Langmuir Limit Research Articles

A variational method for the sheath potential of hypersonic leading edges with space-charge limitations

Electron transpiration cooling for the leading edges (LE) of hypersonic aircraft utilizes thermionic emission; however, space-charge effects limit the electron emission rate, potentially diminishing the efficiency of this cooling mechanism. We develop a variational weak form of the Poisson equation that describes the sheath potential and then numerically solve it using the finite element method. This formulation has two main benefits: (1) the space-charge limit condition can be incorporated as a constraint and (2) it allows for the analysis of three-dimensional geometries with complex boundary conditions. We demonstrate that the current emitted from the surface of an LE is generally a small fraction of the Child–Langmuir limit due to space charge. We then propose several methods to enhance the emitted current from the surface and to boost the cooling effect of thermionic emission. These include increasing the plasma density, applying a negative surface potential, and using fringe fields under suitable geometric conditions. For a LaB6 emitting LE, the total emitted current is shown to be minimal and independent of the temperature of a surface with floating potential. However, when a negative potential is applied and the surface is heated, the emitted current follows the Richardson–Dushman relationship up to a critical temperature, beyond which it remains constant. At an applied surface potential of −5 V, the critical temperature is around 1700 K.

Brightness calculation formula for Kohler illumination beam

A two-lens optical system entails two beams: a crossover beam and a Kohler illumination beam. We observed that upon varying the excitation corresponding to the first lens from large to small values, the crossover beam changed to a Kohler illumination beam with increased brightness. We derived new calculation formulas for the Kohler illumination beam. Brightness B and beam current Ib can be expressed as B = Bco (ϕco/ϕ)2 and Ib = Bco (παϕco)2/4, where Bco, ϕco, ϕ, and α denote the crossover beam brightness, crossover beam size, Kohler illumination beam size, and beam semi-angle, respectively; additionally, ϕco and ϕ are obtained at the same first lens excitation. These two equations were experimentally validated. We obtained a brightness of 2.05 × 108 A/cm2 sr using a beam energy and an emission current of 20 keV and 54 µA, respectively. Notably, this value surpasses the Langmuir limit of 4.11 × 105 A/cm2 sr by 499 times.

Brightness as a function of lens excitation

The Langmuir limit, defined by B. D. Langmuir, was derived from Lambert’s law. For convex cathode or curved trajectories, beams with brightness higher than the Langmuir limit are obtainable. Measured brightness varies dramatically when we vary the excitation in the first lens of a system consisting of an electron gun and two magnetic lenses. The ratios of maximum to minimum brightness were measured as 717, 403, and 310 for beam energies of 10, 20, and 30 keV, respectively. These results indicate that the brightness was not conserved when lens excitation was varied. The maximum brightness for beam energies of 10, 20, and 30 keV were observed to be, respectively, 143, 208, and 92.1 times higher than the Langmuir limit.

A brightness exceeding simulated Langmuir limit

When an excitation of the first lens determines a beam is parallel beam, a brightness that is 100 times higher than Langmuir limit is measured experimentally, where Langmuir limits are estimated using a simulated axial cathode current density which is simulated based on a measured emission current. The measured brightness is comparable to Langmuir limit, when the lens excitation is such that an image position is slightly shorter than a lens position. Previously measured values of brightness for cathode apical radii of curvature 20, 60, 120, 240, and 480 μm were 8.7, 5.3, 3.3, 2.4, and 3.9 times higher than their corresponding Langmuir limits, respectively, in this experiment, the lens excitation was such that the lens and the image positions were 180 mm and 400 mm, respectively. From these measured brightness for three different lens excitation conditions, it is concluded that the brightness depends on the first lens excitation. For the electron gun operated in a space charge limited condition, some of the electrons emitted from the cathode are returned to the cathode without having crossed a virtual cathode. Therefore, method that assumes a Langmuir limit defining method using a Maxwellian distribution of electron velocities may need to be revised. For the condition in which the values of the exceeding the Langmuir limit are measured, the simulated trajectories of electrons that are emitted from the cathode do not cross the optical axis at the crossover, thus the law of sines may not be valid for high brightness electron beam systems.

