The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It arises from the interaction of the magnetic moment associated with the angular momentum of the atomic electrons with the external magnetic field.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of normal Zeeman effect, splitting of spectral lines, role of weak magnetic fields, and selection rules (delta m = 0, +/- 1).
PGT extension
Quantitative derivation of Zeeman shift, Normal vs. Anomalous Zeeman effect, Lande g-factor, Paschen-Back effect, and Stark effect comparison.
Prerequisites
Bohr's theory of hydrogen atom
Electron spin and orbital angular momentum
Spatial quantization
Phenomenology of the Zeeman Effect
When a light source emitting a line spectrum is placed in a uniform magnetic field, individual spectral lines split into multiple equidistant components.
Discovered by Pieter Zeeman in 1896, it provided direct evidence for the spatial quantization of angular momentum in atoms.
The magnitude of the frequency shift is proportional to the magnetic field strength and independent of the initial unperturbed frequency to a first approximation.
Normal Zeeman Effect
Observed in singlet spectral lines (total spin S = 0, so total angular momentum J = L).
A single spectral line splits into precisely three components: an unshifted central pi component (delta m = 0) and two symmetrically shifted sigma components (delta $m = +/- 1)$.
It is adequately explained by classical electron theory combined with quantum mechanical spatial quantization.
Anomalous Zeeman Effect
Observed in doublet, triplet, or higher multiplet lines where electron spin is non-zero (S != 0).
The spectral line splits into more than three components (often 4, 6, or more).
Requires the introduction of the Lande g-factor to account for the differing contributions of orbital and spin angular momenta to the total magnetic moment.
Lande g-Factor and Magnetic Moment
The magnetic moment of an electron is not simply proportional to its total angular momentum J due to the anomalous gyromagnetic ratio for spin (g_s approx 2).
The Lande g-factor is given by $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$.
It determines the precise energy level splitting in a weak magnetic field for the anomalous Zeeman effect.
Derivations
Normal Zeeman Shift Derivation
Conditions: Weak magnetic field, singlet states (S=0, J=L)
Consider an electron orbiting a nucleus with orbital angular momentum L. The circulating charge constitutes a current loop with magnetic dipole moment m$u_L = - \frac{e}{2m} L$.
In the presence of an external magnetic field B directed along the z-axis, the potential energy of interaction is Delta $E = - \mu_L \cdot B = - \mu_{Lz} B_z$.
Using the spatial quantization condition for the z-component of orbital angular momentum, $L_z = m_l \hbar$, we get Delta $E = \frac{e \hbar B}{2m} m_l$.
Expressed in terms of angular frequency shift, Delta $\omega = \frac{eB}{2m} = \omega_L$, where \omega_L is the Larmor frequency.
Result: Delta $E = m_l \mu_B B$, where \mu_B is the Bohr magneton.
Anomalous Zeeman Energy Splitting
Conditions: Weak magnetic field, arbitrary L, S, and J
The total magnetic moment operator is the sum of orbital and spin contributions: $\mu = \mu_B (L + 2S) / \hbar$.
Projecting this magnetic moment onto the total angular momentum vector J, the effective magnetic moment is \m$u_{eff} = -g \frac{e}{2m} J$.
The energy shift in a weak magnetic field B is given by the expectation value Delta $E = - \langle \mu \cdot B \rangle$.
Substituting the Lande g-factor, the energy shift becomes Delta $E = m_j g \mu_B B$, where m_j takes values from -j to +j.
Result: Delta $E = m_j g \mu_B B$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Bohr Magneton
$$\mu_B = \frac{e \hbar}{2m}$$
e = electronic charge, hbar = reduced Planck constant, m = electron mass J/T or eV/T
J = total angular momentum quantum number, S = spin quantum number, L = orbital quantum number Dimensionless
LS coupling scheme
Anomalous Zeeman Energy Shift
$$\Delta E = m_j g \mu_B B$$
$m_j = magnetic quantum number$, $g = Lande g-factor$, m$u_B = Bohr magneton$, B = magnetic field Joules (J) or eV
Weak field Zeeman effect
Larmor Frequency
$$\omega_L = \frac{e B}{2m}$$
e = charge, B = magnetic field, m = mass rad/s
Classical and quantum orbital precession
Selection Rules for Zeeman Transition
$$\Delta m_j = 0, \pm 1$$
$m_j = magnetic quantum number change$ Dimensionless
Electric dipole transitions
Normal Zeeman Wavelength Shift
$$\Delta \lambda = \frac{e \lambda^2 B}{4 \pi m c}$$
lambda = original wavelength, c = speed of light m
Normal Zeeman effect
Interaction Hamiltonian in Magnetic Field
$$H_{\int} = \frac{e}{2m} (L + 2S) \cdot B$$
$H_int = perturbation Hamiltonian$, L = orbital operator, S = spin operator Joules (J)
Weak magnetic field limit
Graphs and experiments
Normal Zeeman Effect Splitting Diagram
Draw: Energy levels splitting into 3 sub-levels for $m_l = -1$, 0, 1 with vertical transitions governed by delta m = 0, +/- 1.
