arXiv subjects

A. Rokhlenko

Publications and source records attributed to A. Rokhlenko.

10 recordsLinked to original sources

arXiv2013

Time evolution of electron flow in a model diode: Non-perturbative analysis

Using a combination of Eulerian and Lagrangian variables we obtain some exact results and good approximation schemes for the time evolution of the electron flow from a no-current state to a final stationary current state in a planar one-dimensional diode. The electrons can be injected externally or generated by the cathode via field emission governed by a current-field law. The case of equipotential electrodes and fixed injection is studied along with a positive anode potential. When the current is fixed externally the approach to the stationary state goes without oscillations if the initial electron velocity is high enough and the anode can absorb the injected flow. Otherwise the accumulated space charge creates a potential barrier which reflects the flow and leads to its oscillations, but our method of analysis is invalid in such conditions. In the field emission case the flow goes to its stationary state through a train of decaying oscillations whose period is of the order of the electron transit time, in agreement with earlier studies based on perturbation techniques. Our approximate method does not permit very high cathode emissivity, though the method works when the stationary current density is only about 10% smaller than the Child-Langmuir limit.

physics.plasm-ph↗

arXiv2006

Exact Results for the Ionization of a Model Quantum System

We prove that a model atom having one bound state will be fully ionized by a time periodic potential of arbitrary strength r and frequency omega. Starting with the system in the bound state, the survival probability is for small r given by exp(-Gamma t) for times of order GAMMA^(-1)~r^(-2n) where n is the minimum number of 'photons' required for ionization (with large modifications at resonances). For late times the decay is as t^(-3) with the power law modulated by oscillations. As r increases, the time over which there is exponential decay becomes shorter and the power law behaviour starts earlier. Results are for a parametrically excited one-dimensional system with zero-range potential but comparison with analyticalworks and with experiments indicates that many features are general.

math-ph↗

arXiv2006

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{ωt}$ with $U(x)=2δ(x-a)$ or $U(x)= 2δ(x-a)-2δ(x+a)$. The particle is initially in a bound state produced by the binding potential $-2δ(x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $ω$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $ω$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,ω$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

math-ph↗

arXiv2004

Space charge limited flow of a thin electron beam confined by a strong magnetic field

An approximate analytic theory is developed and implemented numerically for calculating the space charge limited current and electric field of a thin cylindrical beam or current sheet between two wide parallel electrodes. The flow is confined by a sufficiently strong magnetic field. Assuming that the potential and current density are almost homogeneous in the direction transversal to the flow we compute the beam current and profile by a variational method. We find that the average current density scales as the reciprocal of the beam width when the latter becomes very small. The total cylindrical beam current thus decreases proportionly to its diameter while the total current of a sheet becomes almost independent of the width in this regime.

physics.plasm-ph↗

arXiv2003

Space Charge Limited 2-d Electron Flow between Two Flat Electrodes in a Strong Magnetic Field

An approximate analytic solution is constructed for the 2-d space charge limited emission by a cathode surrounded by non emitting conducting ledges of width Lambda. An essentially exact solution (via conformal mapping) of the electrostatic problem in vacuum is matched to the solution of a linearized problem in the space charge region whose boundaries are sharp due to the presence of a strong magnetic field. The current density growth in a narrow interval near the edges of the cathode depends strongly on Lambda. We obtain an empirical formula for the total current as a function of Lambda which extends to more general cathode geometries.

physics.plasm-ph↗

arXiv2000

Evolution of a model quantum system under time periodic forcing: conditions for complete ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $η(t)$. We show that for generic $η(t)$, which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as $t\to\infty$. This is irrespective of the magnitude or frequency (resonant or not) of $η(t)$. There are however exceptional, very non-generic $η(t)$, that do not lead to full ionization, which include rather simple explicit periodic functions. For these $η(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.

math-ph↗

arXiv2000

On the complete ionization of a periodically perturbed quantum system

We analyze the time evolution of a one-dimensional quantum system with zero range potential under time periodic parametric perturbation of arbitrary strength and frequency. We show that the projection of the wave function on the bound state vanishes, i.e. the system gets fully ionized, as time grows indefinitely.

math-ph↗

arXiv1998

Bounds on the Mobility of Electrons in Weakly Ionized Plasmas

We obtain exact upper and lower bounds on the steady state drift velocity, and kinetic energy of electrons, driven by an external field in a weakly ionized plasma (swarm approximation). The scattering is assumed to be elastic with simplified velocity dependence of the collision cross sections. When the field is large the bounds are close to each other and to the results obtained from the conventional approximation of the Boltzmann equation in which one keeps only the first two terms of a Legendre expansion. The bounds prove rigorously that it is possible to increase the electron mobility by the addition of suitably chosen scatterers to the system as predicted by the Druyvesteyn approximation and found in experiments.

physics.plasm-ph↗

arXiv1998

Hydrodynamical Equation for Electron Swarms

We study the long time behavior of light particles, e.g. an electron swarm in which Coulomb interactions are unimportant, subjected to an external field and elastic collisions with an inert neutral gas. The time evolution of the velocity and position distribution function is described by a linear Boltzmann equation (LBE). The small ratio of electron to neutral masses, $ε$, makes the energy transfer between them very inefficient. We show that under suitable scalings the LBE reduces, in the limit $ε\to 0$, to a formally exact equation for the speed (energy) and position distribution of the electrons which contains mixed spatial and speed derivatives. When the system is spatially homogeneous this equation reduces to and thus justifies, for $ε$ small enough, the commonly used ``two-term'' approximation.

physics.plasm-ph↗

arXiv1997

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time.

physics.plasm-ph↗