Consider the precession of the spin of a muon, initially longitudinally polarized, as the muon moves in a circular orbit in a plane perpendicular to a uniform magnetic field B.

(a) Show that the difference Ω of the spin precession frequency and the orbital gyration frequency is

Ω = eBα/m_μc

independent of the muon's energy, where α = (g – 2)/2 is the magnetic moment anomaly. (Find equations of motion for the components of spin along the mutually perpendicular directions defined by the particle's velocity, the radius vector from the center of the circle to the particle, and the magnetic field.)

(b) For the CERN Muon Storage Ring, the orbit radius is R = 2.5 meters and В = 17 × 10³ gauss. What is the momentum of the muon? What is the time dilatation factor γ? How many periods of precession T = 2π/Ω occur per observed laboratory mean lifetime of the muons? [m_μ = 105.66 MeV τ₀ = 2.2 × 10^–6 s, α ≈ α/2π].

(c) Express the difference frequency П in units of the orbital rotation frequency and compute how many precessional periods (at the difference frequency) occur per rotation for a 300 MeV muon, a 300 MeV electron, a 5 GeV electron (this last typical of the e⁺e^– storage ring at Cornell).