An upper bound to time-averaged space-charge limited diode currents

The Child–Langmuir law limits the steady-state current density across a one-dimensional planar diode. While it is known that the peak current density can surpass this limit when the boundary conditions vary in time, it remains an open question of whether the average current can violate the Child–Langmuir limit under time-dependent conditions. For the case where the applied voltage is constant but the electric field at the cathode is allowed to vary in time, one-dimensional particle-in-cell simulations suggest that such a violation is impossible. Although a formal proof is not given, an upper bound on the time-averaged current density is offered.

Maximizing ion current by space-charge neutralization using negative ions and dust particles

Ion current extracted from an ion source (ion thruster) can be increased above the Child–Langmuir limit if the ion space charge is neutralized. Similarly, the limiting kinetic energy density of the plasma flow in a Hall thruster might be exceeded if additional mechanisms of space-charge neutralization are introduced. Space-charge neutralization with high-mass negative ions or negatively charged dust particles seems, in principle, promising for the development of a high current or high energy density source of positive light ions. Several space-charge neutralization schemes that employ heavy negatively charged particles are considered. It is shown that the proposed neutralization schemes can lead, at best, only to a moderate but nonetheless possibly important increase of the ion current in the ion thruster and the thrust density in the Hall thruster.

A simple physical derivation of Child–Langmuir space-charge-limited emission using vacuum capacitance

The fundamental Child–Langmuir limit on the maximum current density in a vacuum between two infinite parallel electrodes is one of the most well known and often applied rules of plasma physics. We develop a simple model using vacuum capacitance, conservation of energy, and conservation of charge to derive the Child–Langmuir space-charge-limited emission. This capacitive model provides physical insight into the origins of the well known (voltage)3/2/(gap distance)2 scaling of the classical current density and does not require the solution of the nonlinear differential equation normally associated with the Child–Langmuir formulation. In addition, the full spacecharge-limited solution is reproduced without imposing the condition that the electric field be driven to zero at the cathode surface.

Double sheath formed between two electronegative plasmas

The double sheath formed by plasmas containing thermal electrons, negative ions and positive ions is investigated. In the double sheath, the positive ion flux at the sheath edge forms a higher potential plasma and the fluxes of negatively charged particles from the low potential plasma form a counter-stream. The densities of positive ions within both plasmas have a finite ratio. Depending on the potential difference between both plasmas three cases of double sheaths can be distinguished, which we call weak, medium and strong type. The maximum flux density to be extracted and its relation to the Langmuir limit are given. Some results for current densities, potential and the density ratio are presented. The results will be useful to design negative ion sources from which negative ion beams can be extracted. The double sheath in electronegative plasmas could be a mechanism of natural potential formation in the D-layer where discharges do sometimes occur. Some instabilities are also discussed.

Space-charge-limited current in electron diodes under the influence of collisions

An analytical theory is given for the space-charge-limited (SCL) current of an electron diode in the presence of collisions. It extends the improved SCL current formula, which already takes into account the effect of nonzero electron injection velocities v0, by treating collisions within a simple model, with a friction parameter ν. The general current–voltage characteristic is obtained numerically. Within the Child–Langmuir limit v0→0, however, our model not only recovers analytically both the ballistic Child–Langmuir law (as ν≪1) and the nonballistic drift Mott–Gurney law (as ν≫1), but also provides SCL for any ν. Based on the present analysis, the nonreflective nature of SCL is emphasized. Extensive use is made of the Lagrangian description of the cold-fluid equations and from proper knowledge about the correct definition of the SCL.

Theoretical study of a diode with dielectric-gridded cathode

We investigate the I–V characteristics of a vacuum diode with a gridded cathode. The grid is located on top of a dielectric material that on its back side is covered by a uniform electrode. Ignoring space-charge effects, the current density extracted from the grid is proportional to the dielectric coefficient and it is quadratic with the back electrode voltage. Considering only space-charge associated with the back electrode voltage, it is found that the anode current is proportional to the anode voltage. When all space-charge effects are considered, it is shown that the electrostatic energy coupled into the diode gap through the grid is responsible to the excess of current beyond the Child–Langmuir limit.

