One-dimensional Child-Langmuir 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.

Numerical simulation of a relativistic magnetron using a fluid electron model

An approach to numerically modeling relativistic magnetrons, in which the electrons are represented with a relativistic fluid, is described. A principal effect in the operation of a magnetron is space-charge-limited (SCL) emission of electrons from the cathode. We have developed an approximate SCL emission boundary condition for the fluid electron model. This boundary condition prescribes the flux of electrons as a function of the normal component of the electric field on the boundary. We show the results of a benchmarking activity that applies the fluid SCL boundary condition to the one-dimensional Child–Langmuir diode problem and a canonical two-dimensional diode problem. Simulation results for a two-dimensional A6 magnetron are then presented. Computed bunching of the electron cloud occurs and coincides with significant microwave power generation. Numerical convergence of the solution is considered. Sharp gradients in the solution quantities at the diocotron resonance, spanning an interval of three to four grid cells in the most well-resolved case, are present and likely affect convergence.

Ampere-class bright field emission cathode operated at 100 MV/m

High-current bright sources are needed to power the next generation of compact rf and microwave systems. A major requirement is that such sources could be sustainably operated at high frequencies, well above 1 GHz, and high gradients, well above $100\text{ }\text{ }\mathrm{MV}/\mathrm{m}$. Field emission sources offer simplicity and scalability in a high-frequency era of the injector design, but the output rf cycle charge and high-gradient operation remain a great and largely unaddressed challenge. Here, a field emission cathode based on ultra-nano-crystalline diamond, an efficient planar field emission material, was tested at $100\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ in an $L$-band injector. A very high charge of 38 pC per rf cycle was demonstrated (300 nC per rf pulse corresponding to an rf pulse current of 120 mA). This operating condition revealed a two-dimensional space charge limited emission where the one-dimensional Child-Langmuir limit was surpassed. An injector brightness of ${10}^{14}\text{ }\text{ }\mathrm{A}/(\mathrm{rad}\text{ }\text{ }\mathrm{m}{)}^{2}$ was estimated for the given operating conditions.

Determination of the Pulsed Electron Beam Spectrum by Current and Voltage Oscillograms

The algorithm and results of calculating the integral and differential energy spectra of a pulsed electron beam (350–500 keV, 80 ns), generated by direct-action accelerators, are presented. The electron spectrum was calculated using the oscillograms of the accelerating voltage, electron current, total current of the diode, and the one-dimensional Child–Langmuir (1D CL) ratio. It was found that the discrepancy in the integrated electron beam energy spectrum, when measured using the total current in the diode and using the electron beam current, did not exceed 15% for 80%–95% of the electrons generated in a diode with graphite, carbon fabric, and multipoint cathodes. When calculating the electron spectrum using the 1D CL, the error was much higher.

Investigation of the homogeneity of a high-power ion beam formed by a diode with a closed electron drift

The results of an investigation of the energy-density distribution over the cross section of a pulsed ion beam formed with a passive-anode diode in the mode of magnetic insulation and a closed electron drift in the anode–cathode gap are presented. Diodes of two types are studied: with external magnetic insulation (Br diode) on the BIPPAB-450 accelerator (400 kV, 80 ns) and self-magnetic insulation of electrons (spiral diode) on the TEMP-4M accelerator (250 kV, 120 ns). In the investigated diodes, various processes are used to form anode plasma: a breakdown over the surface of a dielectric coating on the anode and ionization of the anode surface with accelerated electrons (Br diode), as well as explosive emission of electrons (spiral diode). To analyze the ion-beam energy density, thermal-imaging diagnostics is used with a spatial resolution of 1–2 mm. The energy-density is calculated from the one-dimensional Child–Langmuir relationship. It is shown that a continuous plasma layer is efficiently formed on the working anode surface for all the investigated diodes. The anode-plasma concentration is rather high, and the beam-energy density is limited by the space charge of ions, but not by the plasma concentration. It is found that, when the magnetic field in the Br-diode anode–cathode gap decreases or the electron current in the spiral diode increases, the energy density of the high-power ion beam rises significantly, but the beam homogeneity decreases.

The influence of a change in the electron trajectory in the vacuum-diode anode–cathode gap on the impedance

The analysis of the influence of the electron-trajectory curvature in the anode–cathode gap of a vacuum diode on its impedance during generation of high-current electron beams is performed, and the applicability of the one-dimensional Child–Langmuir formula for the electron-current calculation is evaluated. The results of an experimental study of a flat diode with explosive-emission graphite, copper, and carbon- fabric cathodes and also with a multipointed cathode are presented. These investigations were performed on the TEU-500 accelerator (350–500 kV, 60 ns). A strip diode with a graphite cathode was also studied in the mode of magnetic self-insulation of electrons. The experiments were performed on the TEMP-4М accelerators (150–200 kV, 400–600 ns). The investigation results showed that the satisfactory coincidence of the experimental values of the total current with those calculated from the one-dimensional Child–Langmuir formula (for the working area of the cathode) is observed in diodes not only in the absence of a change in the electron trajectory in the anode–cathode gap but also upon a deflection of the trajectory from the normal to the cathode surface by an angle of <90°. An increase in the electron current owing to a decrease in the anode–cathode gap and an increase in the emission area of the cathode during the cathode-plasma production and also at the expense of the presence of microirregularities on the cathode is properly described by the onedimensional Child-Langmuir formula, if the above factors are taken into account.

