Anomalous Zeeman Effect and the Lande g-Factor

Derivation

Anomalous Zeeman Effect and the Lande g-Factor

D-302 Home PU-306 Threads energy · fields · matter · symmetry Depends on Fine-Structure Energy Formula, Degenerate Perturbation Theory

Statement

For an atom in a weak, uniform magnetic field \( \vec B = B\,\hat z \) — weak meaning the magnetic interaction is small compared with the fine-structure splitting — each fine-structure level \( ^{2S+1}L_J \) splits into \( 2J+1 \) equally spaced sublevels with first-order energy shift \( \Delta E = g_J\,\mu_B\,B\,m_J \), where the Landé g-factor is \( g_J = 1 + \dfrac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)} \). In the opposite (strong-field, Paschen–Back) limit the interaction decouples \( \vec L \) and \( \vec S \), and the shift becomes \( \Delta E = \mu_B B\,(m_L + 2 m_S) \) with the spin–orbit term reduced to a diagonal correction.

Why it matters

The "anomalous" Zeeman effect — anomalous because the multiplet-dependent \( g_J \neq 1 \) contradicted classical Larmor theory — was one of the sharpest early signatures that the electron carries an intrinsic spin with gyromagnetic ratio \( g_s \approx 2 \). The pattern of split lines is a direct fingerprint of the quantum numbers \( L, S, J \) of the states involved.

Beyond history, the Landé factor is a working tool: it underlies magnetometry and atomic clocks, sets the sensitivity of atomic states to stray fields (which levels to choose for a qubit or a frequency standard), and is read off directly in astrophysics, where the Zeeman splitting of spectral lines measures kilogauss fields on stellar surfaces and in sunspots.

Assumptions

The field is weak compared with fine structureIf \( \mu_B B \) is not \( \ll \) the spin–orbit splitting, \( J \) ceases to be a good quantum number, the projection theorem no longer applies, and the linear-in-\(m_J\) formula fails — one enters the Paschen–Back crossover treated later. The field is uniform over the atomA gradient adds a force \( \nabla(\vec\mu\cdot\vec B) \) that couples internal and centre-of-mass motion; the clean level-splitting picture is replaced by Stern–Gerlach deflection. Electron spin g-factor taken as \( g_s = 2 \)The QED correction \( g_s = 2.00232\ldots \) shifts \( g_J \) at the \( 10^{-3} \) level; dropping the approximation, one replaces the "\(2\)" in \( \vec L + 2\vec S \) by \( g_s \) and every "\(1+\)" numerator gains a small \( (g_s-2) \) term. \( LS \) (Russell–Saunders) coupling holdsThe derivation assumes \( L \) and \( S \) are separately good quantum numbers of the field-free atom. In heavy atoms with strong spin–orbit coupling (\(jj\)-coupling), \( L \) and \( S \) are not sharp and the Landé formula gives only an approximate \( g_J \). Diamagnetic \( A^2 \) term neglectedThe \( \frac{e^2}{2m}A^2 \) piece of the minimal-coupling Hamiltonian scales as \( B^2 \) and is negligible for atomic levels at laboratory fields; it matters only at the enormous fields of white dwarfs and neutron stars, where it deforms and mixes levels.

Derivation

\[ \hat H_Z = -\vec\mu\cdot\vec B = \frac{\mu_B}{\hbar}\left(\vec L + g_s\vec S\right)\cdot\vec B \]

Magnetic moment of an electron with orbital and spin angular momentum; \( \mu_B = e\hbar/2m_e \). A

\[ \hat H_Z = \frac{\mu_B B}{\hbar}\left(\hat L_z + 2\hat S_z\right) = \frac{\mu_B B}{\hbar}\left(\hat J_z + \hat S_z\right) \]

Choose \( \vec B = B\hat z \) so only \( z \)-components survive; set \( g_s = 2 \) and use \( \hat L_z + 2\hat S_z = (\hat L_z+\hat S_z)+\hat S_z = \hat J_z + \hat S_z \). A

\[ \hat H_{\text{FS}} \gg \hat H_Z \;\Rightarrow\; |n,L,S,J,m_J\rangle \text{ are the good zeroth-order states} \]

