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, Am and Bm 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.