Magnetic Moment of the Proton in Units of the Bohr Magneton; the Magnetic Moment of the Electron

Abstract

The details of a previously reported measurement of the proton magnetic moment in units of the Bohr magneton are given. This ratio of moments, which is obtained from common magnetic field observations of the nuclear magnetic resonance frequency of protons in a spherical sample of mineral oil and the cyclotron frequency of free low-energy electrons, is found to be $\frac{{\ensuremath{\mu}}_{p(\mathrm{oil})}}{{\ensuremath{\mu}}_{0}}={(657.462\ifmmode\pm\else\textpm\fi{}0.003)}^{\ensuremath{-}1}$, where the uncertainty represents the estimated 50% probable error. The magnetic moment of the free proton is found, upon application of the appropriate diamagnetic correction factor, to be $\frac{{\ensuremath{\mu}}_{p}}{{\ensuremath{\mu}}_{0}}={(657.442\ifmmode\pm\else\textpm\fi{}0.003)}^{\ensuremath{-}1}$. The present result may be combined with reported values for the ratio of the magnetic moment of the electron to the moment of the proton to yield for the magnetic moment of the free electron in units of the Bohr magneton, ${\frac{{\ensuremath{\mu}}_{e}}{{\ensuremath{\mu}}_{0}},=1.001168\ifmmode\pm\else\textpm\fi{}0.000005}{=1+(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}})+(1.2\ifmmode\pm\else\textpm\fi{}0.9)(\frac{{\ensuremath{\alpha}}^{2}}{{\ensuremath{\pi}}^{2}}),}$ where the uncertainty is the estimated 50% probable error. This result is to be compared with the current theoretically estimated value for this quality, ${\frac{{\ensuremath{\mu}}_{e}}{{\ensuremath{\mu}}_{0}},=1+(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}})\ensuremath{-}0.328(\frac{{\ensuremath{\alpha}}^{2}}{{\ensuremath{\pi}}^{2}})}{=1.0011596.}$