Atomic Physics - III
Purnendu Chakraborty
Basirhat College, North 24 Parganas
15 December, 2022
Magnetic Interactions in Atoms Zeeman Effect
Learning objectives
Learning objective
1 Zeeman effect
2 Spin-orbit interaction.
3 Addition of angular momentum
References
1 Physics for Scientists and Engineers with Modern Physics - Serway, Jewett
2 University Physics with Modern Physics - Young and Freedman
3 Concepts of Modern Physics - Beiser
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
The Cosmic Laboratory
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
The Cosmic Laboratory
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
The Cosmic Laboratory
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Atomic spectra
Spectrum of hydrogen
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Bohr Model and explanation of hydrogen spectrum
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Our nearest star - The Sun
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Our nearest star - The Sun
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Our nearest star - The Sun
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
What is Zeeman effect?
The Zeeman effect is the splitting of atomic energy levels and the associated
spectral lines when the atoms are placed in a magnetic field. This effect confirms
experimentally the quantization of angular momentum.
Assumptions
We assume external magnetic field B constant over the dimension of atomic
system and we assume orbital motion only.
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Magnetic Moment of an Atom in Bohr model
An electron rotating in an circular Bohr orbit implies that it posses an orbital
angular momentum L. ~
An electron rotating in a closed orbit is equivalent to a closed current loop
and hence create a magnetic dipole µ ~ For electron, I = ev/(2πr) and
~ = IdA.
A = πr2 .
For electron, µ = evr/2 = eL/(2m) and as the current is opposite to the
direction of rotation of electron
e ~ ~
L
~ =−
µ L = −µB
2m ~
In Bohr model L = n~. Minimum value of L is ~. For an n = 1 state,
µ = µB = e~/2m is called Bohr magneton. µB = 9.27 × 10−24 JT-1 (or
A.m2 ) = 5.788 × 10−5 eV.-1 .
In general, for a system of electrons possessing a total angular momentum J~
J~
~ = −gµB
µ
~
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Magnetic interaction in hydrogen like systems
Magnetic interaction : Bohr model vs Schroedinger wave function
Bohr model correctly predicts existence and quantization of atomic magnetic
moments but fails to account for magnetic interaction. For example, Bohr model
suggests that the hydrogen atom in ground state will have magnetic moment of
Bohr magneton. But Schroedinger wave function of the hydrogenic ground sate is
an s state (l = 0) and the orbital magnetic moment would be zero.
But note that, in Schroedinger’s formulation the electrons have the same ratio of µ
to L (gyromagnetic ratio).
What happens when an atom is placed in a magnetic field?
An atom with magnetic moment µ ~
~ placed in a magnetic field B.
Interaction energy of the system :
U = −~ ~.
µ·B
The system experience a torque,
~ ~.
~ ×B
Γ=µ
Net force on the system in inhomogeneous field
~ = −∇U
F ~
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
Interaction energy of an atomic dipole in homogeneous field
~ is along z direction,
We assume B
U = −~ ~ = −µz B = ml e~ B = ml µB B
µ·B (ml = 0, ±1, ±2, · · · , ±l)
2m
Since,
e e~
µz = − Lz = −ml
2m 2m
Note carefully : in presence of magnetic degenerate energy states belonging to the
same value of l split into (2l + 1) levels. The adjacent states differing in energy by
µB
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
Normal Zeeman splitting increases linearly with the magnetic field.
Selection rule
Due to angular momentum conservation, (photon ordinarily takes away one unit
of angular momentum), the allowed transitions follow the selection rule :
∆l = 1, ∆ml = 0, ±1
Hence, there would be only three lines.
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
Only nine allowed transitions and three possible energies.
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
Purnendu Chakraborty, Basirhat College Atomic Physics
Magnetic Interactions in Atoms Zeeman Effect
Zeeman effect
Normal and Anomalous Zeeman effect
We have assumed that the spin of initial and final states are both zero. Zeeman
effect for transitions between singlet states is called a “normal” effect. In normal
Zeeman effect a spectral line splits into three components. Whereas the transition
for which the total spin of either the initial or final states, or both, is nonzero is
called the anomalous effect. The spectral line in this splits into more components.
Purnendu Chakraborty, Basirhat College Atomic Physics
