Editing Scattering (section)

===Attenuation due to scattering===
[[Image:Xsection2.png|288px|thumb|left|Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units]]

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 number of particles per unit area per unit time (<math>I</math>), i.e. that

:<math> \frac{dI}{dx}=-QI \,\!</math>

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:

: <math>I = I_o e^{-Q \Delta x} = I_o e^{-\frac{\Delta x}{\lambda}} = I_o e^{-\sigma (\eta \Delta x)} = I_o e^{-\frac{\rho \Delta x}{\tau}} ,</math>

where ''I''<sub>o</sub> is the initial flux, path length Δx&nbsp;≡&nbsp;''x''&nbsp;&minus;&nbsp;''x''<sub>o</sub>, 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 (physics)|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'' =&nbsp;1/''&lambda;'' =&nbsp;''&eta;&sigma;'' =&nbsp;''&rho;/&tau;'', as shown in the figure at left.

In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm<sup>&minus;1</sup>) is variously called [[opacity (optics)|opacity]], [[absorption coefficient]], and [[attenuation coefficient]].  In nuclear physics, area cross-sections (e.g. σ in [[barn (unit)|barn]]s or units of 10<sup>&minus;24</sup> cm<sup>2</sup>), density mean free path (e.g. τ in grams/cm<sup>2</sup>), and its reciprocal the [[mass attenuation coefficient]] (e.g. in cm<sup>2</sup>/gram) or ''area per nucleon'' are all popular, while in electron microscopy the [[inelastic mean free path]]<ref>R. F. Egerton (1996) ''Electron energy-loss spectroscopy in the electron microscope'' (Second Edition, Plenum Press, NY) {{ISBN|0-306-45223-5}}</ref> (e.g. λ in nanometers) is often discussed<ref>Ludwig Reimer (1997) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Fourth Edition, Springer, Berlin) {{ISBN|3-540-62568-2}}</ref> instead.