What causes deviations from the Bier-Lambert law

Beer-Lambert Law - Beer-Lambert Law

Law describing light absorption
A demonstration of the Beer-Lambert law: green laser light in a solution of rhodamine 6B. The radiation power of the beam becomes weaker as it passes through the solution.

The Beer-Lambert law , also known as Beer's law , the Lambert-Beer law or that Beer-Lambert law Bouguer concerns the attenuation of light on the properties of the material through which the light moves. The law is commonly applied to chemical analysis measurements and is used to understand the attenuation in physical optics for photons, neutrons, or dilute gases. In mathematical physics, this law results from the solution of the BGK equation.


The law was discovered by Pierre Bouguer before 1729 while on short vacation in Alentejo, Portugal, while looking at red wine. It is often attributed to Johann Heinrich Lambert, who wrote in his 1760 Photometria Bouguers Essai d'optique sur la gradation de la lumière (Claude Jombert, Paris, 1729) cited - and even cited from it The intensity of light, when it propagates in a medium, is directly proportional to the intensity and the length of the path. Much later, in 1852, August Beer discovered another dampening relationship. Beer's law says that the permeability of a solution remains constant if the product of concentration and path length remains constant. The modern derivation of the Beer-Lambert law combines the two laws and correlates the extinction, which is the negative decadic logarithm of the permeability, both with the concentrations of the attenuating species and with the thickness of the material sample.

Mathematical formulation

A general and practical expression of Beer-Lambert's law relates to the optical attenuation of a physical material containing a single attenuation species in uniform concentration, the optical path length through the sample, and the absorptivity of the species. This expression is:


A more general form of the Beer-Lambert law states that to attenuate species in the material sample

or equivalent that


In the above equations, the transmittance of the material sample is defined by the following definition with its optical depth and its absorption A in Relationship set


  • is the radiation flux , the is transferred from this material sample;
  • is the radiant flux received by this material sample.

The damping cross-section and the molar damping coefficient are related

and density of numbers and concentration of quantities

where is the constant Avogadro.

At more even These relationships are dampening

or equivalent

cases more uneven Attenuation occurs, for example, in atmospheric scientific applications and in radiation protection theory.

The law tends to break down at very high concentrations, especially if the material scatters heavily. The absorption in the range of 0.2-0.5 is ideal for maintaining linearity in the Beer-Lambart law. If the radiation is particularly intense, non-linear optical processes can also cause deviations. The main reason, however, is that the concentration dependence is generally not linear and Beer's law is only valid under certain conditions, as can be seen from the following derivation. In the case of strong oscillators and high concentrations, the deviations are greater. When the molecules are closer together, interactions can occur. These interactions can be roughly divided into physical and chemical interactions. The physical interaction does not change the polarizability of the molecules as long as the interaction is not so strong that light and the molecular quantum state mix (strong coupling), but rather ensures that the attenuation cross-sections via the electromagnetic coupling are not additive. In contrast, chemical interactions change the polarizability and thus the absorption.

Expression with damping coefficient

Beer-Lambert's law can be expressed as a damping coefficient, but in this case it is better known as Lambert's law because the volume concentration is hidden in the damping coefficient according to Beer's law. The (Napierian) damping coefficient and the decadal damping coefficient of a material sample are related to their number densities and quantity concentrations as

in each case by defining the damping cross-section and the molar damping coefficient. Then the Beer-Lambert Law


At more even These relationships are dampening

or equivalent

In many cases the damping coefficient does not vary with it. In this case one does not have to perform an integral and can express the law as follows:

where the attenuation is usually an addition of the absorption coefficient (generation of electron-hole pairs) or of the scattering (e.g. Rayleigh scattering if the scattering centers are much smaller than the incident wavelength). Note also that for some systems we can put (1 over inelastic mean free path) instead of.


Suppose a beam of light enters a sample of material. Define z as an axis parallel to the beam direction. Divide the material sample into thin slices perpendicular to the light beam with a thickness d z that is so small that a particle in one layer cannot obscure another particle in the same layer when it passes along the z- Direction is considered. The radiant flux of the light exiting a disc is reduced compared to that of the light entering from d? e ( z ) = - μ ( z ) Φ e ( z ) d Z , in which μ the (Nepersian) damping coefficient, which gives the following linear ODE of first order:

The attenuation is caused by the photons that did not make it to the other side of the layer due to scattering or absorption. The solution of this differential equation is obtained by multiplying the integration factor

to get anywhere

This is simplified due to the product rule (applied backwards)

Integrate both sides and resolve to Φ e for a material with real thickness , where the radiation flux incident on the pane Φ e i = Φ e (0) and the transmitted radiation flux Φ e t = Φ e ( ) result

and finally

Since the decadal damping coefficient μ 10 with the (Napierian) damping coefficient μ 10 = μ / ln 10 is related, one also has