Re: An idle question...

On Sun, 22 Jul 2007 10:55:25 -0700, Edward Green wrote:

I've come across the assertion that if scattering centers are randomly
distributed in a plane with desnity (of centers) N and diameter D,
that the mean free path is:

mfp = 1/(ND)

I've been think about this, and I can imagine some complicated
integral approaches which might involve magical cancellation, but I
can't think of a simple argument.

Comments?

This looks like cross-sections, except we're in two dimensions.

So how about this:

First we calculate the probability of a collision in a short distance dx.
We want dx much shorter than the mean free path, but much longer than the
diameter of the scattering centers.

(By doing this we're assuming that the mean free path is much larger than
the diameter of the scattering centers; otherwise we'd have to deal with
either overlapping scattering centers, or a less-than-perfectly-random
distribution. I suspect the formula is only supposed to apply when the
mean free path is much larger than the diameter.)

To calculate the probability of a collision in the distance dx, we look at
a rectangular strip of dimensions y*dx (y >> mean free path).

[View in fixed-width font.]
__
| |
| o|
| |
|o |
y | o|
| |
*--->
|o |
|__|
dx

* = particle
o = scattering center

The number of scattering centers in the strip is N*y*dx. We imagine the
particle starts at a random point on one of the long sides of the strip and
crosses the strip in the x-direction. Since dx is small, the chance of one
scattering center obscuring another is negligible. So the portion of the
long side of the strip such that a particle launched from there would be
scattered is the number of scattering centers N*y*dx times their diameter
D. That makes the probability of scattering in length dx

P(dx) = (N*y*dx*D)/y = N*D*dx.

We assume that for each length dx the particle travels, the probability of
scattering remains the same. The probability the particle is *not*
scattered in length x is then

P(x) = (1 - N*D*dx)^(x/dx) = exp(-N*D*x).

The probability a particle is scattered between distances x and x+dx is

P(x) - P(x+dx).

So to get the mean free path, we do the integral

mfp = int_{x=0}^{inf} x [P(x) - P(x+dx)]
= - int_{x=0}^{inf} x dP
= int_{x=0}^{inf} P(x) dx
= int_{x=0}^{inf} exp(-N*D*x) dx
= 1/(ND).

--
Hope this helps,
Jim E. Black
.

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