Scattering in a magnetic field

The fixed target program at Fermilab has come to an end. New projects are in the planning stage. Among them is a muon storage ring. Up to the present, all storage rings in high-energy physics have carried stable particles, namely the electron and proton and their antiparticles. The muon is unstable and decays with a mean lifetime of 2.0 x 10-6 sec. Two types of cooling have been used in the past. One is stochastic cooling where an electrode is used to detect the positions of the particles and send a signal to another position across the ring. Through successive applications of this technique, the phase space is ultimately greatly reduced and beams can be made to collide with a useful event rate. The second type of cooling is electron cooling. Here protons and electrons are made to travel together for a short distance. Equipartition causes transfer of transverse energy of the protons to that of the electrons. Neither of these methods is fast enough to allow acceleration of a sufficient number of muons up to maximum energy before they decay. A new method known as ionization cooling has been proposed.[1] The muons are cooled by passing them through a container of liquid hydrogen. The energy loss reduces both transverse and longitudinal momentum. The longitudinal momentum is restored with RF cavities. The net result is to maintain the longitudinal momentum while cooling the transverse momentum. To minimize the total travel distance of the muons the liquid hydrogen is placed inside the focusing solenoids. The question arises as to whether the presence of the solenoids influences the phase space occupied by the muons. After the muon scatters it has transverse momentum. In a constant longitudinal magnetic field the trajectory wraps around the field lines and coincides in momentum and position with a particle which scatters one cycle later. Here we calculate the change in emittance for both a drift space and a solenoid. We find that the presence of the solenoid does cause a reduction in phase space. Shown below are both a derivation of the behavior of the muon phase space and a plot showing the strength of the effect described