Electron Emission Velocity Research Articles

On the Child–Langmuir law in one, two, and three dimensions

We consider the limiting current from an emitting patch whose size is much smaller than the anode–cathode spacing. The limiting current is formulated in terms of an integral equation. It is solved iteratively, first to numerically recover the classical one-dimensional Child–Langmuir law, including Jaffe's extension to a constant, nonzero electron emission velocity. We extend to two-dimensions in which electron emission is restricted to an infinitely long stripe with infinitesimally narrow stripe width so that the emitted electrons form an electron sheet. We next extend to three-dimensions in which electron emission is restricted to a square tile (or a circular patch) with an infinitesimally small tile size (or patch radius) so that the emitted electrons form a needlelike line charge. Surprisingly, for the electron needle problem, we only find the null solution for the total line charge current, regardless of the assumed initial electron velocity. For the electron sheet problem, we also find only the null solution for the total sheet current if the electron emission velocity is assumed to be zero, and the total maximum sheet current becomes a finite, nonzero value if the electron emission velocity is assumed to be nonzero. These seemingly paradoxical results are shown to be consistent with the earlier works of the Child–Langmuir law of higher dimensions. They are also consistent with, or perhaps even anticipated by, the more recent theories and simulations on thermionic cathodes that used realistic work function distributions to account for patchy, non-uniform electron emission. The mathematical subtleties are discussed.

Space Charge Limited Current and Induced Particle Reflection With a Time-Varying Current Injection

Properties of space charge limited current and induced particle reflection with a time-varying current injection are investigated by an efficient grid-free numerical approach. Considering a planar diode with a single-frequency current emitted parallel to the upstream electrode surface, we study the current transmission characteristics between two dc biased electrodes and the influence of current frequency and magnitude on the limiting current first leading to electrons reflection. It is found that on account of the time-varying current injection, the limiting current strongly depends on the current frequency and magnitude, as well as the emission velocity of electrons. For the low-frequency limit, the dc component of the limiting current is approximately equal to half of that obtained from the time-independent model. The limiting current increases monotonically with increasing the frequency, and the dc component has a maximum equal to the limiting current obtained from the time-independent model.

Calculations of Multipactor Growth in Rectangular Waveguides

Multipactor growth in rectangular waveguides is probed based on a kinetic approach. Unlike most studies relying on the Vaughan model, a probabilitic approach for random multiple secondary particle emissions is used. Spread in electron emission velocities, the angular dependence of secondary emission yields, and an external radio frequency (RF) driving field due to a TE10 mode, were all built in. The calculations predict the secondary emission yield for copper, probe the population growth dynamics, and obtain the susceptibility diagram. Despite a maximum field at the waveguide center from the RF excitation, maximum electron densities are predicted at locations symmetrically displaced from the center. The secondary electron yield (SEY) characteristics, its local maxima, and the role of oblique incident angles, collectively lead to multipactor finding its place at off-center locations.

Multipactor susceptibility chart of coaxial transmission lines with stationary statistical modeling

Multipactor breakdown is a detrimental electromagnetic phenomenon caused by resonant secondary electron emissions synchronizing with field oscillation, which frequently takes place in powerful microwave devices and accelerating structures. Regarded as the principal failure mode of space microwave systems, multipactor may cause the performance to degenerate or even hardware operation to deteriorate catastrophically, thus multipactor becomes a major limitation in promoting the further development of space communication technology. Meanwhile, higher power capacity and volume integration accordingly lead to continuously growing multipactor hazard. In order to prevent multipactor from occurring, the accurate predictive technique to determine multipactor susceptibility has become a key issue for the mechanical design and performance optimization of microwave devices in the ground stage. Compared with the existing approaches to investigating the multipactor, statistical theories are able to conduct multipactor threshold calculation and mechanism analysis, with the stochastic nature of secondary emission fully considered from the probabilistic perspective. Currently, stationary statistical theory of multipactor has been developed for efficient multipactor threshold analysis of the parallel-plate geometry. However, it has not been further extended to the coaxial geometry which is commonly involved in radio frequency (RF) systems. For this reason, the stationary statistical modeling of the coaxial multipactor with all influencing factors considered is detailed in this paper. Due to the field nonuniformity and the secondary emission randomness, analytic equation of electron trajectories in the coaxial geometry is approximately derived by using the perturbation approach. Based on the implicit correlation between electron emission velocity and transit time, the joint probability density function is constructed for the calculation of the probability density distribution of electron transit time. Afterwards, a system of integral equations for depicting electron multiplication process in the coaxial geometry is formulated and solved with a novel and general iteration method. Finally, this stationary statistical theory is applied to the full multipactor susceptibility chart of coaxial transmission lines with typical coating materials in space engineering, such as silver, copper, alumina and alodine. A comparison shows that the calculation results are in reasonable agreement with the experimental measurements provided by the Europe Space Agent. What is more, there exists significant difference between multipactor susceptibility curves of the parallel-plate geometry and the coaxial geometry. This research is of great significance for optimizing the mechanism design and material selection of multipactor-free microwave devices.

