Betatron oscillations

The properties of this X-ray beam are in a good agreement with the synchrotron radiation emitted by the trapped electrons undergoing betatron oscillations in an ion channel, as described by numerical simulations. The... 

A positive value of n implies a focusing effect. The value of n varies with the radius r (it is zero at the center of the magnet) and it can be used to calculate the motion of particles in the radial and axial directions at a particular radius. Ignoring the detailed discussion, particles displaced axially or radially from their equilibrium orbit will sense a force which is proportional to the displacement (like a simple harmonic oscillator) and will perform stable axial and radial oscillations, which are known as betatron oscillations. These oscillations are stable when n lies between 0 and 1. Any ion leaving the ion source in a direction not in the median plane and not on the equilibrium orbit will make axial and radial betatron oscillation. 

The radial betatron oscillation frequency can be determined by numerical calculations and it is usually close to the revolution frequency of the particle ... 

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