The Paschen-Back and Intermediate-Field Regimes

The weak-field Zeeman effect treats the magnetic interaction as a small perturbation on the fine-structure eigenstates. Nothing forces the field to stay small. As $B$ grows, the Zeeman energy $μ_{B} B$ eventually overtakes the spin-orbit splitting, and the hierarchy of couplings inverts: the field now sets the fast precession and the spin-orbit interaction becomes the perturbation. In this strong-field or Paschen-Back regime the good quantum numbers change from $(j, m_{j})$ to $(m_{ℓ}, m_{s})$ , and the anomalous multi-line pattern collapses back to a normal-looking triplet. The genuinely hard case is the intermediate field, where neither coupling dominates and the level positions require an exact diagonalization. For a single valence electron that diagonalization is a two-by-two problem with a closed-form solution.

The competing energy scales

Two interactions compete for control of the spin. The spin-orbit coupling,

H_{so} = A L \cdot S, A = \frac{ζ _{n ℓ}}{ℏ ^{2}},

with $ζ_{n ℓ} = \frac{μ _{0}}{4 π} g_{s} \frac{e ^{2} ℏ ^{2}}{2 m _{e}^{2}} ⟨ r^{- 3} ⟩$ the radial spin-orbit constant, locks $L$ and $S$ into a fixed $J$ . The Zeeman coupling,

H_{Z} = \frac{μ _{B} B}{ℏ} (L_{z} + 2 S_{z}),

torques $L$ and $S$ separately about $\overset{z}{^}$ . Which one wins is set by the dimensionless ratio

x \equiv \frac{μ _{B} B}{Δ _{fs}}, Δ_{fs} = A ℏ^{2} (ℓ + \frac{1}{2}),

where $Δ_{fs}$ is the zero-field splitting between the $j = ℓ + \frac{1}{2}$ and $j = ℓ - \frac{1}{2}$ levels. The weak-field theory is the expansion in $x ≪ 1$ ; the Paschen-Back theory is the expansion in $x ≫ 1$ .

Weak field:

L

and

S

lock into

J

, which precesses about B. Strong field:

L

and

S

decouple and precess separately.

Strong field: the Paschen-Back limit

When $x ≫ 1$ , drop $H_{so}$ at zero order. The eigenstates are the uncoupled product states $∣ ℓ s m_{ℓ} m_{s} ⟩$ , and the Zeeman energy is exact and additive,

E_{Z} = μ_{B} B (m_{ℓ} + 2 m_{s}) .

The spin-orbit term is then a first-order correction. In an $(m_{ℓ}, m_{s})$ eigenstate the transverse pieces average to zero, $⟨ L_{x} S_{x} + L_{y} S_{y} ⟩ = 0$ , because $L_{\pm}$ and $S_{\pm}$ change $m_{ℓ}$ or $m_{s}$ and connect to orthogonal states, leaving only

⟨ H_{so} ⟩ = A ⟨ L_{z} S_{z} ⟩ = A ℏ^{2} m_{ℓ} m_{s} = ζ_{n ℓ} m_{ℓ} m_{s} .

The complete strong-field energy is therefore

E = E_{n ℓ} + μ_{B} B (m_{ℓ} + 2 m_{s}) + ζ_{n ℓ} m_{ℓ} m_{s} .

The dipole selection rules in this basis are $Δ m_{ℓ} = 0, \pm 1$ (the photon acts on the orbital motion) and $Δ m_{s} = 0$ (radiation does not flip the spin). A transition frequency shifts by

h ν = h ν_{0} + μ_{B} B Δ m_{ℓ} + (small spin-orbit term),

so to leading order only three groups survive at $ν_{0}$ and $ν_{0} \pm μ_{B} B / h$ : the anomalous pattern has reverted to the normal triplet. This reversion is the defining experimental signature of the Paschen-Back effect, and its onset marks the field at which $μ_{B} B$ passes $Δ_{fs}$ .

The uncoupled strong-field states arranged by

m_{ℓ} + 2 m_{s}

, with group spacing

μ_{B} B

and a small spin-orbit split

ζ m_{ℓ} m_{s}

within each group; the pattern collapses to a normal triplet.