Multipactor threshold calculation of coaxial transmission lines in microwave applications with nonstationary statistical theory

This paper presents a statistical theory for the initial onset of multipactor breakdown in coaxial transmission lines, taking both the nonuniform electric field and random electron emission velocity into account. A general numerical method is first developed to construct the joint probability density function based on the approximate equation of the electron trajectory. The nonstationary dynamics of the multipactor process on both surfaces of coaxial lines are modelled based on the probability of various impacts and their corresponding secondary emission. The resonant assumption of the classical theory on the independent double-sided and single-sided impacts is replaced by the consideration of their interaction. As a result, the time evolutions of the electron population for exponential growth and absorption on both inner and outer conductor, in response to the applied voltage above and below the multipactor breakdown level, are obtained to investigate the exact mechanism of multipactor discharge in coaxial lines. Furthermore, the multipactor threshold predictions of the presented model are compared with experimental results using measured secondary emission yield of the tested samples which shows reasonable agreement. Finally, the detailed impact scenario reveals that single-surface multipactor is more likely to occur with a higher outer to inner conductor radius ratio.

General Study of Multipactor Between Curved Metal Surfaces

An analytical study of the electron trajectories between two opposite electrodes having curved surfaces is undertaken for the case when the electron transit time exceeds the RF period. The analysis is based on a statistical approach, which makes it possible to calculate the width of an electron bunch after a number of electron transits taking into account the spread of electron emission velocity, and the spatial nonuniformity of the RF field using the concept of the ponderomotive force. The results are used to estimate the multipactor threshold in terms of a value of the secondary emission yield, which is necessary to balance electron losses. Based on the model, it is predicted that multipactor is impossible inside the realistic configuration of a helix antenna where four electrodes are placed on the same cylindrical surface.

Non-resonant multipactor—A statistical model

High power microwave systems operating in vacuum or near vacuum run the risk of multipactor breakdown. In order to avoid multipactor, it is necessary to make theoretical predictions of critical parameter combinations. These treatments are generally based on the assumption of electrons moving in resonance with the electric field while traversing the gap between critical surfaces. Through comparison with experiments, it has been found that only for small system dimensions will the resonant approach give correct predictions. Apparently, the resonance is destroyed due to the statistical spread in electron emission velocity, and for a more valid description it is necessary to resort to rather complicated statistical treatments of the electron population, and extensive simulations. However, in the limit where resonance is completely destroyed it is possible to use a much simpler treatment, here called non-resonant theory. In this paper, we develop the formalism for this theory, use it to calculate universal curves for the existence of multipactor, and compare with previous results. Two important effects that leads to an increase in the multipactor threshold in comparison with the resonant prediction are identified. These are the statistical spread of impact speed, which leads to a lower average electron impact speed, and the impact of electrons in phase regions where the secondary electrons are immediately reabsorbed, leading to an effective removal of electrons from the discharge.

