Mean Free Path - Derivation

Derivation

Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (Figure 1). The atoms (or particles) that might stop a beam particle are shown in red. The magnitude of the mean free path depends on the characteristics of the system the particle is in:

Where is the mean free path, n is the number of target particles per unit volume, and is the effective cross sectional area for collision.

The area of the slab is and its volume is . The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., . The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab.

P(\mathrm{stopping \ within\ d}x) = \frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} = \frac{\sigma n L^{2}\, \mathrm{d}x}{L^{2}} = n \sigma\, \mathrm{d}x

where is the area (or, more formally, the "scattering cross-section") of one atom.

The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab

dI = -I n \sigma dx

This is an ordinary differential equation

\frac{dI}{dx} = -I n \sigma \ \stackrel{\mathrm{def}}{=}\ -\frac{I}{\ell}

whose solution is known as Beer-Lambert law and has the form, where x is the distance travelled by the beam through the target and I0 is the beam intensity before it entered the target; ℓ is called the mean free path because it equals the mean distance traveled by a beam particle before being stopped. To see this, note that the probability that a particle is absorbed between x and x + dx is given by

Thus the expectation value (or average, or simply mean) of x is

\langle x \rangle \ \stackrel{\mathrm{def}}{=}\ \int_0^\infty x dP(x) = \int_0^\infty \frac{x}{\ell} e^{-x/\ell} dx = \ell

The fraction of particles that are not stopped (attenuated) by the slab is called transmission where x is equal to the thickness of the slab x = dx.

Read more about this topic:  Mean Free Path