Alternating-gradient focusing
Such small values of $n$ give rather “weak” focusing. It is clear that much more effective radial focusing would be given by a large positive gradient ($n\gg1$), but then the vertical forces would be strongly defocusing. Similarly, large negative slopes ($n\ll-1$) would give stronger vertical forces but would cause radial defocusing. It was realized about $10$ years ago, however, that a force that alternates between strong focusing and strong defocusing can still have a net focusing force. To explain how alternating-gradient focusing works, we will first describe the operation of a quadrupole lens, which is based on the same principle. Imagine that a uniform negative magnetic field is added to the field of Fig. 29–14, with the strength adjusted to make zero field at the orbit. The resulting field—for small displacements from the neutral point—would be like the field shown in Fig. 29–15. Such a four-pole magnet is called a “quadrupole lens.” A positive particle that enters (from the reader) to the right or left of the center is pushed back toward the center. If the particle enters above or below, it is pushed away from the center. This is a horizontal focusing lens. If the horizontal gradient is reversed—as can be done by reversing all the polarities—the signs of all the forces are reversed and we have a vertical focusing lens, as in Fig. 29–16. For such lenses, the field strength—and therefore the focusing forces—increase linearly with the distance of the lens from the axis. Now imagine that two such lenses are placed in series. If a particle enters with some horizontal displacement from the axis, as shown in Fig. 29–17(a), it will be deflected toward the axis in the first lens. When it arrives at the second lens it is closer to the axis, so the force outward is less and the outward deflection is less. There is a net bending toward the axis; the average effect is horizontally focusing. On the other hand, if we look at a particle which enters off the axis in the vertical direction, the path will be as shown in Fig. 29–17(b). The particle is first deflected away from the axis, but then it arrives at the second lens with a larger displacement, feels a stronger force, and so is bent toward the axis. Again the net effect is focusing. Thus a pair of quadrupole lenses acts independently for horizontal and vertical motion—very much like an optical lens. Quadrupole lenses are used to form and control beams of particles in much the same way that optical lenses are used for light beams.  We should point out that an alternating-gradient system does not always produce focusing. If the gradients are too large (in relation to the particle momentum or to the spacing between the lenses), the net effect can be a defocusing one. You can see how that could happen if you imagine that the spacing between the two lenses of Fig. 29–17 were increased, say, by a factor of three or four. Let’s return now to the synchrotron guide magnet. We can consider that it consists of an alternating sequence of “positive” and “negative” lenses with a superimposed uniform field. The uniform field serves to bend the particles, on the average, in a horizontal circle (with no effect on the vertical motion), and the alternating lenses act on any particles that might tend to go astray—pushing them always toward the central orbit (on the average).  There is a nice mechanical analog which demonstrates that a force which alternates between a “focusing” force and a “defocusing” force can have a net “focusing” effect. Imagine a mechanical “pendulum” which consists of a solid rod with a weight on the end, suspended from a pivot which is arranged to be moved rapidly up and down by a motor driven crank. Such a pendulum has two equilibrium positions. Besides the normal, downward-hanging position, the pendulum is also in equilibrium “hanging upward”—with its “bob” above the pivot! Such a pendulum is drawn in Fig. 29–18. By the following argument you can see that the vertical pivot motion is equivalent to an alternating focusing force. When the pivot is accelerated downward, the “bob” tends to move inward, as indicated in Fig. 29–19. When the pivot is accelerated upward, the effect is reversed. The force restoring the “bob” toward the axis alternates, but the average effect is a force toward the axis. So the pendulum will swing back and forth about a neutral position which is just opposite the normal one. There is, of course, a much easier way of keeping a pendulum upside down, and that is by balancing it on your finger! But try to balance two independent sticks on the same finger! Or one stick with your eyes closed! Balancing involves making a correction for what is going wrong. And this is not possible, in general, if there are several things going wrong at once. In a synchrotron there are billions of particles going around together, each one of which may start out with a different “error.” The kind of focusing we have been describing works on them all. 