Lectures 7-8: Fine and hyperfine structure of hydrogen
o Fine structure
o Spin-orbit interaction.
o Relativistic kinetic energy correction
o Hyperfine structure
o The Lamb shift.
o Nuclear moments.
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Spin-orbit coupling in H-atom
o
Fine structure of H-atom is due to spin-orbit interaction:
E so  
o
Jˆ is a max
Z
Sˆ  Lˆ
2
3
2m cr
If L is parallel to S => J is a maximum => high energy
configuration.
Lˆ

o
+Ze


1
Sˆ  Lˆ  ( Jˆ  Jˆ  Lˆ  Lˆ  Sˆ  Sˆ )
2
 Sˆ  Lˆ 

2
2
-e

Angular momenta are described in terms of quantum
numbers, s, l and j:
Jˆ  Lˆ  Sˆ
Jˆ 2  ( Lˆ  Sˆ )( Lˆ  Sˆ )  Lˆ Lˆ  SˆSˆ  2 Sˆ  Lˆ
Sˆ
Jˆ is a min
Lˆ

+Ze

-e
[ j( j  1)  l(l  1)  s(s  1)]
Sˆ
Z 3 1
E so  
[ j( j  1)  l(l  1)  s(s  1)]
4m 2c r 3

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Spin-orbit effects in H fine structure
o
For practical purposes, convenient to express spin-orbit coupling as
a
j  j 1  l(l 1)  ss 1
2
where a  Ze 20 2 /8m 2 r 3 is the spin-orbit coupling constant. Therefore, for the 2p
electron:
 E  a 3 3  11(1 1)  1 1  1  1 a



 
so



 2
2
2
2
2
2


E so 
a 1 1 
1 1 
E so     11(1 1)    1 a
2 2 2 
2 2 
E
j = 3/2
+1/2a
Angular momenta aligned
j = 1/2
-a
Angular momenta opposite
2p1
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Spin-orbit coupling in H-atom
o
The spin-orbit coupling constant is directly measurable from the doublet structure
of spectra.
o
If we use the radius rn of the nth Bohr radius as a rough approximation for r (from
Lectures 1-2):
n2 2
r  4 0
2
mZe
Z4
 a ~ 6
n
o
Spin-orbit coupling increases sharply with Z. Difficult for observed for H-atom, as
Z = 1: 0.14 Å (H), 0.08 Å (H), 0.07 Å (H).

o
Evaluating the quantum mechanical form,
o
Therefore, using this and s = 1/2:
Z4
a~ 3
n l(l  1)(2l  1)
Z 2 4
2 [ j( j  1)  l(l  1)  3/4]
E so 
mc
2n 3
l(l  1)(2l  1)

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Term Symbols
o
Convenient to introduce shorthand notation to label energy levels that occurs in the LS
coupling regime.
o
Each level is labeled by L, S and J:
o
o
o
o
2S+1L
J
L = 0 => S
L = 1 => P
L = 2 =>D
L = 3 =>F
o
If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2P3/2 and 2P1/2
o
2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin.
o If S = 0 => multiplicity is 1: singlet term.
o If S = 1/2 => multiplicity is 2: doublet term.
o If S = 1 => multiplicity is 3: triplet term.
o
Most useful when dealing with multi-electron atoms.
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Term diagram for H fine structure
o
The energy level diagram can also be drawn as a term diagram.
o
Each term is evaluated using: 2S+1LJ
o
For H, the levels of the 2P term arising from spin-orbit coupling are given below:
2P
3/2
E
+1/2a
Angular momenta aligned
2p1 (2P)
2P
1/2
-a
Angular momenta opposite
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Hydrogen fine structure
o
Spectral lines of H found to be composed of
closely spaced doublets. Splitting is due to
interactions between electron spin S and the
orbital angular momentum L => spin-orbit
coupling.
o
H line is single line according to the Bohr or
Schrödinger theory. Occurs at 656.47 nm for
H and 656.29 nm for D (isotope shift, ~0.2
nm).
o
H
Spin-orbit coupling produces fine-structure
splitting of ~0.016 nm. Corresponds to an
internal magnetic field on the electron of
about 0.4 Tesla.
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Relativistic kinetic energy correction
o
According to special relativity, the kinetic energy of an electron of mass m and velocity v is:
p2
p4
T

2m 8m 3c 2
o
The first term is the standard non-relativistic expression for kinetic energy. The second term is
the lowest-order relativistic correction to this energy.

o
Using perturbation theory, it can be show that
E rel
 1
Z 2 4
3 
  3 mc 2
 
2l  1 8n 
n
o
Produces an energy shift comparable to spin-orbit effect.
o
A complete relativistic 
treatment of the electron involves the solving the Dirac equation.
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Total fine structure correction
o
For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much
smaller than the gross structure.
Enlj = En + EFS
o
Gross structure determined by En from Schrödinger equation. The fine structure is determined
by
Z 4 4
3 
2  1
EFS  E so  E rel  
mc



2l  1 8n 
2n 3
o
As En = -Z2E0/n2, where E0 = 1/22mc2, we can write

o
Z 2 E 0  Z 2 2  1
3 
E H atom   2 1



n 
n j  1/2 4n 
Gives the energy of the gross and fine structure of the hydrogen atom.