Weak-field assumption: fine structure sets the level, so we apply first-order perturbation theory in \( \hat H_Z \) within the \( (2J+1) \)-fold degenerate \( J \)-manifold, using the \( m_J \) basis that already diagonalises \( \hat H_{\text{FS}} \). B

\[ \Delta E = \frac{\mu_B B}{\hbar}\Big(\langle \hat J_z\rangle + \langle \hat S_z\rangle\Big) = \frac{\mu_B B}{\hbar}\Big(\hbar\,m_J + \langle \hat S_z\rangle\Big) \]

First-order shift is the expectation of \( \hat H_Z \) in \( |J,m_J\rangle \); \( \hat J_z \) is diagonal with eigenvalue \( \hbar m_J \). Only \( \langle \hat S_z\rangle \) remains to evaluate. B

\[ \langle \vec S\rangle_{J} = \frac{\langle \vec S\cdot\vec J\rangle}{\langle \vec J^{\,2}\rangle}\,\langle \vec J\rangle \qquad(\text{within fixed }J) \]

Projection theorem (a corollary of the Wigner–Eckart theorem): any vector operator, restricted to a fixed-\(J\) subspace, is proportional to \( \vec J \). Physically, \( \vec S \) precesses rapidly about the conserved \( \vec J \), so only its component along \( \vec J \) survives averaging. C

\[ \vec S\cdot\vec J = \tfrac12\left(\vec J^{\,2} + \vec S^{\,2} - \vec L^{\,2}\right) \]

From \( \vec L = \vec J - \vec S \), square to get \( \vec L^{\,2} = \vec J^{\,2} - 2\,\vec S\cdot\vec J + \vec S^{\,2} \) and solve for \( \vec S\cdot\vec J \). This expresses the dot product entirely through Casimir operators with sharp eigenvalues. B

\[ \langle \hat S_z\rangle = \frac{\langle \vec S\cdot\vec J\rangle}{\langle \vec J^{\,2}\rangle}\,\langle \hat J_z\rangle = \hbar\,m_J\,\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)} \]

Take the \( z \)-component of Step 5 and insert the eigenvalues \( \langle\vec S\cdot\vec J\rangle = \tfrac{\hbar^2}{2}[J(J{+}1)+S(S{+}1)-L(L{+}1)] \), \( \langle\vec J^{\,2}\rangle = \hbar^2 J(J{+}1) \), \( \langle\hat J_z\rangle = \hbar m_J \). B

\[ \Delta E = \mu_B B\,m_J\left[\,1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}\,\right] \equiv g_J\,\mu_B B\,m_J \]

Substitute Step 7 into Step 4; the "\(1\)" comes from \( \hat J_z \), the bracket from \( \hat S_z \). The dimensionless factor is defined as \( g_J \). A

\[ \hat H_Z \gg \hat H_{\text{FS}}:\quad \Delta E = \mu_B B\,(m_L + 2 m_S) + A\,\hbar^2 m_L m_S \]

Strong-field (Paschen–Back) limit: now \( \hat H_Z \) is treated first, so \( |m_L,m_S\rangle \) are the good states and \( \hat H_Z \) is diagonal with eigenvalue \( \mu_B B(m_L+2m_S) \). The spin–orbit term \( \hat H_{\text{FS}} = A\,\vec L\cdot\vec S \) enters as a first-order correction; only its diagonal part \( A\hat L_z\hat S_z \to A\hbar^2 m_L m_S \) contributes, the raising/lowering part having zero diagonal matrix element. C

Result

\[ \boxed{\;\Delta E_{\text{weak}} = g_J\,\mu_B B\,m_J,\qquad g_J = 1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}\;} \]

\[ \Delta E_{\text{strong}} = \mu_B B\,(m_L + 2 m_S) + A\hbar^2 m_L m_S \]

Reading. In a weak field every fine-structure level fans out into \( 2J+1 \) equally spaced sublevels, but the spacing \( g_J\mu_B B \) depends on the multiplet through \( L, S, J \): different terms split by different amounts, which is exactly why the observed line patterns look "anomalous." A pure singlet state (\( S=0 \), so \( J=L \)) gives \( g_J = 1 \) and recovers the normal Zeeman effect. As the field is raised past the fine-structure scale, the \( g_J\)-labelled sublevels smoothly reorganise into the Paschen–Back grid labelled by \( m_L + 2m_S \).

