## Mean free path in radiography

## Mean free path in particle physics

## Mean free path in nuclear physics

## Mean free path in optics

## Mean free path in acoustics

## Examples

## Derivation

## Mean free path in kinetic theory

In physics, the **mean free path** is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties.

The following table lists some typical values for air at different pressures at room temperature.

Vacuum range | Pressure in hPa (mbar) | Pressure in mmHg (Torr) | Molecules / cm | Molecules / m | Mean free path |
---|---|---|---|---|---|

Ambient pressure | 1013 | 759.8 | 2.7 × 10 | 2.7 × 10 | 68 nm |

Low vacuum | 300 – 1 | 225 – 7.501×10 | 10 – 10 | 10 – 10 | 0.1 – 100 μm |

Medium vacuum | 1 – 10 | 7.501×10 – 7.501×10 | 10 – 10 | 10 – 10 | 0.1 – 100 mm |

High vacuum | 10 – 10 | 7.501×10 – 7.501×10 | 10 – 10 | 10 – 10 | 10 cm – 1 km |

Ultra-high vacuum | 10 – 10 | 7.501×10 – 7.501×10 | 10 – 10 | 10 – 10 | 1 km – 10 km |

Extremely high vacuum | <10 | <7.501×10 | <10 | <10 | >10 km |

Mean free path for photons in energy range from 1 keV to 20 MeV for elements with *Z* = 1 to 100. The discontinuities are due to low density of gas elements. Six bands correspond to neighborhoods of six noble gases. Also shown are locations of absorption edges.

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:

- $scriptlevel="0">\ell ={\mu}^{-1}=((\mu /\rho )\rho {)}^{-1},\{\backslash displaystyle\; \backslash ell\; =\backslash mu\; ^\{-1\}=((\backslash mu\; /\backslash rho\; )\backslash rho\; )^\{-1\},\}$

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 NIST databases

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. Because of spectrum hardening, the *mean free path* of the X-ray spectrum changes with distance.

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 37% (1/*e*) of photons. This concept is closely related to half-value layer (HVL): a material with a thickness of one HVL will attenuate 50% of photons. A standard x-ray image is a transmission image, an image with negative logarithm of its intensities is sometimes called a *number of mean free paths* image.

In particle physics the concept of the mean free path is not commonly used, being replaced by the similar concept of attenuation length. 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.

Independent-particle models in nuclear physics require the undisturbed orbiting of nucleons within the nucleus before they interact with other nucleons. Blatt and Weisskopf, in their 1952 textbook "Theoretical Nuclear Physics" (p. 778) wrote: "The effective mean free path of a nucleon in nuclear matter must be somewhat larger than the nuclear dimensions in order to allow the use of the independent particle model. 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." (quoted by Norman D. Cook in "Models of the Atomic Nucleus" Ed. 2 (2010) Springer, in Chapter 5 "The Mean Free Path of Nucleons in Nuclei").

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:

- $scriptlevel="0">l=\frac{2d}{3\mathrm{\Phi}{Q}_{\text{s}}},\{\backslash displaystyle\; l=\{\backslash frac\; \{2d\}\{3\backslash Phi\; Q\_\{\backslash text\{s\}\}\}\},\}$

where *Q*_{s} is the scattering efficiency factor. *Q*_{s} can be evaluated numerically for spherical particles using Mie theory.

In an otherwise empty cavity, the mean free path of a single particle bouncing off the walls is:

- $scriptlevel="0">l=\frac{4V}{S},\{\backslash displaystyle\; l=\{\backslash frac\; \{4V\}\{S\}\},\}$

where *V* is volume of the cavity and *S* is total inside surface area of cavity. This relation is used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation.

The mean free path is used in the design of chemical apparatus, e.g., systems for distillation. The sizes of atoms and molecules can be estimated from their mean free path. MFP can be used to estimate the resistivity of a material from the mean free path of its electrons.

In aerodynamics, the mean free path has the same order of magnitude as the shockwave thickness at Mach numbers greater than one.

Slab of target

Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (see the figure). 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. An expression for the MFP is

- $scriptlevel="0">\ell =(\sigma n{)}^{-1},\{\backslash displaystyle\; \backslash ell\; =(\backslash sigma\; n)^\{-1\},\}$

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 area of the slab is *L*, and its volume is *L* *dx*. The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., *n L* *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:

- $scriptlevel="0">P(\text{stopping within}dx)=\frac{{\text{Area}}_{\text{atoms}}}{{\text{Area}}_{\text{slab}}}=\frac{\sigma n{L}^{2}\phantom{\rule{thinmathspace}{0ex}}dx}{{L}^{2}}=n\sigma \phantom{\rule{thinmathspace}{0ex}}dx,\{\backslash displaystyle\; P(\{\backslash text\{stopping\; within\; \}\}dx)=\{\backslash frac\; \{\{\backslash text\{Area\}\}\_\{\backslash text\{atoms\}\}\}\{\{\backslash text\{Area\}\}\_\{\backslash text\{slab\}\}\}\}=\{\backslash frac\; \{\backslash sigma\; nL^\{2\}\backslash ,dx\}\{L^\{2\}\}\}=n\backslash sigma\; \backslash ,dx,\}$