Conformal mapping analysis of multipactor breakdown in waveguide irises

Multipactor breakdown in a single waveguide iris is analyzed using the quasistatic approximation for the spatial distribution of the rf field in the iris. Based on the conformal mapping approach, an analytical description is given of the rf field structure in the iris. It is shown that in the central part of any iris with a length to height ratio greater than approximately 0.5, the rf field structure is close to that between two parallel plates. The multipactor threshold for the iris is determined mainly by electron losses from the central part of the iris where the losses are due to the tangential component of the emission velocity of secondary electrons. The effective length of the iris central part is determined and an estimate of the multipactor threshold for the iris is found in terms of the conventional parameters: Applied rf voltage, product of rf frequency and iris height, and iris length to height ratio. Numerical simulations are also carried out using the exact analytical description of the quasistatic rf field and taking into account a spread of electron emission velocities.

Multipactor in rectangular waveguides

Multipactor inside a rectangular waveguide is studied using both an analytical approach and numerical simulations. Particular attention is given to an analysis of the role of such effects as the velocity spread of secondary emitted electrons and the action of the rf magnetic field on the electron motion. Conventional resonance theory is shown to give correct predictions for the multipactor threshold in cases where the height of the waveguide is very small and first order resonance multipactor dominates. In cases of higher order resonances, an accurate prediction of the multipactor threshold requires that the spread of the normal component of the electron emission velocity is taken into account. Furthermore, the spread of the tangential component of the electron emission velocity and the action of the rf magnetic field are shown to be very important when the waveguide height exceeds a certain critical value, which depends on the waveguide width. A new theory is developed for predicting the multipactor threshold at higher order resonances and this theory is confirmed by numerical simulations.

Parametric analysis of the soft electron emission in ion-helium collisions

We studied the doubly differential cross section (DDCS) for ion-helium ionization, in the region of near zero emission velocity. We expanded the DDCS in powers of the electron emission velocity, with angle-dependent weight coefficients, which are determined from available experimental data and calculated using the CDW-EIS theory. We also compared this expansion with a previously used Legendre polynomials expansion of the DDCS [C.R. Garibotti and W.R. Cravero, Phys. Rev. A 48 (1993) 2012].

Space charge limited field emission in a plane parallel electrode geometry

Poisson's equation in parallel geometry is solved analytically with the inclusion of the electron emission velocity spectrum predicted by quantum mechanics. The mathematical difficulty of classically defining the cathode surface is overcome, to a first approximation, by postulating a virtual cathode situated in front of the real cathode such that the majority of the emitted electrons have penetrated the potential barrier at that position. The treatment is nonrelativistic, assumes that the emission is exclusively by barrier penetration, and is expressed in parameters that enable the results to be applied to most situations of interest. It is shown that the effect of nonzero emission velocity is negligible except for the cases of cathodes of low work function operated at very high current densities. Particular attention is paid to the departure of the experimentally measured current from that predicted by the Fowler-Nordheim theory and the subsequent approach to the Langmuir-Child fully space charge limited current. It is shown that the point of departure can quantitatively be defined in terms of the fully limited current and the bearing of this on the design of field emission electron guns is considered.

Secondary-emission amplification at microwave frequencies

Two types of photomultipliers exhibiting flat base-band response to ∼4 Gc/s are described. One photomultiplier employs crossed electric and magnetic fields, the other is purely electrostatic. At present both devices consist of eight Cu-Be secondary emission stages and an internally situated, opaque-backed S-1 photocathode. Current gain exceeds 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> , and the transit time dispersion is less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7 \times 10^{-11}</tex> s. Frequency response measurements have used heterodyning between the modes of the Zeeman-split Doppler profiles of a 6328 Å He-Ne laser, and also shot noise in the current from a steadily illuminated photocathode as sources of microwave modulated currents. Transient response measurements have been made using sub-nanosecond light pulses from phase-locked He-Ne and argon ion lasers, the results being consistent with the measured frequency response. A calculation of the frequency response to be expected from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> identical stages of secondary emission when only the secondary electron emission velocities contribute to electron transit time differences, as is true of the geometries used in the experiments, gives results in agreement with experiment, assuming a value of 3 eV for the most probable secondary electron emission energy from BeO. This value is consistent with direct observations by others.

