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Brian Hayes

Bottled Electrons

One good way to measure g is to capture an electron and keep it in a bottle formed out of electric and magnetic fields. In confinement, the electron executes an elegant dance of twirls and pirouettes. The various modes of motion are quantized, meaning that only certain discrete energy states are possible. In some of these states the electron's intrinsic magnetic moment is oriented parallel to the external magnetic field, and in other states it is antiparallel. The energy difference between two such states is proportional to g. Thus a direct approach to determining g is simply to measure the energy of a "spin-flip" transition between parallel and antiparallel states.

The drawback of this straightforward experimental design is that you cannot know g with any greater accuracy than you know the strength of the external field. A clever trick sidesteps this problem: Measure the energies of two transitions, both of which depend on the magnetic field but only one of which involves a spin flip. For the non-flip transition, the constant of proportionality that sets the energy scale is exactly 2, whereas for the spin-flip transition the constant is g. Taking the ratio of the two energies eliminates dependence on the strength of the field.

Experiments with isolated electrons were pioneered by Hans Dehmelt of the University of Washington, who kept them penned up for weeks at a time—long enough that some of them were given names, like family pets. Although the technique may sound simple in its principles, getting results accurate to 11 significant figures is not a project for a high school science fair.

In the case of the muon, measuring g is even more challenging. The muon is an unstable particle, with a lifetime of a few microseconds, and so keeping a pet muon in a cage is not an option. The best muon g–2 measurements come from a 20-year-long experiment designated E821, carried out at the Brookhaven National Laboratory by workers from 11 institutions. Clouds of muons with their spins aligned circulate in a toroidal vacuum chamber immersed in a strong magnetic field. The apparatus is adjusted so that if g were exactly 2, the particles would complete each orbit with the same orientation they had at the outset. But because g differs from 2, the spin axis precesses slowly, drifting about 0.8 degree on each circuit of the ring. When a muon decays, it emits an electron preferentially in the direction of the spin axis. The spatial distribution of these electrons reveals the rate of precession and thus the value of g–2.

The latest value of the muon g factor reported by the E821 group works out to 2.0023318416 ± 0.0000000012. This number differs from the electron g factor in the fifth decimal place, and its precision is only at the parts-per-billion level rather than parts-per-trillion. Despite the lesser precision, however, the confrontation between theory and experiment turns out to be more dramatic in the case of the muon.

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