Interpret: Shows the origin of the unshifted pi line and the two symmetrically shifted sigma lines.
Exam use: Tested in TGT/PGT to identify the number of components in normal Zeeman splitting.
Lummer-Gehrcke Plate / Fabry-Perot Etalon Setup
Draw: An electromagnet with pole pieces containing a discharge tube, followed by lenses and a high-resolution interferometer (Fabry-Perot etalon) projecting rings on a screen.
Interpret: High-resolution optical setup required because the wavelength shifts in Zeeman effect are extremely small.
Exam use: Conceptual questions on experimental apparatus used to observe fine splitting.
Polarization of Zeeman Components
Draw: View parallel to B field showing only sigma components (circularly polarized); view perpendicular to B field showing pi component (plane polarized parallel to B) and sigma components (plane polarized perpendicular to B).
Interpret: Demonstrates the vector nature of radiation emitted by precessing atomic dipoles.
Exam use: Direct factual questions in PGT physics regarding polarization states of pi and sigma lines.
Solved numericals
TGT core
A spectral line of wavelength 5890 Angstrom is subjected to a magnetic field of 1 Tesla. Calculate the separation between the Zeeman components assuming normal Zeeman effect. Take e/$m = 1.76 x 10^11 C/kg$ and $c = 3 x 10^8 m/s$.
Formula: Delta lambd$a = (e * \lambda^2 * B) / (4 * \pi * m * c)$
Convert wavelength to meters: lambd$a = 5890 x 10^-10 m$
Substitute values: Delta lambd$a = (1.76 x 10^11 * (5890 x 10^-10)^2 * 1) / (4 * 3.1416 * 3 x 10^8)$
Calculate numerator: 1.76 x 10^11 * 3.469 x 10^-$17 = 6.105 x 10^-6$
Calculate denominator: 12.566 x 3 x 10^8 = 3.77 x 10^9$
Result: Delta lambd$a = 1.62 x 10^-15 m = 0.0162 Angstrom$
Answer: 0.0162 Angstrom
PGT extension
For a 2P_3/2 to 2S_1/2 transition in an atom, compute the Lande g-factor for the 2P_3/2 state and determine the number of Zeeman components observed in a weak magnetic field.
For 2S_1/2 state: $S = 1/2$, L = 0, $J = 1/2$, g = 2
Find maximum number of transitions using selection rules: m_j changes by 0, +/- 1. Upper level has 4 states ($m_j = 3/2$, 1/2, -1/2, -3/2), lower level has 2 states ($m_j = 1/2$, -1/2). Total allowed transitions give 6 components.
Answer: $g = 4/3$, Number of Zeeman components = 6
PGT extension
Calculate the magnetic field strength B required to produce a Zeeman splitting equal to the natural line width of a spectral line of wavelength 5000 Angstrom having a lifetime ta$u = 10^-8 seconds$.
Natural line width in frequency: Delta n$u = 1 / (2 * \pi * \tau) = 1 / (2 * 3.1416 * 10^-8) = 1.59 x 10^7 Hz$
Normal Zeeman frequency shift: Delta n$u = (e * B) / (4 * \pi * m)$
Equate frequency shift to line width: (e / 4 pi m) * $B = 1.59 x 10^7$
Given e/$m = 1.76 x 10^11 C/kg$, (1.76 x 10^11 / 12.566) * $B = 1.59 x 10^7$
1.40 x 10^10 * $B = 1.59 x 10^7$
$B = (1.59 x 10^7) / (1.40 x 10^10) = 1.13 x 10^-3 Tesla$
Answer: 1.13 x 10^-3 T (or 113 Gauss)
Assumptions and limitations
Assumes the applied magnetic field is weak compared to the internal spin-orbit coupling magnetic field (Zeeman regime, not Paschen-Back regime).
Neglects nuclear spin effects (hyperfine structure is ignored).
Common misconceptions
Common error: All spectral lines split into exactly three components in a magnetic field.
Correct: Only singlet lines (Normal Zeeman effect) split into three components; multi-electron atoms with spin show anomalous Zeeman effect with multiple components.
Electron spin introduces an additional magnetic moment contribution that alters the simple 3-level splitting pattern.
