PIC-simulation resolved period-doubling route to chaos in the classical pierce diode

摘要

The classical Pierce diode [J. R. Pierce, J. Appl. Phys. 15, 721 (1944)] is a one-dimensional, electrostatic plasma-filled system bounded by an emitter and a collector separated by a distance d and externally connected by a short circuit. A mono-energetic electron beam is emitted from the emitter with a constant current density j(0) = -en(0)u(0), where n(0) and u(0) are the initial electron density (corresponding to the plasma frequency omega(pe) = root n(0)e(2)/(epsilon(0)m(e)) and velocity, respectively. The immobile ions form a uniform neutralising background. In this work we use Particle-in-Cell (PIC) simulations performed with the BIT1 code [D. Tskhakaya jr. and S. Kuhn, Contrib. Plasma Phys. 42, 302 (2002)] to investigate the evolution of slightly perturbed uniform equilibria into the related non-linear attractor states for a broad range of a values, where alpha = w(pe)d/u(0) is the diode control parameter. First, to demonstrate the suitability of our code, the period-doubling route to chaos for alpha below 3 pi was re-investigated in detail. In both the linear and non-linear regimes, excellent agreement was found with the results obtained previously by Godfrey [Phys. Fluids 30, 1553 (1987)] from numerical integration of a genuine fluid model, and by Horhager and Kuhn [Phys. Fluids B 2, 2741 (1990)] from a three-harmonic fluid analysis. Having thus established that our method is perfectly adequate for these investigations, we present analogous results for linear and non-linear oscillations in alpha domains in which these phenomena have not been considered in detail before. The present kinetic approach to small-amplitude oscillations in a bounded system can be extended to more complex bounded plasma systems, e.g., for PIC simulations of turbulent fusion plasmas, in which small-amplitude oscillations are known to give rise to anomalous transport. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.