AcademyAtomic Quantum Structure
Academy
Zeeman Effect
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
The Zeeman effect is the splitting of atomic energy levels in an external magnetic field. It occurs because angular momentum gives an atom a magnetic moment that interacts with the field.
The simplest model treats the splitting as proportional to magnetic field strength and magnetic quantum number.
Notation
\(B\)
external magnetic field
\(\mathrm{T}\)
\(\mu_B\)
Bohr magneton
\(\mathrm{J\,T^{-1}}\)
\(m_\ell\)
orbital magnetic quantum number
1
\(m_j\)
total angular momentum magnetic quantum number
1
\(g\)
g-factor
1
\(\Delta E\)
energy shift or spacing
\(\mathrm{J}\)
Method
Derivation 1: Magnetic moment in a field
A magnetic moment has energy that depends on its orientation relative to the field.
Magnetic interaction
\[U=-\vec\mu\cdot\vec B\]
Bohr magneton
\[\mu_B=\frac{e\hbar}{2m_e}\]
Derivation 2: Quantize the projection
The magnetic field selects a direction, so the angular-momentum projection matters.
Normal Zeeman shift
\[\Delta E=m_\ell\mu_BB\]
Adjacent spacing
\[\Delta E_{\mathrm{adj}}=\mu_BB\]
Derivation 3: Convert energy splitting to frequency
Spectral lines split because transition energies split.
Photon energy shift
\[\Delta E=h\Delta f\]
Frequency spacing
\[\Delta f=\frac{\mu_BB}{h}\]
Rules
Magnetic interaction
\[U=-\vec\mu\cdot\vec B\]
Bohr magneton
\[\mu_B=\frac{e\hbar}{2m_e}\]
Normal Zeeman shift
\[\Delta E=m_\ell\mu_BB\]
Frequency shift
\[\Delta f=\frac{\Delta E}{h}\]
Examples
Question
Find the adjacent Zeeman spacing for
\[B=1.50\,\mathrm{T}\]
using \[\mu_B=5.79\times10^{-5}\,\mathrm{eV\,T^{-1}}\]
Answer
\[\Delta E=\mu_BB=(5.79\times10^{-5})(1.50)=8.69\times10^{-5}\,\mathrm{eV}\]
Checks
- No magnetic field means no Zeeman splitting.
- Stronger fields produce larger splittings in the simple linear model.
- Splitting depends on allowed magnetic quantum numbers.
- Real atoms can require spin and total-angular-momentum g-factors.