Magnetic moments, Zeeman splitting and fine structure

Classically, an electron of charge −e circling at speed v on radius r is a current ev/(2πr) running opposite to its motion around an area πr². Its moment is μ = IA = evr/2, and since L = mₑ v r, μ = −(e/2mₑ)L: a gyromagnetic ratio that does not depend on r or v. Quantum mechanics keeps the ratio, because the probability current of ψₙₗₘ circulates with the same e/2mₑ, so quantising L_z = mₗ ħ quantises μ_z = −mₗ μ_B with μ_B = eħ/(2mₑ) = 9.274 × 10⁻²⁴ J T⁻¹. The magnitude is √(l(l+1)) μ_B, so a d electron carries 2.45 μ_B yet can only show 2, 1, 0, −1 or −2 μ_B along any chosen axis. In energy units μ_B is 5.788 × 10⁻⁵ eV T⁻¹: even at 10 T, μ_B B = 5.8 × 10⁻⁴ eV, under 1/17 000 of the 10.2 eV Lyman-α photon. That is why every magnetic shift in this lesson is a small perturbation on the Coulomb levels.

Put B along z. The energy of a moment is U = −μ⋅B = −μ_z B, so an orbital state moves by mₗ μ_B B: 2l + 1 sublevels in equal steps of μ_B B, the same step in every level. If both levels of a transition have total spin S = 0 the moment is purely orbital, and the photon's one unit of angular momentum allows Δm = 0, ±1. However many sublevel pairs there are, only three photon energies occur, E₀ and E₀ ± μ_B B. At 1.00 T the side lines sit μ_B B/h = 14.0 GHz from the centre; for cadmium's 643.8 nm ¹D₂ → ¹P₁ line that is 0.019 nm. Planck's constant cancels, since μ_B B/h = eB/(4πmₑ), which is exactly the Larmor shift Lorentz derived for a classical electron bound by a spring. That correspondence is why the triplet is called normal. Its polarisation is classical too: the Δm = 0 line is linearly polarised along B, and the Δm = ±1 lines are circularly polarised when viewed along B, where the central line vanishes.

Spin adds a moment μₛ, z = −gₛ mₛ μ_B with gₛ ≈ 2: twice as much moment per unit of angular momentum as the orbit. It also has something to couple to with no applied field. In the electron's rest frame the nucleus circles it, and that moving charge makes a field Bᵢₙₜ = (1/(e mₑ c² r))(dU/dr) L, parallel to L. The spin's energy −μₛ⋅Bᵢₙₜ, halved by Thomas precession because that frame accelerates, is H_SO = (1/(2mₑ²c²))(1/r)(dU/dr) L⋅S. It depends on the angle between L and S, so neither L_z nor S_z is conserved any more; only J = L + S and its projection mⱼ are. Writing L⋅S = ½(J² − L² − S²) turns it into E_SO = (A/2)[j(j+1) − l(l+1) − s(s+1)]. A p electron has j = 3/2 at +A/2 and j = ½ at −A, a doublet 3A/2 apart. Sodium's D lines at 588.995 nm and 589.592 nm put that gap at 2.13 × 10⁻³ eV, and equating it to 2μ_B Bᵢₙₜ gives an internal field of about 18 T.

A level is labelled ^(2S+1)L_J: the multiplicity 2S + 1, the letter S, P, D, F for L = 0, 1, 2, 3, and J as a subscript. One electron outside closed shells has S = ½, so its terms are doublets: 2p gives ²P₁/₂ and ²P₃/₂. In hydrogen, spin–orbit coupling arrives with two relativistic corrections of the same size, the (v/c)² correction to kinetic energy and the Darwin term, and together they give Dirac's Eₙⱼ = −(13.6 eV/n²)[1 + (α²/n²)(n/(j + ½) − ¾)]. The spin–orbit part alone, using hydrogen's ⟨1/r³⟩, gives A = α⁴m_ec²/[2n³l(l + ½)(l + 1)] = 3.02 × 10⁻⁵ eV for 2p, so the 2p doublet is 3A/2 = 4.53 × 10⁻⁵ eV (10.9 GHz) wide, matching the formula. Two checks: the correction is of order α² = 5.3 × 10⁻⁵ of the gross structure, as a (v/c)² effect should be when v/c = α in the ground state; and α → 0 returns Bohr's −13.6 eV/n². Because only n and j appear, 2s₁/₂ and 2p₁/₂ are predicted degenerate; the 4.4 × 10⁻⁶ eV Lamb shift that parts them is quantum electrodynamics, outside this model.

While μ_B B is small against the fine-structure gap, the internal coupling wins: L and S precess rapidly about J, and J precesses slowly about B. The total moment μ = −(μ_B/ħ)(L + 2S) = −(μ_B/ħ)(J + S) is not parallel to J, because the spin counts double, and only its component along J survives the fast precession. Projecting with S⋅J = ½(J² + S² − L²) gives ΔE = g_J mⱼ μ_B B with g_J = 1 + [j(j+1) + s(s+1) − l(l+1)]/[2j(j+1)]. For S = 0 this is 1 in every level, and the normal triplet follows. For one electron it is 2 in ²S₁/₂, 2/3 in ²P₁/₂ and 4/3 in ²P₃/₂. Upper and lower levels now split at different rates, so the photon shifts (g′m′ − g″m″)μ_B B, primes marking the upper and lower level, no longer collapse: ²P₁/₂ → ²S₁/₂ gives four lines at ±2/3 and ±4/3 of μ_B B, and ²P₃/₂ → ²S₁/₂ gives six at ±1/3, ±1 and ±5/3. Every hydrogen level has S = ½, so hydrogen has no normal Zeeman effect at all: its weak-field splitting is anomalous.

The coupled picture lasts only while the applied field is weak against the internal one. Once μ_B B exceeds the fine-structure gap, B turns L and S about itself faster than they can turn about each other; J stops being conserved and mₗ and mₛ become the good quantum numbers again. Then ΔE = (mₗ + 2mₛ)μ_B B + A mₗ mₛ, with spin–orbit coupling reduced to a first-order correction. Electric-dipole photons do not act on spin, so Δmₛ = 0 and Δmₗ = 0, ±1: a p → s line returns to a triplet spaced μ_B B, each outer line split into a doublet A apart. The crossover sits near μ_B B ≈ ΔE_fs: 0.78 T for hydrogen's 2p, inside any laboratory magnet, but 37 T for sodium's 3p, so sodium stays anomalous in every ordinary field and hydrogen does not. In between, only mⱼ stays good: neither j nor the pair mₗ, mₛ does, so each pair of states sharing mⱼ must be diagonalised exactly, as the figure does. Nuclear moments are smaller by mₑ/mₚ ≈ 1/1836, so hyperfine structure is a further, finer splitting left out here along with the Lamb shift.