Mean free path

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In physics and kinetic theory, the mean free path of a particle, such as a molecule, is the average distance the particle travels between collisions with other particles.

The formula for calculating the magnitude of the mean free path depends on the characteristics of the system the particle is in. For a particle with a high velocity relative to the velocities of an ensemble of identical particles with random locations, the following relationship applies:

$\ell = (n\sigma)^{-1},$

Where $\ell$ is the mean free path, n is the number of particles per unit volume, and σ is the effective cross sectional area for collision. If, on the other hand, the velocities of the identical particles have a Maxwell distribution of velocities, the following relationship applies:

$\ell = (\sqrt{2}\, n\sigma)^{-1}.\,$

Following table lists some typical values for different pressures.

Vacuum range	Pressure in hPa	Molecules / cm³	mean free path
Ambient pressure	1013	2.7*10¹⁹..	68 nm
Low vacuum	300..1	10¹⁹..10¹⁶	0.1..100 μm
Medium vacuum	1..10^-3	10¹⁶..10¹³	0.1..100 mm
High vacuum	10^-3..10^-7	10¹³..10⁹	10 cm..1 km
Ultra high vacuum	10^-7..10^-12	10⁹..10⁴	1 km..10⁵ km
Extremely high vacuum	<10^-12	<10⁴	>10⁵ km

1 Derivation
2 Examples
3 See also
4 External links

[edit] Derivation

Figure 1: Slab of target

Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (Figure 1). The atoms that might stop a beam particle are shown in red. The area of the slab is $L 2$ and its volume is $L 2 d x$ . The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., $n L 2 d x$ . 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.

$P(\mathrm{stopping \ within\ dx}) = \frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} = \frac{\sigma n L^{2} dx}{L^{2}} = n \sigma dx$

where $σ$ 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 being stopped within the slab

d I = - I n σ d x

This is an ordinary differential equation

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

whose solution is $I = I_{0} e^{-x/\ell}$ , where $x$ is the distance traveled by the beam through the target and $I 0$ is the beam intensity before it entered the target.

$\ell$ 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 the a particle is absorbed between x and x+dx is given by

$dP(x) = \frac{I(x)-I(x+dx)}{I_0} = \frac{1}{\ell} e^{-x/\ell} dx.$

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

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

[edit] Examples

A classic application of mean free path is to estimate the size of atoms or molecules. Another important application is in estimating the resistivity of a material from the mean free path of its electrons.

For example, for sound waves in an enclosure, the mean free path is the average distance the wave travels between reflections off the enclosure's walls.