We demonstrate with numerical WKB solutions that stationary density profiles can exist in a crossed-field amplifier. These are density profiles that can exist in equilibrium with a high-frequency (RF) wave propagating in the slow-wave structure. These density profiles are very different from the usual Brillouin profiles. They have a finite and negative density gradient, and extend from the cathode to the anode. They carry a DC current proportional to the RF power in the slow-wave structure. The source of the DC current is a wave–particle resonance that occurs toward the top of the sheath. This resonance drives any Brillouin electron flow unstable. The electrons then redistribute themselves into a new flow – one that can be stationary in the presence of the wave–particle resonance. The new flow has a spoke-like structure, where the spokes carry the current to the anode. Various plots are given, including the DC voltage operating range, DC current flow, phase shifts, and density profiles of the total electron distribution, for various ambient magnetic fields and DC voltages. Numerical values agree reasonably well with the modelled experimental device.