Mean free path 170097 220845492 2008-06-21T22:03:08Z LatitudeBot 6771835 robot Adding: [[ru:Свободный пробег молекулы]] In [[physics]] the '''mean free path''' of a particle is the average distance covered by a particle ([[photon]], [[atom]] or [[molecule]]) between subsequent impacts.<ref>[http://www.euronuclear.org/info/encyclopedia/m/mean-fee-path.htm Mean free path<!-- Bot generated title -->]</ref> ==Derivation== [[Image:Mean_free_path.png|frame|Figure 1: Slab of target]] 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 '''mean free path''' depends on the characteristics of the system the particle is in: :<math>\ell = (n\sigma)^{-1},</math> Where <math>\ell</math> is the mean free path, ''n'' is the number of target particles per unit volume, and &sigma; is the effective [[cross section (physics)|cross sectional]] area for collision. 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. :<math> P(\mathrm{stopping \ within\ d}x) = \frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} = \frac{\sigma n L^{2} \mathrm{d}x}{L^{2}} = n \sigma \mathrm{d}x </math> where <math>\sigma</math> 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 :<math> dI = -I n \sigma dx </math> This is an [[ordinary differential equation]] :<math> \frac{dI}{dx} = -I n \sigma \ \stackrel{\mathrm{def}}{=}\ -\frac{I}{\ell} </math> whose solution is known as [[Beer-Lambert law]] and has form <math>I = I_{0} e^{-x/\ell}</math>, where <math>x</math> is the distance traveled by the beam through the target and <math>I_{0}</math> is the beam intensity before it entered the target. <math>\ell</math> 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 :<math>dP(x) = \frac{I(x)-I(x+dx)}{I_0} = \frac{1}{\ell} e^{-x/\ell} dx.</math> Thus the expectation value (or average, or simply mean) of x is :<math> \langle x \rangle \ \stackrel{\mathrm{def}}{=}\ \int_0^\infty x dP(x) = \int_0^\infty \frac{x}{\ell} e^{-x/\ell} dx = \ell </math> Fraction of particles that were 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 <math>x=dx</math>. == Mean free path in kinetic theory == In [[kinetic theory]] '''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]] of velocities, the following relationship applies: :<math>\ell = (\sqrt{2}\, n\sigma)^{-1}.\,</math> and it may be shown that<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html Mean Free Path, Molecular Collisions]</ref>: :<math>\ell = \frac{R T}{\sqrt 2 \pi d^2 N_A P}</math> where <math>R</math> is the [[universal gas constant]], <math>T</math> is temperature, <math>N_A</math> is [[Avogadro's number]], <math>P</math> is pressure, and <math>d</math> is the diameter of the gas particles. Following table lists some typical values for different pressures. {| class="wikitable" |----- ! width="25%" | Vacuum range ! width="25%" | [[Pressure]] in [[pascal (unit)|hPa (mbar)]] ! width="25%" | [[Molecules]] / cm<sup>3</sup> ! width="25%" | mean free path |----- | width="25%" | Ambient pressure || width="25%" | 1013 | width="25%" | 2.7*10<sup>19</sup> | width="25%" | 68 nm |----- | width="25%" | Low vacuum || width="25%" | 300-1 | width="25%" | 10<sup>19</sup>-10<sup>16</sup> | width="25%" | 0.1-100 μm |----- | width="25%" | Medium vacuum || width="25%" | 1-10<sup>-3</sup> | width="25%" | 10<sup>16</sup>-10<sup>13</sup> | width="25%" | 0.1-100 mm |----- | width="25%" | High vacuum | width="25%" | 10<sup>-3</sup>-10<sup>-7</sup> | width="25%" | 10<sup>13</sup>-10<sup>9</sup> | width="25%" | 10 cm-1 km |----- | width="25%" | Ultra high vacuum | width="25%" | 10<sup>-7</sup>-10<sup>-12</sup> | width="25%" | 10<sup>9</sup>-10<sup>4</sup> | width="25%" | 1 km-10<sup>5</sup> km |----- | width="25%" | Extremely high vacuum | width="25%" | <10<sup>-12</sup> | width="25%" | <10<sup>4</sup> | width="25%" | >10<sup>5</sup> km |} == Mean free path in [[radiography]] == [[Image: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 [http://physics.nist.gov/PhysRefData/XrayNoteB.html]. 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 [[Compton edge|Compton edges]].]] In [[gamma-ray]] [[radiography]] '''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 material and energy of the photons: :<math>\ell = \mu^{-1} = ( (\mu/\rho) \rho)^{-1},</math> where μ is [[linear attenuation coefficient]], μ/ρ is [[mass attenuation coefficient]] and ρ is [[density]] of the material. [[Mass attenuation coefficient]] can be looked up or calculated for any material and energy combination using [[National Institute of Standards and Technology|NIST]] databases <ref name=NIST1> {{cite web | last =Hubbell | first =J. H. | authorlink =john.hubbell@nist.gov | coauthors =Seltzer, S. M. | title =Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients | work = | publisher =[[National Institute of Standards and Technology]] (NIST) | date = | url =http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html | format = | doi = | accessdate = Sep. 2007 }}</ref> <ref name=NIST2> {{cite web | last =Berger | first =M.J. | authorlink =john.hubbell@nist.gov | coauthors =J.H. Hubbell, S.M. Seltzer, J. Chang, J.S. Coursey, R. Sukumar, and D.S. Zucker | title =XCOM: Photon Cross Sections Database | work = | publisher =[[National Institute of Standards and Technology]] (NIST) | date = | url =http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html | format = | doi = | accessdate = Sep. 2007 }}</ref> In [[x-ray]] [[radiography]] the calculation of '''mean free path''' is more complicated since photons are not mono-energetic, but have some [[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]] '''mean free path''' of [[x-ray]] [[spectrum]] changes with distance. Sometimes people measure thickness of material in '''number of mean free paths'''. Material with thickness of one '''mean free path''' will [[attenuation|attenuate]] 37% (1/e) of photons. This concept is closely related to [[Half-Value Layer]] or (HVL) material with thickness of one HVL will [[attenuation|attenuate]] 50% of photons. Standard [[x-ray]] image is a transmission image, a minus log of it is sometimes referred as '''number of mean free paths''' image. == Mean free path in [[particle physics]] == In particle physics the concept of mean free path is not commonly used, replaced instead 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. ==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 (physics)|reflection]]s off the enclosure's walls. ==See also== *[[Scattering theory]] *[[Vacuum]] *[[Knudsen number]] ==References== {{reflist}} ==External links== *[http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate mean free path for mixtures of gases using VHS model [[Category:Statistical mechanics]] [[de:Mittlere freie Weglänge]] [[es:Camino libre medio]] [[fr:Libre parcours moyen]] [[it:Cammino libero medio]] [[he:מהלך חופשי ממוצע]] [[ja:平均自由行程]] [[pt:Percurso livre médio]] [[ru:Свободный пробег молекулы]] [[uk:Довжина вільного пробігу]]