Cyclotron amplification of whistler waves by electron beams in an inhomogeneous magnetic field

Cyclotron wave‐particle interactions in the case of well‐organized distributions of energetic electrons under an inhomogeneous magnetic field are studied. Step‐like and δ function distributions over the field‐aligned velocity are considered. The one‐hop amplification of whistler waves is calculated by simple analytical solution and numerical computation based on strict approach. The strict consideration, taking into account third‐order expansion of the spatial dependence of the electron phase with respect to the wave, reveals sonic new important features of the one‐hop amplification Γ as a function of frequency and electron beam parameters. The main result is that Γ exhibits a quasi‐periodic structure as a function of wave frequency or characteristic electron parallel velocity, remaining always positive in the case of the step‐like distribution but being sign alternative for δ‐function. Dependence of Γ on the parameters of energetic electrons such as their total energy, characteristic parallel velocity, position of the injection point in relation to the equator, and dispersion in parallel velocity is discussed.