Units check. \( \mu_B \) has units of J T\(^{-1}\) (\( 9.274\times10^{-24} \) J T\(^{-1}\)), \( B \) is in T, so \( \mu_B B \) is in joules; \( g_J \) and \( m_J \) are dimensionless. Hence \( \Delta E \) is an energy, as required. In convenient units \( \mu_B = 5.788\times10^{-5} \) eV T\(^{-1}\) \( = 1.400 \) MHz G\(^{-1}\) (as a frequency \( \mu_B/h \)).

Limiting cases

Singlet terms (\( S=0,\ J=L \)): the numerator becomes \( L(L{+}1)-L(L{+}1)=0 \), so \( g_J=1 \) — the normal Zeeman effect, three lines (Lorentz triplet), independent of \( L \).
Pure spin (\( L=0,\ J=S \)): \( g_J = 1 + \dfrac{2S(S+1)}{2S(S+1)} = 2 \), matching the free-electron \( g_s=2 \); e.g. any \( ^2S_{1/2} \) ground state.
\( g_J = 0 \) levels: when \( 3J(J{+}1)+S(S{+}1)-L(L{+}1)=0 \) (e.g. \( ^4D_{1/2} \)), the level is first-order field-insensitive — valued for clock transitions and "magic" states.
Strong-field (Paschen–Back): \( g_J \) loses meaning; splittings become integer multiples of \( \mu_B B \) via \( m_L+2m_S \), and the pattern collapses back toward a normal-Zeeman-like triplet in the optical spectrum.
Classical/Larmor limit: ignoring spin entirely gives \( g=1 \) and the single Larmor frequency \( \omega_L = \mu_B B/\hbar \); the spin-dependent \( g_J \) is the purely quantum correction.

Breaks when

Intermediate fields (\( \mu_B B \sim \Delta E_{\text{FS}} \)): neither \( |J,m_J\rangle \) nor \( |m_L,m_S\rangle \) diagonalises the Hamiltonian. One must diagonalise \( \hat H_{\text{FS}} + \hat H_Z \) exactly within each \( m_J = m_L+m_S \) block (the Breit–Rabi problem for one electron); the levels bend nonlinearly and the simple linear-in-\(m_J\) formula is wrong.
Breakdown of \( LS \) coupling (heavy atoms): in strong spin–orbit ("\(jj\)") regimes \( L \) and \( S \) are not good quantum numbers, so the term symbol used in \( g_J \) is fictitious; measured g-factors deviate from the Landé value and require intermediate-coupling calculations.
Hyperfine-dominated regime (very weak field): if \( \mu_B B \) is comparable to or smaller than the hyperfine splitting, nuclear spin \( I \) couples in and \( F = J+I \) is the good quantum number; the splitting is governed by \( g_F \), not \( g_J \), and crosses over to \( g_J\)-behaviour only at higher fields (the atomic Breit–Rabi diagram).
Nonlinear diamagnetic regime (astrophysical fields): at \( B \gtrsim 10^{5} \) T the neglected \( A^2 \) term dominates, mixing \( n \) and \( l \), and the spectrum becomes quasi-Landau — the perturbative Zeeman picture fails entirely.

