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| In [[physics]], the '''mean free path''' is the average distance travelled by a moving particle (such as an [[atom]], a [[molecule]], a [[photon]]) between successive impacts (collisions), <ref>{{cite web|author=Author: Marion Brünglinghaus, ENS, European Nuclear Society |url=http://www.euronuclear.org/info/encyclopedia/m/mean-fee-path.htm |title=Mean free path |publisher=Euronuclear.org |date= |accessdate=2011-11-08}}</ref> which modify its direction or energy or other particle properties.
| | 58 yr old Marine Biologist Ramey from Canada, usually spends time with hobbies and interests which include lacemaking, camera accessories and kayaking. Has recently completed a journey to Tsodilo.<br><br>Visit my blog post; [http://www.traveleasy.nu/content/talks-microsoft-other-people canon cameras and accessories] |
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| ==Derivation==
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| [[File:Mean free path.png|frame|Figure 1: Slab of target]]
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| 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 the mean free path depends on the characteristics of the system the particle is in:
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| :<math>\ell = (\sigma n)^{-1},</math>
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| Where <math>\ell</math> is the mean free path, ''n'' is the number of target particles per unit volume, and <math>\sigma</math> is the effective [[cross section (physics)|cross sectional]] area for collision.
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| The area of the slab is <math>L^{2}</math> and its volume is <math>L^{2}dx</math>. The typical number of stopping atoms in the slab is the concentration ''n'' times the volume, i.e., <math>n L^{2}dx</math>. 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.
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| :<math>
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| P(\mathrm{stopping \ within\ d}x) =
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| \frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} =
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| \frac{\sigma n L^{2}\, \mathrm{d}x}{L^{2}} = n \sigma\, \mathrm{d}x
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| </math>
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| where <math>\sigma</math> is the area (or, more formally,
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| the "[[scattering cross-section]]") of one atom.
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| The drop in beam intensity equals the incoming beam intensity
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| multiplied by the probability of the particle being stopped within the slab
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| :<math>
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| dI = -I n \sigma dx
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| </math>
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| This is an [[ordinary differential equation]]
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| :<math>
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| \frac{dI}{dx} = -I n \sigma \ \stackrel{\mathrm{def}}{=}\ -\frac{I}{\ell}
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| </math>
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| whose solution is known as [[Beer-Lambert law]] and has the form <math>I = I_{0} e^{-x/\ell}</math>, where ''x'' is the distance travelled by the beam through the target and ''I<sub>0</sub>'' 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
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| :<math>dP(x) = \frac{I(x)-I(x+dx)}{I_0} = \frac{1}{\ell} e^{-x/\ell} dx.</math>
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| Thus the expectation value (or average, or simply mean) of ''x'' is
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| :<math>
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| \langle x \rangle \ \stackrel{\mathrm{def}}{=}\ \int_0^\infty x dP(x) = \int_0^\infty \frac{x}{\ell} e^{-x/\ell} dx = \ell
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| </math>
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| The fraction of particles that are not stopped ([[attenuation|attenuated]]) by the slab is called [[Transmittance|transmission]] <math>T = \frac{I}{I_{0}} = e^{-x/\ell}</math> where ''x'' is equal to the thickness of the slab ''x = dx''.
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| ==Mean free path in kinetic theory==
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| 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 <math>\ell = (n\sigma)^{-1},</math> 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:<ref>S. Chapman and T.G. Cowling, [http://books.google.com/books?id=Cbp5JP2OTrwC&pg=PA88 ''The mathematical theory of non-uniform gases''], 3rd. edition, Cambridge University Press, 1990, ISBN 0-521-40844-X, p. 88</ref>
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| :<math>\ell = (\sqrt{2}\, n\sigma)^{-1}.\,</math>
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| and it may be shown that the mean free path, in meters, is:<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html |title=Mean Free Path, Molecular Collisions |publisher=Hyperphysics.phy-astr.gsu.edu |date= |accessdate=2011-11-08}}</ref>
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| :<math>\ell = \frac{k_{\rm B}T}{\sqrt 2 \pi d^2 p}</math>
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| where ''k''{{sub|B}} is the [[Boltzmann constant]] in J/K, ''T'' is the temperature in K, ''p'' is pressure in [[Pascals]], and ''d'' is the diameter of the gas particles in meters.
