Crossed-field Diode Research Articles

Critical current in a two-dimensional non-magnetically insulated crossed-field gap with monoenergetic emission

Prior studies have developed theories for the maximum permissible current, or critical current, for one-dimensional planar and cylindrical crossed-field diodes where the magnetic field is below the Hull cutoff, meaning that an electron emitted from the cathode reaches the anode. Here, we develop semi-empirical and analytical models to predict the critical current for a two-dimensional (2D) planar diode with nonzero monoenergetic initial velocity. The semi-empirical method considers the geometry, nonzero initial velocity, and magnetic field as multiplicative corrections to the Child–Langmuir law for space-charge limited current in a one-dimensional planar diode with an initial velocity of zero. These results agree well with 2D particle-in-cell (PIC) simulations using the over-injection method to assess virtual cathode formation for different emission widths, magnetic field strengths, and initial velocities. The analytical solution agrees better with PIC results because it accounts for the coupling of the magnetic field, geometry, and initial velocity that the semi-empirical approach does not.

On the two-dimensional Brillouin flow

The Brillouin flow is a rectilinear, sheared electron fluid flow in a crossed electric field (E) and magnetic field (B), in the E × B direction with zero flow velocity and zero electric field at the surface with which the flow is in contact. It is broadly considered as the equilibrium electron flow in high power crossed-field devices including the magnetron and magnetically insulated transmission line oscillator. This paper provides an examination of Brillouin flow in two dimensions, in a cylindrical geometry where the anode radius changes abruptly at a single axial location, while the cathode surface has a constant radius. Our simulation confirms the proof that there is no equilibrium Brillouin flow solution for such a geometry. It further reveals that this change in the anode radius introduces novel bunching of the electrons within the Brillouin hub. This bunching occurs at low frequencies and is very pronounced if the Brillouin flow is from the small gap region to the large gap region, but is minimal if the Brillouin flow is from the large gap region to the small gap region. New insights are provided into the physical processes that initiate and sustain the bunching processes that are unique for a crossed-field diode, as compared with a non-magnetized diode. We argue that this enhanced bunching, and its concomitant formation of strong vortices, is not restricted to an abrupt change in the anode–cathode gap spacing.

Modifications of Limiting Current and Magnetic Insulation in a Crossed-Field Diode by a Series Resistor

Adding an orthogonal magnetic field to a diode is crucial for multiple applications, including high power generation. These crossed-field diodes (CFDs) have curved electron trajectories that store current in the gap. Above a critical magnetic field referred to as the Hull cutoff, electrons emitted from the cathode fail to cross the anode-cathode gap, leading to magnetic insulation and the storage or transport of all the current. Normally, the Hull cutoff may be easily calculated from diode geometry and boundary conditions; however, dissipative effects in the circuit or the addition of a protective shunt resistor may introduce an external resistance in series that causes a mismatch between the applied voltage and the voltage drop across the gap. For non-magnetically-insulated CFDs, non-zero net current flows in the circuit due to space-charge limited current (SCLC) in the gap. In this paper, we examine several models for crossed-field SCLC below the Hull cutoff to determine the impedance of the CFD. We find that the resistor reduces the voltage drop across the gap, reducing the magnetic field necessary for magnetic insulation below the typical Hull cutoff, which in turn lowers the SCLC. One-dimensional particle-in-cell simulations show that adding the series resistor causes electron trajectories to oscillate between insulated and non-insulated states for CFDs operating within 30% of the conventional Hull cutoff; these findings are validated by further examination of the theory. Extensions to other perturbations, such as AC modulation and magnetic field tilts, are discussed.

Explicit Brillouin Flow Solutions in Magnetrons, Magnetically Insulated Line Oscillators, and Radial Magnetically Insulated Transmission Lines

This article re-examines the Brillouin flow solutions in crossed-field diodes, with applications to magnetrons, magnetically insulated line oscillators (MILOs), and magnetically insulated transmission lines (MITLs). The Brillouin flow solutions are constructed for various geometries, including planar magnetrons, MILOs, and MITLs, cylindrical magnetrons with electrons flowing in the azimuthal direction, cylindrical MITLs and MILOs with electrons flowing in the axial direction, and radial MITLs and MILOs with electrons flowing in the radial direction. A common theme of this analysis is that two main external parameters are used to characterize the Brillouin flow: the anode–cathode voltage ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula> ) and the total magnetic flux within the crossed-field diodes ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A_{a}$ </tex-math></inline-formula> ). These two parameters are equivalent to the gap voltage and a specification of the degree of magnetic insulation, which is approximately equal to the ratio of the magnetic field to the Hull cutoff (HC) magnetic field. The magnetic flux may be provided externally by a magnet (as in a magnetron) or by the wall currents without an external magnet (as in a MILO or MITL), or by some combination of the two, as in the intermediate case of a magnetron–MILO hybrid. Once these two parameters are specified, the electron flow speed at the top of the Brillouin hub is uniquely determined. This immediately yields the Buneman–Hartree (BH) condition according to the Brillouin flow model, whether it be a planar magnetron or a cylindrical MILO. In so doing, we have obtained, for the first time using the Brillouin flow model, the BH condition for a cylindrical MILO, and we show that the same condition is obtained from the single-particle orbit model. We also found that, in general, the electron current within the Brillouin hub contributes only to a very small fraction of the magnetic flux <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A_{a}$ </tex-math></inline-formula> , regardless of the gap voltage <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula> , thereby correcting an erroneous notion that the electron flow within the crossed-field gap could be responsible for the magnetic insulation. Another counter-intuitive finding is that, for a given degree of magnetic insulation, the Brillouin hub height decreases as the gap voltage <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula> increases. These conclusions, and other results, were based on the simple, explicit analytic expressions that we have obtained for the Brillouin flow profiles, including the velocity, electron density, as well as the self-magnetic field and the self-electric field profiles due to the Brillouin hub electrons. From these analytical expressions, we deduce useful scaling laws that are applicable to the prevailing cases where the magnetic field exceeds 1.5 times the HC magnetic field, and they are valid in both relativistic and non-relativistic regimes. Thus, these scaling laws show the contrast between magnetrons and MILOs, and a ready assessment of the viability of building a moderate-current MILO, a low-voltage MILO, and a magnetron–MILO hybrid which might combine the advantages of a magnetron and a MILO. The Brillouin flow profiles in a radial MITL are explicitly calculated. Additional issues are addressed.

Space-charge limited current in planar and cylindrical crossed-field diodes using variational calculus

Crossed-field space-charge limited current (CF-SCLC) represents the maximum stable current that can be produced in crossed-field devices (CFDs); prominent CFDs include magnetrons and cyclotrons, critical technologies in vacuum electronics. While planar crossed-field geometries have the most developed theoretical models, cylindrical geometries are far more common and useful in practical applications. CFDs are characterized by the strength of the externally applied magnetic field B, orthogonal (crossed) to the electric field; this is often normalized to the magnetic insulation condition, the Hull cutoff field BH. We apply variational calculus to derive a new theoretical model for CF-SCLC for planar and cylindrical devices below and above BH. The variational model offers a concise derivation for planar results without transforming to the time domain and gives the first analytic results from first principles for cylindrical CF-SCLC. We implement a fully three-dimensional simulation in CST Particle Studio which, in addition to additional derived simple theoretical models, explains the often overlooked experimental current scaling ∝1−B/BH21/2 which decreases to zero current as B→BH−. These additional simple models reduce the maximum mismatch magnitude between theory and experiments or simulations by up to 68% compared to the variational model, with the most improvement at the critical limit B→BH−. Justification for the variational model and future applications are discussed.

Thermal Electron Flow in a Planar Crossed-Field Diode

A self-consistent model of steady-state electron flow in a planar crossed-field diode with a thermionic cathode is presented. Our formulation is a generalization of the classic work of Fry and Langmuir, to include a constant magnetic field B of arbitrary strength parallel to the electrode surfaces. Some effects of the magnetic field on the electron flow are illustrated in an example.

Loss of Magnetic Insulation in a Crossed-Field Diode: Ion and Collisional Effects

The effect of ion production through ionizing collisions in a magnetically insulated crossed-field gap is studied by using one-dimensional particle-in-cell software. These results are compared with the predictions from previous efforts that assumed immobile sheets of positive charge at different positions within the gap. Our results with mobile ions created via collisions indicate that the diode can lose magnetic insulation of the electron flow at ion densities lower than that predicted from the immobile ion case. Furthermore, we observe that electron scattering plays a significant role in this gap closure. This loss of insulation depends on the background pressure and leads to time-dependent migration of charge across the gap. We characterize both the time-scale and the degree of current transport for cases relevant to the high-power microwave community.

Role of Ions in a Crossed-Field Diode

The effect of ions in a magnetically insulated crossed-field gap is studied using a single particle orbit model, shear flow model, and particle-in-cell simulation. It is found that, in general, the presence of ions in a crossed-field gap always increases the electrons' excursion toward the anode region, regardless of the location of the ions. Thus, the rate at which the electrons migrate toward the anode, which is a measure of the diode closure rate, is related to the rate at which ions are introduced into the crossed-field gap. This anode migration of electrons is unrelated to crossed-field ambipolar diffusion. The implications of these findings are explored, such as pulse shortening in relativistic magnetrons and bipolar flows in pulsed-power systems.

