Radiative transitions and isospin mixing inC12

$\gamma$-ray decays of the 12.7 and 16.1 MeV states of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ are investigated in a coincidence study of the ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}({}_{}{}^3{}_{}{}^{}\mathrm{He}, p\gamma )$ reaction. We obtain $\frac{{\Gamma }_{\gamma 0}}{\Gamma }=(1.93\mathrm{}\pm 0.12)\%$, $\frac{{\Gamma }_{\gamma 1}}{{\Gamma }_{\gamma 0}}=(15.0\mathrm{}\pm 1.8)\%$ and $\frac{{\Gamma }_{\alpha }}{\Gamma }=(97.8\mathrm{}\pm 0.1)\%$ for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(12.7), and $\frac{{\Gamma }_{\gamma 1}}{\Gamma }=(2.42\mathrm{}\pm 0.29)\times {10}^{-3\mathrm{}}$ for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(16.1). Relative $\gamma$-ray branching ratios of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(16.1) were measured using the $E_p=163\mathrm{}$ keV ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}(p, \gamma )$ resonance. We find $\frac{{\Gamma }_{\gamma 0}}{{\Gamma }_{\gamma 1}}=(4.6\mathrm{}\pm 0.7)\%$, $\frac{{\Gamma }_{\gamma }(16.1\mathrm{}\to 9.6)}{{\Gamma }_{\gamma 1}}=(2.4\mathrm{}\pm 0.4)\%$, and $\frac{{\Gamma }_{\gamma }(16.1\mathrm{}\pm 12.7)}{{\Gamma }_{\gamma 1}}=(1.46\mathrm{}\pm 0.25)\%$. This information, together with existing data on $M1\mathrm{}$ transitions and single nucleon transfer reactions, is used to determine the isospin mixing between the 12.7 and 15.1 MeV levels of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$. A charge dependent matrix element of 110 ± 30 keV is deduced.