The Beer–Bouguer–Lambert (BBL) extinction law is an
empirical relationship
In science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural science ...
describing the
attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a Transmission medium, medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and ...
in
intensity
Intensity may refer to:
In colloquial use
* Strength (disambiguation)
*Amplitude
* Level (disambiguation)
* Magnitude (disambiguation)
In physical sciences
Physics
*Intensity (physics), power per unit area (W/m2)
*Field strength of electric, m ...
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.
The extinction law's primary application is in
chemical analysis
Analytical chemistry studies and uses instruments and methods to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separa ...
, where it underlies the Beer–Lambert law, commonly called Beer's law. Beer's law states that a beam of
visible light
Light, visible light, or visible radiation is electromagnetic radiation that can be perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400–700 nanometres (nm ...
passing through a
chemical solution of fixed geometry experiences absorption proportional to the solute
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
. Other applications appear in
physical optics
In physics, physical optics, or wave optics, is the branch of optics that studies Interference (wave propagation), interference, diffraction, Polarization (waves), polarization, and other phenomena for which the ray approximation of geometric opti ...
, where it quantifies
astronomical extinction and the absorption of
photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that ...
,
neutrons, or
rarefied gases.
Forms of the BBL law date back to the mid-eighteenth century, but it only took its modern form during the early twentieth.
History
The first work towards the BBL law began with astronomical observations
Pierre Bouguer
Pierre Bouguer () (16 February 1698, Le Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Career
Bouguer's father, Jean Bouguer, ...
performed in the early eighteenth century and published in 1729. Bouguer needed to compensate for the
refraction
In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commo ...
of light by the
earth's atmosphere
The atmosphere of Earth is composed of a layer of gas mixture that surrounds the Earth's planetary surface (both lands and oceans), known collectively as air, with variable quantities of suspended aerosols and particulates (which create weathe ...
, and found it necessary to measure the local height of the atmosphere. The latter, he sought to obtain through variations in the observed intensity of known stars. When calibrating this effect, Bouguer discovered that light intensity had an exponential dependence on length traveled through the atmosphere (in Bouguer's terms, a
geometric progression
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the s ...
).
Bouguer's work was then popularized in
Johann Heinrich Lambert
Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...
's ''
Photometria'' in 1760. Lambert expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in a mathematical form quite similar to that used in modern physics. Lambert began by assuming that the intensity of light traveling into an absorbing body would be given by the differential equation
which is compatible with Bouguer's observations. The constant of proportionality was often termed the "optical density" of the body. As long as is constant along a distance , the exponential attenuation law,
follows from integration.
In 1852,
August Beer noticed that colored solutions also appeared to exhibit a similar
attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a Transmission medium, medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and ...
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." 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
Ultraviolet radiation, also known as simply UV, is electromagnetic radiation of wavelengths of 10–400 nanometers, shorter than that of visible light, but longer than X-rays. UV radiation is present in sunlight and constitutes about 10% of ...
,
visible, or
infrared
Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
), except at entry and exit. Thus light within a solution is reasonably approximated as due to absorption alone. In Bouguer's context, atmospheric dust or other inhomogeneities could also
scatter light away from the detector. Modern texts combine the two laws because scattering and absorption have the same effect. Thus a scattering coefficient and an absorption coefficient can be combined into a total extinction coefficient .
Importantly, Beer also seems to have conceptualized his result in terms of a given thickness' opacity, writing "If is the coefficient (fraction) of diminution, then this coefficient (fraction) will have the value for double this thickness." Although this geometric progression is mathematically equivalent to the modern law, modern treatments instead emphasize the logarithm of , which clarifies that concentration and path length have equivalent effects on the absorption. An early, possibly the first, modern formulation was given by Robert Luther and Andreas Nikolopulos in 1913.
Mathematical formulations
There are several equivalent formulations of the BBL law, depending on the precise choice of measured quantities. All of them state that, provided that the physical state is held constant, the extinction process is linear in the intensity of radiation and amount of radiatively-active matter, a fact sometimes called the fundamental law of extinction. Many of them then connect the quantity of radiatively-active matter to a length traveled and a
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
or
number density . For concentrations expressed as moles per volume, the latter two are related by
Avogadro's number: .
A collimated beam (directed radiation) with cross-sectional area will encounter particles (on average) during its travel. However, not all of these particles interact with the beam. Propensity to interact is a material-dependent property, typically summarized in absorptivity or
scattering cross-section
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflect ...
. These almost exhibit another Avogadro-type relationship: . The factor of appears because physicists tend to use natural logarithms and chemists decadal logarithms.