Failure modes

Using \( m_L \) instead of \( m_J \) in the weak-field formula. The splitting is \( g_J\mu_B B\,m_J \) with half-integer \( m_J \) for one-electron atoms; students who write \( m_L \) get the wrong number of sublevels and wrong spacings.
Forgetting the "\(1+\)" and quoting only the fraction as \( g_J \). The leading \( 1 \) comes from \( \hat J_z \) and is not optional.
Sign/label swap in the projection numerator: writing \( L(L{+}1)+S(S{+}1)-J(J{+}1) \) instead of \( J(J{+}1)+S(S{+}1)-L(L{+}1) \). The \( -L(L{+}1) \) term is the one with the minus sign.
Assuming the same \( g_J \) for upper and lower levels of a transition. The whole point of the anomalous effect is that the two levels usually have different \( g_J \); the line pattern comes from \( g_{\text{up}}m_{J,\text{up}} - g_{\text{low}}m_{J,\text{low}} \).
Applying the Landé formula in strong fields. Once \( \mu_B B \gtrsim \Delta E_{\text{FS}} \), \( g_J \) is meaningless; students who keep using it overpredict the number of resolved lines.
Ignoring selection rules \( \Delta m_J = 0,\pm1 \) (with \( m_J=0\to0 \) forbidden when \( \Delta J=0 \)), and counting all \( m_J \)-pairs as observable transitions.

Discussion

The physical heart of the weak-field result is a separation of timescales. Spin–orbit coupling locks \( \vec L \) and \( \vec S \) into a resultant \( \vec J \) that precesses slowly about \( \vec B \); meanwhile \( \vec L \) and \( \vec S \) precess rapidly about \( \vec J \). Over the slow Zeeman timescale only the projections of \( \vec L \) and \( \vec S \) onto \( \vec J \) survive averaging — this is precisely the projection theorem in Step 5, and the vector model makes it intuitive. The g-factor is nothing but the effective magnetic moment per unit \( \vec J \), built from the differently-weighted (\(1\times\vec L\), \(2\times\vec S\)) contributions projected onto \( \vec J \).

The Paschen–Back limit inverts the hierarchy: the field wins, \( \vec L \) and \( \vec S \) precess independently about \( \vec B \), and \( m_L, m_S \) become good quantum numbers. The spectrum then simplifies, because the spin contributes \( 2m_S\mu_B B \) — the same shift for the upper and lower optical levels — so most of the spin splitting cancels in the transition energy and the pattern collapses toward the normal-Zeeman triplet. Watching a multiplet march from its anomalous low-field pattern to the Paschen–Back triplet as \( B \) increases is one of the cleanest demonstrations of a quantum crossover between two "good basis" descriptions.

The Landé factor connects far beyond spectroscopy. The same projection logic yields the nuclear \( g_F \) in hyperfine structure, the effective moments of rare-earth ions that Van Vleck used to explain paramagnetic susceptibilities, and the g-tensors of electron spin resonance. The measured deviation of the electron's \( g_s \) from exactly 2 — folded into \( g_J \) at the \( 10^{-3} \) level — is itself a precision test of QED.

A subtle point of rigour: the projection theorem holds only within a fixed-\(J\) subspace, because \( \hat H_Z \) has matrix elements connecting \( J \) to \( J\pm1 \) (it is a vector operator, rank-1). In first-order degenerate perturbation theory those off-diagonal-in-\(J\) elements are ignored — legitimate precisely because the fine-structure gap suppresses them. The intermediate-field regime is where those neglected \( \Delta J = \pm1 \) couplings become important, and diagonalising the full \( m_J \)-block (which mixes \( J \) and \( J{-}1 \) at fixed \( m_J \)) is what produces the smooth, nonlinear Breit–Rabi interpolation between the weak- and strong-field limits. The two "regimes" are thus not distinct physics but two ends of a single exactly-solvable two-level (per \( m_J \)) problem.

Common misconceptions. The effect is called "anomalous" for historical reasons only — it is the generic case (any state with \( S\neq0 \)); the "normal" (\( g=1 \)) Zeeman effect is the special case. And the splitting is not caused by the field acting on orbital motion alone: it is the electron spin magnetic moment, with its factor-of-two anomaly, that makes \( g_J \neq 1 \).

