General Design Equations for Isochronous Cyclotrons

In a multiparticle variable energy cyclotron there exists a unique sector shape which will provide optimum focussing and isochronism for the full range of particles to be accelerated. To generate this shape, general hard edge equations for the orbit properties are presented. The equations for the betatron frequencies hold for the case when i) the entry and exit spiral angles are unequal ii) the hill and valley fields vary with the radius and iii) for separated sectored cyclotrons also, with the valley field equal to zero. A series of mc2 type spiral sectored electron cyclotron magnets have been constructed and the hard edge equations have been evaluated against orbit integration results in the measured fields.