Cross section

In molecular , atomic , nuclear and particle physics, the cross section ( sigma ) is a measure of the probability that there is a certain difference between an incident wave radiation or an incident particle (“projectile”) and another particle ( scattering body or target ) Process such as B. absorption , scattering or a reaction takes place. ${\displaystyle \sigma }$

The cross-section has the dimension area. It is usually given in the following units :

• in nuclear and particle physics in Barn (1 b = 10 ⁻²⁸  m ² = 10 ⁻⁴  pm ² = 100 fm ² )
• in atomic and molecular physics in 10 ⁻²²  m ² = 1 Mb = 10 ⁻⁴  nm ² = 100  pm ² .

The idea of ​​the cross-section as a hit area assigned to each target particle offers a clear measure of the “strength” of the process under consideration: a process that occurs frequently has a large cross-section, and a seldom one has a small cross-section. However, this hit area generally does not correspond to clear ideas about the size, shape and position of the target particle .

The cross section depends on the process of interest, on the type and kinetic energy of the incident particle or quantum and on the type of particle hit, e.g. B. atom , atomic nucleus . The latter dependency means that cross- sections are material properties . For example, the calculation of nuclear reactors or nuclear fusion reactors requires extensive nuclear data libraries that contain the cross sections of the various materials for incident neutrons of different energies for various possible scattering processes and nuclear reactions.

Particularly in the case of nuclear reactions , the cross section, considered as a function of the energy of the incident particle / quantum , is sometimes also referred to as the excitation function.

Special terms

Depending on the type of process under consideration, different terms are used for the cross-section:

• Absorption cross-section for each absorption of the incident particle
• Scattering cross-section for scattering, i.e. deflection of the incident particle
• Extinction cross-section for attenuation or energy extraction, sum of scattering and absorption cross-section
• Capture cross section for a certain absorption, namely neutron capture (the (n, ) -nuclear reaction) ${\displaystyle \gamma }$
• Neutron cross-section for (any) interaction of the atomic nucleus with a free neutron
• Reaction cross section for the chemical reaction that is triggered by the collision of two atoms or molecules
• Elastic effective cross-section (often just “elastic cross-section”) for elastic collision, i.e. an impact in which the entire kinetic energy is retained
• Inelastic effective cross-section ("inelastic cross-section") for inelastic collision, i.e. a collision in which kinetic energy is converted into other forms of energy, e.g. A particle is excited (i.e., placed in a higher energy state) or new particles are created
• Ionization cross section for the ionization of the hit atom
• Fission cross section for the induced nuclear fission
• Radiation pressure cross section for radiation pressure.

definition

The hit probability is the total area of ​​the target particles, i.e. (red) divided by the total area of ​​the target (blue). ${\displaystyle w}$ ${\displaystyle N_{T}}$ ${\displaystyle \sigma \cdot N_{T}}$

In an experiment with uniform irradiation of the target, the target particle (target particle) is assigned an area σ as an imaginary “target”. Their size is chosen in such a way that the number of observed reactions ("interactions") is specified precisely by the number of projectile particles - point-like, i.e. imagined without expansion - that fly through this surface. This area is the cross section of the target in question for the interaction in question at the energy of the projectile particles in question.

The probability that an incident particle interacts with a target particle is calculated from ${\displaystyle w}$

${\displaystyle w=\sigma {\frac {N_{T}}{F}}\quad \Leftrightarrow \quad \sigma =w{\frac {F}{N_{T}}}.}$

In it is

• ${\displaystyle F\,}$ the irradiated target area and
• ${\displaystyle N_{T}\,}$ the number of target particles contained therein;

is also assumed, because otherwise the target particles will shadow each other. ${\displaystyle \sigma N_{T}\ll F\quad \Leftrightarrow \quad w\ll 1}$

If a total of projectile particles enter and each of them has a probability of causing a reaction, then the total number of reactions is given by: ${\displaystyle \,N}$ ${\displaystyle \,w}$

${\displaystyle N_{\text{Reaktion}}=wN\quad \Leftrightarrow \quad w={\frac {N_{\text{Reaktion}}}{N}}.\,}$

Together:

${\displaystyle \sigma ={\frac {N_{\text{Reaktion}}}{N}}\,{\frac {F}{N_{T}}}\quad \Leftrightarrow \quad N_{\text{Reaktion}}=\sigma {\frac {N_{T}}{F}}N.}$

For the experimental determination of the cross section is through suitable detectors measured while , and from design and implementation of the experiment are known. ${\displaystyle \,N_{\text{Reaktion}}}$ ${\displaystyle N_{T}\,}$ ${\displaystyle N\,}$ ${\displaystyle F\,}$

