Probing nuclear forces beyond the drip-line using the mirror nuclei ${}^{16}\mathrm{N}$ and ${}^{16}\mathrm{F}$

Radioactive beams of ${}^{14}\mathrm{O}$ and ${}^{15}\mathrm{O}$ were used to populate the resonant states ${1/2}^{+}$, ${5/2}^{+}$ and $0^{-},1^{-},2^{-}$ in the unbound ${}^{15}\mathrm{F}$ and ${}^{16}\mathrm{F}$ nuclei, respectively, by means of proton elastic scattering reactions in inverse kinematics. Based on their large proton spectroscopic factor values, the resonant states in ${}^{16}\mathrm{F}$ can be viewed as a core of ${}^{14}\mathrm{O}$ plus a proton in the $2s_{1/2}$ or $1d_{5/2}$ shell and a neutron in $1p_{1/2}$. Experimental energies were used to derive the strength of the $2s_{1/2}$-$1p_{1/2}$ and $1d_{5/2}$-$1p_{1/2}$ proton-neutron interactions. It is found that the former changes by 40% compared with the mirror nucleus ${}^{16}\mathrm{N}$, and the latter changes by 10%. This apparent symmetry breaking of the nuclear force between mirror nuclei finds explanation in the role of the large coupling to the continuum for the states built on an $\ell =0$ proton configuration.

Figure 1

Measured excitation functions of proton resonant elastic scattering on the ${}^{14}\mathrm{N}$, ${}^{15}\mathrm{N}$, ${}^{15}\mathrm{O}$, and ${}^{14}\mathrm{O}$ nuclei. The differential cross sections measured at 180${}^{\circ }$ in the c.m. are shown as a function of the laboratory energy $E_{\text{Lab}}$. Several resonances in the compound nuclei ${}^{15}\mathrm{O}$, ${}^{16}\mathrm{O}$, ${}^{16}\mathrm{F}$, and ${}^{15}\mathrm{F}$ can be observed. The lines are results of the $R$-matrix calculations using the parameters from Ref. [13] for the ${}^{15}\mathrm{O}$ and ${}^{16}\mathrm{O}$ nuclei, from Table 1 for the ${}^{16}\mathrm{F}$ nucleus, and from this work for the two resonances observed in ${}^{15}\mathrm{F}$.

Figure 2

Schematic view of the valence proton ($\pi$) and neutron ($\nu$) orbits involved in the mirror systems ${}_9^{16}$F${}_7$ and ${}_7^{16}$N${}_9$ on top of the ${}_8^{14}$O${}_6$ and ${}_6^{14}$C${}_8$ core nuclei. The major difference between these two systems is that the proton $2s_{1/2}$ or $1d_{5/2}$ (not shown) orbits are unbound in ${}_9^{16}$F${}_7$. The level schemes of the two mirror nuclei differ significantly, as shown in the bottom part of the figure.

Figure 3

Calculated $1p_{1/2}$, $1d_{5/2}$, and $2s_{1/2}$ single-particle radial wave functions $u_n\left(r\right)$ and $u_p\left(r\right)$ (see text) for ${}^{16}\mathrm{N}$ (dashed lines) and ${}^{16}\mathrm{F}$ (solid lines). For the scattering states, the unbound proton wave function in the region between $r=0$ and $r=16$ fm is renormalized to 1 to have the proton inside the nucleus before it decays. The $2s_{1/2}$ wave function of the unbound proton in ${}^{16}\mathrm{F}$ is more spatially spread than it is for the bound neutron in ${}^{16}\mathrm{N}$, contrary to the $1d_{5/2}$ unbound proton wave function that is retained inside the nucleus by the centrifugal barrier.