Lecture 12: Atoms in magnetic fields Zeeman Effect
o First reported by Zeeman in 1896. Interpreted by Lorentz. o Topics to be covered: o Interaction between atoms and field can be o Normal Zeeman effect classified into two regimes: B = 0
o Weak fields: Zeeman effect, either o Anomalous Zeeman effect normal or anomalous. B > 0 o Plasma physics applications o Strong fields: Paschen-Back effect.
o Normal Zeeman effect agrees with the classical theory of Lorentz. Anomalous Photograph taken by Zeeman effect depends on electron spin, and is purely quantum mechanical.
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Norman Zeeman effect Norman Zeeman effect o Observed in atoms with no spin. o Interaction energy between magnetic moment and a uniform magnetic field is: o Total spin of an N-electron atom is o Assume B is only in the z-direction: 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 The interaction energy of the atom is therefore, o Even number of electrons can produce S = 0 state (e.g., for two valence electrons, S = 0 or 1).
where ml is the orbital magnetic quantum number. This equation implies that B splits the 2 o All ground states of Group II (divalent atoms) have ns configurations => always have S = 0 degeneracy of the ml states evenly. as two electrons align with their spins antiparallel. o Magnetic moment of an atom with no spin will be due entirely to orbital motion:
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o But what transitions occur? Must consider selections rules for ml: !ml = 0, ±1. o Longitudinal Zeeman effect: Observing along magnetic field, photons must propagate in z-direction. o Consider transitions between two Zeeman-split atomic levels. Allowed transition frequencies are therefore, 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 o Emitted photons also have a polarization, depending are present. on which transition they result from.
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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Norman Zeeman effect transitions Anomalous Zeeman effect
o Discovered by Thomas Preston in Dublin in 1897. o Last two columns of table below refer to the polarizations observed in the longitudinal and o Occurs in atoms with non-zero spin => atoms with odd number of electrons. transverse directions.
o In LS-coupling, the spin-orbit interaction couples the spin and orbital angular o Direction of circular polarization in momenta to give a total angular momentum according to longitudinal case is defined relative to B. Jˆ = Lˆ + Sˆ o Interpretation proposed by Lorentz (1896) o In an applied B-field, J precesses about B at the Larmor frequency. ! " - " " + ( m =-1 ) ( m =0 ) ! l ! l (!ml=+1 ) o L and S precess more rapidly about J to due to spin-orbit interaction. Spin-orbit effect therefore stronger.
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o Interaction energy of atom is equal to sum of interactions of spin and orbital magnetic o The angles #1 and #2 can be calculated from the dot products of the respective vectors: moments with B-field:
which implies that (1)
where gs= 2, and the < … > is the expectation value. The normal Zeeman effect is obtained o Now, using Sˆ = Jˆ " Lˆ implies that by setting S ˆ = 0 and Lˆ = m . z z l h therefore o In the case of precessing atomic magnetic in figure on last slide, neither Sz nor Lz are constant. ! Only! Jˆ = m is well defined. z !j h so that o Must therefore project L and S onto J and project onto ! z-axis =>
o Similarly,
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Anomalous Zeeman effect Anomalous Zeeman effect spectra
o We can therefore write Eqn. 1 as o Spectra can be understood by applying the selection rules for j and mj:
o This can be written in the form o Polarizations of the transitions follow the
where gJ is the Lande g-factor given by same patterns as for normal Zeeman effect.
o For example, consider the Na D-lines o This implies that at right produced by 3p # 3s transition.
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.
PY3004 PY3004 Sunspot magnetography Sunspot 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.
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Tokamak plasma diagnostics o Tokamak produces a toroidal magnetic field for confining a plasma. o Leading candidate for producing stable fusion. o Can diagnose strength of magnetic field in tokomak using Zeeman effect. o The figure at bottom shows the subtraction of the two circularly polarised " components for the HeII ion (ie single ionised). o For further details, see http://www.plasma.ernet.in/~othdiag/zeeman/pram/ pram1.html
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