# Absorption coefficient

### From Online Dictionary of Crystallography

##### Revision as of 16:29, 13 April 2016 by BrianMcMahon (talk | contribs)

## Definition

The X-ray mass absorption coefficient, [math]\Big[{\mu\over\rho}\Big](E)[/math] or [math]\Big[{\mu\over\rho}\Big]_{pe}(E)[/math] follows the Beer-Lambert law for a parallel beam of photons of energy [math]E[/math] in which the transmitted photon intensity [math]I(t)[/math] is related to the incoming photon intensity [math]I_0[/math] such that

[math]I(t) = I_0 e^{-\Big[{\mu\over\rho}\Big]\rho t}[/math]

where[math]t[/math] is the thickness of a uniform sample and the density [math]\rho[/math] is usually given in (g/cm^{3}). The mass absorption coefficient is labeled as such because the absorption exponent is linear in the mass per unit area [math]\rho t[/math], otherwise known as the integrated column density through a sample. Use of [math]\mu[/math] for this term is not recommended because it is highly ambiguous and dimensionally inconsistent. The subscript *pe* emphasizes that this is the photo-electric mass absorption coefficient rather than the mass attenuation coefficient (*q.v.*), which of course does not obey the Beer-Lambert law. Note also that it is rare for SI units to be used in texts on absorption spectroscopy.

The X-ray linear absorption coefficient, [math]\mu(E)[/math] or [math]\mu_{pe}(E)[/math] follows [math]I(t) = I_0 e^{-\mu t}[/math], with units of length^{-1} (conventionally cm^{-1}). [math]\mu[/math] is the product of density [math]\rho[/math] (g/cm^{3}) and the mass absorption coefficient [math]\Big[{\mu\over\rho}\Big][/math] (cm^{2}/g).

It is sometimes convenient to describe the decrease in the beam intensity in terms of the absorption length: the thickness of the material in question at which the beam intensity has fallen to [math](1/e)[/math] of the incident beam intensity: that is when [math]\mu t=1[/math], or when 63% of the flux is absorbed. In soft X-ray spectroscopy one absorption length can be some tens of nm while typical values in hard X-ray spectroscopy are microns or millimetres.

[math]\mu[/math] depends on energy, [math]E[/math], of the incoming photon and the elemental composition of the sample. The XAFS technique measures the variations in [math]\mu(E)[/math].

## Historical Note

Early references are P. Bouguer, *Essai d’Optique sur la Graduation de la Lumière* (Paris, Jombert, 1729); J. H. Lambert, *Photometria sive Mensura et Gradibus Luminus, Colorum et Umbrae* (Augsburg, 1760); A. Beer, *Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten*, *Annalen der Physik*. **86** (1852) pp. 78-87.

Note that all these preceded the discovery of X-rays, and were based on visible optics. From *A History of Light and Colour Measurement* [S. F. Johnston(2001), CRC Press] p18: ‘The logarithm of the quantity of light received is inversely [meaning multiplied by [math]-1[/math]] proportional to the thickness (Bouguer’s Law) and to the chemical composition (Beer’s Law) of an absorbing material, and the quantity of light to the cosine of the angle of incidence of the illuminated sample (Lambert’s Law)’.