A differential cross section is a measure of the probability of scattering processes as a function of specific variables, such as the scattering angle or energy. It quantifies how the scattering intensity varies with these parameters, providing detailed information about the interaction dynamics between particles.
The differential cross section, denoted as dσ/dΩ, is defined as the ratio of the number of scattering events into a specific solid angle dΩ to the incident flux of particles. Mathematically, it can be expressed as:
dσ/dΩ = (dN/dΩ) / (Φ * t)
where dN is the number of scattered particles detected in the solid angle dΩ, Φ is the incident particle flux, and t is the target thickness.
The total cross section (σ) is the integral of the differential cross section over all possible scattering angles or energies. It represents the overall probability of scattering occurring in a given interaction, while the differential cross section provides a more granular view of how that probability varies with specific parameters.
A double differential cross section is an extension of the differential cross section that provides information on scattering as a function of two variables, such as both the scattering angle and the energy of the scattered particles. It is denoted as d²σ/(dΩ dE) and is useful for studying complex interactions where multiple outcomes are possible.
Differential and double differential cross sections are commonly used in particle physics, nuclear physics, and astrophysics to analyze scattering experiments. They are essential for understanding fundamental interactions, such as those between electrons and nuclei, or for characterizing the properties of new particles produced in high-energy collisions.