Reorientation-effect measurement of the first 2${}^{+}$ state in ${}^{12}$C: confirmation of oblate deformation

A Coulomb-excitation reorientation-effect measurement using the TIGRESS $\gamma -$ray spectrometer at the TRIUMF/ISAC II facility has permitted the first determination of the $\langle 2^+_1\mid\mid \hat{E2} \mid\mid 2^+_1\rangle$diagonalmatrixelementin$^{12}$Cfromparticle$-\gamma$ coincidence data. Required state-of-the-art no-core shell model calculations of the nuclear polarizability for the ground and first-excited (2${}_1^{+}$) states in ${}^{12}$C using chiral NN N${}^4$LO500 and NN+3NF350 interactions have been performed. Consistent predictions show a larger polarizability than previously anticipated. The polarizability of the 2${}_1^{+}$ state is introduced into the current and previous Coulomb-excitation reorientation-effect analysis of ${}^{12}$C. Spectroscopic quadrupole moments of $Q_{{}_S}(2_1^{+})=+0.053(44)$ eb and $Q_{{}_S}(2_1^{+})=+0.08(3)$ eb are determined, respectively, yielding a weighted average of $Q_{{}_S}(2_1^{+})=+0.071(25)$ eb, in agreement with recent ab initio calculations. The present measurement confirms that the 2${}_1^{+}$ state of ${}^{12}$C is oblate and emphasizes the important role played by the nuclear polarizability in Coulomb-excitation studies of light nuclei.