Diffusion

Diffusion definition

Charged particles diffusion in gas is defined as <ref name="Mason"/> the "disperse" of the particles in a gas "in which there is a net spatial transport" of the charged particles "produced by a gradient in their relative concentrations". Assuming that the charged particles are localized in a gas with a uniform temperature, pressure and has low n charged particle density to ignore Coulomb force.

[math] J = -D \nabla n [/math]

where D is the diffusion coefficient and J is the number of the charged particle flow per unit time.

Maxwell Boltzmann equation solution describes n as a function of position r and time t, In our case, n represents the number electrons propagating in presence of an electric field. Maxwell Boltzmann equation is "the equation of continuity for the population n f dr dc " where f is velocity distribution function. The equation includes the loss of electrons as they transport across a surface boundary in a volume element dr , and the effect of the uniform electric field in accelerating each n dr electrons which changes dc from point to another in the phase space so the number of point loss in a time dt is [math] dt \nabla ._{c}(\frac{n f e E}{m}) dc dr [/math], in addition to the loss of points [math]\Delta[/math]n in dc as result of the quasi discontinuous change in position [math]\Delta[/math]c in velocity space as the electron meets a molecule.

So Maxwell Boltzmann equation can be written as based on the previous assumptions as the following:<ref name="Huxley"> Huxley, L. G. H. Leonard George Holden, The diffusion and drift of electrons in gases, John Wiley and sons, 1974 , call number QC793.5.E628 H89 </ref>

[math] \frac {\partial{} }{\partial{t}}[/math](nf) + [math]\nabla._{r}[/math](n f c) + [math]\nabla._{c}[/math] (n f [math]\frac{e E}{m})[/math]+ S = 0

The previous equation can be written in terms of the diffusion coefficients and the average velocity of the electrons, it is called the scalar equation of Maxwell Boltzmann equation as shown below:

[math] \frac {\partial{} }{\partial{t}}(n f_0)[/math] + [math]\frac{c}{3}\nabla._{r}[/math] (n f_1)+ [math]\frac{1}{4\pi c^2}\frac{\partial {}}{\partial{c}} (\sigma_E - \sigma_{coll}) = 0 [/math]

Assumptions: <ref name="Huxley"/>

Velocity Shells

The electrons are distributed in the phase space in velocity shell of mean velocity W represented by the following equation

[math] W = \frac {\sum_c n_c W(c)}{n dr} [/math]

Where W(c) is the resultant velocity of the of the velocities of the electrons in the velocity shell 4[math]\pi c^2 sin\theta dc d\theta d\phi [/math], so the population of velocity points in the shell is represented by the n dr [math] [c^2 dc] [f(c,\theta,r,t)sin\theta dc d\theta d\phi] [/math]. it is assumed for a velocity shell the following distribution function:

[math] f(c,\theta,r,t) = f_0(c,\theta,r,t) + \sum_{k=0} ^\infty f_k(c,\theta,r,t) P_k (cos\theta)[/math]

[math]P_k (cos\theta)[/math] is the Legendre polynomial of order k. In the case the mean velocity is independent of the azimuthal angle then its magnitude can be determined by the following:

[math] W(c) = \frac{1}{n_c} (ndr) c^2 dc \int_0^\pi \int_0^{2\pi}(f_0 +\sum_1^\infty f_k P_k (cos\theta)) c \, cos\theta sin\theta d\theta d\phi = \frac {cf_1}{3f_0}[/math]

and the population desity point is :

[math] n_c = (ndr) c^2 dc \int_0^\pi \int_0^{2\pi}(f_0 +\sum_1^\infty f_k P_k (cos\theta)) sin\theta d\theta d\phi = n f_0 4\pi c^2 dc dr[/math]

So the mean velocity is evaluated depending on the former definition is :

[math] W = \frac {\sum_c n_c W(c)}{n dr} = \frac{cf_1}{3f_0}[/math]

It is worth mentioning here that W represent the mean velocity of the electron population and the instantaneous velocity of the of the centroid of n.

The loss and the gain in the number of points

The loss in the number of points from (c,dc) is depending on [math] \sigma_E (c) [/math] and mathematically can be written as [math] dt dr dc \frac{\partial{}}{\partial{c}}\sigma_E (c) [/math]. Simultaneously, the gain in the number of points (c,dc) is evaluated using [math] \sigma_{coll} (c) [/math] such that [math] dt dr dc \frac{\partial{}}{\partial{c}}\sigma_{coll} (c) [/math] is the gain per unit volume (in phase space) per unit time.

So the net change in the number of points in the shell is [math] dt dr dc\frac{\partial{}}{\partial{c}}(\sigma_E (c)-\sigma_{coll} (c)) [/math].

