Child Langmuir Research Articles

Two-dimensional spatially periodic electron flow in various emission regimes

We study the effects of cathode surface curvature on the space charge limited current emitted by a two-dimensional periodic array of field emitters. Each of these emitters has a shape described by a simple analytic function. Linear, quadratic, and Fowler–Nordheim current-field dependences of the cathode emissivity as well as the infinite emissivity Child–Langmuir model are considered. We develop a mathematical anzatz to capture the main features of the potential field structure of this system and supplement it with a set of correction functions with free parameters. A special least square procedure is used for an approximate solution of this nonlinear problem. We find that even a smooth curved cathode can yield significant spatial variations in the current density but for the cases considered it does not change substantially the total current (properly adjusted). When the cathode emissivity and/or the applied voltage are high enough the current density from the top of the cathode bump (where the curvature is maximal) exceeds the current density produced by a flat cathode with infinite emissivity placed at the same distance from the anode. An explanation of this effect is given. The spatial pattern of emission is determined almost solely by the cathode curvature no matter how strong the current is.

Space-charge sheath with ions accelerated into the plasma

The conventional model of near-cathode space-charge sheath with ions entering the sheath from the quasi-neutral plasma may not be applicable to discharges burning in cathode vapour, e.g. vacuum arcs, where ionization of emitted atoms may occur inside the sheath with some of the produced ions returning to the cathode and others moving into the plasma. In this connection, a simple model is considered of a sheath formed by electrons and positive ions injected into the sheath with a very low velocity and moving from the sheath into the plasma. It is shown that such a sheath is possible provided that the sheath voltage is equal to or exceeds approximately 1.256kTe/e. This limitation is due to the space charge in the sheath and is in this sense analogous to the limitation of ion current in a vacuum diode expressed by the Child–Langmuir law. The ions leave the sheath and enter the plasma with a velocity equal to or exceeding approximately 1.585uB.

Investigation of magnetically self-insulated effect in an ion diode with an explosive emission potential electrode

The results of an experimental investigation of a magnetically self-insulated effect in an ion diode in bipolar-pulse mode are presented. The investigations were accomplished at the TEMP-4M accelerator by formation of a first negative pulse (100 ns, 150–200 kV) and a second positive pulse (80 ns, 200–300 kV) [G. E. Remnev et al., Surf. Coat. Technol. 114, 206 (1999)]. Plasma behavior in the anode-cathode gap was analyzed according to the current-voltage characteristics of the diode with a time resolution of 0.5 ns. It is shown that during the discrete emissive surface mode, the magnetic field influence on plasma dynamics is slight. During the space charge limitation mode, the current-voltage characteristics of the diode are well-described by the Child–Langmuir ratio. The drift speed of electrons in the diode exceeds 80 mm/ns and the effect of magnetic insulation is insignificant. It was discovered, when plasma formation at the potential electrode is complete and up until the second positive pulse that the plasma speed is constant and equals to 1.3±0.2 cm/μs. After the voltage polarity at the potential electrode changes (second pulse), plasma breakup at the anode-cathode gap takes place. The impedance of the diode begins to increase and, when the total current is more than 30 kA, the diode impedance exceeds the calculated values by more than three times. The energy efficiency and limiting characteristics of the magnetically self-insulated diode are determined.

Space charge effects in field emission: One dimensional theory

The current associated with field emission is greatly dependent on the electric field at the emitting electrode. This field is a combination of the electric field in vacuum and the space charge created by the current. The latter becomes more important as the current density increases. Here, a study is performed using a modified classical one dimensional (1D) Child–Langmuir description that allows for exact solutions in order to characterize the contributions due to space charge. Methods to connect the 1D approach to an array of periodic three dimensional structures are considered.

On thermionic emission and the use of vacuum tubes in the advanced physics laboratory

Two methods are outlined for measuring the charge-to-mass ratio e/me of the electron using thermionic emission as exploited in vacuum tube technology. One method employs the notion of the space charge in the vacuum tube diode as described by the Child–Langmuir equation; the other method uses the electron trajectories in vacuum tube pentodes with cylindrical electrodes under conditions of orthogonally related electric and magnetic fields (the Hull magnetron method). The vacuum diode method gave e/me=1.782±0.166×10+11 C/kg (averaged over the vacuum diodes studied), and the Hull magnetron method gave e/me=1.779±0.208×10+11 C/kg (averaged over both pentodes and the anode voltages studied). These methods afford opportunities for students to determine the e/me ratio without using the Bainbridge tube method and to become familiar with phenomena not normally covered in a typical experimental methods curriculum.

