Quadrupoles

I mentioned last time that weak focusing is all well and good, but that in many cases, it just doesn't cut it. In such situations, experimentalists go for strong focusing, which involves electric and/or magnetic fields that are not radially symmetric, so that a particle traveling along its trajectory will see a different field as it goes along. In particular, quadrupole magnets or electrostatic quadrupoles can serve to focus a beam in one direction while defocusing it in the other. So for instance, one quadrupole magnet might focus the beam into a thin horizontal strip (vertical focusing, horizontal defocusing), and then another immediately afterwards could do the opposite. It turns out that such a setup can have a net focusing effect in both the horizontal and vertical orientations. This, like weak focusing, allows for thinner, higher-flux beams, critical for colliders and target experiments.

How do these things work? As I understand it, there's this mathematical approach to magnetic fields called the multipole expansion, in which a simple permanent magnet generates primarily a second-order (dipole) term. Other, higher-order terms, tend to be smaller than the low-order ones, especially at larger distances, so they can often be ignored. In a quadrupole, though, four magnetic dipoles (either permanent magnets or electromagnets) are positioned in such a way as to cancel the dipole moment, leaving only the quadrupole moment. This generates an interesting-looking magnetic field that is the source of the curvy bits in the Fermilab logo.

Logo courtesy of fnal.gov (upper left hand corner, when I
pulled it off). The curvy bits represent the quadrupoles in
the various particle accelerators on site, while the straight
lines represent the dipoles used to bend the beams.