  A multi-laboratory collaboration is studying the feasibility of building a
muon collider, the first phase of which maybe a neutrino factory. The phase
space occupied by the muons is very large and needs to be cooled several orders
of magnitude for either machine, 100,000 to 1 million for the collider and ten
to 100 for the factory. Ionization cooling is the base line method for muon
cooling. This scheme uses hydrogen absorbers and rf re-acceleration in a long
series of magnetic focusing channels to cool the muons. At Fermilab two rf
cavity types are under study to provide the required cooling rf
re-acceleration, a 805 MHz high gradient cavity for the collider and a 201 MHz
high gradient cavity for the neutrino factory. The 805 MHz cavity currently
under going cold testing is a non-periodic pi-mode cavity with the iris
openings shaped to follow the contour of the beam. The 201 MHz cavity uses
hollow thin metal tubes over the beam aperture to terminate the field in a
pill-box type mode to increase its shunt impedance. This is possible because
muons have little interactions with thin metal membranes. Details of these
cavities and cold measurement data will be presented.
