Cross section (physics)

The cross section ( sigma) is in the molecular, atomic, nuclear and particle physics is a measure of the probability that between an incident wave radiation or an incident particles ( "projectile " ) and another particle ( scatterer or target) a specific interaction such as absorption, scattering, or a reaction taking place.

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

• In nuclear and particle physics in Barn (1 b = 10-28 m2 = 10-4 pm2 = 100 fm2 )
• In atomic and molecular physics in 10-22 m2 = 1 Mb = 10-4 nm2 = 100 pm2.

The idea of ​​the cross section associated with each target particle as a target area provides a convenient measure for the " strength" of the considered process: a frequently occurring process corresponds to a large cross section, a rarely occurring a small cross section. With intuitive ideas about the size, shape and location of Targetteilchens this target area is true, however do not match in general.

The cross section depends on the particular process of interest, the nature and kinetic energy of the incident particle or photon and of the nature of the particle taken (eg, atom, atomic nucleus ). The latter dependence means that the cross sections are material properties. For example, extensive core data libraries for the calculation of nuclear reactors and nuclear fusion reactors required, which contain the cross-sections of the different materials for various incident neutron energy of different possible scattering and nuclear reactions.

In particular in nuclear reactions, the cross-section, considered as a function of the energy of the incident particle / photon, sometimes referred to as the excitation function.

Special denominations

Depending on the nature of the subject process, different names for the cross section are used:

• Absorption cross-section for each of the absorption of the incident particle
• Capture cross section for a given absorbent, namely, the neutron capture ( the ( n ) - nuclear reaction )
• Reaction cross-section for the chemical reaction that is caused by the collision of two atoms or molecules
• Scattering cross-section for scattering, so deflection of the incident particle
• Elastic cross section ( often just " elastic cross section" ) for elastic collision, ie, a shock, in which get the total kinetic energy remains
• Inelastic cross section ( " inelastic cross section" ) for inelastic collision, ie, a shock, in which the kinetic energy changes into other forms of energy, such as a particle is excited (ie placed into a state of higher energy) or new produced particles
• Ionization cross section for the ionization of the atom taken
• Gap cross section for the induced fission.

Definition

In an experiment with a uniform irradiation of the target to the target particles ( target particle ) is an area σ as intended " target " is assigned. Their size is chosen so that the number of observed responses ( "Interactions " ) is given exactly by the number of projectile particles, the point-like - ie without extension - thought fly through this area. This area is the cross-section of the relevant target for the relevant interaction with the corresponding energy of the incident particles.

The probability that an incident particle interacts with a target particle, calculated from

It is

• The irradiated target surface and
• The number of target particles contained therein;

Also is required because the target particles otherwise mutually shade.

If total shrink projectile particles and each of them caused by the probability of a reaction, then the total number of reactions is given by:

Together:

The experimental determination of the cross section is measured by suitable detectors while, and from design and implementation of the experiment are known.

In the theoretical derivation (eg, in the quantum mechanical scattering theory ) the formula is often still divided by the time, so formed, the reaction rate:

With

• The particle current density of the projectile particles and
• The luminosity of the combination of target and particle beam.

Attenuation of the incident particle in the thick target

For an infinitesimally thin target layer thickness is obtained from the above equation, if you type " particles per area " is replaced by " particle density times thickness":

Here, the particle density of the target material, so the number of target particles per unit volume:

With

• Of the Avogadro constant
• The mass density and
• The molar mass.

Solving the above equation to and shall inform the same, we obtain the differential equation

The solution to this

Interpretation: interacting projectile particles are no longer part of the incident beam with the number of particles, as they ( in reaction) are absorbed or deflected ( in dispersion ) from its original path. That is, after the passage of a target layer of the thickness x of the particles in the beam are only available.

Considering the interactions in a particular volume, then, when the length of this volume is. Substituting this one, you can change the equation to calculate the cross section:

Apparently also applies

Wherein the mean free path length by which the intensity of the incident beam is decreased to its original value.

If more than one type of operation is possible in this equation refers to all together, then, is the total cross -section (see below).

