PY3P05 o Atoms in magnetic fields: o Normal Zeeman effect o Anomalous Zeeman effect o Diagnostic applications PY3P05 o First reported by Zeeman in 1896. Interpreted by Lorentz. o Interaction between atoms and field can be classified into two regimes: o Weak fields: Zeeman effect, either normal or anomalous. 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 B = 0 B > 0
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o Atoms in magnetic fields: o Normal Zeeman effect o ... · PY3P05 o Observed in atoms with no spin. o Total spin of an N-electron atom is o Filled shells have no net spin, so only
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PY3P05
o Atoms in magnetic fields:
o Normal Zeeman effect
o Anomalous Zeeman effect
o Diagnostic applications
PY3P05
o First reported by Zeeman in 1896. Interpreted by Lorentz.
o Interaction between atoms and field can be classified into two regimes:
o Weak fields: Zeeman effect, either normal or anomalous.
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
B = 0
B > 0
PY3P05
o Observed in atoms with no spin.
o Total spin of an N-electron atom is
o 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.
o 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.
o Magnetic moment of an atom with no spin will be due entirely to orbital motion:
!
ˆ S = ˆ s ii=1
N
"
!
ˆ µ = "µB
!ˆ L
PY3P05
o Interaction energy between magnetic moment and a uniform magnetic field is:
o Assume B is only in the z-direction:
o The interaction energy of the atom is therefore,
where ml is the orbital magnetic quantum number. This equation implies that B splits the degeneracy of the ml states evenly.
!
ˆ B =00Bz
"
#
$ $ $
%
&
' ' ' !
"E = # ˆ µ $ ˆ B
!
"E = #µzBz = µBBzml
PY3P05
o 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,
o Emitted photons also have a polarization, depending on which transition they result from. !
h" = h" 0 + µBBz
h" = h" 0h" = h" 0 #µBBz
!
"ml = #1"ml = 0"ml = +1
PY3P05
o Longitudinal Zeeman effect: Observing along magnetic field, photons must propagate in z-direction.
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.
PY3P05
o 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.
o Interpretation proposed by Lorentz (1896)
" - (!ml=-1 )
"
(!ml=0 ) " +
(!ml=+1 )
PY3P05
o Discovered by Thomas Preston in Dublin in 1897.
o 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
o In an applied B-field, J precesses about B at the Larmor frequency.
o L and S precess more rapidly about J to due to spin-orbit interaction. Spin-orbit effect therefore stronger.
!
ˆ J = ˆ L + ˆ S
PY3P05
o Interaction energy of atom is equal to sum of interactions of spin and orbital magnetic moments with B-field:
where gs= 2, and the < … > is the expectation value. The normal Zeeman effect is obtained by setting and
o In the case of precessing atomic magnetic in figure on last slide, neither Sz nor Lz are constant. Only is well defined.
o Must therefore project L and S onto J and project onto z-axis =>
!
"E = #µzBz
= #(µzorbital + µz
spin )Bz
= ˆ L z + gsˆ S z
µB
!Bz
!
ˆ L z = ml !.
!
ˆ S z = 0
!
ˆ J z = m j!
!
ˆ µ = " | ˆ L | cos#1
ˆ J | ˆ J |
+ 2 | ˆ S | cos#2
ˆ J | ˆ J |
µB
!
PY3P05
o The angles #1 and #2 can be calculated from the scalar products of the respective vectors:
which implies that (1)
o Now, using implies that
therefore
so that
o Similarly, and
!
ˆ L " ˆ J =| L || J | cos#1
ˆ S " ˆ J =| S || J | cos#2
!
ˆ µ = "ˆ L # ˆ J | ˆ J |2
+ 2ˆ S # ˆ J | ˆ J |2
µB
!ˆ J
!
ˆ S = ˆ J " ˆ L
!
ˆ S " ˆ S = ( ˆ J # ˆ L ) " ( ˆ J # ˆ L ) = ˆ J " ˆ J + ˆ L " ˆ L # 2 ˆ L " ˆ J
!
ˆ L " ˆ J = ( ˆ J " ˆ J + ˆ L " ˆ L # ˆ S " ˆ S ) /2
!
ˆ L " ˆ J | ˆ J |2
=j( j +1) + l(l +1) # s(s +1)[ ]!2 /2
j( j +1)!2
=j( j +1) + l(l +1) # s(s +1)[ ]
2 j( j +1)
!
ˆ S " ˆ J = ( ˆ J " ˆ J + ˆ S " ˆ S # ˆ L " ˆ L ) /2
!
ˆ S " ˆ J | ˆ J |2
=j( j +1) + s(s +1) # l(l +1)[ ]
2 j( j +1)
PY3P05
o We can therefore write Eqn. 1 as
o This can be written in the form
where gJ is the Lande g-factor given by
o This implies that
and hence the interaction energy with the B-field is
o Classical theory predicts that gj = 1. Departure from this due to spin in quantum picture.
!
ˆ µ = "j( j +1) + l(l +1) " s(s +1)[ ]
2 j( j +1)" 2
j( j +1) + s(s +1) " l(l +1)[ ]2 j( j +1)
#
$ %
&
' (
µB
!ˆ J
!
ˆ µ = "g jµB
!ˆ J
!
g j =1+j( j +1) + s(s+1) " l(l +1)
2 j( j +1)
!
µz = "g jµBm j
!
"E = #µzBz = g jµBBzm j
PY3P05
o Spectra can be understood by applying the selection rules for J and mj:
o Polarizations of the transitions follow the same patterns as for normal Zeeman effect.
o For example, consider the Na D-lines at right produced by 3p # 3s transition.
!
"j = 0,±1"m j = 0,±1
PY3P05
o Measure strength of magnetic field from spectral line shifts or polarization.
o Choose line with large Lande g-factor => sensitive to B.
o Usually use Fe I or Ni I lines.
o Measures field-strengths of ~±2000 G.
PY3P05
o Can measure magnetic field strength and orientation in tokamak plasma.
o Use C2+ and O+ multiplets.
o Can measure temperature from width of Zeeman components.