The intermediate field: an exact two-by-two

Between the limits, neither basis diagonalizes the Hamiltonian $H = H_{so} + H_{Z}$ . What survives is the projection quantum number

m_{j} = m_{ℓ} + m_{s},

conserved because $[J_{z}, H_{so}] = 0$ and $[J_{z}, H_{Z}] = 0$ (both interactions are rotationally invariant about $\overset{z}{^}$ ). For a single electron $s = \frac{1}{2}$ , at most two uncoupled states share a given $m_{j}$ :

∣ 1 ⟩ = m_{ℓ} = m_{j} - \frac{1}{2}, m_{s} = + \frac{1}{2} ⟩, ∣ 2 ⟩ = m_{ℓ} = m_{j} + \frac{1}{2}, m_{s} = - \frac{1}{2} ⟩ .

The Hamiltonian is block-diagonal in $m_{j}$ , with each block a $2 \times 2$ matrix (a $1 \times 1$ block for the two stretched states $m_{j} = \pm (ℓ + \frac{1}{2})$ , which have no partner). The Zeeman part is diagonal:

⟨ 1 ∣ H_{Z} ∣ 1 ⟩ = μ_{B} B (m_{j} + \frac{1}{2}), ⟨ 2 ∣ H_{Z} ∣ 2 ⟩ = μ_{B} B (m_{j} - \frac{1}{2}),

reading $m_{ℓ} + 2 m_{s}$ in each state. The spin-orbit part has both diagonal and off-diagonal pieces. Writing $L \cdot S = L_{z} S_{z} + \frac{1}{2} (L_{+} S_{-} + L_{-} S_{+})$ , the diagonal entries come from $L_{z} S_{z}$ ,

⟨ 1 ∣ H_{so} ∣ 1 ⟩ = A ℏ^{2} \frac{1}{2} (m_{j} - \frac{1}{2}), ⟨ 2 ∣ H_{so} ∣ 2 ⟩ = - A ℏ^{2} \frac{1}{2} (m_{j} + \frac{1}{2}),

and the off-diagonal entry from $\frac{1}{2} L_{-} S_{+}$ connecting $∣ 2 ⟩$ to $∣ 1 ⟩$ ,

V \equiv ⟨ 1 ∣ H_{so} ∣ 2 ⟩ = \frac{A ℏ ^{2}}{2} (ℓ + \frac{1}{2})^{2} - m_{j}^{2},

using the ladder normalizations $L_{-} ∣ m_{ℓ} ⟩ = ℏ ℓ (ℓ + 1) - m_{ℓ} (m_{ℓ} - 1) ∣ m_{ℓ} - 1 ⟩$ and the analogous $S_{+}$ .

Within a fixed

m_{j}

only two uncoupled states mix; spin-orbit is diagonal in the coupled basis, Zeeman in the uncoupled one, and they compete.

Diagonalizing the block

The eigenvalues of a symmetric $2 \times 2$ matrix are the mean of the diagonal plus or minus the radius, $E_{\pm} = \overset{ˉ}{d} \pm (\frac{d _{1} - d _{2}}{2})^{2} + V^{2}$ . The mean diagonal is

\overset{ˉ}{d} = - \frac{A ℏ ^{2}}{4} + μ_{B} B m_{j},

and the diagonal difference is $d_{1} - d_{2} = A ℏ^{2} m_{j} + μ_{B} B$ . Substituting $V$ and simplifying the discriminant (the $m_{j}^{2}$ terms cancel) gives the closed-form Breit-Rabi expression for fine structure,

E_{\pm} = E_{n ℓ} - \frac{A ℏ ^{2}}{4} + μ_{B} B m_{j} \pm \frac{Δ _{fs}}{2} 1 + \frac{2 m _{j}}{ℓ + \frac{1}{2}} x + x^{2} x = \frac{μ _{B} B}{Δ _{fs}} .

The single square root interpolates between the two limits.

The two stretched states carry no square-root structure: with $m_{j} = \pm (ℓ + \frac{1}{2})$ there is a single basis state, and its energy is linear in $B$ at all field strengths,

E = E_{n ℓ} + \frac{A ℏ ^{2} ℓ}{2} + μ_{B} B (m_{ℓ} + 2 m_{s}), m_{ℓ} = \pm ℓ, m_{s} = \pm \frac{1}{2} .

