## Diffusion

Diffusion turns curvatures in distributions into changes in time. The effect on a point on a curve is to take a little away from the point, and add a little from what is on both sides of it. This can be expressed as

Where F(x,t) is a function in space and time and d is the diffusion constant.

As a difference equation, this can be reformulated as:

So, what is D? Let's figure it out for Rutherford scattering pitch angle diffusion.

Assume a band of N thinsat atoms occupying a magnetic equatorial band ( B = 90° ) between magnetic radius L and L + d L . Each atom adds a scattering area of:

For small \theta :

This is in all directions. For diffusion that changes the pitch angle \alpha , we need to look at the component in the direction of the field lines only. That will be an annular ring, radius \theta , width d \theta , solid area 2 \pi \theta d \theta .

Let d \alpha = \cos( \gamma ) d \theta . The contribution to the diffusion is proportional to the integral of {\alpha}^2 over the circle:

The area amount d \sigma is multiplied by d N and divided by the area 2 \pi L d L to make d D .

\theta_0 is a very small angle that is a function of the crystal lattice spacing, about 140 picomenters (WAG) - if the distance s approaches that, then the nucleus is shielded by the electrons and the deflecting charge is zero.

\theta_1 must be less than the small angle approximation.

**This needs reworking**, but the point is that we can derive a diffusion constant D , given the energy and the density of the server sky aluminum atoms over the McIlwain L band.