Normal & Anomalous Zeeman Effect Calculator

Comprehensive Theoretical Guide to the Zeeman Effect

The Zeeman effect, discovered experimentally in 1896 by Dutch physicist Pieter Zeeman and recognized with the 1902 Nobel Prize in Physics, describes the splitting of atomic, ionic, and molecular spectral emission and absorption lines in the presence of an external static magnetic field $vec{B}$. In modern physics and astrophysics, Zeeman spectroscopy serves as an indispensable quantum diagnostic tool, enabling astronomers to measure the magnetic fields of sunspots, white dwarfs, and interstellar clouds, while empowering quantum engineers to control qubits in atomic ion traps.

1. Classical Lorentz Theory & The Normal Zeeman Effect

Hendrik Lorentz first explained the effect classically by modeling bound atomic electrons as damped harmonic oscillators of charge $-e$ and mass $m_e$ orbiting in a central potential. When an external magnetic field $vec{B} = Bhat{z}$ is applied along the $z$-axis, each moving electron experiences the Lorentz magnetic force:

Resolving the electron's motion into linear oscillations parallel to $hat{z}$ and counter-rotating circular motions in the $xy$-plane perpendicular to $vec{B}$, Lorentz showed that:

• Oscillations parallel to $vec{B}$ experience zero Lorentz force ($vec{v} parallel vec{B}$), oscillating at the unperturbed natural frequency $omega_0$. This emits linearly polarized light ($pi$-component) with the electric field $vec{E} parallel vec{B}$.
• Circular orbits in the $xy$-plane experience a radial Lorentz force $mp e v B$, shifting their rotational frequencies by the Larmor angular frequency $omega_L$: $$omega_L = frac{e B}{2 m_e},qquad nu_L = frac{e B}{4pi m_e} = frac{mu_B B}{h}$$ where $mu_B = frac{ehbar}{2m_e} approx 9.27401 times 10^{-24},text{J/T}$ is the Bohr magneton.

The classical or Normal Zeeman effect thus predicts that every spectral line splits into an exact symmetric Lorentz triplet:

In terms of wavelength shift $Delta lambda_N$ around center wavelength $lambda_0$:

Experimentally, the Normal Zeeman triplet is observed only in singlet-to-singlet transitions ($S = 0$) where intrinsic electron spin cancels, such as the Cadmium red line ($^1D_2 to {}^1P_1$ at $lambda = 643.8,text{nm}$) and Zinc singlet lines.

2. Quantum Mechanics & The Anomalous Zeeman Effect

When atoms have non-zero total spin ($S ne 0$), spectral lines split into multiplets of 4, 6, 10, or more components—a phenomenon historically termed the Anomalous Zeeman effect. This behavior baffled classical physicists until Uhlenbeck, Goudsmit, and Dirac incorporated relativistic electron spin angular momentum $vec{S}$ and spin-orbit coupling.

A. Magnetic Dipole Operator & Perturbation Hamiltonian

An atomic electron possesses an orbital magnetic dipole moment $vec{mu}_L = -g_L frac{mu_B}{hbar}vec{L}$ (with $g_L = 1$) and an intrinsic spin magnetic dipole moment $vec{mu}_S = -g_S frac{mu_B}{hbar}vec{S}$ (where the Dirac $g$-factor for electron spin is $g_S approx 2.002319 approx 2$). The total magnetic dipole moment operator $vec{mu}$ is:

Because $g_S approx 2$ while $g_L = 1$, the net magnetic moment $vec{mu}$ is not parallel to the total angular momentum vector $vec{J} = vec{L} + vec{S}$. Under weak external fields where spin-orbit coupling dominates, $vec{L}$ and $vec{S}$ precess rapidly around $vec{J}$, while $vec{J}$ precesses slowly around the magnetic field axis $vec{B} = Bhat{z}$.

B. Landé $g$-Factor Derivation via the Wigner-Eckart Projection Theorem

Only the projection of $vec{mu}$ along the direction of $vec{J}$ has a non-vanishing time average $langle vec{mu} rangle$:

Computing the dot product using the identity $vec{S}^2 = (vec{J} - vec{L})^2 = vec{J}^2 + vec{L}^2 - 2vec{J}cdotvec{L}$:

This is the celebrated Landé $g$-factor for Russell-Saunders ($LS$) coupled states:

• For pure orbital singlet states ($S = 0, J = L$): $g_J = 1 + frac{L(L+1) - L(L+1)}{2L(L+1)} = 1$.
• For pure spin states ($L = 0, J = S$): $g_J = 1 + frac{S(S+1) + S(S+1)}{2S(S+1)} = 2$.
• For mixed states (e.g. Sodium $^2P_{1/2}$ with $L=1, S=1/2, J=1/2$): $g_J = 1 + frac{3/4 + 3/4 - 2}{2(3/4)} = 1 - frac{0.5}{1.5} = frac{2}{3}$.
• For Sodium $^2P_{3/2}$ with $L=1, S=1/2, J=3/2$: $g_J = 1 + frac{15/4 + 3/4 - 8/4}{2(15/4)} = 1 + frac{10}{30} = frac{4}{3}$.