The correlation diagram

Plotting the eigenvalues against $x$ threads every $j$ -multiplet sublevel of the weak-field limit onto an $(m_{ℓ}, m_{s})$ level of the strong-field limit. States of the same $m_{j}$ repel (the off-diagonal $V$ produces an avoided crossing), while states of different $m_{j}$ cross freely, since no matrix element connects them.

Level energies versus field. Weak-field

j

sublevels on the left connect to strong-field

(m_{ℓ}, m_{s})

levels on the right; same-

m_{j}

curves avoid.

Worked example. Take a single $p$ electron, $ℓ = 1$ , so $Δ_{fs} = \frac{3}{2} A ℏ^{2}$ separates $^{2} P_{3/2}$ from $^{2} P_{1/2}$ . The projection $m_{j}$ ranges over $\pm \frac{3}{2}, \pm \frac{1}{2}$ . The stretched states $m_{j} = \pm \frac{3}{2}$ are single and shift linearly, $E = \frac{1}{2} A ℏ^{2} + μ_{B} B (m_{ℓ} + 2 m_{s})$ with $m_{ℓ} = \pm 1$ , $m_{s} = \pm \frac{1}{2}$ . Each $m_{j} = \pm \frac{1}{2}$ block is a genuine $2 \times 2$ with

V = \frac{A ℏ ^{2}}{2} (\frac{3}{2})^{2} - (\frac{1}{2})^{2} = \frac{A ℏ ^{2}}{2} 2,

and the boxed formula gives, for $m_{j} = + \frac{1}{2}$ ,

E_{\pm} = E_{n p} - \frac{A ℏ ^{2}}{4} + \frac{μ _{B} B}{2} \pm \frac{Δ _{fs}}{2} 1 + \frac{2}{3} x + x^{2}, x = \frac{μ _{B} B}{Δ _{fs}} .

At $x = 0$ the two roots are the $^{2} P_{3/2}$ and $^{2} P_{1/2}$ sublevels; as $x \to \infty$ they become the uncoupled $(m_{ℓ}, m_{s})$ levels $(0, + \frac{1}{2})$ and $(1, - \frac{1}{2})$ . The six sublevels of the $^{2} P$ term (two stretched, four from the two $2 \times 2$ blocks) thread continuously from six weak-field $m_{j}$ states to six strong-field $(m_{ℓ}, m_{s})$ states.

The avoided crossing is the direct fingerprint of the off-diagonal spin-orbit coupling: its minimum gap equals $2 V = A ℏ^{2} (ℓ + \frac{1}{2})^{2} - m_{j}^{2}$ at the field where the two diagonal entries coincide, $A ℏ^{2} m_{j} + μ_{B} B = 0$ . That the gap is nonzero for every non-stretched $m_{j}$ is why the transition from $L S$ to $j j$ -like labelling is smooth rather than a set of hard level crossings.

Regime map

Regime	Condition	Good numbers	Energy
Weak (anomalous Zeeman)	$μ_{B} B ≪ Δ_{fs}$	$j, m_{j}$	$g_{J} μ_{B} B m_{j}$
Intermediate	$μ_{B} B \sim Δ_{fs}$	$m_{j}$ only	Breit-Rabi root
Strong (Paschen-Back)	$μ_{B} B ≫ Δ_{fs}$	$m_{ℓ}, m_{s}$	$μ_{B} B (m_{ℓ} + 2 m_{s}) + ζ m_{ℓ} m_{s}$

Because $Δ_{fs}$ scales as $Z^{4} / n^{3}$ , the field needed to reach the Paschen-Back regime rises steeply with nuclear charge and falls with excitation: hydrogen $n = 3$ reaches it in a fraction of a tesla, while the sodium D doublet demands tens of tesla. The same competition of a small internal coupling against an external field reappears for the hyperfine interaction, where the analogous crossover is the original Breit-Rabi problem, and for the electric case treated in the next lesson.¹

Bransden & Joachain, Physics of Atoms and Molecules, §9.2 — the Paschen-Back effect and the exact intermediate-field diagonalization for a single valence electron; the fine-structure Breit-Rabi formula and its two limiting expansions. See also Foot, Atomic Physics, §5.5. ↩