The dispersion equation of the induced Smith–Purcell instability

The simplest model of a magnetized infinitely thin electron beam is considered. For the grating, where the depth of grooves is a small parameter, the dispersion equation of the induced Smith–Purcell instability was obtained. It was found that the condition of the Thompson or the Raman regimes of excitation does not depend on beam current but depends on the height of the beam above the grating surface. The growth rate of instability in both cases is proportional to the square root of the electron beam current.