1/11/12 Mean free path - Wikipedia, the free encyclopedia

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Figure 1: Slab of target

Mean free pathFrom Wikipedia, the free encyclopedia

In physics, the mean free path is the average distance covered 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 particleproperties.

Contents

1 Derivation2 Mean free path in kinetic theory3 Mean free path in radiography4 Mean free path in particle physics5 Mean free path in nuclear physics6 Mean free path in optics7 Mean free path in acoustics8 Examples9 See also10 References11 External links

Derivation

Imagine a beam of particles being shot through a target, and consider aninfinitesimally thin slab of the target (Figure 1). The atoms (or particles) that mightstop a beam particle are shown in red. The magnitude of mean free path dependson the characteristics of the system the particle is in:

Where 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 L2 and its volume is L2dx. The typical number of stoppingatoms in the slab is the concentration n times the volume, i.e., nL2dx. Theprobability that a beam particle will be stopped in that slab is the net area of thestopping atoms divided by the total area of the slab.

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where σ is the area (or, more formally, the "scattering cross­section") of one atom.

The drop in beam intensity equals the incoming beam intensity multiplied by the probability of being stopped withinthe slab

dI = − Inσdx

This is an ordinary differential equation

whose solution is known as Beer­Lambert law and has form , where x is the distance traveled bythe beam through the target and I0 is the beam intensity before it entered the target; ℓ is called the mean free pathbecause it equals the mean distance traveled by a beam particle before being stopped. To see this, note that theprobability that a particle is absorbed between x and x + dx is given by

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

Fraction of particles that were not stopped (attenuated) by the slab is called transmission

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 travelsbetween collisions with other moving particles. The formula still holds for a particle with a highvelocity 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:[2]

and it may be shown that:[3]

where kB is the Boltzmann constant, T is temperature, p is pressure, and d is the diameter of the gas particles.

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Mean free path for photons in energy range from 1 keV to 20 MeVfor Elements Z = 1 to 100. Based on data from.[5] The discontinuitiesare due to low density of gas elements. Six bands correspond toneighborhoods of six noble gases. Also shown are locations ofCompton edges.

Following table lists some typical values for air at different pressures.

Vacuum range Pressure in hPa(mbar) Molecules / cm3 Molecules / m3 Mean free path

Ambient pressure 1013 2.7 × 1019 2.7 × 1025 68 nm[4]

Low vacuum 300 – 1 1019 – 1016 1025 – 1022 0.1 – 100 μm

Medium vacuum 1 – 10−3 1016 – 1013 1022 – 1019 0.1 – 100 mm

High vacuum 10−3 – 10−7 1013 – 109 1019 – 1015 10 cm – 1 km

Ultra high vacuum 10−7 – 10−12 109 – 104 1015 – 1010 1 km – 105 km

Extremely highvacuum <10−12 <104 <1010 >105 km

Mean free path in radiography

In gamma­ray radiography the mean freepath of a pencil beam of mono­energeticphotons, is the average distance a photontravels between collisions with atoms of thetarget material. It depends on material andenergy of the photons:

where μ is linear attenuation coefficient, μ/ρ is mass attenuation coefficient and ρ isdensity of the material. Mass attenuationcoefficient can be looked up or calculatedfor any material and energy combinationusing NIST databases [6] [7]

In X­ray radiography the calculation ofmean free path is more complicated sincephotons are not mono­energetic, but havesome distribution of energies calledspectrum. As photons move through thetarget material they are attenuated withprobabilities depending on their energy, asa result their distribution changes in processcalled Spectrum Hardening. Because of Spectrum Hardening mean free path of X­ray spectrum changes withdistance.

Sometimes people measure thickness of material in number of mean free paths. Material with thickness of one

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mean free path will attenuate 37% (1/e) of photons. This concept is closely related to Half­Value Layer (HVL) ormaterial with thickness of one HVL will attenuate 50% of photons. Standard x­ray image is a transmission image, aminus 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 ofattenuation length. In particular, for high­energy photons, which mostly interact by electron­positron pairproduction, 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 nucleusbefore 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 thenuclear dimensions in order to allow the use of the independent particle model. This requirement seems to be incontradiction to the assumptions made in the theory... We are facing here one of the fundamental problems ofnuclear structure physics which has yet to be solved." (quoted by Norman D. Cook in "Models of the AtomicNucleus" Ed.2 (2010) Springer, in Chapter 5 "The Mean Free Path of Nucleons in Nuclei").[8]

