Mean free path
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),[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 | 220 – 8×10−1 | 1019 – 1016 | 1025 – 1022 | 0.1 – 100 μm |
| Medium vacuum | 1 – 10−3 | 8×10−1 – 8×10−4 | 1016 – 1013 | 1022 – 1019 | 0.1 – 100 mm |
| High vacuum | 10−3 – 10−7 | 8×10−4 – 8×10−8 | 1013 – 109 | 1019 – 1015 | 10 cm – 1 km |
| Ultra-high vacuum | 10−7 – 10−12 | 8×10−8 – 8×10−13 | 109 – 104 | 1015 – 1010 | 1 km – 105 km |
| Extremely high vacuum | <10−12 | <8×10−13 | <104 | <1010 | >105 km |
Contents
1 Radiography
2 Electronics
3 Optics
4 Acoustics
5 Derivation
6 Kinetic theory
7 Nuclear and particle physics
8 See also
9 References
10 External links
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:
- ℓ=μ−1=((μ/ρ)ρ)−1,{displaystyle 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 National Institute of Standards and Technology (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.
Electronics
Electrical mobility, a value directly related to electrical conductivity, is:
- μ=eτm=edm∗vF,{displaystyle mu ={frac {etau }{m}}={frac {ed}{m^{*}v_{rm {F}}}},}
where e is the charge, τ{displaystyle tau }
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:[6]
- l=2d3ΦQs,{displaystyle l={frac {2d}{3Phi Q_{text{s}}}},}
where Qs is the scattering efficiency factor. Qs can be evaluated numerically for spherical particles using Mie theory.
Acoustics
In an otherwise empty cavity, the mean free path of a single particle bouncing off the walls is:
- l=FVS,{displaystyle l={frac {FV}{S}},}
where V is volume of the cavity, S is total inside surface area of cavity and F a constant related to the shape of the cavity. For most simple cavity shapes F is approximately 4.[7]
This relation is used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation.[8]
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
- ℓ=(σn)−1,{displaystyle 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 L2, and its volume is L2 dx. The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., n L2 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(stopping within dx)=AreaatomsAreaslab=σnL2dxL2=nσdx,{displaystyle P({text{stopping within }}dx)={frac {{text{Area}}_{text{atoms}}}{{text{Area}}_{text{slab}}}}={frac {sigma nL^{2},dx}{L^{2}}}=nsigma ,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=−Inσdx.{displaystyle dI=-Insigma ,dx.}
This is an ordinary differential equation:
- dIdx=−Inσ=def−Iℓ,{displaystyle {frac {dI}{dx}}=-Insigma {overset {text{def}}{=}}-{frac {I}{ell }},}
whose solution is known as Beer-Lambert law and has the form I=I0e−x/ℓ{displaystyle I=I_{0}e^{-x/ell }}
- dP(x)=I(x)−I(x+dx)I0=1ℓe−x/ℓdx.{displaystyle 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
- ⟨x⟩=def∫0∞xdP(x)=∫0∞xℓe−x/ℓdx=ℓ.{displaystyle langle xrangle {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/I0=e−x/ℓ{displaystyle T=I/I_{0}=e^{-x/ell }}
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 ℓ=(nσ)−1{displaystyle ell =(nsigma )^{-1}}
- ℓ=(2nσ)−1,{displaystyle ell =({sqrt {2}},nsigma )^{-1},}
and using n=N/V=p/(kBT){displaystyle n=N/V=p/(k_{text{B}}T)}
- ℓ=kBT2πd2p,{displaystyle ell ={frac {k_{text{B}}T}{{sqrt {2}}pi d^{2}p}},}
where kB 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
- ℓ=μpπkBT2m,{displaystyle 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.
Nuclear and 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.
Independent-particle models in nuclear physics require the undisturbed orbiting of nucleons within the nucleus before they interact with other nucleons.[11]
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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.
— John Markus Blatt and Victor Weisskopf, Theoretical nuclear physics (1952)[12]
See also
- Scattering theory
- Vacuum
- Knudsen number
- Optics
References
^ Brünglinghaus, Marion. "Mean free path". European Nuclear Society. Retrieved 2011-11-08..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Jennings, S (1988). "The mean free path in air". Journal of Aerosol Science. 19 (2): 159. Bibcode:1988JAerS..19..159J. 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 19 September 2007.
^ 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 19 September 2007.
^ 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.
^ Young, Robert W. (July 1959). "Sabine Reverberation Equation and Sound Power Calculations". The Journal of the Acoustical Society of America. 31 (7): 918. Bibcode:1959ASAJ...31..912Y. doi:10.1121/1.1907816.
^ 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.
^ Cook, Norman D. (2010). "The Mean Free Path of Nucleons in Nuclei". Models of the Atomic Nucleus (2 ed.). Heidelberg: Springer. p. 324. ISBN 978-3-642-14736-4.
^ Blatt, John M.; Weisskopf, Victor F. (1979). Theoretical Nuclear Physics. doi:10.1007/978-1-4612-9959-2. ISBN 978-1-4612-9961-5.
External links
Gas Dynamics Toolbox: Calculate mean free path for mixtures of gases using VHS model