3. Zeeman Sublevel Energy Shift and Dipole Selection Rules

In an external field $B$, a state characterized by quantum numbers $(L, S, J)$ removes its $(2J+1)$-fold spatial degeneracy. Each magnetic sublevel with magnetic quantum number $M_J in {-J, -J+1, dots, +J}$ experiences an energy shift:

For an electric dipole optical transition between upper level $2$ ($g_2, M_{J2}$) and lower level $1$ ($g_1, M_{J1}$), the emitted photon energy difference is:

Electric dipole selection rules govern allowable radiative transitions:

• $Delta M_J = M_{J2} - M_{J1} = 0$: Produces $pi$-lines. When viewed transverse to $vec{B}$, light is linearly polarized parallel to $vec{B}$. When viewed longitudinal (parallel) to $vec{B}$, $pi$-lines have zero emission intensity. (Transition $M_{J2} = 0 to M_{J1} = 0$ is forbidden if $J_2 = J_1$).
• $Delta M_J = pm 1$: Produces $sigma^pm$-lines. When viewed transverse to $vec{B}$, light is linearly polarized perpendicular to $vec{B}$. When viewed longitudinally along $vec{B}$, $sigma^+$ and $sigma^-$ appear as right- and left-circularly polarized light.

4. The Paschen-Back Effect (Strong Field Limit)

The anomalous Zeeman formulation assumes the weak-field regime, where the Zeeman interaction energy $Delta E_Z ll Delta E_{SO}$ (spin-orbit fine structure splitting). When the external magnetic field becomes intense ($B gg 10text{ to } 50,text{Tesla}$ for light atoms), the magnetic interaction overwhelms spin-orbit coupling.

In this regime—the Paschen-Back effect—$vec{L}$ and $vec{S}$ uncouple from $vec{J}$ and precess independently around $vec{B}$. The good quantum numbers become $M_L$ and $M_S$, with energy shifts:

Because electric dipole radiation requires $Delta M_S = 0$ and $Delta M_L in {0, pm 1}$, the anomalous multiplet collapses back into a classical triplet structure with normal Zeeman frequency shifts $Delta nu = 0, pm frac{mu_B B}{h}$.

5. Step-by-Step Worked Example: The Sodium $D_1$ and $D_2$ Lines

Calculate the Zeeman splitting of the Sodium yellow doublet at $B = 1.50,text{Tesla}$:

• Bohr magneton energy: $mu_B B = (9.274 times 10^{-24},text{J/T}) times 1.50,text{T} approx 1.391 times 10^{-23},text{J} approx 86.83,mutext{eV}$
• Larmor frequency shift: $nu_L = frac{mu_B B}{h} = frac{1.391 times 10^{-23}}{6.626 times 10^{-34}} approx 20.99,text{GHz}$
• Normal shift at $lambda_0 = 589.0,text{nm}$: $$Delta lambda_N = frac{lambda_0^2}{c} nu_L = frac{(589.0 times 10^{-9})^2}{2.998 times 10^8} times (20.99 times 10^9) approx 24.28 times 10^{-12},text{m} = 24.28,text{pm}$$

Step 1: Landé Factors for Sodium $D_2$ Transition ($^2P_{3/2} to {}^2S_{1/2}$)

Step 2: Enumerate Allowed Dipole Transitions ($Delta M_J in {0, pm 1}$)

There are 6 allowed transitions for the Sodium $D_2$ line:

• Two $pi$-lines ($Delta M_J = 0$): $$left(+1/2 to +1/2right): Delta E = left(frac{4}{3}(1/2) - 2(1/2)right)mu_B B = left(frac{2}{3} - 1right)mu_B B = -frac{1}{3}mu_B B implies -8.09,text{pm}$$ $$left(-1/2 to -1/2right): Delta E = left(frac{4}{3}(-1/2) - 2(-1/2)right)mu_B B = left(-frac{2}{3} + 1right)mu_B B = +frac{1}{3}mu_B B implies +8.09,text{pm}$$
• Four $sigma$-lines ($Delta M_J = pm 1$): $$left(+3/2 to +1/2right): Delta E = (2 - 1)mu_B B = +1mu_B B implies -24.28,text{pm}$$ $$left(+1/2 to -1/2right): Delta E = left(frac{2}{3} - (-1)right)mu_B B = +frac{5}{3}mu_B B implies -40.47,text{pm}$$ $$left(-1/2 to +1/2right): Delta E = left(-frac{2}{3} - 1right)mu_B B = -frac{5}{3}mu_B B implies +40.47,text{pm}$$ $$left(-3/2 to -1/2right): Delta E = (-2 - (-1))mu_B B = -1mu_B B implies +24.28,text{pm}$$

Notice that unlike the normal triplet, there is no central unshifted line at $lambda_0 = 589.00,text{nm}$. The entire $D_2$ line splits into a symmetric 6-component sextet spanning from $-40.47,text{pm}$ to $+40.47,text{pm}$ (an overall spectral spread of $80.9,text{pm}$).

6. Astrophysical and Laboratory Applications

Zeeman splitting provides the premier remote-sensing methodology for cosmic magnetometry:

• Solar Astronomy: In 1908, George Ellery Hale utilized the Zeeman effect on iron spectral lines (e.g. Fe I at $630.2,text{nm}$) to discover that sunspots harbor immense magnetic fields exceeding $0.3text{ to } 0.4,text{Tesla}$ ($3,000text{ to } 4,000,text{Gauss}$), establishing that sunspots are magnetic vortices.
• Radio Astronomy: The $21,text{cm}$ neutral hydrogen ($H,text{I}$) hyperfine line undergoes tiny Zeeman frequency shifts in microgauss interstellar magnetic fields ($sim 1text{ to } 10,mutext{G}$), charting galactic magnetic spiral arms.
• Magneto-Optical Trapping (MOT): In ultracold atomic physics, quadrupole magnetic fields combine with circularly polarized laser beams to create spatially dependent Zeeman shifts, cooling and trapping neutral atoms at microkelvin temperatures.