A Note on Anomalous Radial Velocity Spectra from Rough Thermionic Emitters†

ABSTRACT A simple pinhole spectrometer is used to estimate the radial emission velocities of electrons from some typical planar cathodes. It is fairly conclusively shown that rough cathodes can yield high apparent radial velocities (order of 10-20 volts) under temperature limited conditions. These anomalous values disappear when the cathodes are physically smoothed or are operated under space charge limited conditions. A simple theory explains the findings.

Noise in Transverse-Field Traveling-Wave Tubes

From theoretical considerations of the fluctuations in an electron beam, an expression for the noise figure of a transverse-field amplifier is derived. There are found to be three statistically uncorrelated sources of noise in the beam of such a tube—(1) that arising from the fluctuations in the transverse emission velocity of electrons in a direction normal to the surface of the beam, (2) that arising from fluctuations in the transverse emission velocity of electrons in a direction parallel to the surface of the beam, and (3) that arising from fluctuations in the mean position of the beam and depending on its thickness. The analysis shows that the use of a collimator is required to give low noise figures. This is illustrated by the following example: for a tube designed to operate at 1000 mc with a collimator whose width is 0.004 in., the theoretical noise figure is 2 db. Without the collimator, the noise figure would be 11 db.

Limiting Resolution in an Image-Orthicon-Type Pickup Tube

An analysis is made of some of the factors which limit resolution in a television pickup tube of the image-orthicon type. Particular attention is given to effects of the glass-target characteristics and to departure from perfect focus in the image section, resulting from electron-emission velocities. It is shown that, as the width of lines in a resolution pattern is reduced, the signal generated tends to take the form of a small periodic modulation superimposed on a constant signal. This gives, in the reproduced image, a gradual reduction in contrast. A quantitative determination of limiting resolution, expressed as a number of lines, depends upon a physiological evaluation, not as yet available, of the minimum contrast in the reproduced image which will permit the pattern to be recognized as having a multiplicity of separate lines rather than a continuous blur. The general effects of target thickness and of color of incident light discussed herein have been confirmed experimentally.

The apparent inter-electrode capacitance of a planar diode

Various experiments have suggested that the effective capacitance of a diode is probably dependent on the magnitude of the anode current: this effect, however, is not disclosed by the classic analyses of Benham and of Llewellyn. This paper shows that an analysis which ignores the emission velocity of electrons is not competent to describe the capacitance effects in a diode. A steady-state solution is found for a planar diode in which the emission velocity is not ignored. Also it is calculated that the mean-square velocity of electrons crossing the potential barrier is the same as the mean-square velocity of emission from the cathode. By considering the inertia effect of the electrons in conjunction with the displacement current at the barrier, an expression is derived for the phase angle of an electrode system. This expression shows that the phase angle is markedly dependent on the anode current. It is therefore thought that the analysis describes the basic mechanism of certain capacitance effects which have been observed in triodes.An Appendix contains the steady-state solution of the temperature-saturated planar diode, and curves are given which show the force at the cathode and at the anode as a function of the anode current, expressed as a fraction of the space-charge limited current; also a curve which shows the transit time as a function of the anode current

The apparent inter-electrode capacitance of a planar diode

Various experiments have suggested that the effective capacitance of a diode is, probably dependent on the magnitude of the anode current: this effect, however, is not disclosed by the classic analyses of Benham and of Llewellyn. This paper shows that an analysis which ignores the emission velocity of electrons is not competent to describe the capacitance effects in a diode. A steady-state solution is found for a planar diode in which the emission velocity is not ignored. Also it is calculated that the mean-square velocity of electrons crossing the potential barrier is the same as the mean-square velocity of emission from the cathode. By considering the inertia effect of the electrons in conjunction with the displacement current at the barrier, an expression is derived for the phase angle of an electrode system. This expression shows that the phase angle is markedly dependent on the anode current. It is therefore thought that the analysis describes the basic mechanism of certain capacitance effects which have been observed in triodes.An Appendix contains the steady-state solution of the temperature-saturated planar diode, and curves are given which show the force at the cathode and at the anode as a function of the anode current, expressed as a fraction of the space-charge limited current; also a curve which shows the transit time as a function of the anode current.