--- title: The Zeeman Effect module: Atoms in External Fields moduleNumber: 6 lessonNumber: 1 order: 601 summary: > A magnetic field couples to the atom through its magnetic moment, splitting each level into equally spaced sublevels labelled by the projection of the total angular momentum. When spin is present the spacing is not the classical one: it carries the Landé g-factor, a projection of the spin and orbital moments onto the total angular momentum. We derive the weak-field Hamiltonian from minimal coupling, evaluate the shift with the projection theorem, and read off the polarization of the emitted components. topics: [Atoms in External Fields] sources: - book: Foot ref: "Ch. 1 — Early Atomic Physics; Ch. 5 — The LS-coupling scheme, §5.5 The Zeeman effect" - book: Bransden & Joachain ref: "Ch. 9 — Atoms in External Fields; §9.1 The Zeeman effect, §9.2 The Paschen-Back effect" - book: Griffiths & Schroeter ref: "Ch. 7 — Time-Independent Perturbation Theory; §7.4 The Zeeman effect" - book: Demtröder ref: "Ch. 5 — The Hydrogen Atom; §5.2 Atoms in magnetic fields" draft: false --- An external magnetic field breaks the rotational symmetry that makes the $(2j+1)$ orientations of an atomic level degenerate. The field selects an axis, and the energy of a state comes to depend on how its angular momentum projects onto that axis. The resulting splitting of spectral lines, discovered by Zeeman in 1896, was the first laboratory access to the internal magnetic structure of the atom, and the pattern of the split lines encodes the coupling of spin and orbital motion more directly than any zero-field measurement. This lesson treats the **weak-field** regime, where the Zeeman energy is small compared with the fine-structure splitting; the strong-field and intermediate cases are the subject of the [next lesson](/atomic-physics/atoms-in-external-fields/paschen-back-intermediate). ## The atomic magnetic moment A charged particle in a bound orbit carries a magnetic moment proportional to its angular momentum. For the orbital motion of an electron of charge $-e$ and mass $m_e$, the classical gyromagnetic relation gives $$ \vec\mu_L = -\frac{e}{2m_e}\,\vec L = -g_L\,\frac{\mu_B}{\hbar}\,\vec L, \qquad \mu_B \equiv \frac{e\hbar}{2m_e}, $$ with orbital $g$-factor $g_L = 1$ and the **Bohr magneton** $\mu_B$ setting the scale of atomic magnetism. Its CODATA value is $\mu_B = 9.2740\times10^{-24}\ \text{J T}^{-1} = 5.7884\times10^{-5}\ \text{eV T}^{-1}$,[^codata] so a one-tesla field shifts levels by tens of microelectronvolts, or equivalently $\mu_B/h = 13.996\ \text{GHz T}^{-1}$. Spin contributes a moment that is anomalously large by a factor $g_s$: $$ \vec\mu_S = -g_s\,\frac{\mu_B}{\hbar}\,\vec S, \qquad g_s = 2.00231930\ldots $$ The value $g_s \approx 2$ is the prediction of the Dirac equation; the small excess is the QED anomalous moment. Combining the two contributions, and taking $g_s = 2$ for now, $$ \vec\mu = -\frac{\mu_B}{\hbar}\big(g_L\vec L + g_s\vec S\big) = -\frac{\mu_B}{\hbar}\big(\vec L + 2\vec S\big). $$ The factor of two on the spin term is the entire source of the "anomalous" Zeeman effect: were $g_s = 1$, the moment would be proportional to $\vec L + \vec S = \vec J$ and every level would split with the same, classical, spacing. ## The Zeeman Hamiltonian from minimal coupling The interaction is not postulated but follows from