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| Following table lists some typical values for air at different pressures and at room temperature.
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| {| class="wikitable"
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| |-
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| ! style="width:20%;"|Vacuum range
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| ! style="width:20%;"|[[Pressure]] in [[pascal (unit)|hPa (mbar)]]
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| ! style="width:20%;"|[[Molecules]] / cm<sup>3</sup>
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| ! style="width:20%;"|[[Molecules]] / m<sup>3</sup>
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| ! style="width:20%;"|Mean free path
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| |-
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| | Ambient pressure
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| | 1013
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| | 2.7 × 10<sup>19</sup>
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| | 2.7 × 10<sup>25</sup>
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| | 68 [[Nanometre|nm]]<ref>{{cite journal|last1=Jennings|first1=S|title=The mean free path in air|journal=Journal of Aerosol Science|volume=19|page=159|year=1988|doi=10.1016/0021-8502(88)90219-4|issue=2}}</ref>
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| |-
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| | Low vacuum
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| | 300 – 1
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| | 10<sup>19</sup> – 10<sup>16</sup>
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| | 10<sup>25</sup> – 10<sup>22</sup>
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| | 0.1 – 100 [[Micrometre|μm]]
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| |-
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| | Medium vacuum
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| | 1 – 10<sup>−3</sup>
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| | 10<sup>16</sup> – 10<sup>13</sup>
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| | 10<sup>22</sup> – 10<sup>19</sup>
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| | 0.1 – 100 mm
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| |-
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| | High vacuum
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| | 10<sup>−3</sup> – 10<sup>−7</sup>
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| | 10<sup>13</sup> – 10<sup>9</sup>
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| | 10<sup>19</sup> – 10<sup>15</sup>
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| | 10 cm – 1 km
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| |-
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| | Ultra high vacuum
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| | 10<sup>−7</sup> – 10<sup>−12</sup>
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| | 10<sup>9</sup> – 10<sup>4</sup>
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| | 10<sup>15</sup> – 10<sup>10</sup>
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| | 1 km – 10<sup>5</sup> km
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| |-
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| | Extremely high vacuum
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| | <10<sup>−12</sup>
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| | <10<sup>4</sup>
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| | <10<sup>10</sup>
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| | >10<sup>5</sup> km
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| |}
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| ==Mean free path in radiography==
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| [[File:Photon Mean Free Path.png|thumb|right|400px|Mean free path for photons in energy range from 1 keV to 20 MeV for Elements Z = 1 to 100. Based on data from.<ref>{{cite web|url=http://physics.nist.gov/PhysRefData/XrayNoteB.html |title=NIST: Note - X-Ray Form Factor and Attenuation Databases |publisher=Physics.nist.gov |date=1998-03-10 |accessdate=2011-11-08}}</ref> The discontinuities are due to low density of gas elements. Six bands correspond to neighborhoods of six [[w:noble gas|noble gases]]. Also shown are locations of [[w:en:absorption edge|absorption edges]].]]
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| In [[gamma-ray]] [[radiography]] the ''mean free path'' of a [[pencil beam]] of [[mono-energetic]] [[photon]]s 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:
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| :<math>\ell = \mu^{-1} = ( (\mu/\rho) \rho)^{-1},</math>
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| 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
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| <ref name=NIST1>{{cite web
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| |last=Hubbell |first=J. H.
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| |author1-link=John H. Hubbell
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| |last2=Seltzer |first2=S. M
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| |title=Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients
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| |publisher=[[National Institute of Standards and Technology]]
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| |url=http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html
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| |accessdate = September 2007}}</ref>
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| <ref name=NIST2>{{cite web | |
| |last=Berger |first=M. J.
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| |last2=Hubbell |first2=J. H. |author2-link=John H. Hubbell |first3=S. M. |last3=Seltzer |first4=J. |last4=Chang |first5=J. S. |last5=Coursey |first6=R. |last6=Sukumar |first7=D. S. |last7=Zucker
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| |title =XCOM: Photon Cross Sections Database
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| |publisher =[[National Institute of Standards and Technology]] (NIST)
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| |url =http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html
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| |accessdate = September 2007}}</ref>
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| In [[X-ray]] [[radiography]] the calculation of the ''mean free path'' is more complicated, because photons are not mono-energetic, but have some [[Frequency distribution|distribution]] of energies called [[spectrum]]. As photons move through the target material they are [[attenuation|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.
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| 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, a minus log of it is sometimes referred as ''number of mean free paths'' image.
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| ==Mean free path in particle physics==
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| 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.
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| ==Mean free path in nuclear physics==
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| Independent particle models in nuclear physics require the undisturbed orbiting of [[nucleon]]s within the [[Atomic nucleus|nucleus]] before they interact with other nucleons. Blatt and [[Victor Frederick Weisskopf|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").<ref>{{cite book
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| | last = Cook
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| | first = Norman D.
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| | authorlink =
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| | coauthors =
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| | title =Models of the Atomic Nucleus
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| | publisher =[[Springer Science+Business Media|Springer]]
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| | location =Heidelberg
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| | page =324
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| | url =http://www.res.kutc.kansai-u.ac.jp/~cook/NVSIndex.html
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| | doi =
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| | isbn = 978-3-642-14736-4}}</ref>
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| ==Mean free path in optics==
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| If one takes a suspension of non light absorbing particles of diameter d with a volume fraction Φ, the mean free path <ref>{{cite journal|last1=Mengual|first1=O|title=TURBISCAN MA 2000: multiple light scattering measurement for concentrated emulsion and suspension instability analysis|journal=Talanta|volume=50|pages=445–56|year=1999|doi=10.1016/S0039-9140(99)00129-0|issue=2|pmid=18967735|last2=Meunier|first2=G|last3=Cayré|first3=I|last4=Puech|first4=K|last5=Snabre|first5=P}}</ref> of the photons is:
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| :<math>l=\frac{2d}{3\Phi Q_s}</math>
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| where ''Q<sub>s</sub>'' is the scattering efficiency factor. ''Q<sub>s</sub>'' can be evaluated numerically for spherical particles thanks to the [[Mie theory]] calculation
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| ==Mean free path in acoustics==
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| In an otherwise empty cavity, the mean free path of a single particle bouncing off the walls is:
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| :<math>l=\frac{4V}{S}</math>
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| where ''V'' is volume of the cavity and ''S'' is total inside surface area of cavity.
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| This relation is used in the derivation of the [[Reverberation|Sabine equation]] in acoustics, using a geometrical approximation of sound propagation.<ref>Davis, D. and Patronis, E. [http://books.google.com/books?id=9mAUp5IC5AMC&pg=PA173 "Sound System Engineering"] (1997) Focal Press, ISBN 0-240-80305-1 p. 173</ref>
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| ==Examples==
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| A classic application of the 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.
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| For example, for [[sound]] [[wave]]s in an enclosure, the mean free path is the average distance the wave travels between [[reflection (physics)|reflections]] off the enclosure's walls.
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| In aerodynamics, the mean free path is in the same order of magnitude as the shockwave thickness at mach numbers greater than one.
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| ==See also==
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| *[[Scattering theory]]
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| *[[Vacuum]]
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| *[[Knudsen number]]
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| *[[Optics]]
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| *[http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate mean free path for mixtures of gases using VHS model
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| {{DEFAULTSORT:Mean Free Path}}
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| [[Category:Statistical mechanics]]
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| [[Category:Optics]]
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