Child’s Law¶

Child’s Law (or the Child-Langmuir Law or three-halves-power law) gives the maximum space-charge-limited current in a planar diode of infinite radius (i.e. one-dimensional beam) as a function of the length and potential difference between anode and cathode.

$J = K V_d^{3/2} / d^2$

where

$J$ = current density (mA mm^-2)
$V_d$ = potential difference between anode and cathode (V)
$d$ = distance between anode and cathode (mm)
$K$ is a constant given by $K = (4/9) \epsilon_0 (-2q/m)^{1/2}$ with units mA V^-3/2, where
- $\epsilon_0$ = permittivity of free-space ~ 8.854187817*10^-12 F m^-1 [1]
- $q/m$ = charge (C) to mass (kg) ratio of particle (absolute value). For an electron, q/m = 1.758820150*1011 C kg^-1, [2]
- For an electron, $K \approx$ 0.0023340 mA V^-3/2.

Note units on left-side of constant:

$(F m^{-1}) C^{1/2} kg^{-1/2} = (C^2 J^{-1}) m^{-1} C^{1/2} kg^{-1/2} = C^{5/2} J^{-1} (J^{-1/2} s^{-1}) = (C s^{-1}) (J/C)^{-3/2} = A V^{-3/2}$

This can be applied near cathode surfaces in space-charge limited cathode emissions. A correction may be applied for non-zero temperature or non-planar surfaces.

Derivation¶

Child’s Law is a fairly direct result of the poission_equation:

$\bnabla^2 V = -\rho / \epsilon_0$

where charge density $\rho = J/v$ , for velocity $v$ , and where $v$ can be expressed in terms of kinetic energy and related to potential energy: $(1/2) m v^2 = -q V$ , for mass $m$ and charge $q$ . This gives

$\bnabla^2 V = dV'/dx = A V^{-1/2}$

where $A = -J \epsilon_0^{-1} (-(1/2) m/q)^{1/2}$ is a constant and $V' = dV/dx$ .

Substitute $dx = dV/V'$ to obtain a separable differential equation and solve, while applying boundary conditions $V(x=0) = 0$ , $V(x=d) = Vd$ , and $V'(x=0) = 0$ :