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# Mean free path

In physics the mean free path of a particle, is the average distance the particle travels between collisions with other particles.

## 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 mean free path depends on the characteristics of the system the particle is in:

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

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.

The area of the slab is L2 and its volume is L2dx. The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., nL2dx. 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

dI = − Inσdx

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 I0 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

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

Fraction of particles that were not stopped (attenuated) by the slab is called transmission $T = \frac{I}{I_{0}} = e^{-x/\ell}$ where x is equal to the thickness of the slab x = dx.

## Mean free path in kinetic theory

In kinetic theory mean free path of a particle, such as a molecule, is the average distance the particle travels between collisions with other moving particles. The formula $\ell = (n\sigma)^{-1},$ still holds for a particle with a high velocity relative to the velocities of an ensemble of identical particles with random locations. 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

## Mean free path in radiography

In gamma-ray radiography mean free path of a pencil-beam of mono-energetic photons, is the average distance a photon travels between collisions with atoms of the target material. It depends on material and energy of the photons:

$\ell = \mu^{-1} = ( (\mu/\rho) \rho)^{-1},$

where μ is linear attenuation coefficient, μ/ρ is mass attenuation coefficient and ρ is density of the material. Mass attenuation coefficient can be looked up or calculated for any material and energy combination using NIST databases [1] [2]

In x-ray radiography the calculation of mean free path is more complicated since photons are not mono-energetic, but have some distribution of energies called spectrum. As photons move through the target material they are attenuated with probabilities depending on their energy, as a result their distribution changes in process called Spectrum Hardening. Because of Spectrum Hardening mean free path of x-ray spectrum changes with distance.

Sometimes people measure thickness of material in number of mean free paths. Material with thickness of one mean free path will attenuate 37% (1/e) of photons. Standard x-ray image is a transmission image, a minus log of it is sometimes referred as number of mean free paths image.

## 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.