replacing the canonical momentum with the gauge-covariant one, $\vec p \to \vec p + e\vec A$, in the kinetic energy. For a uniform field $\vec B = B\hat z$ the symmetric-gauge vector potential is $\vec A = \tfrac12\,\vec B\times\vec r$, and $$ \frac{1}{2m_e}\big(\vec p + e\vec A\big)^2 = \frac{p^2}{2m_e} + \frac{e}{2m_e}\,\vec B\cdot\vec L + \frac{e^2}{8m_e}\,(\vec B\times\vec r)^2 . $$ The cross term reproduces the orbital moment coupling $-\vec\mu_L\cdot\vec B$. Adding the spin moment coupling $-\vec\mu_S\cdot\vec B$ by hand (it emerges automatically only from the Dirac equation) gives the paramagnetic Zeeman Hamiltonian $$ H_Z = \frac{\mu_B}{\hbar}\big(\vec L + 2\vec S\big)\cdot\vec B = \frac{\mu_B B}{\hbar}\big(L_z + 2S_z\big) = \frac{\mu_B B}{\hbar}\big(J_z + S_z\big). $$ The final rearrangement, using $\vec J = \vec L + \vec S$, isolates the piece that is diagonal in $J_z$ (trivial) from the piece proportional to $S_z$ (the source of all the structure). > **Definition (Diamagnetic term).** The quadratic piece > $H_{\text{dia}} = \dfrac{e^2}{8m_e}(\vec B\times\vec r)^2$ shifts levels by > $\sim \dfrac{e^2 B^2 \langle r^2\rangle}{8m_e}$. For a ground-state atom in a > laboratory field of $1\ \text{T}$ this is of order $10^{-9}\ \text{eV}$, four > orders of magnitude below the paramagnetic term, and is dropped except in > white-dwarf and neutron-star fields or for highly excited states where > $\langle r^2\rangle \propto n^4$ grows enormous. Comparing the two field-dependent terms, the paramagnetic term is linear in $B$ and the diamagnetic term quadratic; their ratio is $H_{\text{dia}}/H_Z \sim \mu_B B \langle r^2\rangle/(2\hbar^2/m_e)$, negligible until $B$ reaches $10^{5}\ \text{T}$ for ground states. $$ % caption: The two magnetic contributions add: the orbital moment tracks $\vec L$, % the spin moment tracks $\vec S$ but is twice as large per unit angular momentum. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % orbital moment \draw[->, black, very thick] (0,0) -- (0,2) node[above] {L}; \draw[->, black, thick] (0,0) -- (0,-1.4); \node[black, anchor=east] at (0,-0.9) {moment from L}; % spin moment, twice per unit \draw[->, acc, very thick] (3.2,0) -- (3.2,1.2) node[above] {S}; \draw[->, acc, thick] (3.2,0) -- (3.2,-2.4); \node[acc, anchor=west] at (3.2,-1.6) {moment from S (double)}; \draw[dashed, black] (-1,0) -- (4.4,0); \end{tikzpicture} $$ ## Weak field: the anomalous Zeeman effect When $H_Z$ is small compared with the fine-structure splitting, the good zero-order states are the fine-structure eigenstates $\lvert n\,\ell\,s\,j\,m_j\rangle$, in which $\vec L$ and $\vec S$ are locked into a fixed $\vec J$ that precesses slowly about the field. First-order perturbation theory gives the shift as the expectation value of $H_Z$ in these states, $$ E_Z^{(1)} = \frac{\mu_B B}{\hbar}\,\big\langle J_z + S_z\big\rangle = \mu_B B\,m_j + \frac{\mu_B B}{\hbar}\,\langle S_z\rangle . $$ The first term is elementary; the second requires $\langle S_z\rangle$ inside a state of definite $j$, where $\vec S$ is not separately conserved. The **projection theorem** (a corollary of the Wigner-Eckart theorem for vector operators) supplies it: within a manifold of fixed $j$, every vector operator $\vec V$ has the same matrix elements as its projection onto $\vec J$, $$ \langle \vec V\rangle_{jm_j} = \frac{\langle \vec V\cdot\vec