The Beer–Bouguer–Lambert (BBL) extinction law is an empirical relationship describing the attenuation in intensity of a radiation beam passing through a macroscopically homogenous medium with which it interacts.
Formally, it states that the intensity of radiation decays exponentially in the absorbance of the medium, and that said absorbance is proportional to the length of beam passing through the medium, the concentration of interacting matter along that path, and a constant representing said matter's propensity to interact.
Other applications appear in physical optics, where it quantifies astronomical extinction and the absorption of photons, neutrons, or rarefied gases.
The first work towards the BBL law began with astronomical observations Pierre Bouguer performed in the early eighteenth century and published in 1729.
Lambert began by assuming that the intensity I of light traveling into an absorbing body would be given by the differential equation
[4] In 1852, August Beer noticed that colored solutions also appeared to exhibit a similar attenuation relation.
In his analysis, Beer does not discuss Bouguer and Lambert's prior work, writing in his introduction that "Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there is no information available.
"[5] Beer may have omitted reference to Bouguer's work because there is a subtle physical difference between color absorption in solutions and astronomical contexts.
Solutions are homogeneous and do not scatter light at common analytical wavelengths (ultraviolet, visible, or infrared), except at entry and exit.
In Bouguer's context, atmospheric dust or other inhomogeneities could also scatter light away from the detector.
[10] There are several equivalent formulations of the BBL law, depending on the precise choice of measured quantities.
A collimated beam (directed radiation) with cross-sectional area S will encounter Sℓn particles (on average) during its travel.
Propensity to interact is a material-dependent property, typically summarized in absorptivity ϵ[12] or scattering cross-section σ.
When considering an extinction law, dimensional analysis can verify the consistency of the variables, as logarithms (being nonlinear) must always be dimensionless.
The simplest formulation of Beer's relates the optical attenuation of a physical material containing a single attenuating species of uniform concentration to the optical path length through the sample and absorptivity of the species.
In situations where length may vary significantly, absorbance is sometimes summarized in terms of an attenuation coefficient
In atmospheric science and radiation shielding applications, the attenuation coefficient may vary significantly through an inhomogenous material.
These formulations then reduce to the simpler versions when there is only one active species and the attenuation coefficients are constant.
where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation:
Integrating both sides and solving for Φe for a material of real thickness ℓ, with the incident radiant flux upon the slice
where NA is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.
[16] These deviations are classified into three categories: There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid.
The main reason, however, is that the concentration dependence is in general non-linear and Beer's law is valid only under certain conditions as shown by derivation below.
Physical interaction do not alter the polarizability of the molecules as long as the interaction is not so strong that light and molecular quantum state intermix (strong coupling), but cause the attenuation cross sections to be non-additive via electromagnetic coupling.
The Beer–Lambert law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample.
In practice it is better to use linear least squares to determine the two amount concentrations from measurements made at more than two wavelengths.
The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.
The Bouguer–Lambert law may be applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere.
where each τx is the optical depth whose subscript identifies the source of the absorption or scattering it describes: m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to