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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.


DERIVATION


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}dx. The typical number of stopping atoms in the slab is the concentration ''n'' times the volume, i.e., n L^{2}dx. 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}) =
rac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} =
rac{\sigma n L^{2} dx}{L^{2}} = n \sigma dx


where \sigma 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 = -I n \sigma dx


This is an Ordinary Differential Equation

:
rac{dI}{dx} = -I n \sigma \ \stackrel{\mathrm{def}}{=}\ - rac{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) = rac{I(x)-I(x+dx)}{I_0} = rac{1}{\ell} e^{-x/\ell} dx.
Thus the expectation value (or average, or simply mean) of x is

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


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 Electron s.

For example, for Sound Wave s in an enclosure, the mean free path is the average distance the wave travels between Reflection s off the enclosure's walls.


SEE ALSO



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