Linear Theory of Betatron Oscillations in Sectorial Cyclotrons

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Abstract
In a sectorial cyclotron the ``force functions'' and the frequencies of the linearized normal and axial betatron oscillations are given as power series in the inverse‐square of the number of sectors. The coefficients in these series are expressed either in terms of parameters characterizing the magnetic field or in terms of those specifying the family of equilibrium orbits. The first‐order terms agree with those in existing approximate expressions. Among other features of the higher order terms, they show that it is possible to design magnetic fields with momentum compactions varying over wide ranges and frequencies of betatron oscillations remaining constant. Together with a scheme proposed earlier of transferring ions continually between circular accelerators, this opens the possibility of accelerating ions continually in several stages of constant‐frequency sectorial cyclotrons to multi‐Bev energies. This method of analysis can also be applied to the study of nonlinear betatron oscillations.

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