*How do the normal and anomalous Zeeman formulas account for the details of
the Zeeman splitting?*

I omitted many transition lines from figures 1 and 2
to avoid cluttering the diagram. Nature seems to share this taste for
simplicity. Of the multitude of transition lines one might draw, most are
forbidden by *selection rules*.

For figures 1 and 2, the appropriate selection rules assert that changes by one unit, and changes by at most one unit: , . Conservation of angular momentum accounts for these rules, as hinted earlier. But before getting into that, let us explore the role of selection rules in the Zeeman effect.

As it happens, the normal and anomalous Zeeman effects call for different selection rules. Four selection rules apply to the normal Zeeman effect:

- .
- .
- .
- .

Spin has dropped out completely! has only three possible values, thus the magnetic perturbation splits each spectral line into three.

For example, consider transitions from states to states. Without the magnetic field, each state belongs to a degenerate quintuplet of states (distinguished by the five possible values of ) - where ``degenerate'', you may recall, simply means that all states in the quintuplet have the same energy. Likewise, each state belongs to a degenerate triplet. This degeneracy stems from the fact that the energy does not depend on , when there is no field.

Had we no selection rules at all, we would have 15 possible transitions from the quintuplet to the triplet. The selection rule cuts this down to 9 (we can draw three lines to each state in the triplet). Since the quintuplet and the triplet are each degenerate, is the same for all 9 quintuplet-triplet transitions. We see a single spectral line from 9 transitions.

Now turn on the magnetic field. The degeneracy is lifted - the energy levels in the quintuplet separate slightly, as do those in the triplet (see figures 2). Group the 9 quintuplet-triplet transitions into three groups of three, according to the value of . Each group has the same , and so 9 transitions give rise to three spectral lines.

For the anomalous Zeeman effect, different selection rules apply:

- .
- .

We want to plug this into formula 2 for the anomalous Zeeman
. We don't need the full complexity of formula 2;
the following is enough:

(5) |

As before, we first turn the field off, then on. Without the field, the energy of a state does not depend on , and so each state belongs to a degenerate multiplet. We label the multiplet with the quantum numbers . Without the field, all transitions from one multiplet to another have the same -- shining out one single spectral line.

Turn the field on, and this overloaded spectral line splits according to the different values of . In contrast to normal Zeeman splitting, transitions with the same generally don't have the same . Count the spectral lines, and you've counted the transitions (apart from accidental coincidences, which are rare).

For example, consider sodium, a historically important case. Sodium has
two bright yellow spectral lines, known as D-lines (D and D, close
together).^{11} In a weak magnetic field, the D line
splits in four, and the D line splits in six.

``Scarcely credible!'' wrote Lorentz of this discovery. Each D-line splits
into an *even* number of lines. It would seem that some multiplet must
contain an even number of states. But there are different 's
for each value of ! Heisenberg and Landé unflinchingly ascribed
half-integer values to .

Here are the details. The D-lines come from three multiplets: a quartet and two doublets (see figure 3). The pair of quantum numbers specify a multiplet, as can be seen from the figure- is for all eight states. The D-lines are indicated by the slanted lines. The D-line actually encompasses all transitions from the higher to the lower doublet; the D-line encompasses all transitions from the quartet to the lower doublet.

Applying the selection rule for , we find that all four conceivable D transitions are permitted; six out of the eight conceivable D transitions are permitted. Turn on the field, and these all separate. (See figure 4. To avoid cluttering the figure, the D-transitions are only partly drawn.) Hence the observed splitting.

I note finally that sodium is a ``hydrogenic'' atom. It has 11 electrons,
one more than the noble gas neon. Ten electrons form a relatively inert
``core''-- they make up two complete ``shells'', which contribute
collectively zero spin and zero orbital angular momentum (all electron
spins are paired in the core.) The remaining electron is called a *valence* electron, and is far more loosely bound. A good approximation
treats sodium as a system with a positively charged, immobile core, and a
single orbiting electron-- rather like hydrogen. The eight states we just
discussed are (to a high degree of approximation) eight different states of
the valence electron (which is why they all have ).

Where do the selection rules come from? Here is one way to view a transition:

atom
atom + photon

where ``atom
'' and ``atom'' refer of course
to the initial and final states of the atom.
, so our
rules for adding angular momenta say that
in steps of 1, or in other words, for the atom.

We must supplement the catalog of spin facts to get the rule for .
Our first ``spin fact'' told us that if we have a composite system:

then , in steps of 1. We now add a new requirment:

in steps of 1.

You may be wondering how to interpret these equations. What does it mean to say ? Modern quantum mechanics gives a clear answer: tensor product. (So would be a better notation than .) But a muddled view hews closer to history. The old quantum theory did offer derivations of the selection rules, via a strange brew of Maxwell's equations and other ingredients. On the other hand, the Paschen-Back effect presented a severe conceptual stumbling block. How does one set of selection rules fade away, and the other take over? Heisenberg discarded the derivations when it suited his purpose, a matter of some controversy at the time (see Cassidy's biography). So I leave a coherent discussion to the textbooks, and continue with cruder arguments.

To get the selection rules for and , we shift gears, and treat light as an electromagnetic wave, not photons. Also, we consider the reverse process, where an atomic state absorbs energy from incident light. An electron is so small (point-like, so far as we know today) that it feels at any moment a uniform electric field, and a uniform magnetic field. Classical physics says that magnetic effects are much smaller than electric effects, so we will ignore the former. (In modern parlance, we are ``deriving'' the selection rule for electric dipole transitions, and ignoring magnetic dipole transitions.)

The uniform electric field can shove the electron this way or that, but
cannot flip it over. So **S** is unaffected, and hence
. On the other hand, if the wavelength of light is short enough,
the electric field will be shoving the electron one way in one part of its
orbit, and another way for another part, and so may change its orbital
angular momentum **L**.

. We've just seen that **S** doesn't change, so
**L** and **J** must change the same way. So the selection rules for
and must be the same as those for and .

© 2001 Michael Weiss