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An experimental atomic physicist whose work concerns the Zeeman effect and the Paschen-Back effect. At sufficiently strong magnetic fields, magnetic coupling competes with the internal coupling of angular momenta, changing the pattern of spectral lines. His book with Alfred Landé joins this atomic interpretation to the practical apparatus and analysis needed to resolve the splittings.^[ref]

Friedrich Paschen investigated atomic spectra in strong magnetic fields. The Paschen-Back effect, associated with his work with Ernst Back, occurs when the external magnetic interaction competes with internal spin-orbit coupling. The weak-field description through total angular momentum then ceases to provide the simplest labels. Separate orbital angular momentum and spin projections become useful, changing the pattern of spectral lines. Comparing weak- and strong-field patterns tests the hierarchy of interactions within an atom.^[ref]

A physicist.^[ref]

The Zeeman effect need not be diagonal between different $J$ multiplets. The familiar Landé g-factor gives the interaction projected into one isolated multiplet. Inter-multiplet matrix elements cause higher-order weak-field shifts and become important in the Paschen-Back effect.

The Zeeman effect comes from $-\mathbf{\mu }\cdot \mathbf{B}$. For an electron, the orbital and spin magnetic moments are approximately $-{\mu }_B\mathbf{L}/\hslash$ and $-2{\mu }_B\mathbf{S}/\hslash$, respectively. This explains both the sign and the relative factor two in the term linear in $B$. The second term is the spin-orbit interaction: the electron's motion through the atomic electric field produces a magnetic interaction in its rest frame. The relativistic correction, including Thomas precession, and the radial expectation value are incorporated into $K$. This effective Hamiltonian retains the linear Zeeman splitting and spin-orbit coupling, while omitting diamagnetic and hyperfine structure corrections.

Define the angular momentum ladder operators by $L_{\pm }=L_x\pm i{L}_y$ and similarly for $S_{\pm }$. Direct expansion givesso adding $L_z{S}_z$ gives the required scalar product. The off-diagonal terms change $m_{\ell }$ and $m_s$ in opposite directions. Consequently $J_z=L_z+S_z$ commutes with $H$, and $m_j$ remains a good quantum number at every magnetic field.

In the given product eigenbasis, the diagonal matrix elements areFor $\ell =1$ and $s=1/2$, the only off-diagonal couplings are between $(1,-1/2)$ and $(0,1/2)$, and between $(0,-1/2)$ and $(-1,1/2)$. Each relevant orbital ladder factor is $\hslash \sqrt{2}$ and each spin ladder factor is $\hslash$. Multiplying by $K/{\hslash }^2$ gives $\sqrt{2}K$. ThereforeAll other off-diagonal entries vanish either by the ladder endpoints or by conservation of $m_j$. This derives every entry of the displayed matrix.

The stretched states have no mixing and yieldThe $m_j=1/2$ block is $\left(\begin{array}{cc}-K& \sqrt{2}K\\ \sqrt{2}K& a\end{array}\right)$, while the $m_j=-1/2$ block is $\left(\begin{array}{cc}-a& \sqrt{2}K\\ \sqrt{2}K& -K\end{array}\right)$. Taking the trace and determinant of each $2\times 2$ block gives the remaining four energy eigenvalues:The subscripts indicate $m_j=\pm 1/2$ and the superscripts indicate the upper or lower root. These formulas are the Zeeman spectrum for orbital angular momentum one and remain valid for either sign of $K$.

For the usual ordering $K>0$, including sodium, the zero-field states follow fromThe $j=3/2$ quartet lies at $K$ and the $j=1/2$ doublet at $-2K$, separated by $3K$. In a weak magnetic field, the Landé g-factors giveThese label the curves near $B=0$. In a strong magnetic field, the Paschen-Back effect makes $(m_{\ell },m_s)$ the appropriate approximate labels:for the mixed states, with the stretched-state formulas already exact. The branch connections for $B>0$ areWithin a fixed $m_j$, the nonzero coupling prevents a crossing; this is an avoided crossing. Curves with different $m_j$ may cross because the Hamiltonian does not couple them.

For the sodium estimate, the zero-field frequency separation fixesThe lower $3s$ doublet has shifts $\pm a$. The Sodium D-line Zeeman pattern gives photon-frequency shifts $a/h$ times $\{\pm 2/3,\pm 4/3\}$ for D${}_1$ and $\{\pm 1/3,\pm 1,\pm 5/3\}$ for D${}_2$. Thus the highest-frequency D${}_1$ component and lowest-frequency D${}_2$ component approach each other by $3a/h$. Extrapolating the weak-field Zeeman splitting givesso the crossover occurs at a field of order $10\mathrm{T}$, where $a\sim K$ and the weak-field approximation is becoming marginal.

