A kinetic theory of the extraordinary-mode perturbations in cylindrical relativistic magnetrons

Stability properties of the extraordinary-mode perturbations in a relativistic electron flow generated inside a cylindrical magnetron are investigated within the framework of the linearized Vlasov--Maxwell equations. The stability analysis is carried out under the assumptions that the layer is thin and relatively tenuous, and that the phase velocity of the perturbed waves is very close to the mean drift velocity of the layer. The perturbed distribution function of the extraordinary-mode perturbations is calculated by integrating along the electron orbit. The eigenvalue equation is derived and solved separately in the vacuum and layer regions by properly matching boundary conditions at the inter-regional surface. Solution in the vacuum region includes the resonator and cylindrical curvature effects. A closed algebraic dispersion relation is obtained for the synchronous modes, whose phase velocity is very close to the mean drift velocity of the layer. Analytical investigation of the dispersion relation is carried out for the resonance modes, whose eigenfrequency is very close to the vacuum modes in the magnetron circuit. The necessary and sufficient condition for instability is obtained for the resonance modes. It is shown that typical growth rates of the instability are a few percent of the electron cyclotron frequency