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 the particle being stopped within the slab:

- $scriptlevel="0">dI=-In\sigma \phantom{\rule{thinmathspace}{0ex}}dx.\{\backslash displaystyle\; dI=-In\backslash sigma\; \backslash ,dx.\}$

This is an ordinary differential equation:

- $scriptlevel="0">\frac{dI}{dx}=-In\sigma \stackrel{\text{def}}{=}-\frac{I}{\ell},\{\backslash displaystyle\; \{\backslash frac\; \{dI\}\{dx\}\}=-In\backslash sigma\; \{\backslash overset\; \{\backslash text\{def\}\}\{=\}\}-\{\backslash frac\; \{I\}\{\backslash ell\; \}\},\}$

whose solution is known as Beer-Lambert law and has the form $scriptlevel="0">I={I}_{0}{e}^{-x/\ell}\{\backslash displaystyle\; I=I\_\{0\}e^\{-x/\backslash ell\; \}\}$, where x is the distance traveled by the beam through the target, and *I*_{0} is the beam intensity before it entered the target; 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

- $scriptlevel="0">dP(x)=\frac{I(x)-I(x+dx)}{{I}_{0}}=\frac{1}{\ell}{e}^{-x/\ell}dx.\{\backslash displaystyle\; dP(x)=\{\backslash frac\; \{I(x)-I(x+dx)\}\{I\_\{0\}\}\}=\{\backslash frac\; \{1\}\{\backslash ell\; \}\}e^\{-x/\backslash ell\; \}dx.\}$

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

- $scriptlevel="0">\u27e8x\u27e9\stackrel{\text{def}}{=}{\int}_{0}^{\mathrm{\infty}}xdP(x)={\int}_{0}^{\mathrm{\infty}}\frac{x}{\ell}{e}^{-x/\ell}\phantom{\rule{thinmathspace}{0ex}}dx=\ell .\{\backslash displaystyle\; \backslash langle\; x\backslash rangle\; \{\backslash overset\; \{\backslash text\{def\}\}\{=\}\}\backslash int\; \_\{0\}^\{\backslash infty\; \}xdP(x)=\backslash int\; \_\{0\}^\{\backslash infty\; \}\{\backslash frac\; \{x\}\{\backslash ell\; \}\}e^\{-x/\backslash ell\; \}\backslash ,dx=\backslash ell\; .\}$

The fraction of particles that are not stopped (attenuated) by the slab is called transmission $scriptlevel="0">T=I/{I}_{0}={e}^{-x/\ell}\{\backslash displaystyle\; T=I/I\_\{0\}=e^\{-x/\backslash ell\; \}\}$, where x is equal to the thickness of the slab *x = dx*.

In kinetic theory the *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 $scriptlevel="0">\ell =(n\sigma {)}^{-1},\{\backslash displaystyle\; \backslash ell\; =(n\backslash sigma\; )^\{-1\},\}$ 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, the following relationship applies:

- $scriptlevel="0">\ell =(\sqrt{2}\phantom{\rule{thinmathspace}{0ex}}n\sigma {)}^{-1},\{\backslash displaystyle\; \backslash ell\; =(\{\backslash sqrt\; \{2\}\}\backslash ,n\backslash sigma\; )^\{-1\},\}$

and using $n=N/V=p/(k_{\text{B}}T)$ (ideal gas law) and $scriptlevel="0">\sigma =\pi (2r{)}^{2}=\pi {d}^{2}\{\backslash displaystyle\; \backslash sigma\; =\backslash pi\; (2r)^\{2\}=\backslash pi\; d^\{2\}\}$ (effective cross-sectional area for spherical particles with radius $r$), it may be shown that the mean free path is

- $\ell ={\frac {k_{\text{B}}T}{{\sqrt {2}}\pi d^{2}p}},$

where *k*_{B} is the Boltzmann constant.

In practice, the diameter of gas molecules is not well defined. In fact, the kinetic diameter of a molecule is defined in terms of the mean free path. Typically, gas molecules do not behave like hard spheres, but rather attract each other at larger distances and repel each other at shorter distances, as can be described with a Lennard-Jones potential. One way to deal with such "soft" molecules is to use the Lennard-Jones σ parameter as the diameter. Another way is to assume a hard-sphere gas that has the same viscosity as the actual gas being considered. This leads to a mean free path

- $\ell ={\frac {\mu }{p}}{\sqrt {\frac {\pi k_{\text{B}}T}{2m}}},$

where *m* is the molecular mass, and *μ* is the viscosity. These different definitions of the molecular diameter can lead to slightly different values of the mean free path.

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