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The Zeeman Effect

When considering the Zeeman effect, it is easiest first to consider the hydrogen atom without hyperfine structure. Then tex2html_wrap_inline1524 is a good quantum number, and the atom has a 2j+1 degeneracy associated with the different possible values of tex2html_wrap_inline1524 . In the presence of an external magnetic field, these different states will have different energies due to having different orientations of the magnetic dipoles in the external field. The splitting of these energy levels is called the Zeeman effect.

Figure 8 illustrates the geometry of the Zeeman effect.

   figure723
Figure 8: Geometry of the Zeeman effect. On the left, the total dipole moment tex2html_wrap_inline1826 precesses around the total angular momentum tex2html_wrap_inline1498 . On the right, tex2html_wrap_inline1498 precesses much more slowly about the magnetic field.

The total magnetic dipole moment of the electron is

equation730

where tex2html_wrap_inline1470 and tex2html_wrap_inline1834 have been used. Because of the difference in the orbital and spin gyromagnetic ratios of the electron, this is not in general parallel to

equation329

So, as tex2html_wrap_inline1438 and tex2html_wrap_inline1440 precess about tex2html_wrap_inline1498 , the total dipole moment tex2html_wrap_inline1826 also precesses about tex2html_wrap_inline1498 . Assuming the external field to be in the z direction, this field causes tex2html_wrap_inline1498 to precess about the z-axis. Typical internal magnetic fields in the hydrogen atom can be shown to be of the order 1 Tesla. If the external field is much weaker than 1 Tesla, which it is for almost all practical purposes, then the precession of tex2html_wrap_inline1498 around the z-axis will take place much more slowly than the precession of tex2html_wrap_inline1826 around tex2html_wrap_inline1498 . The Hamiltonian of the Zeeman effect is

equation753

where tex2html_wrap_inline1860 is the projection of the dipole moment onto the direction of the field, the z-axis. Because of the difference in the precession rates, it is reasonable to evaluate tex2html_wrap_inline1864 by first evaluating the projection of tex2html_wrap_inline1826 onto tex2html_wrap_inline1498 , called tex2html_wrap_inline1870 , and then evaluating the projection of this onto tex2html_wrap_inline1756 , thus giving some average projection of tex2html_wrap_inline1826 onto tex2html_wrap_inline1756 . First, the projection of tex2html_wrap_inline1826 onto tex2html_wrap_inline1498 is

equation764

Then

equation776

Evaluating the dot product using again that tex2html_wrap_inline1882 , this becomes

equation792

So when first order perturbation theory is applied, the energy shift is

  equation798

where

equation801

is called the Landé g factor for the particular state being considered. Note that if s=0, then j=l so g=1, and if l=0, j=s so g=2. The Landé g factor thus gives some effective gyromagnetic ratio for the electron when the total dipole moment is partially from orbital angular momentum and partially from spin. From equation 97, it can be seen that the energy shift caused by the Zeeman effect is linear in B and tex2html_wrap_inline1524 , so for a set of states with particular values of n, l, and j, the individual states with different tex2html_wrap_inline1524 will be equally spaced in energy, separated by tex2html_wrap_inline1912 . However, the spacing will in general be different for a set of states with different n, l, and j due to the difference in the Landé g factor.

Including hyperfine structure with the Zeeman effect is more difficult, since the field associated with the proton magnetic dipole moment is weak, and hence it does not take a particularly strong external field to make the Zeeman effect comparable in magnitude to the strength of the hyperfine interactions. The approximation of small external field is thus not practical when discussing the Zeeman splitting of hyperfine structure. However, it can be treated, and the result for the most important case of the Zeeman splitting of the hyperfine levels in the ground state of hydrogengif is shown in figure 9. The degeneracy of the triplet state is lifted, the three states of tex2html_wrap_inline1922 having different energies in the external field. Notice how the splitting is linear for small external field, but then deviates as the field gets larger. The ``21 cm'' transitions shown on the right will have slightly different energies, and measuring the amount of this splitting is a good tool for radio astronomers to measure magnetic fields in the interstellar medium.

   figure811
Figure 9: On left, Zeeman splitting of the hyperfine levels in the ground state tex2html_wrap_inline1218 of hydrogen. On right, some possible transitions between these states.


next up previous
Next: Conclusions Up: Smaller Effects Previous: Hyperfine Structure

Randy Telfer
Mon Oct 26 15:21:07 EST 1998