Common error: Zeeman effect requires an electric field.
Correct: Zeeman effect is exclusively caused by a magnetic field, whereas the Stark effect is caused by an electric field.
Magnetic dipole interaction with B-field governs Zeeman splitting, not electric dipole interaction.
Quick revision
Normal Zeeman effect occurs for S=0 states and yields exactly 3 components (1 pi, 2 sigma).
Anomalous Zeeman effect occurs when S != 0 and is governed by the Lande g-factor.
Bohr magneton has the standard value m$u_B = e hbar / 2m$.
Selection rules for Zeeman transitions are Delta $m_j = 0$, +/- 1.
Must remember
The Zeeman effect is the splitting of spectral lines in the presence of a uniform magnetic field.
The Normal Zeeman effect is observed in singlet states ($S = 0$) where a line splits into exactly 3 components ($\Delta m = 0, \pm 1$).
The Anomalous Zeeman effect occurs when $S \neq 0$, resulting in more than three components, requiring the Lande $g$-factor.
Bohr magneton is given by $\mu_B = \frac{e\hbar}{2m}$.
Landé $g$-factor is expressed as $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$.
Selection rules for Zeeman transitions are $\Delta m_j = 0, \pm 1$.
The Stark effect is the splitting of spectral lines by an electric field, whereas the Zeeman effect is caused by a magnetic field.
Glossary
Normal Zeeman EffectSplitting of a singlet spectral line into three components when placed in a magnetic field.
Anomalous Zeeman EffectSplitting of spectral lines into more than three components for systems with non-zero total spin.
Landé g-factorA dimensionless factor accounting for the unequal contributions of orbital and spin angular momenta to the magnetic moment.
Bohr MagnetonFundamental unit of magnetic moment for an electron, $\mu_B = \frac{e\hbar}{2m}$.
Larmor FrequencyFrequency of precession of a magnetic moment about an external magnetic field, $\omega_L = \frac{eB}{2m}$.
Normal vs Anomalous Zeeman Effect
Feature
Normal Zeeman Effect
Anomalous Zeeman Effect
Total Spin ($S$)
$S = 0$ (Singlet states)
$S \neq 0$ (Doublet, triplet, etc.)
Number of Components
Precisely 3 components ($\pi$ and $\sigma$)
More than 3 components (4, 6, etc.)
Explanation
Classical electron theory & simple spatial quantization
Requires Landé $g$-factor and vector model
Zeeman vs Stark Effect
Feature
Zeeman Effect
Stark Effect
Perturbing Field
Uniform Magnetic field ($B$)
Uniform Electric field ($E$)
Origin
Magnetic dipole interaction with $B$
Electric dipole interaction with $E$
Practice Quiz
Choose an option to reveal the explanation.
1. Who discovered the splitting of spectral lines in a magnetic field in 1896?
Explanation: Pieter Zeeman discovered the Zeeman effect in 1896, providing direct evidence for spatial quantization.
2. What is the primary condition for observing the Normal Zeeman effect?
Explanation: Normal Zeeman effect is observed in singlet spectral lines where total spin $S = 0$.
3. Into how many components does a spectral line split in the Normal Zeeman effect?
Explanation: A single line splits into precisely three components: an unshifted central $\pi$ component ($\Delta m = 0$) and two symmetrically shifted $\sigma$ components ($\Delta m = \pm 1$).
4. What is the selection rule for magnetic quantum number $m_j$ in Zeeman transitions?
Explanation: For electric dipole transitions in a magnetic field, the selection rules are $\Delta m_j = 0, \pm 1$.
5. What does the Landé $g$-factor account for in the Anomalous Zeeman effect?
Explanation: The Landé $g$-factor accounts for the anomalous gyromagnetic ratio for spin ($g_s \approx 2$) combined with orbital momentum.
6. What is the expression for the Bohr magneton $\mu_B$?
Explanation: The fundamental constant for magnetic moment is $\mu_B = \frac{e\hbar}{2m}$.
7. What is the Larmor frequency $\omega_L$ associated with orbital precession in a magnetic field $B$?
Explanation: Classical and quantum orbital precession yields a Larmor frequency of $\omega_L = \frac{eB}{2m}$.
8. Which of the following is caused by an electric field rather than a magnetic field?
Explanation: The Stark effect is the splitting of spectral lines by an electric field, whereas the Zeeman effect is caused by a magnetic field.
9. What is the formula for the Landé $g$-factor in the LS coupling scheme?
Explanation: The exact Landé $g$-factor formula is $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$.
