A simple physical derivation of Child–Langmuir space-charge-limited emission using vacuum capacitance

Abstract

The fundamental Child–Langmuir limit on the maximum current density in a vacuum between two infinite parallel electrodes is one of the most well known and often applied rules of plasma physics. We develop a simple model using vacuum capacitance, conservation of energy, and conservation of charge to derive the Child–Langmuir space-charge-limited emission. This capacitive model provides physical insight into the origins of the well known (voltage)3/2/(gap distance)2 scaling of the classical current density and does not require the solution of the nonlinear differential equation normally associated with the Child–Langmuir formulation. In addition, the full spacecharge-limited solution is reproduced without imposing the condition that the electric field be driven to zero at the cathode surface.

Discrete space charge affected field emission: Flat and hemisphere emitters

“…(A4)) using the assumption that current density (in [A/cm 2 ]) and number density are related via J ¼ Àqqhvi and that hvi % 0 at the cathode surface. A more intuitive but less exact account is obtained using a transit time model, 17,23,24 which can be used to show that field emission can experience space charge limited (SCL) emission even though (as is common to assume) the potential energy does not pass through a maximum nor does the field at the surface vanish, but rather the field strength is limited, a point emphasized by van der Ziel. 25 It is therefore preferable here to speak of space charge affected field emission (SCAFE) (or "field-emitted vacuum space charge" (FEVSC) as Forbes 19 does), rather than SCL emission, to avoid activating intuitions which do not apply.…”

“…The transit time formalism 17,23,24 relates the field at the surface to the field across the AK gap separation in the absence of space charge (F o ¼ V a =D), the temperature and field dependent current density at the surface J(F, T) (Eq. (A1)), and the transit time (s) measuring how long an electron takes to cross the AK gap by…”

Electron dynamics inside a vacuum tube diode through linear differential equations

“…Since the derivation of this fundamental law, many important and useful variations on the classical Child-Langmuir law have been investigated to account for special geometries Blodgett 1923, 1924;Page and Adams 1945), relativistic electron energies (Jory and Trivelpiece 1969), non-zero initial electron velocities (Langmuir 1923;Jaffé 1944), quantum mechanical effects (Lau et al 1991;Ang et al 2003;Jensen et al 2012), non-zero electric field on the cathode surface (Barbour et al 1953), and using vacuum capacitance (Umstattd et al 2005). In technology, spacecharge effects can help to stabilize emission current and to inhibit sharp focusing of particle beams (Forbes 2008;Rokhlenko et al 2010).…”

On thermionic emission and the use of vacuum tubes in the advanced physics laboratory

“…10 Several articles have used the Richardson-Dushman equation and the Child-Langmuir equation ͑vide infra͒ to determine either fundamental constants or the work-function of the metal cathode. [11][12][13][14] This article is not intended to further this work but will discuss a method for measuring the charge-to-mass ratio of the electron. The unique aspect of this article is that we will use standard vacuum tube diodes rather than diodes designed and constructed specifically for pedagogical use.…”