Proton maser gain

It is shown that in the low beam space-charge limit the small-signal gain for a proton maser, calculated from a dispersion relation, is proportional to (${\omega }_p$/${\omega }_0$${)}^2$, and agrees to this order with that derived from orbital velocity perturbations, to order (${\mathrm{qE}}_x$/m${\omega }_0$$U_0$${)}^2$. The rf current density in the interaction space is derived; ‘‘self-bunching’’ and the underlying physics of the gain (‘‘self-catching’’) are explored and a comparison is made between the collective formulation and that of single-particle orbits.