In
physicsPhysics 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
atomThe 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
moleculeA 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
photonIn 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 sectionalA 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 BeerLambert 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 crosssectionThe scattering crosssection, σscat, is a hypothetical area which describes the likelihood of light being scattered by a particle. In general, the scattering crosssection is different from the geometrical crosssection 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 equationIn 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
BeerLambert lawIn 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
I_{0} is the beam intensity before it entered the target;
ℓ is called the mean free path because
it equals the
meanIn 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 (
attenuatedIn physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, Xrays 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
transmissionIn 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 theoryThe 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
moleculeA 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 
PressurePressure 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)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 / cm^{3} 
Molecules / m^{3} 
Mean free path 
Ambient pressure 
1013 
2.7 × 10^{19} 
2.7 × 10^{25} 
68 nm 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 
10^{19} – 10^{16} 
10^{25} – 10^{22} 
0.1 – 100 μmA micrometer , is by definition 1×106 of a meter .In plain English, it means onemillionth of a meter . Its unit symbol in the International System of Units is μm...

Medium vacuum 
1 – 10^{−3} 
10^{16} – 10^{13} 
10^{22} – 10^{19} 
0.1 – 100 mm 
High vacuum 
10^{−3} – 10^{−7} 
10^{13} – 10^{9} 
10^{19} – 10^{15} 
10 cm – 1 km 
Ultra high vacuum 
10^{−7} – 10^{−12} 
10^{9} – 10^{4} 
10^{15} – 10^{10} 
1 km – 10^{5} km 
Extremely high vacuum 
<10^{−12} 
<10^{4} 
<10^{10} 
>10^{5} km 
Mean free path in radiography
In gammaray
radiographyRadiography is the use of Xrays to view a nonuniformly 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 beamIn 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 monoenergetic
photonIn 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 coefficientThe 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
densityThe 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 coefficientThe 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
NISTThe 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 nonregulatory agency of the United States Department of Commerce...
databases
In
XrayXradiation is a form of electromagnetic radiation. Xrays 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...
radiographyRadiography is the use of Xrays to view a nonuniformly 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 monoenergetic, but have some
distributionIn 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
spectrumA 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
attenuatedIn physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, Xrays 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
XrayXradiation is a form of electromagnetic radiation. Xrays 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...
spectrumA 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
attenuateIn physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, Xrays 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
HalfValue LayerHalfvalue 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
attenuateIn physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, Xrays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...
50% of photons. Standard
xrayXradiation is a form of electromagnetic radiation. Xrays 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 lengthIn 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 highenergy photons, which mostly interact by electronpositron pair production, the
radiation lengthIn physics, the radiation length is a characteristic of a material, related to the energy loss of high energy, electromagneticinteracting particles with it.Definition:Highenergy 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
nucleonIn 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
nucleusThe 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
WeisskopfVictor Frederick Weisskopf was an Austrianborn 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
Q_{s} is scattering efficiency factor.
Q_{s} can be evaluated numerically for spherical particles thanks to the
Mie theoryThe 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 equationReverberation 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
resistivityElectrical 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
electronThe 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
soundSound 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...
waveIn 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
reflectionReflection 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.
External links