Lecture 7: Fine and hyperfine structure of one-electron
atoms
Relativistic kinetic energy correction
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(1)
o Relativistic kinetic energy correction
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It was shown in Lecture 6 that fine-structure due to spin-orbit effects is:
Fine structure
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But, from Lectures 1-2, the velocity of the electron is
o The Lamb shift
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Therefore as the spin-orbit correction (!Eso) is of the order of "4 => !Eso ~ v4.
Hyperfine structure
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The KE used when solving the Schrodinger equation had the form
which, using Eqn 2 = > KE ~ "2.
o Nuclear moments
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If KE is expressed in its relativistic form:
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Second term is of order p4 => Relativistic KE will contribute of order "4
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Relativistic kinetic energy correction
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(2)
Relativistic kinetic energy correction
In non-relativistic Schrodinger theory, we wrote
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The total energy correction correction for spin-orbit and relativistic effects can be
obtained by combining Eqns. 1 and 3 and using the fact that j = l -1/2, l + 1/2:
where Hrel includes relativistic effects:
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Energy correction depends on n and j.
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Using
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Is of the order of 10-4 times smaller
than principle energy splitting .
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Using perturbation theory, it can be shown that
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All levels are shifted down from
Bohr energies.
where
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Including relativistic effects, H is replaced with
(3)
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Gross and fine structure of hydrogen atom
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Fine structure of hydrogen
For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude,
but much smaller than the gross structure.
Gross structure determined by En from Schrödinger equation. As En = -Z2E0/n2,
where E0 = 1/2"2mc2, we can write
Gives the energy of the gross and fine
structure of the hydrogen atom.
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Energy correction only depends on j, which is
of the order of 10-4 times smaller that the
principle energy splitting.
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All levels are shifted down from the Bohr
energies.
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For every n>1 and l, there are two states
corresponding to j = l ± 1/2.
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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.
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Using incorrect assumptions, this fine structure
was derived by Sommerfeld by modifying
Bohr theory => right results, but wrong
physics!
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Fine structure: Lamb shift
Fine structure: Lamb shift
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Spectral lines can give info on nucleus via isotope shift and hyperfine structure.
1. Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly
decay to 12S1/2 optically.
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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.
2. Cause transitions to 22P1/2 state using tunable microwaves. Transitions only occur when
microwaves tuned to transition frequency. These atoms then decay emitting H" line.
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Experiment proved that even states with the same total angular momentum J are
energetically different.
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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Fine structure: Lamb shift
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Experiment showed that the 2P1/2 state is slightly
lower than the 2S1/2 state.
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The Lamb shift is due to the quantised EM field - a
quantum electrodynamic effect.
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Using "E"t # mc "t $ h the range over which
force operates is:
!
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Hyperfine structure: Nuclear moments
2
The proton-electron force is slightly weakened
when the electron is very close to nucleus => causes
the 2S1/2 level to be higher in energy than 2P1/2 level.
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Hyperfine structure can also result from magnetic interaction between the electron’s
total angular momentum (J) and the nuclear spin (I).
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Angular momentum of electron creates a magnetic field at the nucleus which is
proportional to J.
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Interaction energy is therefore
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Magnitude is very small as nuclear dipole is ~2000 smaller than electron (µ~1/m).
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Hyperfine splitting is about three orders of magnitude smaller than splitting due to
fine structure.
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Hyperfine structure: Nuclear moments
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Hyperfine structure: Nuclear moments
Like electron, the proton has a spin angular momentum and an associated intrinsic
dipole moment
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The proton dipole moment is weaker than the electron dipole moment by M/m ~
2000 and hence the effect is small.
Total angular momentum including nuclear spin, orbital angular momentum and
electron spin is
where a is the hyperfine structure constant.
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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).
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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.
where
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The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has
spin 1/2.
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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The energy splitting of the hyperfine interaction
is given by
F=1
F=0
21 cm radio map of the Milky Way
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Gross, fine and hyperfine structure of hydrogen
Selection rules for transitions
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Summary of atomic energy scales
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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 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.
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Fine structure:
o Spectral lines often come as multiplets. E.g., H" line.
=> smaller interactions, called spin-orbit interaction.
o Relativistic effects have similar magnitude.
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Hyperfine structure:
o Fine-structure lines are split into more multiplets.
o Caused by interaction between electron spin and nucleus spin.
o E.g., 21-cm line in radio astronomy.
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Selection rules determine the allowed transitions between terms.
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