Space-charge-limited current in cylindrical diodes with finite-length emitter

Abstract

In the present paper we extend to two dimensions the classical Langmuir-Blodgett law, which determines the maximum current I LB that can be accelerated by a given voltage across two infinitely long coaxial cylinders. Generalization of the I LB law is established by performing two-dimensional 2D particle-in-cell numerical experiments on a variety of cylindrical diode configurations with a finite-length emitter. It is found that the limiting current in two dimensions I 2D follows a monotonically decreasing function of the emitter length-to-outer electrode radius ratio L/R as expressed by the fitting function I 2D /I LB ϭ1ϩ0.1536/(L/R)ϩ0.0183/(L/R) 2 within 2.5% accuracy for simulation data in which the ratio of the outer to inner radii exceeds 3. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1446876͔The effect of space charge on the flow of charged particles between two parallel electrodes in vacuum is of great importance in microwave electronics, crossed-field devices, and high-power diodes. In particular, if too much charge is injected into a gap, the resulting electric field becomes sufficiently high to reflect the injected particles back to the emitter, forming a virtual cathode. Limiting currents of vacuum diodes are classical subjects with early works dating back to the beginning of the last century when onedimensional space-charge-limited flow solutions were derived. Only for simple cases, however, such as plane parallel plates, infinitely long coaxial cylinders, and concentric sphere there exist analytical solutions.1 For a planar gap with gap separation D and gap voltage V, the maximum electron current density in a space-charge-limited diode is given by the well-known Child-Langmuir law,where e and m are, respectively, the charge and the mass of the emitted particles, and ⑀ 0 is the free space permittivity. Equation ͑1͒ is derived by assuming zero electron emission velocity and neglecting relativistic effects. Later an analytical solution for limiting current of relativistic electron flow in the planar diode was found, 3 but with the generalization of Child-Langmuir law in two dimensions still remaining a formidable task analytically. So an alternative approach was adopted, namely, the use of particle-in-cell ͑PIC͒ computer simulation to investigate two-dimensional space-chargelimited flow in a planar diode.4,5 About works on cylindrical diodes, particle simulation method was successfully applied for studying the stability of divergent and convergent electron flows in 1D cylindrical Pierce diode. 6 Also numerical experiments were performed to examine electrical properties of charged-particle flows in high-power cylindrical pinchedbeam diode over a wide range of voltages. 7The electron flow between coaxial cylinders is another important topic on beam formation theory, which has laid the basis of many useful gun designs. High-power cylindrical diodes have also found application in cylindrical vircators 8 and in high-resolution radiography sources.9 Again, as for the planar diode, the specialized lit...

Three-dimensional Child–Langmuir law for uniform hot electron emission

“…This finding agrees qualitatively to the 2D PIC simulation results, 14 which has showed that the enhancement of the 2D LB law for convergent flow is larger than divergent flow. In the range of ⌫ =3-10 and 0.2Ͻ W / r 2 Ͻ 5, their PIC simulation results for both divergent and convergent flows were numerically fitted with an empirical formula of 1 + 0.1536/ ͑W / r 2 ͒ + 0.0183/ ͑W / r 2 ͒ 2 to within 2.5% for all simulation data.…”

“…The solid lines are the calculated results at ⑀ = 0 for the convergent flow ͑top͒ and the divergent flow ͑bottom͒. The dash lines are the numerical fitting of 2D PIC simulation,14 which is 1 + 0.1536/ ͑W / r 2 ͒ + 0.0183/ ͑W / r 2 ͒ 2 . It is obvious that the numerical fitted 2D LB law ͑dashed line͒ is no longer accurate at small ⌫ Ϸ 1, which should recovers to 2D CL law ͑for a planar case͒.…”

Absence of saturation for finite injected currents in axially symmetric cavity diode

“…Most studies in two-dimensions have centred around cold emission from a finite emission area in an otherwise infinite parallel plate geometry [4][5][6]. Note that the Child-Langmuir current is defined in these studies as the current at which oscillatory behaviour sets in (the critical current, J CR ).…”

Coaxial foilless diode

“…As the current density is a function of the dimensionless ration of beam width to anode-cathode gap, the two-dimensional effect has a strong influence on diode current when the emitting plane thickness is much less than the A-K gap. 10,11 Neglecting the influence of the two-dimensional effect, through a simple analytic analysis we know that the current has the relationship with cathode radius and the ratio of anode radius to cathode radius. So the diode current does not increase according to r 2 when d keeps constant in Fig.…”