Non-linearities and resonances
This chapter shows how multipole magnets generate field shapes corresponding to the terms of a Taylor Series expansion about the centre line of the magnet. Sextupoles produce a parabolic field shape in the median plane, octupoles a cubic, etc. Each term in the expansion drives dangerous lines in the working diagram. An example is the four, third order, resonance lines that converge on points in the working diagram at one third integer values of Q. The effects of these resonances are derived with the help of the circle diagram introduced in Chapter 6. The effect of an n-th order resonance is to produce n islands in phase space, and beyond these, an unstable non-linear growth in amplitude that causes the beam to be lost. Multipoles can however be used to positive effect by introducing a Q that depends on amplitude, contributing to Landau damping of beam instabilities.
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