Isospin symmetry in $B\left(E2\right)$ values: Coulomb excitation study of ${}^{21}\mathrm{Mg}$

The $T_z=-\frac{3}{2}$ nucleus ${}^{21}\mathrm{Mg}$ has been studied by Coulomb excitation on ${}^{196}\mathrm{Pt}$ and ${}^{110}\mathrm{Pd}$ targets. A 205.6(1)-keV $\gamma$-ray transition resulting from the Coulomb excitation of the ${\frac{5}{2}}^{+}$ ground state to the first excited ${\frac{1}{2}}^{+}$ state in ${}^{21}\mathrm{Mg}$ was observed for the first time. Coulomb excitation cross-section measurements with both targets and a measurement of the half-life of the ${\frac{1}{2}}^{+}$ state yield an adopted value of $B(E2;{\frac{5}{2}}^{+}\to {\frac{1}{2}}^{+})=13.3\left(4\right)$ W.u. A new excited state at 1672(1) keV with tentative ${\frac{9}{2}}^{+}$ assignment was also identified in ${}^{21}\mathrm{Mg}$. This work demonstrates a large difference in the $B(E2;{\frac{5}{2}}^{+}\to {\frac{1}{2}}^{+})$ value between $T=\frac{3}{2}, A=21$ mirror nuclei. The difference is investigated in the shell-model framework employing both isospin conserving and breaking USD interactions and using modern ab initio nuclear structure calculations, which have recently become applicable in the $sd$ shell.

Figure 1

Low-lying level schemes of ${}^{21}\mathrm{Mg}$ and ${}^{21}\mathrm{F}$ with known $\gamma$-ray transitions, branching ratios [blue (dark grey) numbers] and level half-lives [green (light grey) numbers]. Data were obtained from Refs. [7, 8, 9]. Spectroscopic information obtained in the present work is indicated in red (gray).

Figure 2

Energy spectra of $\gamma$ rays gated by the $A=21$ recoils detected in the downstream Si detector and scattered from (a) ${}^{196}\mathrm{Pt}$ and (b) ${}^{110}\mathrm{Pd}$ target. The black curves are with the Doppler correction, while the red (gray) curves are without it. The peaks at 205.6(1) keV [solid red (gray) diamonds] and 1672(1) keV (solid black diamonds) are the first direct observations of the ${\frac{1}{2}}^{+}\to {\frac{5}{2}}^{+}$ and ${\frac{9}{2}}^{+}\to {\frac{5}{2}}^{+}\gamma$-ray transitions in ${}^{21}\mathrm{Mg}$, respectively.

Figure 3

Two-dimensional ${\chi }^2$ surfaces obtained from the gosia2 analysis performed with (a) ${}^{196}\mathrm{Pt}$ and (b) ${}^{110}\mathrm{Pd}$ target data with applied ${\chi }^2<{\chi }_{\min }^2+1$ criterion.

Figure 4

(a) Experimental decay curve of the ${\frac{1}{2}}^{+}$ state in ${}^{21}\mathrm{Mg}$ together with the best fit simulated decay curve. The prompt time difference and time random background sampling distributions are also presented. (b) The obtained ${\chi }^2$ values as a function of $t_{1/2}$ of the simulated activity.

Figure 5

Experimental and theoretical $B(E2;{\frac{5}{2}}^{+}\to {\frac{1}{2}}^{+})$ values for (a) $T_z=-\frac{3}{2}$ and (b) $T_z=+\frac{3}{2}$ mirror nuclei including the new experimental value for ${}^{21}\mathrm{Mg}$. Theoretical $B\left(E2\right)$ values are obtained from the shell-model calculations using USDB and ${\mathrm{USD}}_{2,3}^m$ [6] interactions in addition to the SA-NCSM (only for $A=21$), CCEI, and IM-SRG ab initio calculations.

Figure 6

Experimental (solid symbols) and theoretical (open symbols) $B(E2;{\frac{5}{2}}^{+}\to {\frac{9}{2}}^{+})$ values for ${}^{21}\mathrm{Mg}$ and ${}^{21}\mathrm{F}$.