# On the Space‐Charge‐Limited Current between Nonsymmetrical Electrodes

01 Aug 1955-Journal of Applied Physics (American Institute of Physics)-Vol. 26, Iss: 8, pp 1034-1040

TL;DR: In this article, the validity of employing an equivalent symmetrical anode for expressing the space-charge-limited current between nonsymmetrical electrodes is examined, and it is shown that when the equivalent anode is determined by the capacitance, the error in the expression for the current is usually quite small and its approximate magnitude may often be anticipated.

Abstract: The validity of employing an equivalent symmetrical anode for expressing the space‐charge‐limited current between nonsymmetrical electrodes is examined. It is shown that when the equivalent anode is determined by the capacitance, the error in the expression for the current is usually quite small and its approximate magnitude may often be anticipated. Exact expressions for the capacitance of some basic electrode arrangements are given, followed by a brief resume of corrections required for interpretation of experimental data.

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TL;DR: In this paper, the exact analytic solutions for space-charge-limited currents (SCLCs) are derived for concentric cylindrical and spherical geometries using variational calculus (VC).

Abstract: While exact analytic solutions for space-charge-limited currents (SCLCs) are well-established for parallel plate geometries, they have only recently been derived for concentric cylindrical and spherical geometries using variational calculus (VC). However, actual diode systems and slow-wave structures are usually more complicated, making SCLC calculations more difficult. In this article, we apply conformal mapping to derive the analytical solutions for SCLC for various complicated geometries exhibiting curvilinear flow. We first replicate the exact solution of SCLC for concentric cylindrical electrodes from VC using conformal mapping to transform from the Child–Langmuir (CL) law for a planar geometry. We then derive SCLC in other geometries using conformal transformations to either the planar or the concentric cylinder solution. Because the SCLC calculated using such conformal mappings depends only on the CL law, this may permit future incorporation of relativistic or quantum corrections to determine the appropriate relationships for more complicated geometries.

26 citations

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TL;DR: In this paper, the principal methods by which these two problems can be solved and provides a critical review of the various suggestions that have been made to improve these methods or to replace them by new techniques.

Abstract: Publisher Summary The incorporation of a high-density electron beam in an electron tube poses two problems: the design of an electron gun capable of providing a sufficient number of electrons, and the selection of an efficient method of holding the resulting electron beam together in the face of space-charge forces and other effects leading to divergence This chapter describes the principal methods by which these two problems can be solved and provides a critical review of the various suggestions that have been made to improve these methods or to replace them by new techniques It describes the Pierce method and evaluates several methods of arriving at a Pierce gun by alternate design procedures, which are thought to be more systematic than that proposed originally Other interesting configurations that differ from Pierce's are evaluated in the chapter To describe the methods of counteracting beam spreading three schemes are presented in the chapter: (1)the magnetic field surrounds the entire tube, so that the cathode is immersed in the field; (2) the magnetic field is excluded from the cathode by a ferromagnetic shield; and (3) the magnetic focusing field is periodic in nature, being produced by a series of magnetic lenses of alternating polarities

8 citations

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TL;DR: In this article, a grid of spherical shape is used in front of the cathode for lowvoltage gating of the electron beam, and a means for calculating these displacements as a function of distance along the axis is developed.

Abstract: In a conventional Pierce-type gun, the anode aperture causes a potential reduction in the cathode-anode region from the ideal Langmuir potential distribution. For low-voltage gating of the electron beam, a mesh grid of spherical shape (conforming to an equipotential surface) is used in front of the cathode. When this grid is operated at the Langmuir potential depicted by its relative position, there is a difference in the potential gradients on its two sides. This difference causes a lens action at each mesh element which results in a displacement of the actual electron trajectory from the ideal laminar trajectory in the region beyond the anode. A means for calculating these displacements as a function of distance along the axis is developed. As the grid lenses are divergent, the images of the mesh elements in any plane beyond the anode are larger than those for ideal laminar flow, resulting in a current density distribution which differs from that of the ideal beam. A means of calculating the current density profile by summing the effects of the grid lenses is devised, and the method is applied to a sample gun design to illustrate the effect on the current density distribution.

4 citations

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General Electric

^{1}TL;DR: In this article, Adams et al. considered the effect of space charge and cathode temperature on thermionic current and potential distribution on parallel plane electrodes and provided an approximate solution for the current.

