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Neutrino factory and muon collider cooling lattices require both high gradient rf cavities and strong focusing solenoids. Experiments have shown that there may be serious problems operating rf in the required magnetic fields. Experimental observations using vacuum rf cavities in magnetic fields are discussed, current published models of breakdown with and without magnetic fields are briefly summarized, and some of their predictions compared with observations. A new theory of magnetic field dependent breakdown is presented. It is proposed that electrons emitted by field emission on asperities on one side of a cavity are focused by the magnetic field to the other side where they induce mechanical fatigue leading to cavity surface damage in small spots. Metal is then electrostatically drawn from the molten spots, becomes vaporized and ionized by field emission from the remaining damage, and causes breakdown. The theory is fitted to existing 805 MHz data and predictions are made for performance at 201 MHz. The model predicts breakdown gradients significantly below those specified for either the International Scoping Study neutrino factory or a muon collider. Possible solutions to these problems are discussed, including designs for magnetically insulated rf in which the cavity walls are designed to be parallel to chosen magnetic field contour lines and consequently damage from field emission is expected to be suppressed. An experimental program that could study these problems and their possible solution is outlined. We also mention the use of high pressure gas as an alternative possible solution

rf breakdown with external magnetic fields in 201 and 805 MHz cavities