Space-charge effects and current self-quenching in a metal/CdS/LaS cold cathode

We analyze the importance of space-charge effects in the cathode to anode gap region of a recently proposed metal/CdS(cadmium sulfide)/LaS(lanthanum sulfide) cold cathode. Our approach is based on an ensemble Monte Carlo description of electron transport assuming ballistic injection across the CdS and LaS layers. Under this approximation, the energy spectrum of the injected beam entering the air gap can be determined exactly as a function of the applied bias across the CdS layer. The effects of shot noise in the injected current are taken into account. For some of the biasing conditions considered here, space-charge effects are quite drastic and lead to dynamical effects which are responsible for the onset of current self-quenching similar to the Child–Langmuir regime of operation of thermionic cathodes. The limiting anode current density is found to be much larger than the Child–Langmuir limit. In the presence of strong space-charge effects, large oscillations in the minimum of the electrostatic potential in front of the cathode lead to oscillations in the measured anode current within the tens of GHz frequency range.

The double sheath associated with electron emission into a plasma containing negative ions

The double sheath formed by thermal electrons and negative ions in a plasma and electrons emitted from an electrode is investigated. The ion energy at the sheath edge and the electric field at the electrode surface are calculated for several values of the ratio of negative-ion density to electron density. The maximum beam density above which a virtual cathode appears is given. The relation to the Langmuir limit is shown. The numerical results for the electric potential, the electric field and the space charge density are presented. The floating potential is calculated from the current balance of all components.

Transient current collection and closure for a laboratory tether

The motion of tethered electrodes creates time‐varying plasma currents. A similar current system has been studied in a large laboratory plasma with a stationary, pulsed tether. It is shown that in the appropriate electron MHD regime, the transient current is transported in the whistler mode. The current exhibits helicity. The insulated tether induces electron Hall currents which provide the cross‐field current closure. The transient current increases with voltage and exhibits no saturation at the Langmuir limit due to electron heating. Superposition of delayed pulses emitted from displaced tether positions predicts that tethered satellite currents close along a short whistler wing rather than a long phantom loop.

Micropulsed ion source via self-sputtering avalanche

In the proposed pulsed ion source, a steady-state sputtering avalanche sustained by secondary ions locked in phase with an ac voltage applied between plates creates high current bunches with energy monochromaticity and ion purity. In a two-surface cascade, the average current exhibits a Child–Langmuir-like scaling, due to the dynamical constraints on the normalized ac voltage. A single-surface cascade, using a bending magnetic field, operates with unconstrained voltage and exhibits much lower heat load for given current, as well as the ability for average current in excess of the Child–Langmuir limit. The pulsed source is particularly attractive for ion beams of solid materials with high vaporization temperature.

Pulsed currents carried by whistlers. VI. Nonlinear effects

In a large magnetized laboratory plasma (n≂1011 cm−3, kTe≥1 eV, B0≥10 G, 1 m × 2.5 m), current pulses in excess of the Langmuir limit (150 A, 0.2 μs) are drawn to electrodes in a parameter regime characterized by electron magnetohydrodynamics (ωci≪ω≪ωce). The transient plasma current is transported by low-frequency whistlers forming wave packets with topologies of three-dimensional vortices. The generalized vorticity, Ω, is shown to be frozen into the electron fluid drifting with velocity v, satisfying ∂Ω/∂t≂∇×(v×Ω). The nonlinearity in v×Ω is negligible since v and Ω(r,t) are found to be nearly parallel. However, large currents associated with v≥(2kTe/me)1/2 lead to strong electron heating which modifies the damping of whistlers in collisional plasmas. Heating in a flux tube provides a filament of high Spitzer conductivity, which permits a nearly collisionless propagation of whistler pulses. This filamentation effect is not associated with density modifications as in modulational instabilities, but arises from conductivity modifications. The companion paper [Stenzel and Urrutia, Phys. Plasmas 3, 2599 (1996)] shows that, after the decay of the transient wave magnetic field, magnetic helicity remains in the plasma due to temperature-gradient driven currents.

