Muon cooling channels

A procedure uses the equations that govern ionization cooling, and leads to the most important parameters of a muon cooling channel that achieves assumed performance parameters. First, purely transverse cooling is considered, followed by both transverse and longitudinal cooling in quadrupole and solenoid channels. Similarities and differences in the results are discussed in detail, and a common notation is developed. Procedure and notation are applied to a few published cooling channels. The parameters of the cooling channels are derived step by step, starting from assumed values of the initial, final and equilibrium emittances, both transverse and longitudinal, the length of the cooling channel, and the material properties of the absorber. The results obtained include cooling lengths and partition numbers, amplitude functions and limits on the dispersion at the absorber, length, aperture and spacing of the absorber, parameters of the RF system that achieve the longitudinal amplitude function and bucket area needed. Finally, I compute the merit factor that describes the enhancement of the density in 6D phase space. The consequences of changes in the input parameters are discussed. The lattice parameters needed to achieve the assumed performance are summarized. The design proper of such a lattice, i.e. finding the precise arrangement of magnets, RF cavities, absorbers, etc., which has these properties is well beyond the scope of this document