In the theoretical derivation (e.g. in the quantum mechanical scattering theory ) the formula is often divided by the time, i.e. the reaction rate ( reactor physics : nuclear reaction rate ): ${\displaystyle W}$ ${\displaystyle R}$

${\displaystyle W={\frac {N_{\text{Reaktion}}}{t}}=\sigma N_{T}\,j=\sigma L}$

With

• the particle flux density of the projectile particles and ${\displaystyle j={\frac {N}{F\,t}}}$
• the luminosity of the combination of target and particle beam . ${\displaystyle L=N_{T}\,j}$

Attenuation of the incident particle beam in the thick target

For an infinitesimally thin target layer of thickness , one obtains from the above equation, if one substitutes the product “ particle density times thickness ” for “particles per area” : ${\displaystyle \mathrm {d} x}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \mathrm {d} x}$

${\displaystyle {\frac {\mathrm {d} N_{\text{Reaktion}}}{N}}=\sigma \cdot (\rho _{T}\cdot \mathrm {d} x)}$.

Here is the particle density of the target material, i.e. the number of target particles per unit volume: ${\displaystyle \rho _{T}}$

${\displaystyle \rho _{T}={\frac {N_{\text{A}}\cdot \rho }{M}}}$

With

• ${\displaystyle N_{\text{A}}}$the Avogadro constant ,
• ${\displaystyle \rho }$the mass density and
• ${\displaystyle M}$the molar mass .

If you solve the above equation and equate it , you get the differential equation ${\displaystyle N_{\text{Reaktion}}}$ ${\displaystyle -\mathrm {d} N}$

${\displaystyle -\mathrm {d} N=N(x)\,\rho _{T}\,\sigma \,\mathrm {d} x\quad \Leftrightarrow \quad {\frac {\mathrm {d} N}{\mathrm {d} x}}=-N(x)\,\rho _{T}\,\sigma .}$

The solution to this is

${\displaystyle N(x)=N_{0}\cdot e^{-\sigma \rho _{T}x}.}$

Interpretation: the interacting projectile particles are no longer part of the incident beam with the number of particles because they have been absorbed (in the case of reaction) or (in the case of scattering) have been deflected from their original path. In other words, after passing through a target layer of thickness x, only particles are left in the beam. ${\displaystyle N_{\text{Reaktion}}}$ ${\displaystyle N_{0}}$ ${\displaystyle N(x)=N_{0}-N_{\text{Reaktion}}}$

Looking at the interactions in a given volume , then if is the length of that volume. If you use this, you can change the equation to calculate the cross-section: ${\displaystyle x=l}$ ${\displaystyle l}$

${\displaystyle \Rightarrow \quad \sigma ={\frac {1}{l\cdot \rho _{T}}}\cdot \ln \left({\frac {N_{0}}{N_{0}-N_{\text{Reaktion}}}}\right)}$

Apparently also applies

${\displaystyle \sigma \cdot \rho _{T}={\frac {1}{\lambda }}\quad \Leftrightarrow \quad \sigma ={\frac {1}{\lambda \cdot \rho _{T}}},}$

where is the mean free path after which the intensity of the incident beam has dropped to its original value. ${\displaystyle \lambda }$ ${\displaystyle {\frac {1}{e}}}$

If more than one type of process is possible, this equation refers to all of them together, i.e. it is the total cross-section (see below). ${\displaystyle \sigma }$

Total cross section

Cross-sections for six nuclear reactions of neutron and atomic nucleus ²³⁵ U and their sum, the total cross-section, as a function of the kinetic energy of the neutrons. In the legend, z sometimes stands for neutron instead of the usual symbol n (data source: JEFF, graphic representation: core data viewer JANIS 4 [1] )

The term "total cross section" is used in two meanings:

1. Sometimes it means the cross section for any one of several possible events to occur, e.g. B. Absorption or scattering of the incident particle. For processes that are mutually exclusive, the total cross-section is the sum of the individual cross-sections. The figure shows the cross-sections of the six types of nuclear reactions dominating the energy interval (10 ⁻¹¹ to 20) MeV of neutron and atomic nucleus ²³⁵ U and the sum of these cross-sections, the total cross-section. It is required, for example, when it is only a matter of weakening the incident particle flow or the mean free path .
2. Sometimes “total cross section” is only used in the sense of the cross section defined above for a certain process in order to distinguish it from the differential cross section (see below); a better term in this case is “integral cross-section”. The following applies: ${\displaystyle {\frac {d\sigma }{d\Omega }}}$