Investigation of mechanism of anode plasma formation in ion diode with dielectric anode

The results of investigation of the anode plasma formation in a diode with a passive anode in magnetic insulation mode are presented. The experiments have been conducted using the BIPPAB-450 ion accelerator (350–400 kV, 6–8 kA, 80 ns) with a focusing conical diode with Br external magnetic field (a barrel diode). For analysis of plasma formation at the anode and the distribution of the ions beam energy density, infrared imaging diagnostics (spatial resolution of 1–2 mm) is used. For analysis of the ion beam composition, time-of-flight diagnostics (temporal resolution of 1 ns) were used. Our studies have shown that when the magnetic induction in the A-C gap is much larger than the critical value, the ion beam energy density is close to the one-dimensional Child-Langmuir limit on the entire working surface of the diode. Formation of anode plasma takes place only by the flashover of the dielectric anode surface. In this mode, the ion beam consists primarily of singly ionized carbon ions, and the delay of the start of formation of the anode plasma is 10–15 ns. By reducing the magnetic induction in the A-C gap to a value close to the critical one, the ion beam energy density is 3–6 times higher than that calculated by the one-dimensional Child-Langmuir limit, but the energy density of the ion beam is non-uniform in cross-section. In this mode, the anode plasma formation occurs due to ionization of the anode material with accelerated electrons. In this mode, also, the delay in the start of the formation of the anode plasma is much smaller and the degree of ionization of carbon ions is higher. In all modes occurred effective suppression of the electronic component of the total current, and the diode impedance was 20–30 times higher than the values calculated for the mode without magnetic insulation of the electrons. The divergence of the ion beam was 4.5°–6°.

Features of 200 kV, 300 ns reflex triode vircator operation for different explosive emission cathodes

AbstractTo study the effect of explosive field emission cathodes on high power microwave generation, experiments were conducted on a reflex triode virtual cathode oscillator. Experimental results with cathodes made of graphite, stainless steel nails, and carbon fiber (needle type) are presented. The experiments have been performed at the 1 kJ Marx generator (200 kV, 300 ns, and 9 kA). The experimentally obtained electron beam diode perveance has been compared with the one-dimensional Child-Langmuir law. The cathode plasma expansion velocity has been calculated from the perveance data. It was found that the carbon fiber cathode has the lowest cathode plasma expansion velocity of 1.7 cm/μs. The radiated high power microwave has maximum field strength and pulse duration for the graphite cathode. It was found that the reflex triode virtual cathode oscillator radiates a single microwave frequency with the multiple needle cathodes for a shorter (&lt;200 ns full width at half maximum) voltage pulse duration.

Theory and simulation of nonrelativistic elliptic-beam formation with one-dimensional Child-Langmuir flow characteristics

A theory of nonrelativistic, laminar, elliptic-beam formation is presented. The beam is shown to have the one-dimensional Child-Langmuir flow form. An analytic expression for the electrostatic potential outside the beam is derived. Equipotentials corresponding to electrode surfaces are computed numerically. The effectiveness of the electric field formed by the electrodes in focusing and preserving the elliptic, 1D Child-Langmuir flow beam is verified via 3D ray-tracing simulations.

Application of eclipse laser photodetachment technique to electron sheath thickness and collection region measurements

The laser photodetachment (LPD) technique, which has been used for negative-ion density measurements, is applied for the measurement of electron sheath thickness and the collection region of photodetached electrons (PDE's). By forming a thin laser shadow in the laser beam channel, the electron sheath can be observed in the temporal evolution of the LPD signals. The collection region of PDE's is determined from the response of the signal peak value in scanning the shadow position. The measurement is applied to the electron sheath formed around a cylindrical probe and a plane probe. The experimentally obtained thickness of the sheath in front of the plane probe agrees well with the one-dimensional Child-Langmuir sheath when the magnetic field exists. Further, the results of the sheath thickness around the plane and cylindrical probe at different magnetic field strengths indicate that the effect of magnetic field on the sheath structure is significant even in weakly magnetized plasmas. The length of the collection region of PDE's was measured, and it was confirmed that the region was in the laser beam channel under our experimental conditions. It is proposed that this method be applied to check the validity of the laser photodetachment technique.

Beyond the Child–Langmuir law: A review of recent results on multidimensional space-charge-limited flow

Space-charge-limited (SCL) flows in diodes have been an area of active research since the pioneering work of Child and Langmuir in the early part of the last century. Indeed, the scaling of current density with the voltage to the 3/2’s power is one of the best-known limits in the fields of non-neutral plasma physics, accelerator physics, sheath physics, vacuum electronics, and high power microwaves. In the past five years, there has been renewed interest in the physics and characteristics of SCL emission in physically realizable configurations. This research has focused on characterizing the current and current density enhancement possible from two- and three-dimensional geometries, such as field-emitting arrays. In 1996, computational efforts led to the development of a scaling law that described the increased current drawn due to two-dimensional effects. Recently, this scaling has been analytically derived from first principles. In parallel efforts, computational work has characterized the edge enhancement of the current density, leading to a better understanding of the physics of explosive emission cathodes. In this paper, the analytic and computational extensions to the one-dimensional Child–Langmuir law will be reviewed, the accuracy of SCL emission algorithms will be assessed, and the experimental implications of multidimensional SCL flows will be discussed.

Current and current density of a finite-width, space-charge-limited electron beam in two-dimensional, parallel-plate geometry

The current carried by a steady-state, finite-width beam produced by space-charge-limited emission in two-dimensional parallel plate geometry is known to diverge strongly from estimates based on the classic one-dimensional Child–Langmuir problem. The two-dimensional problem presents formidable analytic difficulties, but a numerical approach to this problem has been developed. The approach simultaneously determines the electrostatic potential and the unknown current density profile. Calculations show that the total current is a function of the dimensionless ratio of beam width (w) to anode–cathode gap (d), but that the current density profile varies with both w and d separately.