Where [math] \sigma_{coll} (c) = 4\pi n c^2 \nu_{el} (\frac {m}{M} c f_0 + \frac{\bar{C^2}}{3} \frac{\partial{f_0}}{\partial{c}}) [/math] and [math] \sigma_E (c) = \frac{4\pi}{3} c^2 \frac {eE}{m} n f_1 [/math] , [math] \nu_{el} = Ncq_{el}(c) [/math], N is the molecular density, M is the mass of the molecule, [math] q_{el}(c) [/math] is the momentum transfer cross section for elastic encounters. and [math] \bar {C^2}[/math] is the mean square speed of the molecules.

Drift velocity and diffusion coefficients

The electron density number, diffusion coefficients and drift velocity relationship is studied for a close chamber contained a travelling swam of electrons in a uniform electric field that directed the swarm toward the +z axis. Mathematically the relationship is assumed as the follwoing:

[math] \frac{\partial{}}{\partial{t}}[/math]n - D [math] \nabla^2[/math]n ) + W [math]\frac{\partial{ }}{\partial{z} }[/math] n = 0

where

[math] D = 4 \pi \int_0^{\infty} \frac{c^2}{3\nu} f_0 c^2 dc [/math]

[math]\nu[/math] represents "effective collision frequency for the momentum transfer"<ref name="Huxley"/>, [math]f_0[/math] is the is independent on r for a uniform stream that elastic collisions. In this case, [math]f_0[/math] is a special form of a general form represented by the following equation:

[math] f_0 = A \exp{\int_0^c \frac{c dc }{V^2 + \bar{C^2}} }[/math]

As a result substituting the main formula in the scalar form of Maxwell Boltzmann , in the absence of the magnetic field ( vanishes) and in a uniform electric field E, we get the following formula:

[math] \frac{\partial{}}{\partial{t}}[/math]n - D [math] ( \frac {\partial^2{} }{\partial{x^2} }[/math] + [math] \frac {\partial^2{}} {\partial{ y^2}}[/math]) n - [math] D_L \frac {\partial^2{}} {\partial{z^2} } [/math] n + W [math]\nabla_r n[/math] = 0

D is not isotropic as the eE force is applied and is defined as in the equation above, W is represented by the following:

[math] W = -\frac{4\pi}{3}(\frac{e}{m}) (\frac{E}{N}) \int_0^{\infty} \frac{c^2}{q_m (c)} \frac {df_0}{dc} dc [/math]

For electrons moving along the z-axis:

[math] - \frac{\partial{}}{\partial{t}}[/math]n + D [math] ( \frac {\partial^2{} }{\partial{x^2} }[/math]n + [math] \frac {\partial^2{}} {\partial{ y^2}}[/math]n ) + [math] D_L \frac {\partial^2{}} {\partial{z^2} } [/math] - W [math]\frac{\partial{ }}{\partial{z} }[/math] n = 0

[math] W = -\frac{4\pi}{3}(\frac{e}{m}) (\frac{E}{N}) \int_0^{\infty} \frac{c^2}{q_m (c)} \frac {df_0}{dc} dc [/math]

[math] \sigma_{coll} (c) = 4\pi n c^2 \nu_{el} (\frac {m}{M} c f_0 + \frac{\bar{C^2}}{3} \frac{\partial{f_0}}{\partial{c}}) [/math]

[math] \sigma_E (c) = \frac{4\pi}{3} c^2 \frac {eE}{m} n f_1 [/math]

[math] \nu_{el} = Ncq_{el}(c) [/math]

The solution of the Boltzmann Equation for a steady stream of electrons originated from a small hole in a metal <ref name="Huxley"/>

Boltzmann equation has an asymtotic solution for a stream of electrons originated from a hole in a metal plate extends over the plane z=0, and the hole is at the origin. The solution has considers all the conditions mentioned previously, in addition to have S = 0. In the solution, the electron stream travels toward +z axis, in a uniform electric, the velocity distribution has only the first two terms [math] f_0+f_1 [/math].

Scalar Boltzmann equation can be written for electron stream moving in the +z-direction as :

[math] (\frac {\partial^2{}}{\partial{x^2}} +\frac {\partial^2{}}{\partial{x^2}})[/math]n + [math] D_L \frac {\partial^2{}}{\partial{z^2}}[/math] - [math] W \frac {\partial{}}{\partial{z}}[/math] n = 0

The solution is :

[math] n = \exp{\lambda_L z}\sum_{k=0}^{Infinity} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) [/math]

[math] r'^2 = \sqrt {x'^2+y'^2+z'^2} [/math] with x' = [math] \frac^{1/2} {D_L}{D} x[/math]

The asymptotic solution is only valid for a distance r from the source, where the spatial gradient have become relatively small.

The electron stream has a close surface that contains all the electrons; [math] n_0 = \int n(r,t) dr [/math] , so n(r,t) anywhere else is zero.

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