Mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a result of one or more successive collisions with other particles.
The magnitude of the mean free path depends on the characteristics of the system.
Assuming that all the target particles are at rest but only the beam particle is moving, that gives an expression for the mean free path: where ℓ is the mean free path, n is the number of target particles per unit volume, and σ is the effective cross-sectional area for collision.
The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., n L2 dx.
The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab: This is an ordinary differential equation: whose solution is known as Beer–Lambert law and has the form
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 The fraction of particles that are not stopped (attenuated) by the slab is called transmission
The derivation above assumed the target particles to be at rest; therefore, in reality, the formula
relative to the velocities of an ensemble of identical particles with random locations.
In that case, the motions of target particles are comparatively negligible, hence the relative velocity
(effective cross-sectional area for spherical particles with diameter
), it may be shown that the mean free path is[3] where kB is the Boltzmann constant,
In fact, the kinetic diameter of a molecule is defined in terms of the mean free path.
One way to deal with such "soft" molecules is to use the Lennard-Jones σ parameter as the diameter.
The following table lists some typical values for air at different pressures at room temperature.
Note that different definitions of the molecular diameter, as well as different assumptions about the value of atmospheric pressure (100 vs 101.3 kPa) and room temperature (293.17 K vs 296.15 K or even 300 K) can lead to slightly different values of the mean free path.
In gamma-ray radiography the 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 the material and the energy of the photons: where μ is the linear attenuation coefficient, μ/ρ is the mass attenuation coefficient and ρ is the density of the material.
The mass attenuation coefficient can be looked up or calculated for any material and energy combination using the National Institute of Standards and Technology (NIST) databases.
[7][8] In X-ray radiography the calculation of the mean free path is more complicated, because photons are not mono-energetic, but have some distribution of energies called a 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.
Sometimes one measures the thickness of a material in the number of mean free paths.
Material with the thickness of one mean free path will attenuate to 37% (1/e) of photons.
is the mean free time, m* is the effective mass, and vF is the Fermi velocity of the charge carrier.
In thin films, however, the film thickness can be smaller than the predicted mean free path, making surface scattering much more noticeable, effectively increasing the resistivity.
In such scenarios electrons alter their motion only in collisions with conductor walls.
If one takes a suspension of non-light-absorbing particles of diameter d with a volume fraction Φ, the mean free path of the photons is:[9] where Qs is the scattering efficiency factor.
Qs can be evaluated numerically for spherical particles using Mie theory.
[10] This relation is used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation.
In particular, for high-energy photons, which mostly interact by electron–positron pair production, the radiation length is used much like the mean free path in radiography.
This requirement seems to be in contradiction to the assumptions made in the theory ... We are facing here one of the fundamental problems of nuclear structure physics which has yet to be solved.
