Weakly nonlinear steady-state oscillations in the Pierce diode

Abstract

An analytic theory of weakly nonlinear oscillations in the uniform Pierce diode [J. Appl. Phys. 15, 721 (1944)] with nontrivial external-circuit elements is presented. The theory is applicable to situations where linear stability analysis predicts just one unstable eigenmode which is oscillatory and whose growth rate is ‘‘sufficiently’’ small. All oscillating quantities are expanded in time Fourier series with fundamental frequency ω, and the n+1st harmonic is assumed to be one order smaller than the nth one. It is shown that determination of the nonlinear amplitudes and fundamental frequency requires calculation of all quantities up to at least the third order, and all spatial profiles and current amplitudes, as well as the nonlinear oscillation frequency, are calculated explicitly for period-one oscillations in the ‘‘three-harmonic approximation.’’ Representative numerical results are presented for the special cases when the external circuit is (a) short, (b) purely inductive, (c) purely resistive, and (d) purely capacitive. These are spot-checked against particle simulations run with PDW1 [J. Comput. Phys. 80, 253 (1989)] and, in the short-circuit case, are also compared with the genuinely numerical results of Godfrey [Phys. Fluids 30, 1553 (1987)]. In all cases considered, the fundamental frequency from the three-harmonic approximation is found to be in excellent agreement with the peak-to-peak frequency from the simulations. In the short-, resistive-, and capacitive-circuit cases, the three-harmonic amplitudes agree well with the simulation results only for ac current amplitudes well below 10% of the dc current. For large-amplitude oscillations found in the inductive-circuit case, however, the three-harmonic amplitudes are correct even for ac amplitudes many times larger than the dc current. To the authors’ knowledge, this is the first analytic theory describing self-consistent nonlinear steady-state oscillations in a bounded-plasma system.