The exact eigenvalues refine the estimate within this Hamiltonian. The relevant photon energies, with a common zero-field reference omitted, are $E_{+}^{(-)}+a$ and $E_{-}^{(+)}-a$. The corresponding electric dipole transitions have $\Delta m_j=+1$ and $-1$, respectively. Equating them givesIts positive solution is $a/K=\sqrt{10}/3$, hencein the retained model. The crossing is between optical transition frequencies; the participating upper states themselves need not be degenerate.

For $5s5d$, addition of angular momentum gives $L=2$ and $S=0,1$. The even-parity atomic terms and atomic fine-structure levels areThe electric-dipole selection rules exclude all connections to the even-parity $5s^2$ and $5s6s$ electron configurations. To the odd-parity $5s5p$ electron configuration, all seven allowed connections in pure LS coupling areHere a connection is listed independently of whether the $5s5d$ energy level is above or below its partner.

In a weak magnetic field, use the Landé g-factorFor the three ${}^3{D}_J$ energy levels, this gives $1/2,7/6,4/3$; for ${}^3{P}_1,{}^3{P}_2$ it gives $3/2$. The six-component spectral line is thereforeA component has frequency displacement $({\mu }_{B}B/h)(m_D/2-3m_P/2)$. The electric dipole transition with $m_D=m_P=0$ has zero matrix element, since the relevant $J=1\leftrightarrow 1$ rank-one Clebsch-Gordan coefficients vanish there. The remaining allowed pairs giveThus all six frequency positions $\pm 1/2,\pm 1,\pm 3/2$ are distinct. Perpendicular to the magnetic field, both the $\pi$ and $\sigma$ components are visible, with mutually perpendicular linear polarizations. Along the magnetic field the $\pi$ components disappear. This is the weak-field Zeeman effect, before the Paschen-Back effect changes the coupling scheme.

For a nucleus of spin $I>0$ and electronic $J=1/2$, the magnetic-dipole hyperfine interaction is $A\mathbf{I}\cdot \mathbf{J}$, using dimensionless angular-momentum operators. The electric-quadrupole hyperfine contribution vanishes for $J=1/2$. In zero field, $F=I\pm 1/2$ labels the levels:Their signed separation is $\Delta =A(I+1/2)$. For $A>0$, the larger-$F$ multiplet lies higher; negative $A$ reverses this order. If $I=0$ there is no such hyperfine splitting.

Let the field define $z$ and writeHere $g_I$ is the signed nuclear Zeeman coefficient expressed in Bohr-magneton units, including its nuclear-to-Bohr-magneton factor; it is not an unconverted nuclear $g$ factor. Total projection $m=m_I+m_J$ is conserved, while $F$ need not be. In the basis $|m-1/2,+1/2⟩$, $|m+1/2,-1/2⟩$, the two-state block isFor $A>0$, define $x=(g_J-g_I){\mu }_{B}B/\Delta$. Diagonalization gives the Breit-Rabi formula,For either sign of $A$, the block itself remains valid; its ordered eigenvalues are $-A/4+g_I{\mu }_{B}Bm\pm \frac{1}{2}\sqrt{{\Delta }^2+2Am(g_J-g_I){\mu }_{B}B+(g_J-g_I{)}^2{\mu }_B^2{B}^2}$.

For a stretched hyperfine Zeeman state at projection $m=\pm (I+1/2)$ there is only one state each. Their energies are exactlyAssigning two branches to each stretched projection would double-count states.

In weak fields the shift is $g_F{\mu }_{B}mB$, with $g_F=g_I\pm (g_J-g_I)/(2I+1)$ for $F=I\pm 1/2>0$; an $F=0$ state has no first-order Zeeman shift. At strong hyperfine-decoupling fields the eigenstates approach product states withThis is the hyperfine regime of the Paschen-Back effect, still assuming the field is weak enough not to invalidate the electronic $J=1/2$ manifold. States of the same $m$ mix and avoid crossing when their off-diagonal coupling is nonzero; different-$m$ levels can cross.

In the weak-field Zeeman effect, spin-orbit coupling keeps $L,S,J$ good and a level shifts by ${\mu }_B{g}_{J}B{m}_J$. At fields large compared with fine-structure coupling but still within the linear electronic Zeeman approximation, $L$ and $S$ decouple: the Paschen-Back effect gives shifts ${\mu }_{B}B(m_L+2m_S)$. Electric-dipole transitions obey $\Delta m_L=0,\pm 1$, $\Delta m_S=0$ in that limit, recovering an orbital triplet pattern. Intermediate fields mix the coupled states.

For the weak-field sodium D-line Zeeman pattern, the given Landé g-factor formula givesAllowed photon shifts are $\delta E={\mu }_{B}B(g_u{m}_u-2m_l)$ with $m_u-m_l=0,\pm 1$. ThusThese are the four- and six-component patterns. Both $\pi$ and $\sigma$ components are represented, so an observation strictly along the field, which removes $\pi$ emission, would not show this complete pattern. Hyperfine structure is neglected.

