An electron storage ring injects a beam whose horizontal emittance is (300 nm rad)
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure
Action and Symmetry, Ch Poisson brackets, phase flow and Liouville's theorem
Problem
An electron storage ring injects a beam whose horizontal emittance is E0=3.0×10−7mrad (300 nm rad), with a waist half-width at injection of xmax,0=1.0mm. After many revolutions the ring settles into an equilibrium in which the horizontal emittance has fallen to Eeq=5.0×10−9mrad (5.0 nm rad), sixty times smaller, a figure typical of a modern synchrotron light source.
((a)) Compute the angular half-spread the beam would have, at a waist of the same size xmax,0=1.0mm, immediately after injection and once the beam has reached equilibrium.
((b)) An accelerator physicist argues: "the magnetic lattice of bending and focusing magnets in the ring is entirely magnetic, so it does exactly what any lens system does: it reshapes the beam's ellipse in phase space without changing its area. A sequence of such magnets, run for long enough, must therefore squeeze the emittance down to whatever value is wanted." Say what is wrong with this argument and name the physical process, absent from a purely magnetic lattice, that is actually responsible for the sixtyfold shrinkage.
((c)) State, in one sentence, why that process is compatible with Liouville's theorem rather than a violation of it.