Assume that a beam of light enters a material sample. Define  as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness  sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the  direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by  where  is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation:
The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factorthroughout to obtainwhich simplifies due to the product rule (applied backwards) to
Integrating both sides and solving for  for a material of real thickness , with the incident radiant flux upon the slice  and the transmitted radiant flux  givesand finally
Since the decadic attenuation coefficient  is related to the (Napierian) attenuation coefficient by  we also have
To describe the attenuation coefficient in a way independent of the number densities  of the  attenuating species of the material sample, one introduces the attenuation cross section   has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the species  in the material sample:
One can also use the molar attenuation coefficients  where  is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations  of the attenuating species of the material sample: