The Paschen-Back Limit

How does the anomalous Zeeman formula turn into the normal Zeeman formula in the Paschen-Back limit of strong magnetic fields?

The previous section puts classical physics in a rather positive light. A thoroughly classical picture appears to account for fine structure, normal Zeeman splitting, anomalous Zeeman splitting, and the passage from one to the other as the field strength increases.

On closer examination, we find cracks in the picture. Several non-classical ingredients played a crucial role:

1. The Bohr-Sommerfeld ``quantization'' prescription.
2. The selection rules.
3. The spin one-half of the electron.
4. The ``mysterious factor 2''.

The old quantum theory had problems with all of these:

1. The quantization rules apply only to ``stationary'' states, but just what constitutes a stationary state? A stationary state is not absolutely stable, since it can make transitions.
2. I've noted already the muddle over selection rules as we pass from the anomalous to the normal Zeeman effect.
3. Heisenberg and Landé had half-integer spin forced upon them by counting spectral lines. Now the quantum number $L$ assumes only integer values; Sommerfeld had a derivation of sorts for this fact (which I have not given). Why should this derivation work for $L$ but not for $S$? Neither Heisenberg nor anyone else had an answer at the time.
4. This is related to the previous point, but is a distinct issue. The ratio of magnetic moment to angular momentum depends on what kind of momentum we're dealing with: orbital or spin.

The question of ``stationary'' states comes to the fore when we look at the Paschen-Back effect. Modern progress in chaos theory has clarified the subject further. For weak fields, we have one kind of approximately stationary state (L and S precess about J); for strong fields, a different kind (L and S precess about B.) For intermediate strength fields, the motion becomes chaotic (classically speaking). The Bohr-Sommerfeld quantization procedure throws up its hands when faced with chaotic motion. Yet quantization does not cease when the motion becomes chaotic, as we can tell by gazing at spectral lines.

Quantum mechanics broke the logjam. The stimulus for the fundamental shift in viewpoint did not come from the Zeeman effect, it is true. But the new approach unlocked this and many other mysteries within a year or two.

The next few sections sketch the resolution of these problems. I will do a pretty good job for (1) and (2), an overbrief treatment of (3), but I will barely touch on (4), regretfully leaving the real explanation to the standard textbooks.

© 2001 Michael Weiss

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