Cross section (physics)

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In nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles. It can therefore characterize the probability that a particular nuclear reaction will take place, or the statistical nature of scattering events. The cross section is expressed in units of area, usually in barn.

1 Scattering
- 1.1 Relation to the S matrix
2 Nuclear physics
3 See also
4 Bibliography
5 External links

[edit] Scattering

In scattering, a differential cross section is defined by the probability to observe a scattered particle in a given quantum state per solid angle unit, such as within a given cone of observation, if the target is irradiated by a flux of one particle per surface unit:

${d \sigma \over d \Omega}={\hbox{Scattered flux / Unit of solid angle} \over \hbox{Incident flux / Unit of surface}}$

The integral cross section is the integral of the differential cross section on the whole sphere of observation (4 $π$ steradian):

$\sigma=\int d\Omega \, {d \sigma \over d \Omega}.$

A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. Usual units are the cm², the barn (1 b = 10⁻²⁴ cm²) and the corresponding submultiples: the millibarn (1 mb = 10⁻³ b), the microbarn (1 $μ$ b = 10⁻⁶ b), the nanobarn ( 1 nb = 10⁻⁹ b), and the picobarn (1 pb = 10⁻¹² b). The cross section of two particles (i.e. observed when the two particles are colliding with each other) is a measure of the interaction event between the two particles.

[edit] Relation to the S matrix

If the reduced masses and momenta of the colliding system are m_i, $\vec{p}_i$ and m_f, $\vec{p}_f$ before and after the collision respectively, the differential cross section is given by

${d\sigma \over d\Omega} = (2\pi)^4 m_i m_f {p_f \over p_i} |T_{fi}|^2,$

where the on-shell T matrix is defined by

$S_{fi} = \delta_{fi} - 2\pi i \delta(E_f -E_i) \delta(\vec{p}_i-\vec{p}_f) T_{fi}$

in terms of the S matrix. The $δ$ function is the distribution called the Dirac delta function. The computation of the S matrix is the main aim of the scattering theory.

[edit] Nuclear physics

It has been suggested that nuclear cross section be merged into this article or section. (Discuss)

In nuclear physics, it is found convenient to express probability of a particular event by a cross section. Statistically, the centers of the atoms in a thin foil can be considered as points evenly distributed over a plane. The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance $r$ of one of these points. In fact, if there are $n$ atomic centers in an area $A$ of the plane, this probability is $(n π r 2) / A$ , which is simply the ratio of the aggregate area of circles of radius $r$ drawn around the points to the whole area. If we think of the atoms as impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this ratio is the probability that the bullet will strike a steel disc, i.e., that the atomic projectile will be stopped by the foil. If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms. This notion can be extended to any interaction between the impinging particle and the atoms in the target. For example, the probability that an alpha particle striking a beryllium target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

In nuclear physics it is conventional to consider that the impinging particles have negligible diameter. Cross sections can be computed for any sort of process, such as capture scattering, production of neutrons, etc. In many cases, the number of particles emitted or scattered in nuclear processes is not measured directly; one merely measures the attenuation produced in a parallel beam of incident particles by the interposition of a known thickness of a particular material. The cross section obtained in this way is called the total cross section and is usually denoted by a σ or σ_T.

The typical nuclear radius is of the order of 10⁻¹² cm. We might therefore expect the cross sections for nuclear reactions to be of the order of $π r 2$ or roughly 10⁻²⁴ cm² and this unit is given its own name, the barn, and is the unit in which cross sections are usually expressed. Actually the observed cross sections vary enormously. Thus for slow neutrons absorbed by the (n, gamma) reaction the cross section in some cases is as much as 1,000 barns, while the cross sections for transmutations by gamma-ray absorption are in the neighborhood of 0.001 barns.