page_title: Scattering theory
section_title: Composite targets and range equations
hierarchical_section_title: Scattering theory / Conceptual underpinnings / Composite targets and range equations
caption_attribution_description: This figure plots equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units, onto a single plot. The grey dots are for inelastic scattering of a 300 keV electron passing through a specimen of solid Silicon, as well as for an electron near its most strongly interacting energy of around 70 eV in a metal. The gold dot shows how a 300 keV electron has a shorter distance mean free path but about the same area cross-section. The mean free path between collisions of nitrogen molecules in air at STP is shown by the darker blue-green dot, while the cyan dot shows approximately where the mean free path for light scattering by fog might plot. The black and yellow dots denote nuclear cross-sections, e.g. in the latter case for Rutherford backscattering of a He nucleus from a gold foil.
context_page_description: In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Scattering also includes the interaction of billiard balls on a table, the Rutherford scattering of alpha particles by gold nuclei, the Bragg scattering of electrons and X-rays by a cluster of atoms, and the inelastic scattering of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of partial differential equations, propagating freely "in the distant past", come together and interact with one another or with a boundary condition, and then propagate away "to the distant future". The direct scattering problem is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. The inverse scattering problem is the problem of determining the characteristics of an object from measurement data of radiation or particles scattered from the object.
context_section_description: When the target is a set of many scattering centers whose relative position varies unpredictably, it is customary to think of a range equation whose arguments take different forms in different application areas.  In the simplest case consider an interaction that removes particles from the "unscattered beam" at a uniform rate that is proportional to the incident flux  of particles per unit area per unit time, i.e. that
  where Q is an interaction coefficient and x is the distance traveled in the target.
  The above ordinary first-order differential equation has solutions of the form:
  where Iₒ is the initial flux, path length Δx ≡ x − xₒ, the second equality defines an interaction mean free path λ, the third uses the number of targets per unit volume η to define an area cross-section σ, and the last uses the target mass density ρ to define a density mean free path τ.   Hence one converts between these quantities via Q = 1/λ = ησ = ρ/τ, as shown in the figure at left.
  In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm⁻¹) is variously called opacity, absorption coefficient, and attenuation coefficient.  In nuclear physics, area cross-sections (e.g. σ in barns or units of 10⁻²⁴ cm²), density mean free path (e.g. τ in grams/cm²), and its reciprocal the mass attenuation coefficient (e.g. in cm²/gram) or area per nucleon are all popular, while in electron microscopy the inelastic mean free path (e.g. λ in nanometers) is often discussed instead.
caption: Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units.
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