Precision Measurement of the g Factor of the Free Electron

The measurement reported is essentially a refinement of an earlier measurement in which the $g$ factor was found to \ifmmode\pm\else\textpm\fi{}2.4 parts in ${10}^{6}$. The method is as follows: 100-keV electrons in 0.2-\ensuremath{\mu}sec bunches move parallel to a magnetic field and strike a gold foil. The part of an electron bunch which is scattered at right angles, and which, consequently, is partially polarized, is trapped in the magnetic field and held for a measured length of time (up to 1.9 msec). The bunch is then released from the trap and allowed to strike a second gold foil. The cycle is repeated 500 times per sec. The part of the bunch scattered at right angles by the second foil strikes a thin-window Geiger counter. The fraction of the bunch scattered into the counter depends upon its final direction of polarization. A plot of the intensity vs trapping time is a cosine curve whose frequency is the difference between the orbital frequency and the spin precession frequency. This is related to the $g$ factor as follows: $\frac{{\ensuremath{\omega}}_{D}\mathrm{mc}}{\mathrm{Be}}=a$, where $g$ is $2(1+a)$, ${\ensuremath{\omega}}_{D}$ is $2\ensuremath{\pi}$ times the difference frequency, $B$ is the magnetic induction, and $m$, $c$, and $e$ have the usual meaning. Thus the "anomaly," $a$, is measured directly. The present experiment is an advance over the earlier one in four main respects: (a) Separation of the polarization effect from the background. Alternate groups of 64 electron bunches were held in the trap for times $t$ and $t+\frac{1}{2}{T}_{D}$, where ${T}_{D}$ is the period of alternation of intensity or "difference period." Counts from the alternate bunches were accumulated in separate scalers, and the ratio was used as the measure of polarization. This eliminated virtually all instrumental asymmetries associated with counting. (b) Electrostatic effects. A new vacuum chamber in which all material was eliminated from inside the electron orbits and removed to a greater distance from the orbits on all sides greatly reduced the effects of stray electric fields due to surface charges. (c) Magnetic field. A new solenoid of increased dimensions, and new proton-resonance field measuring apparatus were used, to obtain a significant improvement in the mapping of the magnetic field in the trap. ($B$ appears in the formula for $a$, and in order to have a trap, $B$ must be slightly nonuniform; hence the necessity of mapping $B$ in the trap.) (d) The beginning of the measured trapping interval was moved out to about 300 \ensuremath{\mu}sec after injection and a time difference method was used. This eliminated all erros associated with initial structure (bunching) of the electron cloud, and eliminated errors in the knowledge of the time of injection and capture. The final result is $a=0.001159622\ifmmode\pm\else\textpm\fi{}0.000000027$. In terms of a series in the fine structure constant, the experiment gives $a=\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}\ensuremath{-}\frac{(0.327\ifmmode\pm\else\textpm\fi{}0.005){\ensuremath{\alpha}}^{2}}{{\ensuremath{\pi}}^{2}}$ and theory gives $a=\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}\ensuremath{-}\frac{0.328{\ensuremath{\alpha}}^{2}}{{\ensuremath{\pi}}^{2}}$.