Two-dimensional axisymmetric Child–Langmuir scaling law

The classical one-dimensional (1D) Child–Langmuir law was previously extended to two dimensions by numerical calculation in planar geometries. By considering an axisymmetric cylindrical system with axial emission from a circular cathode of radius r, outer drift tube radius R>r, and gap length L, we further examine the space charge limit in two dimensions. Simulations were done with no applied magnetic field as well as with a large (100 T) longitudinal magnetic field to restrict motion of particles to 1D. The ratio of the observed current density limit JCL2 to the theoretical 1D value JCL1 is found to be a monotonically decreasing function of the ratio of emission radius to gap separation r/L. This result is in agreement with the planar results, where the emission area is proportional to the cathode width W. The drift tube in axisymmetric systems is shown to have a small but measurable effect on the space charge limit. Strong beam edge effects are observed with J(r)/J(0) approaching 3.5. Two-dimensional axisymmetric electrostatic particle-in-cell simulations were used to produce these results.

Validating the collision-dominated Child–Langmuir law for a dc discharge cathode sheath in an undergraduate laboratory

In this paper, we propose a simple method of observing the collision-dominated Child–Langmuir law in the course of an undergraduate laboratory work devoted to studying the properties of gas discharges. To this end we employ the dc gas discharge whose properties are studied in sufficient detail. The undergraduate laboratory work itself is reduced to registering the voltage drop across the electrodes, the discharge current as well as the cathode sheath thickness. We can easily perform the measurements of all three quantities with sufficient accuracy in a laboratory equipped with vacuum pumps.

The dependence of vircator oscillation mode on cathode material

This paper presents the effects of cathode materials on the oscillation mode of a virtual cathode oscillator (vircator). In the case of the stainless steel cathode, an oscillation mode hopping appeared with two separate frequencies. Interestingly, the vircator using the carbon fiber cathode exhibited an almost unchanged microwave frequency throughout the microwave pulse. To understand this phenomenon, several parameters are compared, including the diode voltage, accelerating gap, emitting area, and beam uniformity. It was found that a flat-top voltage and a relatively stable gap will provide a possibility of generating a constant microwave frequency. Further, the cathode operated in a regime where the beam current was between the space-charge limited current determined by Child–Langmuir law and the bipolar flow. On the cathode surface, the electron emission is initiated from discrete plasma spots and next from a continuing area, while there is a liberation process of multilayer gases on the anode surface. The changes in the emitting area of carbon fiber cathode showed a self-quenching process, which is not observed in the case of stainless steel cathode. The two-dimensional effect of microwave frequency is introduced, and the obtained results supported the experimental observations on the oscillation mode. By examining the cross section of electron beam, the electron beam for carbon fiber cathode was significantly centralized, while the discrete beam spots appeared for stainless steel cathode. These results show that the slowed diode closure, high emission uniformity, and stable microwave frequency tend to be closely tied.

Simple solutions for relativistic generalizations of the Child–Langmuir law and the Langmuir–Blodgett law

In this paper, the Child–Langmuir law and Langmuir–Blodgett law are generalized to the relativistic regime by a simple method. Two classical laws suitable for the nonrelativistic regime are modified to simple approximate expressions applicable for calculating the space-charge-limited currents of one-dimensional steady-state planar diodes and coaxial diodes under the relativistic regime. The simple approximate expressions, extending the Child–Langmuir law and Langmuir–Blodgett law to fit the full range of voltage, have small relative errors less than 1% for one-dimensional planar diodes and less than 5% for coaxial diodes.