Initial velocity effect on space-charge-limited currents

The characteristics of a monopolar space-charge region under the effect of charge injection with a uniform initial velocity are studied. Poisson’s equation has been solved as a Dirichlet boundary-value problem, other than by conventional serial integrations, to yield exact analytical expressions for the potential, field, charge density, and current. The structure of the space-charge region is characterized by a potential well and an effective diode gap, and can be explained in terms of the initial kinetic energy of the charge. The Child–Langmuir limit, when calculated by using the effective diode gap, is a firm limit.

Performance of two helium ion diodes

Two magnetically insulated gas-puff diodes were tested. In one design the plasma source was a fast inductive coil; in the other a coaxial gun created the plasma. The plasma current density from both sources, as well as the accelerated beam from each, was comparable to the Child–Langmuir limit of ∼10 A/cm2.

High-purity ion beam production at high current densities with a liquid-helium-cooled series-field-coil extraction ion diode

Experiments are described in which a high-purity, high-power (0.15 TW, 1 MeV) proton beam is generated from an ion source consisting of H2 gas frozen onto a liquid-helium-cooled copper anode at 4.2 K in a series-field-coil extraction diode on the 0.7 TW HydraMITE-II accelerator. Peak anode proton current densities of 2 kA/cm2 were measured. This current density is a factor of 100 higher than those obtained in previous liquid-helium-cooled cryogenic diode experiments on small accelerators and is in the range required for high-power ion beam applications. Thomson parabola, Faraday cup, and carbon activation measurements indicate an ion beam proton fraction close to 100% for the cryogenic source, compared to 50–70% for the standard hydrocarbon anode tested. The cryogenic proton source is believed to consist of no more than a few monolayers of molecular hydrogen. The hydrogen-coated cryogenic anode shows a faster initial anode turn-on than other materials. However, source-limited emission from the thin hydrogen layer results in a somewhat longer current risetime, reduced ion diode efficiency, lower proton current enhancement over the Child–Langmuir limit, and a proton spectrum of lower average energy than for the hydrocarbon anode. Techniques to overcome these limitations are discussed. Cryogenic ion sources consisting of frozen N2, CH4, and Ne have also been studied. In each case, high intensity beams consisting predominantly of components of the refrigerated gas were produced.

Production of pulsed sodium ion beams from anode plasma initiated by photon irradiation

Diode experiments that trigger the anode plasma formation by external radiation have been carried out with the goal of producing a pure beam of metallic ions. The anode of a race-track-type MID, which is coated with a sodium film in situ by vapor deposition, is synchronously irradiated by radiation emitted from a vacuum diode. Pure beams of Na+ are extracted only under operation with the radiation trigger. The current density reaches the Child–Langmuir limit.

The operation of a convergent-flow electron gun for thin beams

The primary object of this investigation was to undertake a practical survey of the operational features of a scaled-down electron gun with an LM-cathode. In order to accomplish this, a study has been made on the effect of reducing the gap between the anode and the cathode-grid assembly on the current density profile across the section of the beam beyond the anode. In particular the value of the peak current density jm, and the agreement this value has with Langmuir's limit, has been studied. The other parameters which are also included in the discussion are the transmission efficiency of the current through the anode and the perveance of the gun. At each of these electrode gaps, the practical values of the above mentioned parameters have been observed for different anode-to-cathode voltage in the range 50 to 400 V, in steps of 50 V. For easy reference, these values have been plotted on graphs. The profiles of the current density distribution across the section of the beam in the post-anode region, have been found to be very similar to Gaussian, and the parameters of the Gaussian in each case have been computed. The interesting feature of this study has been the fact that with this convergent-flow electron gun it is possible to produce beams with peak current densities of the order of 10 A cm−2 with a low dc voltage of the order of 400 V between the anode and the cathode. The effect of applying a bias on the grid with respect to the cathode is to cause a general deterioration of the performance. A slight improvement is, however, noticed with a small bias of the order of 0·5 V.