Cross section (physics)

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles.

For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus.

Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns.

When two discrete particles interact in classical physics, their mutual cross section is the area transverse to their relative motion within which they must meet in order to scatter from each other.

If the particles are hard inelastic spheres that interact only upon contact, their scattering cross section is related to their geometric size.

If the particles interact through some action-at-a-distance force, such as electromagnetism or gravity, their scattering cross section is generally larger than their geometric size.

Thus, specifying the cross section for a given reaction is a proxy for stating the probability that a given scattering process will occur.

Differential and total scattering cross sections are among the most important measurable quantities in nuclear, atomic, and particle physics.

It is not uncommon for the actual cross-sectional area of a scattering object to be much larger or smaller than the cross section relative to some physical process.

For example, plasmonic nanoparticles can have light scattering cross sections for particular frequencies that are much larger than their actual cross-sectional areas.

The volumetric number density of scattering centers is designated by n. Solving this equation exhibits the exponential attenuation of the beam intensity: where Φ0 is the initial flux, and z is the total thickness of the material.

The differential size of the cross section is the area element in the plane of the impact parameter, i.e. dσ = b dφ db.

The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deflection.

In cylindrically symmetric situations (about the beam axis), the azimuthal angle φ is not changed by the scattering process, and the differential cross section can be written as In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle.

The differential cross section is extremely useful quantity in many fields of physics, as measuring it can reveal a great amount of information about the internal structure of the target particles.

For example, the differential cross section of Rutherford scattering provided strong evidence for the existence of the atomic nucleus.

Differential cross sections in inelastic scattering contain resonance peaks that indicate the creation of metastable states and contain information about their energy and lifetime.

The arrow indicates that this only describes the asymptotic behavior of the wave function when the projectile and target are too far apart for the interaction to have any effect.

For Nσ ≪ 1 we have If the reduced masses and momenta of the colliding system are mi, pi and mf, pf before and after the collision respectively, the differential cross section is given by[clarification needed] where the on-shell T matrix is defined by in terms of the S-matrix.

To avoid the need for conversion factors, the scattering cross section is expressed in cm2, and the number concentration in cm−3.

In the interaction of light with particles, many processes occur, each with their own cross sections, including absorption, scattering, and photoluminescence.

The absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance T:[3] Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in this area.

Cross sections commonly calculated using Mie theory include efficiency coefficients for extinction

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 the Beer–Lambert law, which says that absorbance is proportional to concentration: Aλ = Clσ, 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 T:[3] 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.

Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

The beam consists of a uniform density of parallel rays, and the beam-circle interaction is modeled within the framework of geometric optics.

Then the increase of the length element perpendicular to the beam is The reflection angle of this ray with respect to the incoming ray is 2α, and the scattering angle is The differential relationship between incident and reflected intensity I is The differential cross section is therefore (dΩ = dθ) Its maximum at θ = π corresponds to backward scattering, and its minimum at θ = 0 corresponds to scattering from the edge of the circle directly forward.

The total cross section is equal to the diameter of the circle: The result from the previous example can be used to solve the analogous problem in three dimensions, i.e., scattering from a perfectly reflecting sphere of radius a.

In any plane of the incoming and the reflected ray we can write (from the previous example): while the impact area element is In spherical coordinates, Together with the trigonometric identity we obtain The total cross section is