Worked examples

Landé factor and sublevel spacing for the sodium \( 3p\,^2P_{3/2} \) level in \( B=0.50 \) T

\[ L=1,\quad S=\tfrac12,\quad J=\tfrac32 \]

\[ g_J = 1 + \frac{\tfrac32\cdot\tfrac52 + \tfrac12\cdot\tfrac32 - 1\cdot2}{2\cdot\tfrac32\cdot\tfrac52} = 1 + \frac{\tfrac{15}{4}+\tfrac34-2}{\tfrac{15}{2}} = 1 + \frac{\tfrac{10}{4}}{\tfrac{30}{4}} = 1 + \frac13 = \frac43 \]

\[ \delta E = g_J\,\mu_B B = \tfrac43\,(5.788\times10^{-5}\ \text{eV T}^{-1})(0.50\ \text{T}) = 3.86\times10^{-5}\ \text{eV} \]

Adjacent sublevels differ by \( \Delta m_J=1 \), so the uniform spacing is \( g_J\mu_B B \). The four sublevels \( m_J=+\tfrac32,+\tfrac12,-\tfrac12,-\tfrac32 \) sit at \( \pm\tfrac32\delta E \) and \( \pm\tfrac12\delta E \). A

\[ g_J = \tfrac43,\qquad \delta E = 3.86\times10^{-5}\ \text{eV} = 38.6\ \mu\text{eV}\;\;(\equiv 9.34\ \text{GHz}) \]

Reading. The \( ^2P_{3/2} \) level fans into four equally spaced sublevels separated by about 39 μeV at half a tesla — tiny beside the optical transition energy (~2.1 eV) but comfortably resolvable spectroscopically.

Units check. eV T\(^{-1}\) × T = eV; dividing 38.6 μeV by \( h=4.136\times10^{-15} \) eV s gives 9.3 GHz, a sensible microwave-scale splitting.

Estimating the Paschen–Back crossover field for the sodium \( 3p \) doublet

\[ \text{Crossover when } \mu_B B \sim \Delta E_{\text{FS}},\qquad \Delta E_{\text{FS}}(3p) \approx 2.1\times10^{-3}\ \text{eV}\;(17.2\ \text{cm}^{-1}) \]

\[ B_{\times} \sim \frac{\Delta E_{\text{FS}}}{\mu_B} = \frac{2.1\times10^{-3}\ \text{eV}}{5.788\times10^{-5}\ \text{eV T}^{-1}} \approx 36\ \text{T} \]

The condition for entering the strong-field regime is that the magnetic energy overtakes the fine-structure splitting that binds \( \vec L \) and \( \vec S \) into \( \vec J \). Rearranging symbolically first, then inserting numbers. B

\[ B_{\times} \approx 36\ \text{T} \]

Reading. Ordinary laboratory fields (up to a few tesla) satisfy \( \mu_B B \ll \Delta E_{\text{FS}} \) by an order of magnitude or more, so sodium's optical Zeeman spectrum is firmly in the weak-field, Landé-\(g_J\) regime. Reaching full Paschen–Back for the sodium D lines needs tens of tesla — only pulsed or hybrid magnets. (By contrast, hydrogen's much smaller \( n=2 \) fine structure crosses over near a few tesla.)

Units check. eV ÷ (eV T\(^{-1}\)) = T, correctly a magnetic field.

