Heating of microprotrusions in accelerating structures

The thermal and field emission of electrons from protrusions on metal surfaces is a possible limiting factor on the performance and operation of high-gradient room temperature accelerator structures. We present here the results of extensive numerical simulations of electrical and thermal behavior of protrusions. We unify the thermal and field emission in the same numerical framework, describe bounds for the emission current and geometric enhancement, then we calculate the Nottingham and Joule heating terms and solve the heat equation to characterize the thermal evolution of emitters under rf electric field. Our findings suggest that heating is entirely due to the Nottingham effect. The time dependence of the rf field leads to a time dependent tip temperature with excursion that depends weakly on rf frequency. We build a phenomenological model to account for the effect of space charge and show that space charge eliminates the possibility of copper tip melting for tips with radii less than $10\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ with vacuum fields on their surface less than $12\text{ }\text{ }\mathrm{GV}/\mathrm{m}$, and for rf frequencies above 1 GHz.