Cross section (physics)

In nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles.

The term is derived from the purely classical picture of (a large number of) point-like projectiles directed to an area that includes a solid target. Assuming that an interaction will occur (with 100% probability) if the projectile hits the solid, and not at all (0% probability) if it misses, the total interaction probability for the single projectile will be the ratio of the area of the section of the solid (the cross section) to the total targeted area. This basic concept is then extended to the cases where the interaction probability in the targeted area assumes intermediate values - because the target itself is not homogeneous, or because the interaction is mediated by a non-uniform field.

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

To put it another way, it is the rate of scattering events ($$N_s/\Delta t$$) normalized to the beam intensity ($$N_b/\Delta t$$), the target density ($$\rho_t$$), the length of the beam-target interaction region ($$l_{bt}$$), the geometrical "size" of detector ($$(\Delta\Omega)_d$$), and the "counting" efficiency of the detector ($$f_d$$).
 * $${d \sigma \over d \Omega}={ {(N_s / \Delta t)} \over {(N_b / \Delta t) \times \rho_t \times l_{bt} \times (\Delta \Omega)_d \times f_d}}$$

If the detector is small and sufficiently far from the target, then the geometrical "size" of the detector is given by:
 * $$ (\Delta \Omega)_d = { {A_d} \over {4 \pi r_{td}^2} } = {\hbox{physical area of the face of the detector} \over \hbox{surface area of a sphere with a radius equal to the target-detector distance} } $$

The integral cross section is the integral of the differential cross section on the whole sphere of observation (4$$\pi$$ 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&minus;28 m2) and the corresponding submultiples: the millibarn (1 mb = 10&minus;3 b), the microbarn (1 $$\mu$$b = 10&minus;6 b), the nanobarn ( 1 nb = 10&minus;9 b), the picobarn (1 pb = 10&minus;12 b), and the shed (1 shed = 10&minus;24 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.

Relation to the S matrix
If the reduced masses and momenta of the colliding system are mi, $$\vec{p}_i$$ and mf, $$\vec{p}_f$$ 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 $$\delta$$ function is the distribution called the Dirac delta function. The computation of the S matrix is the main aim of the scattering theory.

Nuclear physics
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 $$(n \pi r^2)/A$$, 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.