# Cross section (physics)

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A cross section is the effective area that governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer–Lambert law.

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

When particles in a beam are thrown against a foil made of a certain substance, the cross section $\sigma$ is a hypothetical area measure around the target particles of the substance (usually its atoms) that represents a surface. If a particle of the beam crosses this surface, there will be some kind of interaction.

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, represented by $\sigma$) 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. A particular case is scattering.

## Scattering

The scattering cross-section, σscat, is a hypothetical area which describes the likelihood of light (or other radiation) being scattered by a particle. In general, the scattering cross-section is different from the geometrical cross-section of a particle, and it depends upon the wavelength of light and the permittivity, shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present. In terms of area, the total cross-section (σ) is the sum of the cross-sections due to absorption, scattering and luminescence

$\sigma = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L}.\$

The total cross-section is related to the absorbance of the light intensity through Beer-Lambert's law, which says absorbance is proportional to concentration: $A_\lambda = C \,\ell\, \sigma$, where C is the concentration as a number density, Aλ is the absorbance at a given wavelength λ, and $\ell$ is the path length. The extinction or absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance:[1]

$A_\lambda = - \log \mathcal{T}.\$

## Nuclear physics

In nuclear physics, it is convenient to express the 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.

## References

• J.D.Bjorken, S.D.Drell, Relativistic Quantum Mechanics, 1964
• P.Roman, Introduction to Quantum Theory, 1969
• W.Greiner, J.Reinhardt, Quantum Electrodinamics, 1994
• R.G. Newton. Scattering Theory of Waves and Particles. McGraw Hill, 1966.
1. ^ Bajpai, P.K. "2. Spectrophotometry". Biological Instrumentation and Biology. ISBN 81-219-2633-5.