Cross section (physics)

If a beam of particles enters a thin layer of material (thickness dx), then the particle number N will be reduced by dN = −μN dx, where μ is the attenuation coefficient. To describe the attenuation coefficient in a way independent of the material density, one introduces the cross section σ = μ/n, where n is the numerical density (number of atoms per volume) of the material. σ has the dimension of an area; it expresses the likelihood of interaction between particles.

The attenuation may be due to scattering, or to absorption, or (in nuclear physics) to transformation into a different particle.

Integrating dN = −μN dx leads to the decreasing exponential function N = N0 exp(−μx) where N0 is the initial particle number. For light, this is called the Beer–Lambert law.

The term "cross section" may be naively understood from the purely classical picture of (a large number of) point-like projectiles directed to an area that includes a solid target, a circular disc of area σ. Assuming that an interaction will occur (with 100% probability) if the projectile hits the disc, 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 disc (the cross section, represented by σ) 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

The scattering cross section, σs, 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

$\sigma = \sigma_\mathrm{a} + \sigma_\mathrm{s} + \sigma_\mathrm{l}.$

The total cross-section is related to the absorbance of the light intensity through Beer–Lambert law, which says absorbance is proportional to concentration:

$A_\lambda = C \ell \sigma,$

where Aλ is the absorbance at a given wavelength λ, C is the concentration as a number density, and $\ell$ is the path length. The 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

Main article: neutron cross section

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.

Rate (particle physics)

For the similar quantity in chemical kinetics, see reaction rate.

In scattering theory, particle physics and nuclear physics, the rate at which a specific subatomic particle reaction occurs is a physical quantity measuring the number of reactions per unit time.

Partial cross section

For a particle beam (say of neutrons, pions) incident on a target (liquid hydrogen), for each type of reaction in the scattering process labelled by an index r = 1, 2, 3..., it is calculated from:[2]

$W_r = JN\sigma_r,$

where N is the number of target particles, illuminated by the beam containing n particles per unit volume in the beam (number density of particles) traveling with flow velocity u in the rest frame of the target, and these two quantities combine into the flux of the beam J = nu. The cross section of the reaction is σr. Since the beam flux has dimensions of [length]−2·[time]−1 and σr has dimensions of [length]2 while N is a dimensionless number, the rate W has the dimensions of reciprocal time - which intuitively represents a frequency of recurring events.

The above formula assumes the following:

• the beam particles all have the same kinetic energy,
• the number density of the beam particles is sufficiently low: allowing the interactions between the particles within the beam to be neglected,
• the number density of target particles is sufficiently low: so that only one scattering event per particle occurs as soon as the beam is incident with the target, and multiple scattering events within the target can be neglected,
• the de Broglie wavelength of the beam is much smaller than the inter-particle separations within the target, so that diffraction effects through the target can be neglected,
• the collision energy is sufficiently high allowing the binding energies in the target particles to be neglected.

These conditions are usually met in experiments, which allows for a very simple calculation of rate.

Sometimes the rate per unit target particle, or rate density, is more useful. For reaction r:[3]

$W_r/N = J\sigma_r.$

Total cross section

The cross section σr is specifically for one type of reaction, and is called the partial cross section. The total cross section, and corresponding total rate of the reaction, can be found by summing over the cross sections and rates for each reaction:[2]

$W = \sum_r W_r = JN \sum_r \sigma_r = JN \sigma.$

Differential cross section

In terms of the differential cross section dσr/dΩ, which is a function of angular coordinates θ and φ, the differential rate for a reaction r is:[2]

$\mathrm d W_r = J N \mathrm d \sigma_r = J N \frac{\mathrm d \sigma_r}{\mathrm d \Omega} \, \mathrm d \Omega$,

where dΩ = sin θ dθ dφ is the solid angle element in the vicinity of the event with vertex at the point of scattering. Integrating over θ and φ yields the total rate of the reaction r:

$W_r = J N \int_0^{2\pi} \int_0^\pi \frac{\mathrm d \sigma_r}{\mathrm d \Omega} \sin \theta \, \mathrm d \theta \, \mathrm d \phi$.