Mean free path

Mean free path

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the mean free path is the average distance covered by a moving particle (such as an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

, a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

, a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

) between successive impacts (collisions) which modify its direction or energy or other particle properties.

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.
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Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the mean free path is the average distance covered by a moving particle (such as an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

, a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

, a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

) between successive impacts (collisions) which modify its direction or energy or other particle properties.

Derivation


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:


Where is the mean free path, n is the number of target particles per unit volume, and is the effective cross sectional
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

 area for collision.

The area of the slab is and its volume is . The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., . 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.


where is the area (or, more formally,
the "scattering cross-section
Scattering cross-section
The scattering cross-section, σscat, is a hypothetical area which describes the likelihood of light being scattered by a particle. In general, the scattering cross-section is different from the geometrical cross-section of a particle, and it depends upon the wavelength of light and the...

") of one atom.

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


This is an ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....




whose solution is known as Beer-Lambert law
Beer-Lambert law
In optics, the Beer–Lambert law, also known as Beer's law or the Lambert–Beer law or the Beer–Lambert–Bouguer law relates the absorption of light to the properties of the material through which the light is travelling.-Equations:The law states that there is a logarithmic dependence between the...

 and has form ,
where x is the distance traveled by the beam through the target and I0 is the beam intensity before it entered the target; is called the mean free path because
it equals the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population 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
Thus the expectation value (or average, or simply mean) of x is


Fraction of particles that were not stopped (attenuated
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

) by the slab is called transmission
Transmittance
In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, or absorption factor, which is the fraction of radiation absorbed by a sample at a specified wavelength...

  where x is equal to the thickness of the slab x = dx.

Mean free path in kinetic theory


In kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

 the mean free path of a particle, such as a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

, is the average distance the particle travels between collisions with other moving particles. The formula 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:


and it may be shown that:


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

Following table lists some typical values for air at different pressures.
Vacuum range Pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 in hPa (mbar)
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

Molecules / cm3 Molecules / m3 Mean free path
Ambient pressure 1013 2.7 × 1019 2.7 × 1025 68 nm
Nanometre
A nanometre is a unit of length in the metric system, equal to one billionth of a metre. The name combines the SI prefix nano- with the parent unit name metre .The nanometre is often used to express dimensions on the atomic scale: the diameter...

Low vacuum 300 – 1 1019 – 1016 1025 – 1022 0.1 – 100 μm
Micrometre
A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is 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 high vacuum <10−12 <104 <1010 >105 km

Mean free path in radiography


In gamma-ray radiography
Radiography
Radiography is the use of X-rays to view a non-uniformly composed material such as the human body. By using the physical properties of the ray an image can be developed which displays areas of different density and composition....

 the mean free path of a pencil beam
Pencil beam
In optics, a pencil or pencil of rays is a geometric construct used to describe a beam or portion of a beam of electromagnetic radiation or charged particles, typically in the form of a narrow cone or cylinder....

 of mono-energetic photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

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:


where μ is linear attenuation coefficient, μ/ρ is mass attenuation coefficient
Mass attenuation coefficient
The mass attenuation coefficient is a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per unit mass...

 and ρ is density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

 of the material. Mass attenuation coefficient
Mass attenuation coefficient
The mass attenuation coefficient is a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per unit mass...

 can be looked up or calculated for any material and energy combination using NIST
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

 databases
In X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

 radiography
Radiography
Radiography is the use of X-rays to view a non-uniformly composed material such as the human body. By using the physical properties of the ray an image can be developed which displays areas of different density and composition....

 the calculation of mean free path is more complicated since photons are not mono-energetic, but have some distribution
Frequency distribution
In statistics, a frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of...

 of energies called spectrum
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

. As photons move through the target material they are attenuated
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

 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
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

 spectrum
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

 changes with distance.

Sometimes people measure thickness of material in number of mean free paths. Material with thickness of one mean free path will attenuate
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

 37% (1/e) of photons. This concept is closely related to Half-Value Layer
Half-Value Layer
Half-value layer is the thickness of specified material which reduces the intensity of radiation entering the material by half...

 (HVL) or material with thickness of one HVL will attenuate
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

 50% of photons. Standard x-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

 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
Attenuation length
In physics, the attenuation length or absorption length is the distance \lambda into a material when the probability has dropped to 1/e that a particle has not been absorbed...

. In particular, for high-energy photons, which mostly interact by electron-positron pair production, the radiation length
Radiation length
In physics, the radiation length is a characteristic of a material, related to the energy loss of high energy, electromagnetic-interacting particles with it.-Definition:High-energy electrons predominantly lose energy in matter...

 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 nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...

s within the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

 before they interact with other nucleons. Blatt and Weisskopf
Victor Frederick Weisskopf
Victor Frederick Weisskopf was an Austrian-born Jewish American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli and Niels Bohr...

, 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").

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 free path of the photons is:
where Qs is scattering efficiency factor. Qs can be evaluated numerically for spherical particles thanks to the Mie theory
Mie theory
The Mie solution to Maxwell's equations describes the scattering of electromagnetic radiation by a sphere...

 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 of the Sabine equation
Reverberation
Reverberation is the persistence of sound in a particular space after the original sound is removed. A reverberation, or reverb, is created when a sound is produced in an enclosed space causing a large number of echoes to build up and then slowly decay as the sound is absorbed by the walls and air...

 in acoustics, using a geometrical approximation of sound propagation.

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
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...

 of a material from the mean free path of its electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s.

For example, for sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

 wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s in an enclosure, the mean free path is the average distance the wave travels between reflection
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

s off the enclosure's walls.

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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