Stability problems in a conducting crossed-field diode

Current work on excess noise and chaos in the presence of a magnetic field justifies further inquiry into the more general question of stability of such systems. We have shown that the electron flow in a conducting, crossed-field diode is fundamentally unstable for certain values of the magnetic field and the injection current density. In the process, we have systematised some of the earlier investigations of electron flow stability in the absence of the magnetic field.

Loading and Injection of Maxwellian Distributions in Particle Simulations

Existing and new particle loading and injection algorithms for particle simulations are analyzed to determine numerical accuracy and computational efficiency. Emphasis has been placed on loading and emission of Maxwellian, drifting Maxwellian, and cutoff Maxwellian velocity distributions. Once a velocity distribution has been inverted for loading or injection, time-centering of the position and velocity is necessary in order to maintain second-order accuracy. Here, the accuracy of these methods is determined and compared to three analytic test cases with spatially varying, time-dependent, and time-independent electric fields in a homogeneous magnetic field and a self-consistent crossed-field diode. The initial push is shown to be important in calculating the correct electric field at the boundary where particles are injected, in relaxing constraints on the time step, and in providing reliable field fluctuations due to particle statistics.

Surface wave enhanced collisionless transport in a bounded crossed-field non-neutral plasma

Electron transport across the magnetic field in a cutoff planar smooth-bore diode is described on the basis of surface waves perpendicular to the magnetic field and along the cathode. A self-consistent (two spatial dimensions and three velocity components) electrostatic particle-in-cell (PIC) simulation of a crossed-field diode produces a near-Brillouin flow which slowly expands across the diode, punctuated by sudden transport across the diode. The theory of slow transport across the diode is explained by the addition of perturbed orbits to the Brillouin shear flow motion of the plasma in the diode. The wave results from the sheared flow instability, so that a definite mechanism behind the surface waves is established. The growth and wavelength of this wave are compared to PIC simulations. A slow drift compared to the shear flow is described which results from an electrostatic ponderomotive-like force in a dc external magnetic field. The slow drift obtained from single particle motion compares well to the expansion of the hub in PIC simulations. The slow expansion of the Brillouin hub leads to configurations that are not stable and quickly decay via a large transport of charge to the anode. Both the slow transport of electrons and current spikes have been observed in experiments.

Reduction of crossed-field diode transmitted current due to anode secondary emission

The limiting current theory for planar crossed-field diodes has long been studied extensively for various emission energies and temperatures. However, experimental measurements of transmitted current have shown significant departure from theory. This paper attempts to explain the reduction in transmitted current from that expected in theory in terms of secondary electron emission created by electrons hitting the anode. It is proposed that the presence of the secondary electrons increases the charge density in the gap, thereby reducing the amount of current transmitted. A detailed secondary emission model is implemented in a particle-in-cell code to study current reduction. The effect of secondary electrons on charge density, and on the resultant electric field and potential is also presented.

Measurement and interpretation of current transmission in a crossed-field diode below cutoff

Measurements on the current-voltage-magnetic field characteristics of a space-charge-limited cylindrical cross-field diode below cutoff are presented. The measured current is found to be lower than predicted by simple cold-fluid theory. This reduction combined with observed oscillations in the current can be explained by secondary electron emission from the anode, leading to an increase of space charge in the diode.

Rapid current transition in a crossed-field diode

The transmitted current in a crossed-field gap has been characterized analytically by a number of authors. Using a one dimensional PIC simulation, we explore the behavior of the crossed-field diode at B=BHull. For mono-energetic (cold) emission, a rapid reduction of transmitted current is observed when the injected current exceeds the critical current by just 1%. The addition of a small electron temperature normal to the cathode eliminates the transition, even for kT/V∼10−5 (V=10 kV, gap = 1 cm, B=337 G, J=1.69 A/cm2), while an isotropic velocity distribution accelerates the transition.

Transit-time effects on noise in planar crossed-field electron guns

A theory is developed for the noise in an unbounded parallel-plate crossed-field diode. The high-frequency regime is studied, where transit-time effects must be included. The diode is assumed to be operating in the space-charge-limited regime. Maxwell's equations and the Vlasov equation are expanded to first order. A key consideration in this expansion is the fluctuation in the boundaries of velocity space appearing in the integrals for the charge and current density. The first-order terms describe the linear response of the electric field and transmitted current resulting from the injection of a small additional cathode emission composed of electrons of a specified velocity. Using a statistical approach, these perturbations are combined to determine the noise factor. The special case of a parabolic average-potential distribution in space is considered.

Self-consistent theory for crossed-field diode current fluctuations

Applying a perturbation expansion to the self-consistent Poisson equation for a planar, space-charge-limited crossed-field diode, we have derived four universal diode functions (independent of any particular diode specification), in addition to the usual Fry-Langmuir function for unperturbed, unmagnetized planar diodes. A crossed-field diode is completely specified by three external parameters: its normalized width X = xa/λDb, anode voltage V = φa/φTb, and magnetic field Ω = ωc/ωpb. Here a and b, respectively, denote anode and cathode quantities, λDb is the electronic Debye length, ωpb is the plasma frequency, φTb is the electron temperature in volts, and ωc is the electron gyrofrequency. Evaluating the universal functions at abscissas appropriate for a particular set of X, V, and Ω then allows the calculation of the anode noise power (mean square current fluctuation at the anode) as a function of X, V, and Ω. The range of external parameters for which the theory is applicable is specified by a simple constraint relation for X, V, and Ω, which ensures that the diode is space-charge limited both with and without the applied magnetic field. The results of our analysis show that the noise level of a space-charge-limited diode increases monotonically from the reduced level at zero magnetic field. These results are for a range of magnetic fields well below the threshold where the diode enters the regime of magnetic-field-limited flow in which the noise power reaches the temperature-limited level. Thus the range of the space-charge-limited regime studied is always associated with a reduced noise level which is, nevertheless, greater than the unmagnetized space-charge-limited level.

Weak magnetic-field expansion for the self-consistent steady-state potential in a planar crossed-field diode

In a planar crossed-field diode, the steady-state flow in the space-charge-limited régime is governed by a nonlinear integrodifferential equation. We have derived a generalized Fry-Langrnuir equation for the potential which is self-consistent to second order in the normalized magnetic field µ. The small-µ crossed-field potential may be considered as the Fry-Langmuir potential of the corresponding unmagnetized diode, i.e., with the same diode parameters but with zero magnetic field, plus the steady-state second-order perturbation produced by the magnetic field. Owing to the symmetry of the unbounded, planar geometry, there is no first-order perturbation. From four universal functions (the Fry-Langrnuir function and three additional functions we have derived), one can determine the potential and density profiles of any space-charge-limited planar crossed-field diode for which µ is small. It is shown that the crossed-field potential minimum is deepened and shifted toward the anode with respect to the unmagnetized potential minimum.

Noise in planar crossed-field electron guns—II: Numerical solution

Theory and numerical results for the shot effect in an unbounded, parallel-plate, crossed-field diode are presented. The low-frequency régime is studied, where transit-time effects are negligible. It is further assumed that short-circuit conditions prevail, and that the diode is operating in the space-charge-limited régime. Previous theory is first developed into a form suitable for automatic computation. Problems and approaches associated with the numerical solution are then discussed. Finally, results of the computation are presented in normalized form, giving the noise factor for a representative diode as a function of magnetic field. The results show this factor ranging through values less than unity. It follows that, in the present model, feedback effects due to the potential minimum continue to smooth the noise in the presence of magnetic field, just as in the case of no magnetic field.

Noise in planar crossed-field electron guns—I: Theory

A theory is developed for the shot effect in an unbounded parallel-plate crossed-field diode. The low-frequency regime is studied, where transit-time effects are negligible. It is further assumed that short-circuit conditions prevail, and that the diode is operating in the space-charge-limited regime. The theory begins with the derivation of general equations for the potential and the transmitted current. These are then subjected to a perturbation expansion. The zero-order terms in this expansion describe the steady-state potential and transmitted current. The first-order terms describe the linear response of the potential and transmitted current resulting from the injection of a small additional emission composed of electrons of a specified velocity. The theory shows how these perturbations are combined to determine the current noise power at the anode and thereby the noise factor. The special case where the average potential shows a parabolic distribution in space is considered.

Comments on "Evaluation of the DC parameters and noise transport in the gun region of an injected-beam crossed-field diode"

A half-Maxwellian y -component velocity distribution and a full-Maxwellian z -component velocity distribution are assumed in order to evaluate the position and depth, y m and V m , of the potential minimum as a boundary-value problem. The various dc parameters such as the voltage distribution, space-charge density, velocity and current density components and trajectories are then evaluated as an initial-value problem. The results obtained in this manner agree closely with the results obtained from the Kino gun model except that the y -component current density is not constant, as is usually assumed in the Kino gun model. The steady-state parameters are calculated here for both temperature-limited and space-charge-llmited conditions. The Kino gun results are shown to be essentially those for space-charge-limited operation. Even though the injection conditions under the two types of operation are identical, the formation of a potential minimum considerably changes the electron trajectories and the corresponding velocity components. The growth rate of a hybrid wave is reduced as ω p is decreased and/or \omega/\omega_{c} is increased, and the propagation constants of the two conventional space-charge waves are modified, the over-all growth rate of the slow wave being greater than that of the fast wave. For large values of ω p the conventional fast space-charge wave is a backward wave, although it becomes a forward wave if \omega/\omega_{c} is large. It is noticed that the conditions in the gun region are more favorable to the existence of low-frequency perturbations. Based upon these results several experimental observations made at various laboratories are explained qualitatively.