A periodic focusing lattice in the horizontal plane of a beamline is built from…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Linear algebra and vector spaces

The Nucleus Resolved, Ch Beams and the edges of the chart: emittance, heavy ions, high spin and the drip lines

Problem

A periodic focusing lattice in the horizontal plane of a beamline is built from alternating thin quadrupoles of focal length +f+f and −f-f, each pair separated by a field-free drift of length dd, repeated indefinitely: +f+f, drift dd, −f-f, drift dd, +f+f, ... . Treat the lenses as thin and work in the paraxial approximation. ((a)) Multiply the four matrices of one period (+f+f, drift dd, −f-f, drift dd, in that order) to find the one-period transfer matrix MM, and show that its trace is Tr M=2−d2/f2\mathrm{Tr}\,M = 2 - d^2/f^2. ((b)) A particular lattice uses f=1.25 mf = 1.25\,\mathrm{m} quadrupoles spaced by drifts of d=1.6 md = 1.6\,\mathrm{m}. Is it stable? If so, find the phase advance per cell. ((c)) For the same f=1.25 mf = 1.25\,\mathrm{m}, find the drift length dmax⁡d_{\max} at which the lattice reaches the edge of stability.

Related theory

Browse the problem library