Download o  Atoms in magnetic fields: Normal Zeeman effect Anomalous Zeeman effect

o  Atoms in magnetic fields:
o  Normal Zeeman effect
o  Anomalous Zeeman effect
o  Diagnostic applications
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First reported by Zeeman in 1896. Interpreted by
Lorentz.
o 
Interaction between atoms and field can be
classified into two regimes:
B=0
o  Weak fields: Zeeman effect, either normal or
anomalous.
B>0
o  Strong fields: Paschen-Back effect.
o 
Normal Zeeman effect agrees with the classical
theory of Lorentz. Anomalous effect depends on
electron spin, and is purely quantum mechanical.
Photograph taken by Zeeman
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Observed in atoms with no spin.
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Total spin of an N-electron atom is Sˆ = " sˆi
N
i=1
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Filled shells have no net spin, so only consider valence electrons. Since electrons have spin
1/2, not possible to obtain S = 0 from atoms with odd number of valence electrons.
!
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Even number of electrons can produce S = 0 state (e.g., for two valence electrons, S = 0 or 1).
o 
All ground states of Group II (divalent atoms) have ns2 configurations => always have S = 0
as two electrons align with their spins antiparallel.
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Magnetic moment of an atom with no spin will be due entirely to orbital motion:
µˆ = "
µB ˆ
L
!
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!
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Interaction energy between magnetic moment and a uniform magnetic field is:
"E = #µˆ $ Bˆ
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Assume B is only in the z-direction:
!
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"0%
$ '
ˆ
B =$ 0 '
$ '
# Bz &
The interaction energy of the atom is therefore,
"E = #µz Bz = µB Bz ml
!
where ml is the orbital magnetic quantum number. This equation implies that B splits the
degeneracy of the ml states evenly.
!
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But what transitions occur? Must consider selections rules for ml: !ml = 0, ±1.
o 
Consider transitions between two Zeeman-split atomic levels. Allowed transition frequencies
are therefore,
h" = h" 0 + µB Bz
h" = h" 0
h" = h" 0 # µB Bz
o 
"ml = #1
"ml = 0
"ml = +1
Emitted photons also have
! a polarization, depending
!
on which transition they result from.
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Longitudinal Zeeman effect: Observing along magnetic field, photons must propagate in zdirection.
o  Light waves are transverse, and so only x and y polarizations are possible.
o  The z-component (!ml = 0) is therefore absent and only observe !ml = ± 1.
o  Termed !-components and are circularly polarized.
o 
Transverse Zeeman effect: When observed at right angles to the field, all three lines are
present.
o  !ml = 0 are linearly polarized || to the field.
o  !ml = ±1 transitions are linearly polarized at right angles to field.
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Last two columns of table below refer to the
polarizations observed in the longitudinal and
transverse directions.
o 
The direction of circular polarization in the
longitudinal observations is defined relative to B.
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Interpretation proposed by Lorentz (1896)
" (!ml=-1 )
"
(!ml=0 )
" +
(!ml=+1 )
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Discovered by Thomas Preston in Dublin in 1897.
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Occurs in atoms with non-zero spin => atoms with odd
number of electrons.
o 
In LS-coupling, the spin-orbit interaction couples the spin
and orbital angular momenta to give a total angular
momentum according to
Jˆ = Lˆ + Sˆ
o 
In an applied B-field, J precesses about B at the Larmor
frequency.
!
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L and S precess more rapidly about J to due to spin-orbit
interaction. Spin-orbit effect therefore stronger.
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Interaction energy of atom is equal to sum of interactions of spin and orbital magnetic
moments with B-field:
"E = #µz Bz
= #(µzorbital + µzspin )Bz
µ
= Lˆ z + gsSˆ z B Bz
!
where gs= 2, and the < … > is the expectation value. The normal Zeeman effect is obtained
by setting Sˆ z = 0 and Lˆ z = ml !.
!
o 
In the case of precessing atomic magnetic in figure on last slide, neither Sz nor Lz are constant.
! Jˆz = m
Only
!j ! is well defined.
o 
Must therefore project L and S onto J and project onto
! z-axis =>
Jˆ
Jˆ µB
µˆ = " | Lˆ | cos #1
+ 2 | Sˆ | cos # 2
ˆ
ˆ
|J | !
|J |
!
o 
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The angles #1 and #2 can be calculated from the scalar products of the respective vectors:
Lˆ " Jˆ =| L || J | cos #1
Sˆ " Jˆ =| S || J | cos #
2
which implies that
o 
Lˆ # Jˆ
Sˆ # Jˆ µ
µˆ = "
+ 2 2 B Jˆ
2
ˆ
|J |
| Jˆ | !
!
Now, using Sˆ = Jˆ " Lˆ implies that
(1)
Sˆ " Sˆ = ( Jˆ # Lˆ ) " ( Jˆ # Lˆ ) = Jˆ " Jˆ + Lˆ " Lˆ # 2 Lˆ " Jˆ
!
therefore Lˆ " Jˆ = (Jˆ " Jˆ + Lˆ " Lˆ # Sˆ " Sˆ ) /2
!
!
Lˆ " Jˆ
[ j( j + 1) + l(l + 1) # s(s + 1)] ! 2 /2
so that
=
!
j( j + 1)! 2
| Jˆ |2
=
o 
Similarly,
!
!
[ j( j + 1) + l(l + 1) # s(s + 1)]
2 j( j + 1)
Sˆ " Jˆ = ( Jˆ " Jˆ + Sˆ " Sˆ # Lˆ " Lˆ ) /2
and
Sˆ " Jˆ
[ j( j + 1) + s(s + 1) # l(l + 1)]
=
2 j( j + 1)
| Jˆ |2
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!
o 
We can therefore write Eqn. 1 as
# [ j( j + 1) + l(l + 1) " s(s + 1)]
[ j( j + 1) + s(s + 1) " l(l + 1)] & µB Jˆ
µˆ = "%
"2
(
2 j( j + 1)
2 j( j + 1)
$
' !
o 
This can be written in the form
µˆ = "g j
!
µB ˆ
J
!
where gJ is the Lande g-factor given by
!
o 
This implies that
g j = 1+
j( j + 1) + s(s + 1) " l(l + 1)
2 j( j + 1)
µz = "g j µB m j
and hence the interaction!energy with the B-field is
!
o 
"E = #µz Bz = g j µB Bz m j
Classical theory predicts that gj = 1. Departure from this due to spin in quantum picture.
!
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Spectra can be understood by applying the selection rules for J and mj:
"j = 0,±1
"m j = 0,±1
o 
Polarizations of the transitions follow the
! as for normal Zeeman effect.
same patterns
o 
For example, consider the Na D-lines
at right produced by 3p # 3s transition.
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Measure strength of magnetic field from spectral line shifts or polarization.
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Choose line with large Lande g-factor => sensitive to B.
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Usually use Fe I or Ni I lines.
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Measures field-strengths of ~±2000 G.
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Can measure magnetic field strength
and orientation in tokamak plasma.
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Use C2+ and O+ multiplets.
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Can measure temperature from width
of Zeeman components.
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J D Hey et al 2002 J. Phys. B
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