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 as it depends upon the wavelength of light and the permittivity in addition to the 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
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: , where Aλ is the absorbance at a given wavelength λ, C is the concentration as a number density, and l 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:
Relation to physical size
There is no simple relationship between the scattering cross-section and the physical size of the particles, as the scattering cross-section depends on the wavelength of radiation used. This can be seen when driving in foggy weather: the droplets of water (which form the fog) scatter red light less than they scatter the shorter wavelengths present in white light, and the red rear fog light can be distinguished more clearly than the white headlights of an approaching vehicle. That is to say that the scattering cross-section of the water droplets is smaller for red light than for light of shorter wavelengths, even though the physical size of the particles is the same.
The scattering cross-section is related to the meteorological range, LV:
The quantity C σscat is sometimes denoted bscat, the scattering coefficient per unit length.
When the scattered radiation is visible light, it is conventional to measure the path length in centimetres. To avoid the need for conversion factors, the scattering cross-section is expressed in cm² (1 cm2 = 10−4 m2) and the number concentration in cm−3 (1 cm−3 = 10−6 m−3). The measurement of the scattering of visible light is known as nephelometry, and is effective for particles of 2–50 µm in diameter: as such, it is widely used in meteorology and in the measurement of atmospheric pollution.
Scattering cross-sections are also widely used in the description of the scattering of neutrons by atomic nuclei. In this case, the conventional unit is the barn, b, where 1 b = 10−28 m2 = 100 fm2.
In classical mechanics, the differential cross section is defined as follows: let a beam of intensity (measured in number of particles per area per time) be incident on a scattering center. In general, the angle at which a particle is scattered is a function of impact parameter. The number of scattered particles per solid angle per time (the radiant intensity), is therefore well defined. We define the differential cross section to be
Note that this quantity has units of area. Furthermore, it depends only on the geometry of the scattering center, and not on the incident flux or distance of the detector from the scattering center. The geometric interpretation is as follows: consider particles that scatter through a solid angle and ask what values of impact parameter produced them. These impact parameters form a differential area, in space. The differential cross section is simply
In quantum mechanics, the wave function of the incident particle is a plane-wave with amplitude 1, that is, . In general the scattered wave is of the form
We then have as the definition of differential cross section
This has the simple interpretation of the probability of finding a scattered particle within a given solid angle.
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 = 10−28 m2) and the corresponding submultiples: the millibarn (1 mb = 10−3 b), the microbarn (1 b = 10−6 b), the nanobarn ( 1 nb = 10−9 b), the picobarn (1 pb = 10−12 b), and the shed (1 shed = 10−24 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. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles ) depends only on the number of incident particles per unit of time (current of incident particles ), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction. For we have
Relation to the S matrix
where the on-shell T matrix is defined by
- Bajpai, P.K. "2. Spectrophotometry". Biological Instrumentation and Biology. ISBN 81-219-2633-5.
- IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Scattering cross-section, σscat".
- International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 127–28, ISBN 92-822-2213-6