Mass attenuation coefficient

The mass attenuation coefficient can be thought of as a variant of absorption cross section where the effective area is defined per unit mass instead of per particle.

Mathematical definitions

The mass attenuation coefficient is defined as $${\displaystyle \frac{\mu }{{\rho }_m},}$$ where

• μ is the attenuation coefficient (linear attenuation coefficient);
• ρ_m is the mass density.

When using the mass attenuation coefficient, the Beer–Lambert law is written in alternative form as $${\displaystyle I=I_0{e}^{-(\mu /{\rho }_m)\lambda }}$$ where $${\displaystyle \lambda ={\rho }_m\ell }$$ is the area density, also known as mass thickness, and ${\displaystyle \ell }$ is the length over which the attenuation takes place.

Mass absorption and scattering coefficients

When a narrow (collimated) beam passes through a volume, the beam will lose intensity to two processes: absorption and scattering.

The mass absorption coefficient and mass scattering coefficient are defined as $${\displaystyle \frac{{\mu }_{\mathrm{a}}}{{\rho }_m},\frac{{\mu }_{\mathrm{s}}}{{\rho }_m},}$$ where

• μ_a is the absorption coefficient;
• μ_s is the scattering coefficient.

In solutions

In chemistry, mass attenuation coefficients are often used for a chemical species dissolved in a solution. In that case, the mass attenuation coefficient is defined by the same equation, except that the "density" is the density of only that one chemical species, and the "attenuation" is the attenuation due to only that one chemical species. The actual attenuation coefficient is computed by $${\displaystyle \mu =(\mu /\rho {)}_1{\rho }_1+(\mu /\rho {)}_2{\rho }_2+\dots ,}$$ where each term in the sum is the mass attenuation coefficient and density of a different component of the solution (the solvent must also be included). This is a convenient concept because the mass attenuation coefficient of a species is approximately independent of its concentration (as long as certain assumptions are fulfilled).

A closely related concept is molar absorptivity. They are quantitatively related by

(mass attenuation coefficient) × (molar mass) = (molar absorptivity).