Abstract: Effect of space charge and cathode temperature on thermionic current and potential distribution.---I. Case of parallel plane electrodes. (a) Current limited by space charge. The results obtained by E. Q. Adams (unpublished), Epstein, Fry and Laue are discussed and summarized, certain errors are pointed out, and the equations are put in a form adapted to easy numerical calculation. Assuming the normal components of the velocities of the emitted electrons have the Maxwell distribution, the integration of Poisson's equation between proper limits leads to a numerical relation between the new variables $\ensuremath{\xi}=2(x\ensuremath{-}{x}_{m}){[\frac{2{\ensuremath{\pi}}^{3}{e}^{2}{i}^{2}m}{{k}^{3}{T}^{3}}]}^{\frac{1}{4}}$ and $\ensuremath{\eta}=\frac{e(V\ensuremath{-}{V}_{m})}{\mathrm{kT}}$, where ${x}_{m}$ and ${V}_{m}$ give the position and voltage of the plane of minimum potential, and $k$ is the Boltzmann gas constant. Denoting values at the cathode by the subscript 1, and inserting values of constants: ${\ensuremath{\eta}}_{1}=log (\frac{{i}_{0}}{i})$, where ${i}_{0}$ is the saturation current; $V\ensuremath{-}{V}_{1}=\frac{T(\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}_{1})}{11,600}$, $\ensuremath{\xi}\ensuremath{-}{\ensuremath{\xi}}_{1}=9.180\ifmmode\times\else\texttimes\fi{}{10}^{5}{T}^{\ensuremath{-}\frac{3}{4}}{i}^{\frac{1}{2}}(x\ensuremath{-}{x}_{1})$. These equations and the tables of $\ensuremath{\xi}(\ensuremath{\eta})$ for various values of $\ensuremath{\eta}$ enable, for a given cathode temperature $T$, the potential distribution for a given current $i$, or vice versa, to be computed. An approximate solution for the current is: $i=[\frac{(\frac{{2}^{\frac{1}{2}}}{9\ensuremath{\pi}}){(\frac{e}{m})}^{\frac{1}{2}}{(V\ensuremath{-}{V}_{m})}^{\frac{3}{2}}}{{(x\ensuremath{-}{x}_{m})}^{2}}](1+2.66{\ensuremath{\eta}}^{\ensuremath{-}\frac{1}{2}})$, which reduces to the usual three halves power law equation if we neglect ${V}_{m}$ and ${x}_{m}$ and the correction factor in $\ensuremath{\eta}$. (b) Equilibrium condition with anode at great distance, current zero. If the only retarding field is that of the space charge, the density of charge is: $\ensuremath{\rho}=\frac{\mathrm{kT}{\ensuremath{\rho}}_{1}}{{[{(\mathrm{kT})}^{\frac{1}{2}}+x{(2\ensuremath{\pi}e{\ensuremath{\rho}}_{1})}^{\frac{1}{2}}]}^{2}}$ where ${\ensuremath{\rho}}_{1}={i}_{0}{[\frac{2\ensuremath{\pi}m}{\mathrm{kT}}]}^{\frac{1}{2}}$. Except near the cathode this is approximately equal to $\frac{\mathrm{kT}}{2\ensuremath{\pi}e{x}^{2}}$; hence $\ensuremath{\rho}$ is proportional to the absolute temperature of the cathode and inversely proportional to the square of the distance away. The potential gradient at the cathode is ${X}_{1}={[\frac{8\ensuremath{\pi}{\ensuremath{\rho}}_{1}\mathrm{kT}}{e}]}^{\frac{1}{2}}$. Equations are also given for the case where an external retarding field ${X}_{\ensuremath{\infty}}$ is applied. II. In the case of concentric cylindrical electrodes, the current is: $i=\frac{(\frac{{8}^{\frac{1}{2}}}{9}){(\frac{e}{m})}^{\frac{1}{2}}{[V\ensuremath{-}{V}_{m}+\frac{1}{4}{V}_{0}{{log (\frac{V}{\ensuremath{\lambda}{V}_{0}})}}^{2}]}^{\frac{3}{2}}}{r}$, where ${V}_{0}$ is the initial energy of the electrons expressed in volts ($\frac{3kT}{2e}$), $r$ is the radius of the anode, and $\ensuremath{\lambda}$ is a constant between 1 and 2, not yet experimentally determined. The deviations from the three halves power law are not more than one quarter as much as for parallel planes and amount to only about 3 per cent at 130 volts.

526 citations