J\rangle}{j(j+1)\hbar^2}\,\langle \vec J\rangle_{jm_j}. $$ The physical content is the **vector model**: $\vec S$ precesses rapidly about $\vec J$, so only its component along $\vec J$ survives the time average, and that surviving component points along $\vec J$ with weight $(\vec S\cdot\vec J)/J^2$. $$ % caption: In the vector model $\vec L$ and $\vec S$ precess about their fixed % resultant $\vec J$; only the projections onto $\vec J$ survive the average. \begin{tikzpicture}[scale=1.05, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % J axis \draw[->, acc, very thick] (0,0) -- (0,3.2) node[above] {J}; % L along a slanted direction \draw[->, black, very thick] (0,0) -- (-1.5,2.2) node[left] {L}; % S continuing to J tip region \draw[->, black, very thick] (-1.5,2.2) -- (0,3.0) node[right] {S}; % precession cones (ellipse hints) \draw[dashed, black] (0,2.2) ellipse (1.5 and 0.42); \draw[dashed, black] (0,3.0) ellipse (0.55 and 0.16); % projection of S onto J \draw[black, thick] (-0.75,2.6) -- (0,2.6); \node[black, anchor=west] at (0.15,1.2) {J precesses slowly about B}; \node[black, anchor=west] at (0.15,3.0) {B axis}; \end{tikzpicture} $$ To evaluate $\vec S\cdot\vec J$, write $\vec L = \vec J - \vec S$ and square: $L^2 = J^2 + S^2 - 2\,\vec S\cdot\vec J$, so $$ \vec S\cdot\vec J = \tfrac12\big(J^2 + S^2 - L^2\big), \qquad \langle \vec S\cdot\vec J\rangle = \frac{\hbar^2}{2}\big[j(j+1) + s(s+1) - \ell(\ell+1)\big]. $$ Then $\langle S_z\rangle = \dfrac{\langle\vec S\cdot\vec J\rangle}{j(j+1)\hbar^2}\,m_j\hbar$, and the shift takes the form $E_Z^{(1)} = g_J\,\mu_B B\,m_j$ with the **Landé g-factor** $$ \;g_J = 1 + \frac{j(j+1) + s(s+1) - \ell(\ell+1)}{2\,j(j+1)}\; $$ for $g_s = 2$; keeping the exact $g_s$ replaces the $1$ by $\tfrac12(1+g_s)$ and scales the second term correspondingly.[^lande] > **Theorem (Landé interval within a Zeeman multiplet).** In a weak field a > level of fixed $(\ell,s,j)$ splits into $2j+1$ equally spaced sublevels, > $E_Z^{(1)} = g_J\,\mu_B B\,m_j$ with $m_j = -j,\dots,+j$, the spacing > $g_J\mu_B B$ set entirely by the Landé factor. The centre of gravity of the > multiplet is unshifted, since $\sum_{m_j} m_j = 0$. The equal spacing is the signature of the weak-field regime: the field enters only through $J_z$, whose eigenvalues are equally spaced, and the coefficient $g_J$ is a pure number fixed by the angular-momentum quantum numbers. $$ % caption: A level of total angular momentum $j$ splits into $2j+1$ equally % spaced sublevels; for $j=\tfrac32$ the four are, from the top, % $m_j = +\tfrac32, +\tfrac12, -\tfrac12, -\tfrac32$, spaced by $g_J\mu_B B$. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % zero-field level \draw[black, very thick] (-3.2,0) -- (-1.6,0); \node[anchor=east] at (-3.2,0) {B = 0}; % split levels \foreach \k/\yy in {1/1.5, 2/0.5, 3/-0.5, 4/-1.5}{ \draw[acc, very thick] (0.6,\yy) -- (2.6,\yy); } % connectors \foreach \yy in {1.5,0.5,-0.5,-1.5}{ \draw[dashed, black] (-1.6,0) -- (0.6,\yy); } % spacing bracket \draw[<->, black] (2.9,0.5) -- (2.9,1.5); \node[black, anchor=west] at (3.0,1.0) {equal spacing}; % right labels for the four sublevels (values in caption) \node[acc, anchor=west] at (2.65,1.5) {top}; \node[acc, anchor=west] at (2.65,-1.5) {bottom}; \node[black, anchor=north] at (1.6,-2.1) {four equally spaced sublevels}; \end{tikzpicture} $$ ### Worked Landé factors for the