${\displaystyle \sigma =\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega }$

Differential cross section

If the reaction between the incident primary radiation and the target creates secondary radiation (scattered primary radiation or another type of radiation), its intensity distribution over the spatial directions is described by the differential (also differential) cross-section ${\displaystyle \Omega }$ ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}:}$

${\displaystyle j_{\text{sek.}}(\Omega )={\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\;j_{\text{prim.}}\quad \Leftrightarrow \quad {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}={\frac {j_{\text{sek.}}(\Omega )}{j_{\text{prim.}}}}.}$

In it is

• ${\displaystyle \,j_{\text{sek.}}(\Omega )}$the current density of the secondary radiation escaping in the direction of Ω in the presence of a single target particle ( see definition ), given in particles per solid angle unit and time unit ${\displaystyle \,N_{T}=1}$
• ${\displaystyle \,j_{\text{prim.}}}$ the current density of the (parallel incoming) primary radiation in particles per unit area and time unit.

Therefore the dimension has area per solid angle and as a unit of measurement z. B. Millibarn per steradian . (Physically, the solid angle of a size of the number of dimensions and the differential cross section , therefore, the same dimension of surface as the cross section itself.) ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}}$ ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}}$ ${\displaystyle \sigma }$

In order to obtain the correct hit area for the generation of the secondary radiation in the direction , the entire secondary radiation is considered in a small solid angle element . As a first approximation it is given by ${\displaystyle \Omega }$ ${\displaystyle \Delta \Omega }$

${\displaystyle j_{\text{sek.}}(\Omega )\cdot \Delta \Omega =\left({\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\;\Delta \Omega \right)\;j_{\text{prim.}}.}$

The expression on the left corresponds exactly to the reaction rate as mentioned above (with N _T = 1), imagine an experiment with a detector of exactly the size that responds to every incoming secondary particle. Therefore, on the right-hand side, in front of the incoming current density with the factor ${\displaystyle \Delta \Omega }$ ${\displaystyle j_{\text{prim.}}}$

${\displaystyle \Delta \sigma :={\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\Delta \Omega \qquad \left(\Rightarrow \quad {\frac {j_{\text{sek.}}(\Omega )}{j_{\text{prim.}}}}={\frac {\Delta \sigma }{\Delta \Omega }}\right)}$

exactly the hit area (correct with dimension area ) which belongs to the reactions observed in this experiment.

The integral of the differential cross section over all directions is the total (or integral ) cross section for the type of reaction observed:

${\displaystyle \sigma =\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega }$

The differential cross section depends on

• like the cross section itself: on the type of reaction (type of target, type and energy of the particles of the primary and secondary radiation)
• additionally from the direction , which can be specified by two angles. Usually only the deflection angle relative to the direction of the primary beam is of interest; then the differential cross section is also called the angular distribution for short . ${\displaystyle \Omega }$

The term "differential effective cross-section" without any further addition is almost always meant. Other differential cross sections are: ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}}$

Secondary energy distribution

The cross-section derived from the energy of the secondary particle, i.e. the scattered particle or reaction product, which describes the energy distribution of the secondary particles, is required less often . It depends on the primary and secondary energy. ${\displaystyle E_{s}}$ ${\displaystyle {\frac {d\sigma }{dE_{s}}}}$

Double differential cross section

For complex processes such as the penetration ( transport ) of fast neutrons into thick layers of matter, where a neutron can participate in various scattering processes and nuclear reactions one after the other, the double differential cross section is also considered, as it allows the most detailed physical description. ${\displaystyle {\frac {d^{2}\sigma }{d\Omega \cdot dE_{s}}}}$

Geometric cross section

Illustration of the geometric cross-section
(for example 1):
if the center of particle  b penetrates the blue circle, it collides with particle  a .
The area of ​​the blue circle is thus the geometric cross section, its radius is the sum of the particle radii.

In classical mechanics , all particles fly on well-defined trajectories . For reactions that require projectile and target particles to come into contact, the term geometric cross-section is used, because here not only the size of the cross-section as the hit surface, but also its shape and position (relative to the target particle) have a simple geometric meaning: all particles that fly through this area on their trajectory trigger the observed reaction, all others do not.

