Coulomb excitation of radioactive<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">Na</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>and its stable mirror<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">Ne</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>

The low-energy structures of the mirror nuclei $^{21}\mathrm{Ne}$ and radioactive $^{21}\mathrm{Na}$ have been examined by using Coulomb excitation at the TRIUMF-ISAC radioactive ion beam facility. Beams of $~5\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ions/s were accelerated to 1.7 MeV/$A$ and Coulomb excited in a 0.5 mg/${\mathrm{cm}}^{2}$ ${}^{\mathrm{nat}}$Ti target. Scattered beam and target particles were detected by the segmented Si detector BAMBINO, while $\ensuremath{\gamma}$ rays were observed by using two TIGRESS HPGe clover detectors perpendicular to the beam axis. For each isobar, Coulomb excitation from the ${\frac{3}{2}}^{+}$ ground state to the first excited ${\frac{5}{2}}^{+}$ state was observed and $B(E2)$ values were determined by using the ${2}^{+}\ensuremath{\rightarrow}{0}^{+}$ de-excitation in $^{48}\mathrm{Ti}$ as a reference. The \ensuremath{\phi} segmentation of BAMBINO was used to deduce tentative assignments for the signs of the mixing ratios between the $E2$ and $M1$ components of the transitions. The resulting $B(E2)\ensuremath{\uparrow}$ values are $131\ifmmode\pm\else\textpm\fi{}9 {e}^{2}$ ${\mathrm{fm}}^{4}$ ($25.4\ifmmode\pm\else\textpm\fi{}1.7$ W.u.) for $^{21}\mathrm{Ne}$ and $205\ifmmode\pm\else\textpm\fi{}14 {e}^{2}$ ${\mathrm{fm}}^{4}$ ($39.7\ifmmode\pm\else\textpm\fi{}2.7$ W.u.) for $^{21}\mathrm{Na}$. The fit to the present data and the known lifetimes determined $E2/M1$ mixing ratios and $B(M1)\ensuremath{\downarrow}$ values of $\ensuremath{\delta}=(\ensuremath{-})0.0767\ifmmode\pm\else\textpm\fi{}0.0027$ and $0.1274\ifmmode\pm\else\textpm\fi{}0.0025 {\ensuremath{\mu}}_{N}{\text{}}^{2}$ and $\ensuremath{\delta}=(+)0.0832\ifmmode\pm\else\textpm\fi{}0.0028$ and $0.1513\ifmmode\pm\else\textpm\fi{}0.0017 {\ensuremath{\mu}}_{N}{\text{}}^{2}$ for $^{21}\mathrm{Ne}$ and $^{21}\mathrm{Na}$, respectively (with Krane and Steffen sign convention). By using the effective charges ${e}_{p}=1.5e$ and ${e}_{n}=0.5e$, the $B(E2)$ values produced by the $p$-$\mathit{sd}$ shell model are 30.7 and 36.4 W.u. for $^{21}\mathrm{Ne}$ and $^{21}\mathrm{Na}$, respectively. This analysis resolves a significant discrepancy between a previous experimental result for $^{21}\mathrm{Na}$ and shell-model calculations.

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