When an electron is placed in a magnetic field, it experiences a force at right-angles to the direction of the field and also at right angles to its line of motion, and this force is to be superposed on the other forces acting on the electron. The effect is most con veniently described by Larmor's theorem which asserts that the field is equivalent to a rotation. This means that the actual motion of the electron in the field is the same as the apparent motion it would have without the field, but now as seen by an observer who is himself rotating at a suitable speed about an axis in the direction of the magnetic force. The angular velocity of the Larmor rotation is eH / 2mc, where H is the strength of the field, e and m the charge and mass of the electron and c the velocity of light. Compared to ordinary standards this is a very high speed even for quite weak fields (for the earth's magnetic field is about a million rotations a second), but it is much smaller than the speed with which it is to be compared, the electron's motion in the atom.
The equivalence of magnetism and rotation is true whatever other forces may act on the electron. In our case the effect can best be seen by considering three special types of orbits ; the general motion is merely a superposition of these. The first is a motion along the line of the magnetic field, and this is evidently unaffected by the rotation. The others are circular motions, both in a plane perpendicular to the field, but in opposite directions. The Larmor rotation must be added to one of these, and sub tracted from the other. If the original frequency was zi there will now be three frequencies, p and v ±eH/4 7r mc, and we con clude that the magnetic field will split one line into three. More over each of the three will have a characteristic polarization, asso ciated with the corresponding motion of the electron. These po larizations vary according to the position of the observer, and are most easily described by saying that the light-vector (electric force) behaves like the perspective view that the observer has of the motion of the emitting electron. Thus the light of frequency corresponds to the electron vibrating along the line of the field. Viewed from anywhere it appears to describe a straight line, and so the associated light is plane-polarized. From the poles (in the direction of the magnetic field) the electron will appear motion less, and so no light is emitted in this direction. At the equator the apparent motion is a maximum and the light is polarized with vector in the same direction as the field ; for this reason the corn ponent is called parallel. Next consider one of the circular mo tions; on the same principle it emits circularly polarized light towards the poles, and plane-polarized light towards the equator where the electron's motion is seen edgewise. From the direction of its polarization at the equator this component is called per pendicular. One of the first tasks of Zeeman was to examine the light in the polar direction, for, by finding whether the bluer com ponent gives right- or left-handed circularly polarized light, it is possible to fix the sign of the Larmor rotation, and so to determine the sign of the electron's charge e. Once this is done it is most convenient to observe the effect from the equator, since in that direction all three components are plane-polarized, and this is much more convenient for investigation.
placement always bears a simple numerical ratio to that given by the simple theory. The general rules are very complicated and we may be content to describe a single example, the yellow lines, and D2, of sodium. These two lines break up into 4 and 6 components respectively, in the manner shown in figure. This is the simplest case of the anoma lous effect, and theory now shows it to be the most primitive effect of all, much more so than the normal Lorentz triplet.
The anomalous Zeeman effect is connected with another im portant phenomenon, called the Paschen-Back effect after its dis coverers. As the magnetic field is increased, the components get further apart, and for a strong enough field those of and D2 ought to overlap one another. This does not occur, but a compli cated rearrangement takes place ; some lines weaken in intensity and disappear, others melt together and finally, when the magnetic field is very strong indeed, a totally new pattern is observed in the form of a single Lorentz triplet. A trace of the original two lines remains, in that each component has a fine structure and is not a simple line. The actual transition to the Paschen-Back effect can not be observed for the D-lines, because it would need quite un attainable strengths of field, but it is safely inferred from the be haviour of other lines of the same type which are originally much closer together. Though we have only described one particular example, it is universally true that in very strong fields every mu/tip/et of a spectrum is replaced by a single Lorentz pattern. The disentangling of the very complicated patterns was made possible by the quantum theory of spectra, according to which there is a spectrum of levels underlying the spectrum of lines. Every line is given in frequency by the difference in "height" be tween two levels, and the levels have much simpler characteristics than the lines. The analysis was worked out with the help of quasi-dynamical models, and its result was to express the dis placement, polarization and intensity of every line algebraically in terms of the quantum numbers which describe the two asso ciated levels. (See QUANTUM THEORY.) From this analysis it emerged that in the atom there are two kinds of system, one of which exhibits the Larmor rotation, while the other shows a rotation just twice as great. By the interaction of the two sys tems it is possible to explain both the anomalous Zeeman effect and the Paschen-Back effect. For some time the existence of this doubled rotation was mysterious, but it was finally traced to the electron itself ; the electron in addition to its electric charge is a magnet and rotates in the magnetic field with twice the Larmor speed. Even this is not the last word, for it has been shown that it is only possible to make a picture of electrical phenomena which rigorously reconciles the quantum theory with the theory of rela tivity by endowing the electron with magnetism in just the way required for the Zeeman effect. Einstein's conception of relativity was developed to explain a totally different category of phenom ena, and it is one of the most remarkable syntheses in the history of physics that it should be possible to make it responsible for the intricacies of the Zeeman effect.
We have described the most interesting aspect of the Zeeman effect, and need only touch on a few others. There is the inverse Zeeman effect where light is absorbed by matter in a magnetic field; this follows exactly the same rules as the direct effect. Faraday's magnetic gyration (see LIGHT) is another aspect of the same thing. The theory of the effect is still very incomplete for band spectra, and indeed for some classes of line spectra. In con clusion we may refer to a more practical use to which the Zeeman effect has been put : by its means it is known that there are very powerful magnetic fields in sun-spots, and also that the sun as a whole has a magnetic field like the earth.