To determine the magnetic field from a Zeeman diagram, the unsplit separation provides its energy scale:The smallest adjacent split spacing $s_E=2{\mu }_B{B}_1/3$ is approximately $0.070$ of that separation in the sketch. ThereforeThis is a graphical estimate; the drawing does not warrant high precision. Its splitting energy is small compared with $\Delta E_D$, consistent with the weak-field approximation.

For Doppler resolution of a Zeeman pattern, thermal velocities give Gaussian Doppler broadening with widthAt $350\mathrm{K}$, the photon-energy FWHM near $589\mathrm{nm}$ is $5.88\times 10^{-6}\mathrm{eV}$. Equating the smallest adjacent Zeeman spacing to this width gives $B\sim 0.15\mathrm{T}$ for loss of individually resolved components. There is no sharp Gaussian resolution threshold; using whole-pattern width instead gives a smaller field of order $0.03$–$0.04\mathrm{T}$ for complete disappearance into a single envelope. Instrumental and collisional broadening would also affect the estimate.

In the regime of weak-field Zeeman splitting, $J,m_J$ remain good labels and $\Delta E={\mu }_B{g}_J{m}_{J}B$. For the 393.4 nm line, $J_u=3/2$ in $4p{}^2{P}_{3/2}$ and $J_l=1/2$ in $4s{}^2{S}_{1/2}$. The Landé g-factors are $g_u=4/3$ and $g_l\simeq 2$. Apply $\Delta m_J=0,\pm 1$ to the four upper and two lower sublevels. The allowed pairs $(m_u,m_l)$ areTheir frequency displacements in units of ${\mu }_{B}B/h$ are respectively $1,-1/3,-5/3,5/3,1/3,-1$, so none coincide. Looking perpendicular to the field permits both the $\pi$ radiation from $q=0$ transitions and the $\sigma$ radiation from $q=\pm 1$ transitions. ConsequentlyA linear polarizer parallel to the field isolates the two $\pi$ components; a transverse polarizer selects the four $\sigma$ components. Without polarization selection both sets are present. This count assumes a field small compared with fine structure coupling and neglects isotope-dependent hyperfine structure; the strong-field Paschen-Back effect or a resolved nonzero-spin isotope would require a different model. The spectral spacings are unnecessary for the count, but the distinct shifts verify that counting allowed transitions does not inadvertently count degenerate lines twice.

In 1925, Wolfgang Pauli connected two puzzles: alkali metal doublets and their anomalous Zeeman effect, and the capacities of closed electron shells. His paper assigned the extra two-valued property to the optically active electron, rather than an angular momentum of the closed core. Its Paschen-Back effect classification used four quantum numbers; these historical labels should not be identified literally with modern $n,\ell ,m_{\ell },m_s$. The exclusion rule forbade matching all four. This connected resolved one-electron states with closed-group populations. An additional test was the missing triplet counterpart of the equivalent two-$s$-electron ground state in alkaline earth metals. Pauli explicitly claimed counts of allowed states, not their energies or ordering.^[ref]

The modern question is especially concrete: if a Quantum Hamiltonian has a lowest one-particle state, why cannot every electron occupy it? Minimizing energy answers which of the allowed quantum states is preferred; the Pauli exclusion principle determines which quantum states are allowed in the first place. Spectral lines then test energy differences between allowed multiparticle quantum states. A line count is not itself a direct count of occupied spin orbitals: transitions also depend on selection rules and the prepared quantum state populations. The observed anomalous Zeeman effect supplied constraints for the historical classification; the two-particle antisymmetrization below supplies the modern mathematical explanation of exclusion.

The spin-orbit interaction couples orbital angular momentum to spin. In chemical atoms it helps determine fine structure and the coupling scale that separates weak-field Zeeman splitting from the Paschen-Back effect.

For orbital angular momentum $\ell =1$ and spin $s=1/2$, considerThe quantum number $m_j=m_{\ell }+m_s$ is conserved. The stretched states with $m_j=\pm 3/2$ have energies $K\pm 2a$. The remaining eigenvalues follow from two $2\times 2$ blocks with off-diagonal entries $\sqrt{2}K$:The subscripts denote $m_j=\pm 1/2$. For $K>0$, the zero-field multiplets are $j=3/2$ at $K$ and $j=1/2$ at $-2K$. Their weak-field Zeeman splitting is $g_{j}a{m}_j$, with Landé g-factors $4/3$ and $2/3$, respectively. When $a\gg |K|$, the Paschen-Back effect gives approximate product states $(m_{\ell },m_s)$ and energies $a(m_{\ell }+2m_s)+2K{m}_{\ell }{m}_s$. Mixing within a fixed $m_j$ causes an avoided crossing; different $m_j$ sectors can cross because the Hamiltonian does not couple them.