A Resonance Effect in the Synchrotron

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Abstract
The effect of the resonance occurring at n=−d(logH)/d(logr)=¾ between the second harmonic of the frequency of radial oscillations and the frequency of revolution of the particles in a synchrotron is analyzed. It is shown that this resonance will lead to an increase in the amplitude of the radial oscillations if the first Fourier coefficient of the azimuthal variation of n is too large. The effect may occur even if the value of n at the equilibrium orbit is different from ¾, because the orbit will drift back and forth periodically due to the phase oscillations and may cross the place where n=¾. A criterion is derived for the condition that the increase in the amplitude of the oscillations per unit time due to this resonance be less than the decrease in the amplitude due to the magnetic damping, i.e. that there be a net damping effect. It turns out that this condition reduces to the requirement that the first Fourier component of the azimuthal inhomogeneities of the magnetic field be less than about 10−3. If this condition is not satisfied the resonance effect will reduce the intensity of the beam unless the equilibrium value of n is sufficiently far from ¾.

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