Measurement of the Muonium Hfs Splitting and of the Muon Moment by "Double Resonance," and a New Value ofα

We have determined the hyperfine interval $\Delta \nu$ and the muon moment ${\mu }_{\mu }=\frac{g_{\mu }^{}{}_{}{}^{\prime }{\mu }_{\mathrm{B}}}{2}$ from the frequencies of the two alternative Zeeman transitions in muonium, working at that "magic" field where $\frac{\partial {\nu }_1}{\partial B}=\frac{\partial {\nu }_2}{\partial B}=0\mathrm{}$. Extrapolating determinations of $\Delta \nu$ in Kr and Ar to zero pressure, we obtain consistent results, yielding jointly $\Delta \nu \mathrm{}\left(0\right)=4463.3022\mathrm{}\left(89\right)\mathrm{}$ MHz. Assuming that $g_j\mathrm{}\left(M\right)\mathrm{}$ is unshifted by collisions, we obtain $\frac{g_e}{g_{\mu }^{}{}_{}{}^{\prime }}=206.765\mathrm{}09\left(80\right)\mathrm{}$, which corresponds to $\frac{f_{\mu }}{f_p}=3.183\mathrm{}373\left(13\right)\mathrm{}$. The combination of these results gives ${\alpha }^{-1\mathrm{}}-137=0.036\mathrm{}54\left(30\right)\mathrm{}$ (2.1 ppm in ${\alpha }^{-2\mathrm{}}$), in good agreement with the recommended value of 0.036 02(21).