On the construction of core-less coils for FFAG accelerators

A calculation method for the determination of the core-less coil of an accelerator having a constant field and radial sectors is proposed. In the free space without current, reserved for the vacuum chamber, the Laplace equation ΔA = 0 was solved, where A is the potential vector (defined in terms of the magnetic induction B as B = rot A). In the coil itself on the other hand, the equation that has to be resolved is that of Maxwell, rot(rot) A = μ0j, where j is the current density that is considered to be the unknown of the problem. The supplementary condition is that the magnetic induction becomes zero in the same way outside the coil and outside the effective vacuum chamber volume. Our calculation consists in the determination of A from A using a relationship of the form F (x,y,z,λ).A, where F is either a squared diagonal matrix of the 3. order, or a scalar function, as in the particular case studied; λ is a parameter or a set of parameters linked to the geometry of the coil. A coil having a simple shape was studied, and the lines of current were traced. (authors)