The use of toroidal boundary conditions in the program POISSON

The authors consider the use of toroidal coordinates in constructing the boundary to a relaxation mesh for use in analyzing the magnetic fields of circular devices (such as accelerators and spectrometers), and in formulating the boundary conditions that then may be usefully imposed at such boundaries. The procedure adopted makes use internally of the characteristic form of the vector-potential function in a source-free region, when expressed in toroidal coordinates. The relevant properties of associated Legendre functions of half-integral degree are used in this connection and their introduction into the program POISSON is outlined. Results of some test cases are included to illustrate the application of this technique for configurations with median-plane symmetry.