Problems

Compute the Landé g-factor of the \( ^3P_2 \) term (\( L=1,\ S=1,\ J=2 \)).
Solution
\( g_J = 1 + \dfrac{J(J{+}1)+S(S{+}1)-L(L{+}1)}{2J(J{+}1)} = 1 + \dfrac{2\cdot3 + 1\cdot2 - 1\cdot2}{2\cdot2\cdot3} = 1 + \dfrac{6}{12} = \dfrac32. \) The level splits into \( 2J+1=5 \) sublevels with spacing \( \tfrac32\mu_B B \).
Show that the \( ^4D_{1/2} \) term (\( L=2,\ S=\tfrac32,\ J=\tfrac12 \)) has \( g_J = 0 \), and explain the physical significance.
Solution
Numerator: \( J(J{+}1)+S(S{+}1)-L(L{+}1) = \tfrac12\cdot\tfrac32 + \tfrac32\cdot\tfrac52 - 2\cdot3 = \tfrac34 + \tfrac{15}{4} - 6 = \tfrac{18}{4} - 6 = -\tfrac32. \) Denominator: \( 2J(J{+}1) = 2\cdot\tfrac12\cdot\tfrac32 = \tfrac32. \) So \( g_J = 1 + \dfrac{-3/2}{3/2} = 1 - 1 = 0. \) With \( g_J=0 \) the level has no first-order Zeeman shift: it is field-insensitive to linear order (the orbital and spin projected moments cancel). Such levels are prized for clock transitions and "magic" states because their frequencies are immune to small stray fields to first order.
Sodium's D\(_1\) line is the transition \( 3p\,^2P_{1/2}\to 3s\,^2S_{1/2} \). Using the selection rule \( \Delta m_J = 0,\pm1 \), how many Zeeman components appear in a weak field?
Solution
Both levels have \( J=\tfrac12 \), so \( m_J=\pm\tfrac12 \) in each. The g-factors are \( g(^2P_{1/2}) = 1 + \dfrac{\tfrac34+\tfrac34-2}{3/2} = 1 - \tfrac13 = \tfrac23 \) and \( g(^2S_{1/2}) = 2. \) The four \( m_J^{\text{up}}\to m_J^{\text{low}} \) pairs all obey \( \Delta m_J=0,\pm1 \): \( +\tfrac12\to+\tfrac12 \) (\(\Delta m=0\)), \( +\tfrac12\to-\tfrac12 \) (\(-1\)), \( -\tfrac12\to+\tfrac12 \) (\(+1\)), \( -\tfrac12\to-\tfrac12 \) (\(0\)). All four are allowed (\( m_J=0\to0 \) forbidden rule doesn't apply since there is no \( m_J=0 \) here), giving four distinct components. Their energies are \( h\nu_0 + (g_{\text{up}}m^{\text{up}} - g_{\text{low}}m^{\text{low}})\mu_B B \), i.e. shifts \( \{\tfrac23\cdot\tfrac12 - 2\cdot\tfrac12,\ \ldots\}\mu_B B = \{-\tfrac23, +\tfrac43, -\tfrac43, +\tfrac23\}\mu_B B\times\tfrac12 \) — the classic four-line anomalous pattern of D\(_1\).
Estimate the field at which the hydrogen \( n=2 \) levels enter the Paschen–Back regime, given the \( n=2 \) fine-structure splitting \( \Delta E_{\text{FS}} \approx 4.5\times10^{-5} \) eV.
Solution
Crossover when \( \mu_B B \sim \Delta E_{\text{FS}} \): \( B_{\times} \sim \dfrac{4.5\times10^{-5}\ \text{eV}}{5.788\times10^{-5}\ \text{eV T}^{-1}} \approx 0.78\ \text{T}. \) So for hydrogen \( n=2 \), fields of order 1 T already drive the crossover — dramatically smaller than sodium's ~36 T, because hydrogen's fine structure (scaling as \( Z^4 \) but with no valence screening enhancement) is small. This is why hydrogen was historically a convenient system for observing Paschen–Back behaviour.
In the strong-field (Paschen–Back) limit, list the distinct energy shifts \( \Delta E/\mu_B B = m_L + 2m_S \) for a \( p \) electron (\( l=1,\ s=\tfrac12 \)), and identify the degeneracies.
Solution
\( m_L \in \{-1,0,+1\} \), \( m_S \in \{-\tfrac12,+\tfrac12\} \), so \( m_L+2m_S \) takes: for \( m_S=+\tfrac12 \) (\(2m_S=+1\)): \( 0,+1,+2 \); for \( m_S=-\tfrac12 \) (\(2m_S=-1\)): \( -2,-1,0 \). Collecting: values \( \{-2,-1,0,+1,+2\} \) with \( 0 \) doubly degenerate (\( m_L=+1,m_S=-\tfrac12 \) and \( m_L=-1,m_S=+\tfrac12 \)) and the rest singly. The six \( |m_L,m_S\rangle \) states thus form five equally spaced levels in units of \( \mu_B B \) — a normal-Zeeman-like triplet structure once the spin-diagonal spin–orbit correction \( A\hbar^2 m_L m_S \) lifts the residual \( 0 \)-degeneracy.