Mean free path

The mean free path is the path length that a particle (e.g. atom , molecule , ion or electron ) travels on average in a given material before colliding (of any kind) with another particle. If a particle flow in a material has passed the mean free path, almost 2/3 of the particles have already carried out an impact, the remaining third (exactly a fraction 1 / e ) has not yet carried out a collision . ${\displaystyle \lambda }$

Calculation from cross section and particle density

The mean free path is related to the particle density   (number of particles per volume) and the total cross section : ${\displaystyle n}$ ${\displaystyle \sigma }$

${\displaystyle \lambda ={\frac {1}{n\cdot \sigma }}}$

The size of the target that one particle offers the other particles for a collision is clear. After it has flown a free path , it has swept over the volume with this area , that is the volume in which there is an average particle. The particle has collided with another particle on average once. ${\displaystyle \sigma }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda \sigma =1/n}$

Estimation for gases

The geometric cross-section when two spherical particles collide with the same diameter is given by ${\displaystyle d}$

${\displaystyle \sigma _{\text{geom}}=\pi \,d^{2}}$

The geometric mean free path follows from this

${\displaystyle \Rightarrow \lambda _{\text{geom}}={\frac {1}{\pi \,n\,d^{2}}}}$

The above interpretation applies when the collision partners of the flying particle are at rest.

If, however, all particles move in a disorderly manner (i.e. also the collision partners of the flying particle), equilibrium considerations, assuming a Maxwellian velocity distribution, lead to a free path that is shorter by the factor : ${\displaystyle {\tfrac {1}{\sqrt {2}}}}$

${\displaystyle \lambda ={\frac {1}{{\sqrt {2}}\,\pi \,n\,d^{2}}}}$

Definition of two types of particles

In an area of ​​space that contains two types of particles, three types of collision are possible:

• two particles of type 1 collide
• two particles of type 2 collide
• A type 1 particle and a type 2 particle are involved in the collision.

Let the particle densities of the particle types be or and the cross sections , and . ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle \sigma _{1}}$ ${\displaystyle \sigma _{2}}$ ${\displaystyle \sigma _{12}}$

The mean free path lengths for collisions between particles in each case are already defined with the above formula:

${\displaystyle \lambda _{1}={\frac {1}{n_{1}\cdot \sigma _{1}}}}$

or.

${\displaystyle \lambda _{2}={\frac {1}{n_{2}\cdot \sigma _{2}}}}$

The mean free path of a particle of type 2 in the medium of type 1 is defined accordingly:

${\displaystyle \lambda _{21}={\frac {1}{n_{1}\cdot \sigma _{12}}}}$

or analogously when a particle of type 1 collides with a medium of type 2: ${\displaystyle \lambda _{12}}$

${\displaystyle \lambda _{12}={\frac {1}{n_{2}\cdot \sigma _{12}}}}$

the cross-section being the same in both cases . ${\displaystyle \sigma _{12}}$

The cross section and the mean free path of two different particles are mostly without an index or other markup characters written so and : ${\displaystyle \sigma =\sigma _{12}}$ ${\displaystyle \lambda =\lambda _{21}}$

${\displaystyle \lambda ={\frac {1}{n_{1}\cdot \sigma }}}$

The number density index is also usually omitted when only the mean free path of particles in any medium with type 1 particles is concerned. Then the definition of the mean free path given at the beginning of this article and the equation given last appear to be formally the same (and are therefore sometimes confused). The number densities of the particles and are in reality different sizes: at is the number density of the particles of only one type, but at the number density of the particles of the medium. ${\displaystyle n_{1}}$ ${\displaystyle n}$ ${\displaystyle n_{1}}$ ${\displaystyle n}$ ${\displaystyle n_{1}}$

Similar to the game of billiards , the elastic collision of two particles causes changes in the direction of both particles, however, in contrast to the game of billiards, in three-dimensional physical space and with particles of different sizes. A collision between two different types of particles often also means that they react with one another. For example, if the collision partners are two atoms , a molecule can be formed; if the collision partners are a neutron and an atomic nucleus , another nuclide can arise or an atomic nucleus can be split .

We implicitly assume that both collision partners are particles of a gas , because only there can both collision partners move freely, which suggests that we speak of a free path. But there are also collisions and mean free paths if one collision partner is a particle of a solid or a liquid (particle of type 1) and only the second collision partner behaves like a particle of a gas (particle of type 2). As a rule, collisions between particles of type 1 and particles of type 2 are of interest here (and not collisions of the two respective particle types with their own kind).

