Mean free path

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),^[1] 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 / cm³	Molecules / m³	Mean free path
Ambient pressure	1013	759.8	2.7 × 10¹⁹	2.7 × 10²⁵	68 nm^[2]
Low vacuum	300 – 1	225 – 7.501×10⁻¹	10¹⁹ – 10¹⁶	10²⁵ – 10²²	0.1 – 100 μm
Medium vacuum	1 – 10⁻³	7.501×10⁻¹ – 7.501×10⁻⁴	10¹⁶ – 10¹³	10²² – 10¹⁹	0.1 – 100 mm
High vacuum	10⁻³ – 10⁻⁷	7.501×10⁻⁴ – 7.501×10⁻⁸	10¹³ – 10⁹	10¹⁹ – 10¹⁵	10 cm – 1 km
Ultra-high vacuum	10⁻⁷ – 10⁻¹²	7.501×10⁻⁸ – 7.501×10⁻¹³	10⁹ – 10⁴	10¹⁵ – 10¹⁰	1 km – 10⁵ km
Extremely high vacuum	<10⁻¹²	<7.501×10⁻¹³	<10⁴	<10¹⁰	>10⁵ km

Mean free path in radiography

Mean free path for photons in energy range from 1 keV to 20 MeV for elements with Z = 1 to 100.^[3] 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:

\ell =\mu ^{{-1}}=((\mu /\rho )\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^[4]^[5]

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.

Mean free path in particle physics

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.

Mean free path in nuclear physics

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").^[6]

Mean free path in optics

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:^[7]

l={\frac {2d}{3\Phi Q_{\text{s}}}},

where Q_s is the scattering efficiency factor. Q_s can be evaluated numerically for spherical particles using Mie theory.

Mean free path in acoustics

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

l={\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.^[8]

Examples

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.

Derivation

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

\ell =(\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 $L 2$ , 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({\text{stopping within }}dx)={\frac {{\text{Area}}_{\text{atoms}}}{{\text{Area}}_{\text{slab}}}}={\frac {\sigma nL^{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 the particle being stopped within the slab:

dI=-In\sigma \,dx.

This is an ordinary differential equation:

{\frac {dI}{dx}}=-In\sigma {\overset {\text{def}}{=}}-{\frac {I}{\ell }},

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

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 {\overset {\text{def}}{=}}\int _{0}^{\infty }xdP(x)=\int _{0}^{\infty }{\frac {x}{\ell }}e^{-x/\ell }\,dx=\ell .

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

Mean free path in kinetic theory

In kinetic theory 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 $\ell =(n\sigma )^{{-1}},$ 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, the following relationship applies:^[9]

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

and using $n=N/V=p/(k_{\text{B}}T)$ (ideal gas law) and $\sigma =\pi (2r)^{2}=\pi d^{2}$ (effective cross-sectional area for spherical particles with radius $r$ ), it may be shown that the mean free path is^[10]

\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.

References

↑ Brünglinghaus, Marion. "Mean free path". European Nuclear Society. Retrieved 2011-11-08.
↑ Jennings, S (1988). "The mean free path in air". Journal of Aerosol Science. 19 (2): 159. doi:10.1016/0021-8502(88)90219-4.
↑ Based on data from "NIST: Note - X-Ray Form Factor and Attenuation Databases". Physics.nist.gov. 1998-03-10. Retrieved 2011-11-08.
↑ Hubbell, J. H.; Seltzer, S. M. "Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients". National Institute of Standards and Technology. Retrieved September 2007. Check date values in: |access-date= (help)
↑ Berger, M. J.; Hubbell, J. H.; Seltzer, S. M.; Chang, J.; Coursey, J. S.; Sukumar, R.; Zucker, D. S. "XCOM: Photon Cross Sections Database". National Institute of Standards and Technology (NIST). Retrieved September 2007. Check date values in: |access-date= (help)
↑ Cook, Norman D. Models of the Atomic Nucleus. Heidelberg: Springer. p. 324. ISBN 978-3-642-14736-4.
↑ Mengual, O.; Meunier, G.; Cayré, I.; Puech, K.; Snabre, P. (1999). "TURBISCAN MA 2000: multiple light scattering measurement for concentrated emulsion and suspension instability analysis". Talanta. 50 (2): 445–56. doi:10.1016/S0039-9140(99)00129-0. PMID 18967735.
↑ Davis, D. and Patronis, E. "Sound System Engineering" (1997) Focal Press, ISBN 0-240-80305-1 p. 173.
↑ S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases, 3rd. edition, Cambridge University Press, 1990, ISBN 0-521-40844-X, p. 88.
↑ "Mean Free Path, Molecular Collisions". Hyperphysics.phy-astr.gsu.edu. Retrieved 2011-11-08.

External links

Gas Dynamics Toolbox: Calculate mean free path for mixtures of gases using VHS model

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