Shot to shot variation in perveance of the explosive emission electron beam diode

The shot to shot variation in perveance of a planar diode with explosive emission graphite cathode in a range of accelerating gaps 3–12 mm is investigated experimentally. The typical electron beam parameters were 200 kV, 12 kA, 100 ns, with a few hundreds of A/cm2 current density. The diode perveance remains less than the Child–Langmuir value, indicating that only a fraction of the cathode take part in the emission process. A simple statistical analysis of the diode perveance shows that the shot to shot variation is more pronounced for the later part of the accelerating pulse. The cathode plasma expansion velocity and the effective initial emission area have been calculated from the perveance data. It was found that the plasma expansion velocity varies from 3 to 6.5 cm/μs. The mean expansion velocity and the standard deviation increase with the increase in the accelerating gap. The initial emission areas also varies randomly on a shot to shot basis and at the beginning of the accelerating pulse only 4%–35% of the cathode area take part in the emission process. The mean initial emission area and the standard deviation also increase with the increase in the accelerating gap. Experimental result indicates that the larger gaps and lower electric fields suggest a path to more uniform emission.

Analytic model of near-field radio-frequency sheaths. I. Tenuous plasma limit

An analytic model is derived for electromagnetic radio-frequency (rf) wave propagation in a waveguide filled by a tenuous plasma with a slightly tilted equilibrium magnetic field B, i.e., by=By/B⪡1. The calculation includes the self-consistent coupling between the rf fields and the sheaths at the sheath-plasma interface and can be used to describe antenna sheath formation in the ion cyclotron range of frequencies. The sheaths are treated as thin vacuum regions separating the plasma and metal wall. It is shown that (i) the launched fast wave is coupled parasitically to the slow wave by the magnetic field structure when by≠0 and by the sheath boundary condition, (ii) the sheath voltage Vsh is dependent on the wave parity (the “antenna phasing”), and (iii) integrating the vacuum rf fields, Vvac=−∫dzE∥(vac), gives an overestimate of the sheath voltage. An expression for the self-consistent Vsh including plasma effects and satisfying the Child–Langmuir law is obtained.

Generalization of the two-dimensional Child–Langmuir law for non-zero injection velocities in a planar diode

AbstractA simple analytic expression of the two-dimensional Child–Langmuir law is derived for non-zero injection velocities and Lau's result is obtained in our model by setting the injection velocity equal to zero. The calculation results show that the modify term in our model is larger than Lau's with a non-zero electronic initial energy, and it is twice as large as Lau's when the electronic initial energy is much greater than the potential energy of the gap.

Analytical expression for sheath edge around corner cathodes

A simple analytical expression for the position of the sheath edge around a two-dimensional corner cathode with included angle θc has been discovered. This expression is valid for weakly collisional sheaths in the Child–Langmuir regime ϕc ≫ kTe/e, where −ϕc < 0 is the cathode bias and Te is the electron temperature. In polar coordinates (r, θ), the sheath edge is given by (r/s0)sin[πθ/(2π − θc)] = [π/(2π − θc)] where s0 is the planar sheath width far from the vertex of the corner. This result is verified by comparison with previous numerical solutions (Watterson P A 1989 J. Phys. D: Appl. Phys. 22 1300) for the knife edge (θc = 0) and convex square corner (θc = π/2). The observed agreement suggests that this expression gives the sheath edge for all corner angles, both concave and convex. The utility of this result is demonstrated by computing the full sheath solution for a knife-edge cathode with ϕc = 100kTe/e.

Two-dimensional expansion of finite-size barium photoplasma in an electrostatic field

Two-dimensional evolution of finite-size barium photoplasma, produced using multistep-resonant ionization is experimentally investigated in an externally applied electrostatic field. Several processes like bulk motion, ambipolar diffusion, Coulomb repulsion, Child–Langmuir flux, bounded diffusion, etc. that contribute to its expansion, have been identified. They are quantified with the help of signals recorded by Faraday cups, electrodes and plates and by two-dimensional particle-in-cell simulation. These processes are superimposed and their relative magnitudes decide the evolution of the photoions. When external field is dominant, a significant fraction of ions reach the cathode with negligible vertical spread and the plasma motion can be considered as one-dimensional. However, when plasma collective effects are dominant, then the different mechanisms become comparable and the photoplasma expands in two dimensions. The spread of photoions at different locations in parallel plate geometry is determined as a function of plasma density and compared with simulation.

The Child–Langmuir law and analytical theory of collisionless to collision-dominated sheaths

This paper is concerned with summarizing simple analytical models of space-charge sheaths and tracing their relation to the Child–Langmuir model of an ion sheath. The topics discussed include the Child–Langmuir law and model of a collisionless ion sheath, the Mott–Gurney law and model of a collision-dominated ion sheath, the Bohm model of a collisionless ion–electron sheath, the Su–Lam–Cohen model of a collision-dominated ion–electron sheath, ion sheaths with arbitrary collisionality, high-accuracy boundary conditions for the Child–Langmuir and Mott–Gurney models of an ion sheath and the mathematical sense of Child–Langmuir type models of an ion sheath from the point of view of modern theoretical physics.