Estimation for the collision of two types of particles

The geometric cross section for the elastic collision of two rigid spheres with radii or is ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

${\displaystyle \sigma _{\text{geom}}=\pi \,(r_{1}+r_{2})^{2}}$

The geometric mean free path thus becomes:

${\displaystyle \Rightarrow \lambda _{\text{geom}}={\frac {1}{\pi \,n_{1}\,(r_{1}+r_{2})^{2}}}}$

The geometric cross-section and thus the geometric mean free path of the elastic scattering do not depend on the kinetic energies of the spheres. Real effective cross-sections and thus the mean free path lengths, on the other hand, can strongly depend on the kinetic energy of the collision partners and are consequently not necessarily determined by the above. calculate simple geometric model. Even in the case of real cross-sections, however, it can be helpful to use the geometric cross-section as a reference value, as follows: the real cross-section is 10 times larger than the geometric, then the real mean free path is only a tenth of the geometric.

Collisions of neutrons and atomic nuclei

Collisions from neutrons and atomic nuclei are (currently) the most important case in physics for collisions from two types of particles, they shape reactor physics .

When the mean free path is mentioned in reactor physics , the second definition of this quantity is always meant, i.e. the mean free path of neutrons in matter. Free neutrons (particles of type 2 with a particle density ) move in a solid or a liquid (“host medium”) i. generally as chaotic as molecules in a gas. We assume that the host medium consists of only one type of particle or one type of atom (particles of type 1), think of graphite atoms , for example . Since each atom has only one atomic nucleus, the particle density of the atoms is the same as that of their atomic nuclei. ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$

The reciprocal of the mean free path of this type is one of the most important quantities in reactor physics under the name of macroscopic cross section :

${\displaystyle \Sigma ={\frac {1}{\lambda }}=n_{1}\cdot \sigma }$

Cross-sections of nuclear reactions depend extremely strongly on the energy and can therefore no longer be explained geometrically. Only in the case of the elastic scattering of neutrons on the atomic nuclei of common moderators does the above geometric model lead to mean free path lengths that are in the order of magnitude of the measured values; this at least for neutrons with kinetic energies in a certain mean interval.

Neutrons also collide with one another. This is a case for the first definition of the mean free path. The number density of neutrons is comparatively low even in the high flux reactor . The cross section for the collision of two neutrons is also small. Therefore, probably no textbook on reactor or neutron physics even mentions the mean free path for this type of collision.

Atomic nuclei cannot collide with one another. Even if the neutrons move in helium gas, as in a high-temperature reactor , at most helium atoms collide. However, if necessary, the mean free path for such atomic collisions can be calculated with the above. Calculate the first definition formula and the geometric cross section, which exceeds that of the atomic nucleus by orders of magnitude.

Examples

Gas molecules

The mean free path of a gas molecule in air under standard conditions is about 68  nanometers .

The following table lists approximate numbers for free path lengths for gas molecules at different pressures:

Pressure area	Pressure in hPa	Particle density in molecules per cm³ ${\displaystyle n}$	mean free path ${\displaystyle \lambda }$
Ambient pressure	1013	2.7 · 10 19	68 nm
Low vacuum	300… 1	10 19 … 10 16	0.1 ... 100 μm
Fine vacuum	1… 10 −3	10 16 … 10 13	0.1 ... 100 mm
High vacuum (HV)	10 −3 ... 10 −7	10 13 … 10 9	10 cm ... 1 km
Ultra high vacuum (UHV)	10 −7 ... 10 −12	10 9 … 10 4	1 km ... 10 5 km
extr. Ultra high voltage (XHV)	<10 −12	<10 4	> 10 5 km

Electrons

Universal curve for the inelastic mean free path of electrons in elements based on equation (5) in

The mean free path of free electrons is important when using electron beams in a vacuum (e.g. for certain surface-sensitive analytical methods or in Braun tubes ). It depends on the kinetic energy of the electron.

The inelastic free path in the solid can be estimated for most metals with a “universal curve” (see fig.): At energies around 100  eV it is lowest for most metals, since processes in the solid can be excited here, e.g. . B. plasmons ; at higher and lower energies the mean free paths in the solid are greater. The effective cross-section for elastic collisions is usually smaller than that for inelastic collisions, so the elastic free path is greater than the inelastic path given the same particle density .

In gaseous insulating materials (e.g. sulfur hexafluoride ) the mean free path influences the dielectric strength .

For electrons in momentum space (see Fermi sphere ) one considers the mean free flight time instead of the path length .

Individual evidence

1. ^ William C. Hinds: Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles . Wiley-Interscience, New York 1999, ISBN 0-471-19410-7
2. ^ Paul Reuss: Neutron physics . EDP ​​Sciences, Les Ulis, France 2008, ISBN 978-2-7598-0041-4 , pp. xxvi, 669 . , P. 50
3. ↑ ^{a b} M. P. Seah, WA Dench: Quantitative electron spectroscopy of surfaces: A standard data base for electron inelastic mean free paths in solids. In: Surface and Interface Analysis. 1, 1979, p. 2, doi: 10.1002 / sia.740010103 .
4. ↑ Wolfgang SM Werner: Electron transport in solids for quantitative surface analysis. In: Surface and Interface Analysis. 31, 2001, p. 141, doi: 10.1002 / sia.973 .