Based on a microscopic cluster model that reproduces well the ${}^{17}\mathrm{C}$ spectrum below the neutron threshold, we study the spectrum of its mirror nucleus ${}^{17}\mathrm{Na}$. We find that the ${}^{17}\mathrm{Na}$ ground state should be located at $2.4$ MeV above the ${}^{16}\mathrm{Ne}$($0_1^{+})+p$ threshold, being a broad $1/2^{+}$ resonance with a width of $1.4$ MeV. However, there should exist at least four narrow excited states in ${}^{17}\mathrm{Na}$; three of them, $3/2_1^{+}$, $7/2_1^{+}$, and $9/2_1^{+}$, decay into the ${}^{16}\mathrm{Ne}$($2_1^{+})+p$ channel, and the other, $5/2_1^{+}$, decays mainly into the ${}^{16}\mathrm{Ne}$($0_1^{+})+p$ channel. Because the daughter nucleus ${}^{16}\mathrm{Ne}$ is a two-proton emitter, the narrow ${}^{17}\mathrm{Na}$ states must undergo a sequential three-proton decay via intermediate states in ${}^{16}\mathrm{Ne}$.

• Received 5 February 2010

DOI:https://doi.org/10.1103/PhysRevC.81.051301