PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous...

14
PY3P05 Lecture 12: The Zeeman Effect Lecture 12: The Zeeman Effect o Atoms in magnetic fields: o Normal Zeeman effect o Anomalous Zeeman effect o Astrophysical applications QuickTime™ and a YUV420 codec decompressor are needed to see this picture.
  • 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...

Page 1: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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.

Page 2: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 3: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 4: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 5: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 6: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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.

Page 7: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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 )

Page 8: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 9: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 10: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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)

Page 11: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 12: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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

Page 13: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

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.

Page 14: PY3P05 Lecture 12: The Zeeman Effect oAtoms in magnetic fields: oNormal Zeeman effect oAnomalous Zeeman effect oAstrophysical applications.

PY3P05

Sunspot magnetographySunspot magnetography

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.