Quadrupole Moments of Neutron-Deficient ${}^{20,21}$Na

The electric-quadrupole coupling constant of the ground states of the proton drip line nucleus ${}^{20}$Na($I^{\pi }$ = 2${}^{+}$, $T_{1\mathrm{/}2}$ = 447.9 ms) and the neutron-deficient nucleus ${}^{21}$Na($I^{\pi }$ = 3/2${}^{+}$, $T_{1\mathrm{/}2}$ = 22.49 s) in a hexagonal ZnO single crystal were precisely measured to be $|eqQ/h| = 690 \pm 12$kHzand939$\pm$14kHz,respectively,usingthemulti-frequency$\beta$-ray detecting nuclear magnetic resonance technique under presence of an electric-quadrupole interaction. A electric-quadrupole coupling constant of ${}^{27}$Na in the ZnO crystal was also measured to be $\mathrm{\mid }eqQ\mathrm{/}h\mathrm{\mid }=48.4\pm 3.8$ kHz. The electric-quadrupole moments were extracted as $\mathrm{\mid }Q{(}^{20}$Na)$\mathrm{\mid }$ = 10.3 $\pm$ 0.8 $e$ fm${}^2$ and $\mathrm{\mid }Q{(}^{21}$Na)$\mathrm{\mid }$ = 14.0 $\pm$ 1.1 $e$ fm${}^2$, using the electric-coupling constant of ${}^{27}$Na and the known quadrupole moment of this nucleus as references. The present results are well explained by shell-model calculations in the full $sd$-shell model space.