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 freepath [9] of the photons is:

where Qs is scattering efficiency factor. Qs can be evaluated numerically for spherical particles thanks to the Mietheory calculation

Mean free path in acoustics

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

where V is volume of the cavity and S is total inside surface area of cavity. This relation is used in the derivation ofthe Sabine equation in acoustics, using a geometrical approximation of sound propagation.[10]

Examples

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A classic application of mean free path is to estimate the size of atoms or molecules. Another important applicationis in estimating the resistivity of a material from the mean free path of its electrons.

For example, for sound waves in an enclosure, the mean free path is the average distance the wave travels betweenreflections off the enclosure's walls.

In aerodynamics, the mean free path is in the same order of magnitude as the shockwave thickness at machnumbers greater than one.

See also

Scattering theoryVacuumKnudsen numberOptics

References

1. ^ Author: Marion Brünglinghaus, ENS, European Nuclear Society. "Mean free path"(http://www.euronuclear.org/info/encyclopedia/m/mean­fee­path.htm) . Euronuclear.org.http://www.euronuclear.org/info/encyclopedia/m/mean­fee­path.htm. Retrieved 2011­11­08.

2. ^ S. Chapman and T.G. Cowling, The mathematical theory of non­uniform gases (http://books.google.com/books?id=Cbp5JP2OTrwC&pg=PA88) , 3rd. edition, Cambridge University Press, 1990, ISBN 052140844X, p. 88

3. ^ "Mean Free Path, Molecular Collisions" (http://hyperphysics.phy­astr.gsu.edu/hbase/kinetic/menfre.html) .Hyperphysics.phy­astr.gsu.edu. http://hyperphysics.phy­astr.gsu.edu/hbase/kinetic/menfre.html. Retrieved 2011­11­08.

4. ^ Jennings, S (1988). "The mean free path in air". Journal of Aerosol Science 19 (2): 159. doi:10.1016/0021­8502(88)90219­4 (http://dx.doi.org/10.1016%2F0021­8502%2888%2990219­4) .

5. ^ "NIST: Note ­ X­Ray Form Factor and Attenuation Databases"(http://physics.nist.gov/PhysRefData/XrayNoteB.html) . Physics.nist.gov. 1998­03­10.http://physics.nist.gov/PhysRefData/XrayNoteB.html. Retrieved 2011­11­08.

6. ^ Hubbell, J. H.; Seltzer, S. M.. "Tables of X­Ray Mass Attenuation Coefficients and Mass Energy­AbsorptionCoefficients" (http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html) . National Institute of Standards andTechnology (NIST). http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html. Retrieved September 2007.

7. ^ Berger, M.J.; J.H. Hubbell, S.M. Seltzer, J. Chang, J.S. Coursey, R. Sukumar, and D.S. Zucker. "XCOM:Photon Cross Sections Database" (http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html) . NationalInstitute of Standards and Technology (NIST). http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html.Retrieved September 2007.

8. ^ Cook, Norman D.. Models of the Atomic Nucleus (http://www.res.kutc.kansai­u.ac.jp/~cook/NVSIndex.html) .Heidelberg: Springer. p. 324. ISBN 978­3­642­14736­4. http://www.res.kutc.kansai­u.ac.jp/~cook/NVSIndex.html.

9. ^ Mengual, O; Meunier, G; Cayré, I; Puech, K; Snabre, P (1999). "TURBISCAN MA 2000: multiple light scatteringmeasurement for concentrated emulsion and suspension instability analysis". Talanta 50 (2): 445–56.doi:10.1016/S0039­9140(99)00129­0 (http://dx.doi.org/10.1016%2FS0039­9140%2899%2900129­0) .PMID 18967735 (http://www.ncbi.nlm.nih.gov/pubmed/18967735) .

10. ^ Davis, D. and Patronis, E. "Sound System Engineering" (http://books.google.com/books?id=9mAUp5IC5AMC&pg=PA173) (1997) Focal Press, ISBN 0240803051 p. 173

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External links

Gas Dynamics Toolbox (http://web.ics.purdue.edu/~alexeenk/GDT/index.html) Calculate mean free pathfor mixtures of gases using VHS model

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