We report the first experimental determination of the hyperfine quenching rate of the $6s^2 ^1!S_0 (F=1/2) - 6s6p ^3!P_0 (F=1/2)$ transition in $^{171}$Yb with nuclear spin $I=1/2$. This rate determines the natural linewidth and the Rabi frequency of the clock transition of a Yb optical frequency standard. Our technique involves spectrally resolved fluorescence decay measurements of the lowest lying $^3!P_{0,1}$ levels of neutral Yb atoms embedded in a solid Ne matrix. The solid Ne provides a simple way to trap a large number of atoms as well as an efficient mechanism for populating $^3!P_0$. The decay rates in solid Ne are modified by medium effects including the index-of-refraction dependence. We find the $^3!P_0$ hyperfine quenching rate to be $(4.42pm0.35)times10^{-2} mathrm{s}^{-1}$ for free $^{171}$Yb, which agrees with recent ab initio calculations.
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