• Example impact of two balls (radii and , see figure): A contact with the target ball a takes place precisely for the projectile balls b, whose center would not pass the center of the target ball further than is indicated by the sum of their two radii. For the center of the moving sphere, the hit area is a circular disk around the center of the stationary sphere with a radius . The (total) cross section is the area of ​​this circle: ${\displaystyle R_{a}}$ ${\displaystyle R_{b}}$ ${\displaystyle R_{\sigma }=R_{a}+R_{b}}$

${\displaystyle \sigma _{\text{geom}}=\pi \,(R_{a}+R_{b})^{2}.}$

• Example football (radius ) and goal wall (radius of the hole ), flight direction perpendicular to the wall. What is asked is the geometric cross section for the (audience) reaction TOOR !! , so for free flying through: If applies, is . In this case , the ball fits through, but the trajectory of the center of the ball may miss the center of the hole by the distance at most . The hit area (for the center of the ball) is a circular disk with a radius around the center of the hole. The geometric cross section is ${\displaystyle R_{\text{Ball}}}$ ${\displaystyle R_{\text{Loch}}}$ ${\displaystyle R_{\text{Ball}}>R_{\text{Loch}}}$ ${\displaystyle \sigma _{\text{geom}}=0}$ ${\displaystyle R_{\text{Ball}}\leq R_{\text{Loch}}}$ ${\displaystyle R_{\sigma }=R_{\text{Loch}}-R_{\text{Ball}}}$ ${\displaystyle R_{\sigma }}$

${\displaystyle \sigma _{\text{geom}}=\pi \,(R_{\text{Loch}}-R_{\text{Ball}})^{2}}$.

Both examples show that one cannot even identify the geometric cross-section with the size of one of the bodies involved (unless the projectile including the range of the force is viewed as point-shaped). The second also shows how wide the scope of the term cross-section can be.

In the case of wave phenomena, the geometric interpretation is not possible. In principle, no deterministic statements can be made about individual projectile or target particles in quantum mechanics either.

Macroscopic cross section

In the physics of nuclear reactors, in addition to the microscopic cross-section defined above (i.e. based on 1 target particle, usually 1 atom), the macroscopic cross-section based on 1 cm ^{3 of}  material with the symbol (large sigma) is used. It results from the microscopic cross section by multiplying it by the atomic number density, i.e. the number of the respective atoms per cm ³ . It thus corresponds to the reciprocal of the mean free path introduced above. The usual unit of the macroscopic cross section is cm ² / cm ³ = 1 / cm. In this area of ​​application, the energies of the two reaction partners are generally not defined uniformly, so that the kinetic energy in their center of gravity varies within a certain frequency distribution. The variable of interest is then the average value of the macroscopic cross-sections determined with this distribution. This can e.g. B. be temperature dependent. ${\displaystyle \Sigma }$

Temperature-dependent cross section

In thermodynamic equilibrium, the atoms and molecules of matter have a low kinetic energy compared to the particles at a given temperature. In a thermal reactor , a neutron reaches the “temperature” of the medium after a very short time (on the order of microseconds), mainly due to elastic scattering on the proton of the water molecule. Then the cross section will no longer depend on the speed of the particle alone, but on the relative speed of the atomic nucleus and the particle. The cross-section becomes temperature-dependent and one speaks of a temperature-dependent cross-section or a temperature-dependent macroscopic cross-section .

Cross section and Fermi's golden rule

Fermi's Golden Rule states that the reaction rate (number of reactions per time) is: ${\displaystyle W}$

${\displaystyle W={\frac {2\pi }{\hbar }}|M_{fi}|^{2}\rho }$

With

• ${\displaystyle \hbar ={\frac {h}{2\pi }}}$the reduced Planck quantum of action
• ${\displaystyle M_{fi}}$the transition matrix element or the probability amplitude (given in Born's approximation by the form factor of the potential of the interaction)
• ${\displaystyle \rho }$the phase space factor .

Since the reaction rate is also directly proportional to the (differential) cross section

${\displaystyle W=L\,\sigma }$( see above : as the luminosity of the particle beam), ${\displaystyle L\,}$

consequently:

${\displaystyle \sigma ={\frac {2\pi }{\hbar }}{\frac {1}{L}}\cdot |M_{fi}|^{2}\rho .}$

See also

• Nuclear reaction rate
• Rutherford scattering
• Mott scatter
• Rosenbluth formula
• Structure function

Web links

• The cross section (geometric interpretation)

Individual evidence

1. ^ ^{A b} HC van de Hulst: Light Scattering by Small Particle . Dover Publication Inc., 1981, ISBN 0-486-64228-3 , pp. 13 .
2. ^ William M. Irvine: Light Scattering by Spherical Particles: Radiation Pressure, Asymmetry Factor, and Extinction Cross Section. In: Journal of the Optical Society of America. No. 55 (1), 1965, pp. 16-19