucleon-nucleon interaction using the ${}^{21}\mathrm{F}(d,p){}^{22}\mathrm{F}$ 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 $\pi 0d_{5/2}$ orbital coupled to a neutron in either the $\nu 0d_{5/2}$, $\nu 1s_{1/2}$, or $\nu 0d_{3/2}$ orbitals were observed up to an excitation energy of $\sim 5$ 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 $\nu 0d_{5/2}$ and $\nu 1s_{1/2}$ 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 $\nu 0d_{3/2}$ strength. Diagonal two-body matrix elements of the ${\left(0d_{5/2}\right)}_{J=0–5}^2$ 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.

Figure 1

Measured proton energies ($E_{\text{lab}}$) as a function of the distance from the target ($Z$) for the ${}^{21}\mathrm{F}(d,p){}^{22}\mathrm{F}$ reaction in inverse kinematics at 10 MeV/u with a magnetic field strength of 2.5 T. The proton data shown required a coincidence with ${}^{22}\mathrm{F}$ in the recoil detector telescope. Final states identified in ${}^{22}\mathrm{F}$, appearing as tilted parallel lines, are indicated by arrows and are labeled by their corresponding excitation energy.

Figure 2

The excitation-energy spectrum of ${}^{22}\mathrm{F}$ as determined from the ${}^{21}\mathrm{F}(d,p)$ reaction from the same data set as presented in Fig. 1. States clearly identified in the present work are labeled with their corresponding spin assignments. The states that have been observed in previous measurements are shown in the upper panel for comparison [18].

Figure 3

Experimental (black points) and calculated (lines) cross sections as a function of the center-of-mass angle for various final states in the reaction ${}^{21}\mathrm{F}(d,p){}^{22}\mathrm{F}$ at 10 MeV/u. Only statistical uncertainties are shown for the experimental data, and there is a systematic uncertainty of $\sim 30\%$ on the absolute cross section scale. The dashed and dot-dashed curves are the calculated DWBA cross sections for $\ell =2$ and $\ell =0$ neutron transfer, respectively. The red solid curves are the resulting fits which incorporated mixed $\ell =0$ and $\ell =2$ angular distributions. Note the multiplication factors used for the ease of plotting.

Figure 4

Experimental (a) and calculated (b)–(d) $0d_{5/2}$ (blue cross-hatched bars), $1s_{1/2}$ (green bars), $0d_{3/2}$ (red slashed bars) single-neutron strengths as a function of excitation energy in ${}^{22}\mathrm{F}$ using the (b) USD, (c) USDA, and (d) USDB interactions. The weaker strengths extracted assuming only $\ell =2$ neutron transfer are represented by the white unfilled bars. States with calculated $S\le 0.04$ were not included in the plots. Spin-parity labels are shown only for those levels which are relevant to the discussions in the text.