10. What is the energy shift $\Delta E$ in a weak-field Zeeman effect?
Explanation: The weak-field Zeeman energy shift is given by $\Delta E = m_j g \mu_B B$.
11. The frequency shift $\Delta \nu$ in the Normal Zeeman effect is proportional to:
Explanation: $\Delta \nu = \frac{eB}{4\pi m}$, showing direct proportionality to magnetic field strength $B$.
12. Why do multi-electron atoms with $S \neq 0$ exhibit the Anomalous Zeeman effect?
Explanation: Spin introduces an anomalous gyromagnetic ratio ($g_s \approx 2$), altering the splitting pattern from the classical triplet.
13. What is the normal Zeeman wavelength shift $\Delta \lambda$ proportional to?
Explanation: From $\Delta \lambda = \frac{e\lambda^2 B}{4\pi mc}$, the wavelength shift is proportional to $\lambda^2$.
14. What is the interaction Hamiltonian $H_{\int}$ for an electron in a weak magnetic field $B$?
Explanation: The perturbation Hamiltonian accounts for orbital and twice the spin magnetic moment: $H_{\int} = \frac{e}{2m}(L + 2S) \cdot B$.
15. In a Normal Zeeman triplet, what is the quantum number change corresponding to the unshifted central $\pi$ component?
Explanation: The central $\pi$ component corresponds to $\Delta m = 0$, which experiences no energy shift in the magnetic field.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. In the Normal Zeeman effect, a spectral line of frequency $\nu$ splits into three components. The frequency separation between adjacent components is given by:
Explanation: The frequency shift for normal Zeeman components is $\Delta \nu = \frac{eB}{4\pi m}$, representing the separation between adjacent lines.
2. For a state with $L=1$, $S=1$, and $J=2$, what is the Landé $g$-factor?
3. Which of the following transitions violates the selection rules for the Zeeman effect?
Explanation: Electric dipole selection rules permit $\Delta m_j = 0, \pm 1$; hence $\Delta m_j = \pm 2$ is forbidden.
4. The Anomalous Zeeman effect is observed when the total spin quantum number $S$ is:
Explanation: Anomalous Zeeman effect requires non-zero spin ($S \neq 0$), occurring in doublets, triplets, etc.
5. What is the unit of the Bohr magneton $\mu_B$ in SI units?
Explanation: Bohr magneton has units of magnetic moment, expressed as J/T or $\text{Am}^2$.
6. In a weak magnetic field, the energy levels of an atom split. This phenomenon is known as:
Explanation: Splitting of energy levels in a weak magnetic field is the hallmark of the Zeeman effect.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If a magnetic field of $1\text{ T}$ is applied to a normal Zeeman setup, what is the approximate magnitude of frequency shift $\Delta \nu$ (using $e/m \approx 1.76 \times 10^{11}\text{ C/kg}$)?
2. For a singlet state ($S=0, J=L$), what is the value of the Landé $g$-factor?
Explanation: When $S=0$, $J=L$, substituting into the Landé formula yields $g = 1$.
3. Which component of the Zeeman triplet has polarization parallel to the external magnetic field?
Explanation: The central $\pi$ component ($\Delta m = 0$) is linearly polarized parallel to $B$, while $\sigma$ components are polarized perpendicular to $B$.
4. What happens to the Zeeman splitting when the external magnetic field strength $B$ is increased?
Explanation: The frequency shift and energy shift are directly proportional to the magnetic field strength $B$.
5. In the vector model of an atom, the magnetic moment vector $\mu_J$ is antiparallel to:
Explanation: Due to the negative charge of the electron, the magnetic moment $\mu_J$ is antiparallel to the total angular momentum $J$.
6. What is the Landé $g$-factor for a pure spin magnetic moment where $L=0$ and $J=S$?
Explanation: For a pure spin state ($L=0, J=S$), the Landé formula gives $g = g_s \approx 2$.
7. The phenomenon where spectral lines split under the influence of an electric field is called:
Explanation: Electric field splitting of spectral lines is the Stark effect.
8. The Larmor frequency $\omega_L$ represents the frequency of precession of:
Explanation: Larmor frequency is the classical and quantum precession frequency of the magnetic moment about the external magnetic field axis.
9. What is the primary distinction between the Normal and Anomalous Zeeman effects regarding line splitting?
Explanation: Normal Zeeman effect always splits into 3 components, whereas Anomalous splits into multiple components depending on $J$ and $S$.
10. In terms of wavelengths, the Normal Zeeman wavelength shift $\Delta \lambda$ is directly proportional to:
Explanation: From $\Delta \lambda = \frac{e\lambda^2 B}{4\pi mc}$, the shift is proportional to the square of the original wavelength $\lambda^2$.