PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous...
-
date post
20-Dec-2015 -
Category
Documents
-
view
264 -
download
2
Transcript of PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous...
PY3P05
Lecture 12: The Zeeman EffectLecture 12: The Zeeman Effect
o Atoms in magnetic fields:
o Normal Zeeman effect
o Anomalous Zeeman effect
o Astrophysical applications
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
PY3P05
Zeeman EffectZeeman Effect
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.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Photograph taken by Zeeman
B = 0
B > 0
PY3P05
Norman Zeeman effectNorman Zeeman effect
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
hˆ L
PY3P05
Norman Zeeman effectNorman Zeeman effect
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 =
0
0
Bz
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟€
ΔE = − ˆ μ ⋅ ˆ B
€
ΔE = −μ zBz = μB Bzml
PY3P05
Norman Zeeman effect transitionsNorman Zeeman effect transitions
o But what transitions occur? Must consider selections rules for ml: Δml = 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.€
hν = hν 0 + μB Bz
hν = hν 0
hν = hν 0 − μB Bz
€
Δml = −1
Δml = 0
Δml = +1
PY3P05
Norman Zeeman effect transitionsNorman Zeeman effect transitions
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 (Δml = 0) is therefore absent and only observe Δml = ± 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.
Δml = 0 are linearly polarized || to the field.
Δml = ±1 transitions are linearly polarized at right angles to field.
PY3P05
Norman Zeeman effect transitionsNorman Zeeman effect transitions
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)
-
(Δml=-1 )
(Δml=0 )
+
(Δml=+1 )
PY3P05
Anomalous Zeeman effectAnomalous Zeeman effect
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
Anomalous Zeeman effectAnomalous Zeeman effect
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 =>
€
ΔE = −μ zBz
= −(μ zorbital + μ z
spin )Bz
= ˆ L z + gsˆ S z
μB
hBz
€
ˆ L z = ml h.
€
ˆ S z = 0
€
ˆ J z = m jh
€
ˆ μ = − | ˆ L | cosθ1
ˆ J
| ˆ J |+ 2 | ˆ S | cosθ2
ˆ J
| ˆ J |
μB
h
PY3P05
Anomalous Zeeman effectAnomalous Zeeman effect
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 | cosθ1
ˆ S ⋅ ˆ J =| S || J | cosθ2
€
ˆ μ = −ˆ L ⋅ ˆ J
| ˆ J |2+ 2
ˆ S ⋅ ˆ J
| ˆ J |2μB
hˆ 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)[ ]h2 /2
j( j +1)h2
=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
Anomalous Zeeman effectAnomalous Zeeman effect
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
hˆ J
€
ˆ μ = −g j
μB
hˆ J
€
g j =1+j( j +1) + s(s +1) − l(l +1)
2 j( j +1)
€
μz = −g jμB m j
€
ΔE = −μ zBz = g jμB Bzm j
PY3P05
Anomalous Zeeman effect spectraAnomalous Zeeman effect spectra
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.
€
Δj = 0,±1
Δm j = 0,±1
PY3P05
Sunspot magnetographySunspot magnetography
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.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
PY3P05
Sunspot magnetographySunspot magnetography
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.