Measurement of the Muonium Hfs Splitting and of the Muon Moment by "Double Resonance," and a New Value of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>

Abstract

We have determined the hyperfine interval $\ensuremath{\Delta}\ensuremath{\nu}$ and the muon moment ${\ensuremath{\mu}}_{\ensuremath{\mu}}=\frac{g_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{'}}{\ensuremath{\mu}}_{\mathrm{B}}}{2}$ from the frequencies of the two alternative Zeeman transitions in muonium, working at that "magic" field where $\frac{\ensuremath{\partial}{\ensuremath{\nu}}_{1}}{\ensuremath{\partial}B}=\frac{\ensuremath{\partial}{\ensuremath{\nu}}_{2}}{\ensuremath{\partial}B}=0$. Extrapolating determinations of $\ensuremath{\Delta}\ensuremath{\nu}$ in Kr and Ar to zero pressure, we obtain consistent results, yielding jointly $\ensuremath{\Delta}\ensuremath{\nu}(0)=4463.3022(89)$ MHz. Assuming that ${g}_{j}(M)$ is unshifted by collisions, we obtain $\frac{{g}_{e}}{g_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{'}}}=206.76509(80)$, which corresponds to $\frac{{f}_{\ensuremath{\mu}}}{{f}_{p}}=3.183373(13)$. The combination of these results gives ${\ensuremath{\alpha}}^{\ensuremath{-}1}\ensuremath{-}137=0.03654(30)$ (2.1 ppm in ${\ensuremath{\alpha}}^{\ensuremath{-}2}$), in good agreement with the recommended value of 0.036 02(21).