sodium doublet The sodium D lines connect the ground term $3s\ ^2S_{1/2}$ to the two fine-structure components $3p\ ^2P_{1/2}$ and $3p\ ^2P_{3/2}$. Their Landé factors follow directly: | Term | $\ell$ | $j$ | $g_J$ | | --- | --- | --- | --- | | $^2S_{1/2}$ | $0$ | $\tfrac12$ | $1 + \dfrac{3/4+3/4-0}{2\cdot 3/4} = 2$ | | $^2P_{1/2}$ | $1$ | $\tfrac12$ | $1 + \dfrac{3/4+3/4-2}{2\cdot 3/4} = \tfrac23$ | | $^2P_{3/2}$ | $1$ | $\tfrac32$ | $1 + \dfrac{15/4+3/4-2}{2\cdot 15/4} = \tfrac43$ | Because the three factors differ, the split sublevels of the upper and lower terms are not uniformly spaced relative to one another, and the transition resolves into more than three lines. This is the **anomalous** Zeeman pattern, which puzzled spectroscopists for three decades until spin supplied the extra factor of two. > **Worked example.** Count the Zeeman components of the two sodium D lines in a > weak field, in units of $\mu_B B$. A sublevel sits at $g_J m_j$; a component > frequency is the upper shift minus the lower shift, allowed when > $\Delta m_j = m_j' - m_j \in \{0, \pm1\}$. > > For **D$_1$** ($^2P_{1/2} \to {}^2S_{1/2}$) the upper sublevels are at > $g_J m_j = \tfrac23(\pm\tfrac12) = \pm\tfrac13$ and the lower at > $2(\pm\tfrac12) = \pm1$. The four allowed differences are > $$ > \pm\tfrac13 - (\pm1) \ \Rightarrow\ \Big\{-\tfrac23,\ +\tfrac43,\ -\tfrac43,\ +\tfrac23\Big\}, > $$ > four components at $\pm\tfrac23$ and $\pm\tfrac43$: no unshifted line at all. > > For **D$_2$** ($^2P_{3/2} \to {}^2S_{1/2}$) the upper sublevels are at > $\tfrac43 m_j = \{+2, +\tfrac23, -\tfrac23, -2\}$ for > $m_j = +\tfrac32, \dots, -\tfrac32$. Applying $\Delta m_j \in \{0,\pm1\}$ to > each and subtracting the lower $\pm1$ gives six distinct components at > $\pm\tfrac13,\ \pm1,\ \pm\tfrac53$. The sodium doublet therefore splits into a > four-line and a six-line pattern, the textbook anomalous signature, and > neither is the classical triplet. ## The normal Zeeman triplet When both the upper and lower levels have $g_J = 1$, which happens for pure singlet states ($s = 0$, so $j = \ell$ and the Landé formula collapses to unity), the sublevel spacing is $\mu_B B$ in every level. The dipole selection rule $\Delta m_j = 0, \pm 1$ then produces transition frequencies $$ h\nu = h\nu_0 + \mu_B B\,\Delta m_j, \qquad \Delta m_j \in \{-1,0,+1\}, $$ so the line splits into exactly three components at $\nu_0$ and $\nu_0 \pm \mu_B B/h$, regardless of the individual $m_j$ values. The shift in wavenumbers is the **Lorentz unit** $\mu_B B/hc = 0.4669\ \text{cm}^{-1}\text{T}^{-1}$. This is the **normal Zeeman effect**, and it reproduces exactly the classical Lorentz calculation of an electron oscillator in a field, in which the field splits the oscillation into one unshifted linear mode and two counter-rotating circular modes. $$ % caption: With equal g-factors the line splits into three (normal triplet); % with unequal g-factors many components appear (anomalous pattern). \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % axis \draw[->, black] (-3.2,0) -- (3.2,0) node[right] {frequency}; \draw[->, black] (-3.2,2.6) -- (3.2,2.6) node[right] {frequency}; % normal triplet (top) \node[anchor=east, black!70] at (-3.4,2.6) {normal}; \foreach \x in {-1.0,0,1.0}{ \draw[acc, very thick] (\x,2.6) -- (\x,3.3); } \node[acc, anchor=south] at (0,3.35) {3 lines}; % anomalous (bottom) \node[anchor=east, black!70] at (-3.4,0) {anomalous}; \foreach \x in {-1.6,-1.1,-0.6,0.6,1.1,1.6}{ \draw[black, very thick] (\x,0) -- (\x,0.7); } \node[black, anchor=south] at (0,0.75) {many lines}; \node[black, anchor=north] at (0,-0.4) {unshifted centre marked by dashes}; \draw[dashed, black] (0,-0.1) -- (0,0.8); \draw[dashed, black] (0,2.5) -- (0,3.4); \end{tikzpicture} $$ ## Polarization and the transverse/longitudinal patterns The three families of $\Delta m_j$ carry distinct polarizations, fixed by the angular momentum the emitted photon must remove. - **$\pi$ components ($\Delta m_j = 0$).** The dipole operator component is $z$, the emission is linearly polarized parallel to $\vec B$, and it vanishes when the atom is viewed along the field (a dipole radiates nothing along its own axis). - **$\sigma^{\pm}$ components ($\Delta m_j = \pm 1$).** The dipole operator components are $x \pm iy$; the photon carries $\pm\hbar$ of angular momentum along $\vec B$. Viewed **along** the field (longitudinal) these appear circularly polarized in opposite senses; viewed **across** the field (transverse) they appear linearly polarized perpendicular to $\vec B$. $$ % caption: Viewed along the field only the two circular $\sigma$ components % appear; viewed across it all three appear, linearly polarized. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % field axis \draw[->, black!70, very thick] (0,-1.6) -- (0,1.6) node[above] {B}; % source \fill[acc] (0,0) circle (3pt); % longitudinal observer (up along B) \draw[->, black, thick] (0.2,1.0) -- (1.7,1.6); \node[black, anchor=west] at (1.75,1.6) {along B: two circular}; % transverse observer \draw[->, black, thick] (0.3,0) -- (2.2,0); \node[black, anchor=west] at (2.25,0) {across B: three linear}; \node[black, anchor=north] at (0,-1.9) {a dipole emits nothing along its own axis}; \end{tikzpicture} $$ The selection rules and polarizations are not independent additions: both follow from the same dipole matrix element $\langle f\lvert \vec r\rvert i\rangle$ decomposed into spherical components, the $q = 0$ component driving $\pi$ and the $q = \pm 1$ components driving $\sigma^{\pm}$. A measurement of the polarization of each Zeeman component therefore fixes the sign of $\Delta m_j$ and, through the line positions, the Landé factors of both levels. ## Magnitude and regime of validity The whole treatment rests on $H_Z \ll H_{\text{fs}}$. Fine-structure splittings run from $\sim 10^{-4}\ \text{eV}$ in hydrogen $n=2$ to $\sim 2\times10^{-3}\ \text{eV}$ for the sodium D doublet, while $\mu_B B$ reaches $10^{-4}\ \text{eV}$ only near $B \approx 2\ \text{T}$. Weak-field theory therefore holds for the sodium D lines up to a few tesla but fails much earlier for the small hydrogen fine structure, where the Paschen-Back regime is reached in ordinary laboratory fields. The crossover, and the diagonalization that bridges the two limits, is developed next. [^codata]: CODATA 2018 recommended values: Bohr magneton $\mu_B = 9.2740100783(28)\times10^{-24}\ \text{J T}^{-1}$, electron $g$-factor $g_e = -2.00231930436256(35)$. NIST, . [^lande]: **Foot**, _Atomic Physics_, §5.5 — the Landé $g$-factor from the vector model and the projection of the magnetic moment onto $\vec J$; and **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_, §7.4, for the weak-field perturbation calculation of $\langle S_z\rangle$.