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A Coulomb-excitation reorientation-effect measurement using the TIGRESS$\gamma-$ray spectrometer at the TRIUMF/ISAC II facility has permitted thefirst determination of the $\langle 2^+_1\mid\mid \hat{E2} \mid\mid2^+_1\rangle$ diagonal matrix element in $^{12}$C from particle$-\gamma$coincidence data. Required state-of-the-art no-core shell model calculations ofthe 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 beenperformed. Consistent predictions show a larger polarizability than previouslyanticipated. The polarizability of the 2$^+_1$ state is introduced into thecurrent and previous Coulomb-excitation reorientation-effect analysis of$^{12}$C. Spectroscopic quadrupole moments of $Q_{_S}(2_1^+)= +0.053(44)$ eband $Q_{_S}(2_1^+)= +0.08(3)$ eb are determined, respectively, yielding aweighted average of $Q_{_S}(2_1^+)= +0.071(25)$ eb, in agreement with recent abinitio calculations. The present measurement confirms that the 2$^+_1$ state of$^{12}$C is oblate and emphasizes the important role played by the nuclearpolarizability in Coulomb-excitation studies of light nuclei.

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