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 σ 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]]