Radiative instability of a relativistic electron beam moving in a photonic crystal

The radiative instability of a beam moving in a photonic crystal of finite dimensions is studied. The dispersion equation is obtained. The law $\Gamma\sim \rho ^{1/\left( {s + 3} \right)}$ is shown to be valid and caused by the mixing of the electromagnetic field modes in the finite volume due to the periodic disturbance from the photonic crystal.