Yields and production rates of $^9$Li and $^8$He measured with the Double Chooz near and far detectors

The yields and production rates of the radioisotopes $^9$Li and $^8$He created by cosmic muon spallation on $^{12}$C have been measured by the Double Chooz experiment. Comparing the data to a detailed simulation of the $^9$Li and $^8$He decays, the contribution of the $^8$He radioisotope is found to be compatible with zero. The observed $^9$Li yields in the near ($\sim$120 m.w.e overburden) and far ($\sim$300 m.w.e.) detectors are $4.37\pm0.49$ and $7.98\pm0.52$ respectively, in units of $10^{-8}\,\mathrm{\mu ^{-1}g^{-1}cm^{2} }$. The low overburdens of the near and far detectors gives a unique insight when combined with measurements by KamLAND and Borexino to give the first multi-experiment, data driven relationship between the $^9$Li yield and the mean muon energy according to the power law $Y = Y_0\left^{\overline{\alpha}}$, giving $\overline{\alpha}=0.79\pm0.06$ and $Y_0=(0.29\pm0.09)\times 10^{-8}\,\mathrm{\mu ^{-1}g^{-1}cm^{2} }$. This relationship gives future liquid scintillator based nuclear reactor experiments the ability to predict their cosmogenic $^9$Li background rates.