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Fine structure of hydrogen
o
Energy correction only depends on j, which is
of the order of 2 ~ 10-4 times smaller that the
principle energy splitting.
o
All levels are shifted down from the Bohr
energies.
o
For every n>1 and l, there are two states
corresponding to j = l ± 1/2.
o
States with same n and j but different l, have
the same energies (does not hold when Lamb
shift is included). i.e., are degenerate.
o
Using incorrect assumptions, this fine structure
was derived by Sommerfeld by modifying
Bohr theory => right results, but wrong
physics!
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Hyperfine structure: Lamb shift
o
Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine
structure.
o
According to Schrödinger and Dirac theory, states with same n and j but different l are
degenerate. However, Lamb and Retherford showed in 1947 that 22S1/2 (n = 2, l = 0, j = 1/2)
and 22P1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate.
o
Experiment proved that even states with the same total angular momentum J are energetically
different.
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Hyperfine structure: Lamb shift
1. Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly
decay to 12S1/2 optically.
2. Cause transitions to 22P1/2 state using tunable microwaves. Transitions only occur when
microwaves tuned to transition frequency. These atoms then decay emitting Ly line.
3. Measure number of atoms in 22S1/2 state from H-atom collisions with tungsten (W) target.
When excitation to 22P1/2, current drops.
4. Excited H atoms (22S1/2 metastable state) cause secondary electron emission and current from
the target. Dexcited H atoms (12S1/2 ground state) do not.
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Hyperfine structure: Lamb shift
o
According to Dirac and Schrödinger theory, states with the same n and j quantum
numbers but different l quantum numbers ought to be degenerate. Lamb and
Retherford showed that 2 S1/2 (n=2, l=0, j=1/2) and 2P1/2 (n=2, l=1, j=1/2) states of
hydrogen atom were not degenerate, but that the S state had slightly higher energy
by E/h = 1057.864 MHz.
o
Effect is explained by the theory of quantum electrodynamics, in which the
electromagnetic interaction itself is quantized.
o
For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html
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Hyperfine structure: Nuclear moments
o
Hyperfine structure results from magnetic interaction between the electron’s total
angular momentum (J) and the nuclear spin (I).
o
Angular momentum of electron creates a magnetic field at the nucleus which is
proportional to J.
o
ˆ nucleus  Bˆ electron  Iˆ  Jˆ
Interaction energy is therefore E hyperfine  
o
Magnitude is very small as nuclear dipole is ~2000 smaller than electron (~1/m).

o
Hyperfine splitting is about three orders of magnitude smaller than splitting due to
fine structure.
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Hyperfine structure: Nuclear moments
o
o
o
o
Like electron, the proton has a spin angular momentum and an associated intrinsic dipole
moment
e
ˆ p  gp Iˆ

M
The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and
hence the effect is small.
 be shown to be:
Resulting energy correction can
gpe 2 ˆ ˆ
E p 
IJ
mMc 2 r 3
Total angular momentum including nuclear spin, orbital angular momentum and electron spin
is
Fˆ  Iˆ  Jˆ

where F  f ( f  1)
Fz  m f
o The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,.
o Hence every energy level associated with a particular set of quantum numbers n, l, and j will
be split into two levels of slightly different energy, depending on the relative orientation of the
proton magnetic dipole with the electron state.
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Hyperfine structure: Nuclear moments
o
The energy splitting of the hyperfine interaction is
given by
a
E HFS   f  f 1  i(i 1)  j  j 1
2
where a is the hyperfine structure constant.
o
o

E.g., consider the ground state of H-atom. Nucleus
consists of a single proton, so I = 1/2. The hydrogen
ground state is the 1s 2S1/2 term, which has J = 1/2.
Spin of the electron can be parallel (F = 1) or
antiparallel (F = 0). Transitions between
these levels

occur at 21 cm (1420 MHz).
For ground state of the hydrogen atom (n=1), the
energy separation between the states of F = 1 and F =
0 is 5.9 x 10-6 eV.
F=1
F=0
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
21 cm radio map of the Milky Way
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Selection rules
o
Selection rules determine the allowed transitions between terms.
n = any integer
l = ±1
j = 0, ±1
f = 0, ±1
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Summary of Atomic Energy Scales
o
Gross structure:
o Covers largest interactions within the atom:
o Kinetic energy of electrons in their orbits.
o Attractive electrostatic potential between positive nucleus and negative electrons
o Repulsive electrostatic interaction between electrons in a multi-electron atom.
o Size of these interactions gives energies in the 1-10 eV range and upwards.
o Determine whether a photon is IR, visible, UV or X-ray.
o
Fine structure:
o Spectral lines often come as multiplets. E.g., H line.
=> smaller interactions within atom, called spin-orbit interaction.
o Electrons in orbit about nucleus give rise to magnetic moment
of magnitude B, which electron spin interacts with. Produces small shift in energy.
o
Hyperfine structure:
o Fine-structure lines are split into more multiplets.
o Caused by interactions between electron spin and nucleus spin.
o Nucleus produces a magnetic moment of magnitude
~B/2000 due to nuclear spin.
o E.g., 21-cm line in radio astronomy.
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