Chaotic Dynamics and Coherent Structure in Electron Beam with Virtual Cathode in the Diode with Local Neutralization

The paper consider a complex dynamics of electron beam with virtual cathode and local neutralization of the beam charge density near anode. Different types of nonlinear behaviour, including deterministic chaos, were treated. It is shown that chaotic dynamics arises as results of spatiotemporal structures interaction. Introduction and model In the present paper we consider a electrostatic shortcircuited diode model. The immobile ion background with concentration np locates near left boundary (anode plasma). The electron beam with overcritical current injected into diode with nonpurtubated velocity v0 and charge density ρ0. In this case virtual cathode (VC) forms in the beam as a result of the electrostatic instability [1], and some beam part is reflected from VC to injected boundary. VC oscillates in the diode region. The electron beam with VC demonstrates wide diversity of nonlinear phenomenons, including chaotic behaviour [2], synchronization [3] and other. Investigation of complex dynamics attracts many researchers, that is, such behaviour is the characteristic property of the beam with VC. That counts is study the structure formation, since it is well known, that chaotic dynamics in distributed systems is connected with pattern formation (see [4] and references therein). The such simplest model of device with VC as a planar diode with overcritical current describes a different nonlinear phenomenons in the real vircator systems. Our model with local neutralisation is a simple model of vircator with injected plasma [2]. The behaviour of system is determined by the dimensionless parameter related current α = ωpL/v0, where ωp is the beam plasma frequency, L is the distance between diode planes and neutralisation parameter n = np/n0. Hence n0 = ρ0/e and value of plasma region length xp is constant (xp = 0.25L). The effect of neutralisation degree of anode plasma on VC dynamics was investigated with the aid of particle–in–cell simulation. The macroparticles in the simulation obey the non-relativistic equations of motion dx/dt = v, dv/dt = −(q/m)∂φ/∂x, where x is the position of particles, v is the velocity of of the particles, q is the charge and m is the mass of the macroparticles. The code integrates the equation of motion forward in time using a leapfrog scheme. The potential φ is computed by the Poisson’s equation in one dimension ∂φ(x) ∂x = −α(ρ(x)− ρp(x)). Hence ρ(x) is the spatial distribution of beam charge density and ρp(x) is the distribution of immobile ion background. In our case