Total cross section

The term "total effective cross-section " is used in two senses:

Differential cross section

If the reaction between the incident primary radiation and the secondary radiation, a target is formed ( scattered primary radiation or a different type of radiation), the intensity distribution is described by the spatial directions by the differential cross section

It is

• The current density in the direction of the outgoing Ω secondary radiation in the presence of a single Targetteilchens ( see definition ), given in parts per solid angle unit and time unit
• The current density of the (parallel incoming ) primary radiation in particles per surface unit and time unit.

Therefore, the dimension of area per solid angle and the unit of measurement eg millibarn per steradian. ( Mathematically, the angle space is seen dimensionless and the differential cross-section, therefore, of the same dimension as the cross section area of ​​yourself )

In order to obtain the correct target area for generating the secondary radiation in the direction of looking at the whole secondary radiation into a small solid angle. It is given in a first approximation by

The expression on the left-hand side is exactly the same reaction rate as mentioned above (for NT = 1), we think, about an experiment with a detector of exactly the size that is responsive to each incoming offspring. Therefore, standing on the right side before the incoming power density by a factor

Exactly the target area ( right with dimension surface ), which belongs to the observed responses in this experiment.

The integral of differential cross -section over all directions, the total ( or integral ) cross-section for the type of reaction observed.

The differential cross section depends

• As the cross-section itself, from the nature of the response ( type of the target, the type and energy of the particles of the primary and the Sekundärstahlung )
• In addition, the direction can be specified by two angles. Most are only interested in deflection angle relative to the direction of the primary beam; then called the differential cross section also briefly angular distribution.

By the term " Differential cross section ", without further addition is almost always meant. Other differential cross section are:

Secondary energy distribution

Is rarely needed after the energy of the secondary particle, that is the scattered particle or the reaction product, derived from the cross section, which describes the energy distribution of the secondary particles. It depends on the primary and secondary energy.

Double differential cross section

For complex operations, such as the penetration (transport ) of fast neutrons into thick layers of matter, where a neutron scattering processes and in various nuclear reactions can take successively and the double differential cross section is considered, since it allows the most detailed physical description.

Geometric cross section

In classical mechanics, all particles flying on well-defined trajectories. For reactions that require a touch of projectile and target particles, the concept of geometrical cross section is used, because here not only the size of the cross section as a hit area, but also their shape and position ( relative to the target particle ) a simple geometric meaning: all particles flying on their trajectory through this area, do not trigger the reaction under consideration, all the others.

• Example, collision of two spheres ( radii ): A contact with the target ball takes place exactly for the projectile balls whose center would pass no further from the center of the target sphere as indicated by the sum of the two radii. The target area is the center of the moving ball that is a disc centered at the center of the stationary sphere of radius. The (total ) cross-section, the area of this circle:

• For football ( radius) and Goal Wall (radius of the hole ), flight direction perpendicular to the wall. Asked whether the geometric cross section for the (viewers ) reaction TOOR, thus for free passage of flying:! Falls holds is. In the case of ball to pass through though, but may the trajectory of the ball center to miss the hole center to the maximum distance. The target area ( for the center of the ball ) is a circular disk with radius around the center of the hole. The geometric cross-section is

Both examples show that you do not even allowed to identify the geometric cross section with the size of the bodies involved (except when the projectile including the range of the force is viewed as a point ). The second also shows may be the scope of the term cross section as large.

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

Macroscopic and temperature-dependent cross section

In the physics of nuclear reactors, in addition to the above-defined microscopic ( i.e. one target particle, mostly related 1 atom ) and the macroscopic cross-section, related to 1 cm3 material cross-section with the symbols (large Sigma) was used. It is clear from the microscopic cross-section by multiplication with the atomic number density, ie the number of the respective atoms per cm3. The usual unit of the macroscopic cross section is cm2/cm3 = cm -1. In this application, the energies of the two reactants are generally determined not uniform, but vary within certain frequency distributions, and the quantity of interest is the average of the macroscopic cross section. This can then be temperature dependent, for example.

Cross section and Fermi's Golden Rule

Fermi's golden rule states that for the reaction rate is considered ( number of reactions per unit time ):

With

• The reduced Planck 's constant
• The transition probability matrix element or the amplitude ( in the Born approximation is given by the form factor of the potential of interaction)
• The phase space factor.

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

Consequently, the following applies: