Appendix B - Accelerator magnets

Summary

In Chapters 2 and 3 the analysis of the particle motion is developed in the environment of sharply defined regions of constant dipole and quadrupole fields. Real-world magnets are more complicated.

Multipole expansion of a 2-dimensional magnetic field

First consider a purely 2-dimensional field, which is a rather reasonable approximation since the majority of accelerator magnets are long compared with their aperture. In the source-free region of the magnet gap, the field can be derived from a scalar potential φ. In the local cylindrical coordinate system for the magnet, the general Fourier expansion of this scalar potential will be,

where, A_m and B_m are constants. The zero order term is a constant and can be disregarded, since it will make no contribution to the fields derived later. Equation (1B) contains two orthogonal sets of multipoles;

• the sine terms are designated normal or right multipoles, and

• the cosine terms are designated skew multipoles.

An alternating-gradient lattice has two normal modes for particle oscillations and by using only normal lenses these modes are made horizontal and vertical. As discussed in Chapter 5 the inclusion of skew lenses in a normal lattice will cause coupling between these modes. It is clear from (1B) that the skew multipoles can be made by simply rotating the normal multipoles by π/(2m).

Equation (1B) is a natural starting point in a mathematical sense, but is slightly inconvenient for certain applications.