Well away from the interaction point of a hadron collider
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
Well away from the interaction point of a hadron collider, one transverse phase plane of the beam has a waist of half-width xmax=0.20mm and angular half-spread xmax′=1.0×10−5rad (orders of magnitude typical of a large collider before its final squeeze). To raise the collision rate, the final focusing magnets squeeze this beam to as small a spot as possible at the interaction point, but non-linear aberrations in those magnets limit the usable angular half-spread there to at most xmax,IP′=2.0×10−4rad.
((a)) Estimate the beam's emittance.
((b)) Estimate the smallest spot half-width achievable at the interaction point consistent with that emittance and the stated angular limit.
((c)) An engineer proposes redesigning the final magnets to tolerate an angular half-spread ten times larger. Using conservation of emittance, say by what factor the spot half-width would then shrink, and give the resulting spot half-width.