Measurement of potential distributions in sheath and presheath near a mesh and metal plate

The potential distributions in the boundary regions of a mesh and solid metal plate were measured using a fine structured emissive probe and compared with theoretical models. It is shown that the sheath region near the mesh is thinner than that near the plate, owing to the fact that the mesh is partially transparent to the ions. The measured electric field at the Bohm sheath edge is in agreement with the asymptotic result in the transition region, but the field at the Child–Langmuir (CL) sheath edge is larger than the values of several models found in the literature. The potential distribution in the CL sheath region agrees with the theoretical curve in the case of the plate, but deviates from the theoretical curve in the case of the mesh.

Experimental study of space-charge-limited flows in a nanogap

An experimental investigation of space-charge-limited flow of current in a nanogap is presented. Electrodes with gap size d∼70–110nm corresponding to d∕λ0∼(1−5)×103, where λ0 is the de Broglie wavelength of the space-charge electrons are experimented. Unlike classical Child–Langmuir’s (CL) law, the current density varies as square root of applied voltage (V1∕2), when d becomes comparable to λ0. Additionally, a transition regime has been found for the 90nm gap size where the CL law appears at voltages &amp;gt;45V. At d=110nm, the system is found to exhibit purely classical behavior.

Child–Langmuir flow in a planar diode filled with charged dust impurities

The Child–Langmuir (CL) flow in a planar diode in the presence of stationary charged dust particles is studied. The limiting electron current density and other diode properties, such as the electrostatic potential, the electron flow speed, and the electron number density, are calculated analytically. A comparison of the results with the case without dust impurities reveals that the diode parameters mentioned above decrease with the increase of the dust charge density. Furthermore, it is found that the classical scaling of D−2 (the gap spacing D) for the CL current density remains exactly valid, while the scaling of V3∕2 (the applied gap voltage V) can be a good approximation for low applied gap voltage and for low dust charge density.

Electron emission from ferroelectric plasma cathodes

Recent and not so recent experimental data are analyzed to show that the reason for strong electron emission from dielectric cathodes is the incomplete discharge occurring on the dielectric surface due to the electric field there being tangentially nonzero. The places of origin of such discharges are the metal–dielectric–vacuum triple junctions (TJs). As the discharge plasma moves over the surface of the dielectric electrode, the bias current arises, and an electric microexplosion occurs at a TJ. If the number of TJs is large, as it is for a metal grid held tightly to a ferroelectric, electron currents of up to 104 A with densities of more than 102 A cm−2 can be achieved. A surface discharge is initiated by applying a triggering pulse to the metal substrate deposited beforehand onto the opposite side of the ferroelectric. If this pulse leads the accelerating voltage pulse, the electron current is many times the Child–Langmuir current. The reason for the ferroelectric effect is the large permittivity (ε > 103) of the materials used (BaTiO3, PLZT, PZT). Although these devices have come to be known as ferroelectric cathodes, we believe ferroelectric plasma cathodes would be a better term to use to emphasize the key role of plasma effects.

Intense gigawatt relativistic electron beam generation in the presence of prepulse. Part II

Intense relativistic electron beam diode has been operated without a prepulse switch. Our previously reported results demonstrated that the large pulse power systems in the presence of prepulse can deliver gigawatt power pulses into a matched load if the anode cathode gap is set larger than that estimated by the Child Langmuir relation. This article reports some more experimental results on the current and voltage characteristics, the shot to shot reproducibility of the diode in the presence of prepulse. Intense electron beam diode behavior was studied for various anode cathode gaps and voltages in presence of prepulse. It has been observed that for small gap, prepulse generated plasma completely fills the anode cathode gap and the diode behaves as plasma filled diode. At a large gap the beam parameters obtained are 420 keV, 22 kA, 100 ns. From the experimentally obtained values of perveance an upper limit was set up for the Marx voltage (or anode cathode voltage) and lower limit for the anode cathode gap in order to avoid the gap closure problem and the diode can be operated with a better shot to shot reproducibility.