The spin g factor and magnetic resonance

Send a charge −e round a loop of radius r with period T. The current is −e/T and the moment is μ = IA = −eπr²/T, while the angular momentum is L = mₑvr = 2πmₑr²/T. Divide, and the geometry cancels: μ/L = −e/2mₑ. Promote it to operators and orbital motion gives μₗ = −(e/2mₑ)L = −μ_B L/ħ, which is gₗ = 1 by the definition of the Bohr magneton μ_B = eħ/2mₑ = 5.7884×10⁻⁵ eV T⁻¹. Now take the Stern–Gerlach result at face value. Silver's beam splits into two, and the deflection measures a moment projection of one Bohr magneton. But the two states carry S_z = ±ħ/2, half the orbital unit. So the spin moment runs at twice the classical rate, and the only honest response is to put the factor in explicitly: μₛ = −gₛ μ_B S/ħ, with gₛ ≈ 2 read off the data. Nothing in the loop argument predicts that, and nothing in it forbids it — μ/L is a property of how charge and mass are distributed through a body, and spin is not a distribution of anything.

In a uniform field B₀ẑ the energy is H = −μ⋅B₀ = +gₛ μ_B B₀ S_z/ħ, so E(mₛ) = gₛ μ_B B₀ mₛ and the gap is ΔE = gₛ μ_B B₀. The sign matters and it is routinely got backwards: because the electron's charge is negative, μ points against S, so mₛ = +½ — spin along the field — is the upper state, and mₛ = −½ is the ground state. Numerically gₛμ_B = 115.90 µeV T⁻¹, so at 0.340 T the gap is 39.4 µeV. Set that against room temperature: kT at 295 K is 25.4 meV, 645 times larger. Boltzmann then gives a fractional excess in the lower state of only ΔE/2kT = 7.8×10⁻⁴, about one spin in 1290. Inductively detected magnetic resonance is a bulk technique by necessity — the signal is the small imbalance across a very large number of spins, not the flipping of one.

Ehrenfest's theorem gives d⟨S⟩/dt = (i/ħ)⟨[H, S]⟩. With H = −μ⋅B₀ linear in S, and the spin commutators [Sᵢ, Sⱼ] = iħ εᵢⱼₖ Sₖ, the algebra closes on itself: every ħ cancels and what is left is d⟨S⟩/dt = ⟨μ⟩ × B₀, the same torque equation a classical magnetised top obeys. So the expectation value of the spin precesses about B₀ at a fixed angle, at the Larmor rate ω_L = γB₀ with γ = gₛ μ_B/ħ = 1.7609×10¹¹ rad s⁻¹ T⁻¹. This is not an approximation and not a picture — it is exact for a uniform field, and it is the one place in spin physics where a classical vector is safe. In ordinary frequency the rate is γ/2π = 28.025 GHz per tesla, so at 0.340 T the moment turns once every 105 ps. What the picture still does not license is definite transverse components: ⟨Sₓ⟩ and ⟨S_y⟩ trace a circle, but Sₓ and S_y have no simultaneous values.

Add a small field B₁ rotating in the xy plane at frequency f, and move to a frame turning with it. In that frame the static field shrinks to B₀ − 2πf/γ along z while B₁ stands still along x. On resonance, where 2πf = γB₀, the z part vanishes and the only field left is B₁ — so the spin precesses about B₁ and flips completely, however weak B₁ is. Off resonance the effective field tilts back towards z, and the largest flip probability falls to ω₁²/(ω₁² + Δω²), with ω₁ = γB₁ and Δω the detuning. Detune by much more than γB₁ and the transition dies. Neglecting relaxation, which adds its own width in a real sample, that is why the line is narrow: with B₁ = 0.10 mT, ω₁/2π is 2.8 MHz, three parts in 10⁴ of a 9.5 GHz carrier. In practice the carrier is pinned by a resonant cavity and B₀ is swept through the line, because sweeping a magnet is far easier than sweeping a cavity.

One (f, B₀) pair gives g = hf/(μ_B B₀), and it swallows every calibration error in the magnet. Take several fields instead and fit f against B₀: the slope is gμ_B/h, and a non-zero intercept exposes a residual field offset rather than hiding it inside g. A slope of 28.025 GHz T⁻¹ divided by μ_B/h = 13.9962 GHz T⁻¹ returns g = 2.0023. Dirac's equation, applied to a spin-½ point particle in an electromagnetic field, produces exactly 2 with nothing inserted by hand — the strongest reason to stop calling gₛ = 2 an empirical fudge. The measured free-electron value is 2.00231930436, and the anomaly aₑ = (g − 2)/2 = 0.00115965 is the QED correction from the electron emitting and reabsorbing virtual photons. Schwinger's one-loop term α/2π = 0.00116141 reproduces it to 0.15%, and the full calculation against the full measurement is one of the sharpest tests any physical theory has passed.

gₛ = 2.0023 belongs to a free electron. Bind it and orbital angular momentum leaks back in through spin–orbit mixing with excited states, shifting g by roughly the ratio of the spin–orbit coupling to the orbital level spacing. DPPH's unpaired electron sits at g = 2.0036, 0.064% above free — small, because the radical's orbital moment is well quenched — while a transition-metal ion can run from about 1.4 to 6 and depend on which crystal axis lies along B₀. A measured g is therefore a probe of the electron's surroundings, not a universal constant. Nothing in the resonance argument was electronic, either. A proton has spin ½ with μ = gₚ μ_N I/ħ, where μ_N = μ_B mₑ/mₚ = 3.1525×10⁻⁸ eV T⁻¹ and gₚ = 5.5857 — measured, not 2, because the proton is composite. That gives γₚ/2π = 42.577 MHz T⁻¹, so in a 1.5 T scanner protons resonate at 63.9 MHz while electrons in the same magnet need 42.0 GHz, a factor of 658.