Beam intensity can also be described in terms of multiple variables: the
intensity
Intensity may refer to:
In colloquial use
* Strength (disambiguation)
*Amplitude
* Level (disambiguation)
* Magnitude (disambiguation)
In physical sciences
Physics
*Intensity (physics), power per unit area (W/m2)
*Field strength of electric, m ...
or
radiant flux
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the ...
. In the case of a collimated beam, these are related by , but is often used in non-collimated contexts. The ratio of intensity (or flux) in to out is sometimes summarized as a
transmittance
Electromagnetic radiation can be affected in several ways by the medium in which it propagates. It can be Scattering, scattered, Absorption (electromagnetic radiation), absorbed, and Fresnel equations, reflected and refracted at discontinui ...
coefficient .
When considering an extinction law,
dimensional analysis can verify the consistency of the variables, as logarithms (being nonlinear) must always be dimensionless.
Formulation
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. This expression is:
The quantities so equated are defined to be the
absorbance , which depends on the
logarithm base. The
Naperian absorbance is then given by and satisfies
If multiple species in the material interact with the radiation, then their absorbances add. Thus a slightly more general formulation is that
where the sum is over all possible radiation-interacting ("translucent") species, and indexes those species.
In situations where length may vary significantly, absorbance is sometimes summarized in terms of an
attenuation coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient val ...
In
atmospheric science
Atmospheric science is the study of the Atmosphere of Earth, Earth's atmosphere and its various inner-working physical processes. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather forecasting. Clima ...
and
radiation shielding applications, the attenuation coefficient may vary significantly through an inhomogenous material. In those situations, the most general form of the Beer–Lambert law states that the total attenuation can be obtained by integrating the attenuation coefficient over small slices of the beamline:
These formulations then reduce to the simpler versions when there is only one active species and the attenuation coefficients are constant.
Derivation
There are two factors that determine the degree to which a medium containing particles will attenuate a light beam: the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light.
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
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient val ...
, which yields the following first-order
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
,
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
:
The attenuation is caused by the photons that did not make it to the other side of the slice because of
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
or
absorption. The solution to this differential equation is obtained by multiplying the
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivari ...
throughout to obtain
which simplifies due to the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
(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
gives
and 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 coefficient
In chemistry, the molar absorption coefficient or molar attenuation coefficient () is a measurement of how strongly a chemical species absorbs, and thereby attenuates, light at a given wavelength. It is an intrinsic property of the species. The In ...
s
where is the
Avogadro constant
The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles.
It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
, to describe the attenuation coefficient in a way independent of the
amount concentrations
of the attenuating species of the material sample:
Validity
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of
analyte
An analyte, component (in clinical chemistry), titrand (in titrations), or chemical species is a substance or chemical constituent that is of interest in an analytical procedure. The remainder of the sample is called the matrix. The procedure ...
. These deviations are classified into three categories:
# Real—fundamental deviations due to the limitations of the law itself.
# Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
# Instrument—deviations which occur due to how the attenuation measurements are made.
There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid. These are:
# The attenuators must act independently of each other.
# The attenuating medium must be homogeneous in the interaction volume.
# The attenuating medium must not scatter the radiation—no
turbidity
Turbidity is the cloudiness or haziness of a fluid caused by large numbers of individual particles that are generally invisible to the naked eye, similar to smoke in air. The measurement of turbidity is a key test of both water clarity and wa ...
—unless this is accounted for as in
DOAS.
# The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
# The incident radiation should preferably be
monochromatic
A monochrome or monochromatic image, object or palette is composed of one color (or values of one color). Images using only shades of grey are called grayscale (typically digital) or black-and-white (typically analog). In physics, mon ...
, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the power is needed instead of a photodiode which cannot discriminate between wavelengths.
# The incident flux must not influence the atoms or molecules; it should only act as a non-invasive probe of the species under study. In particular, this implies that the light should not cause optical saturation or optical pumping, since such effects will deplete the lower level and possibly give rise to stimulated emission.
If any of these conditions are not fulfilled, there will be deviations from the Beer–Lambert law.
The law tends to break down at very high concentrations, especially if the material is highly
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
. Absorbance within range of 0.2 to 0.5 is ideal to maintain linearity in the Beer–Lambert law. If the radiation is especially intense,
nonlinear optical
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typicall ...
processes can also cause variances. 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. For strong oscillators and at high concentrations the deviations are stronger. If the
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s are closer to each other interactions can set in. These interactions can be roughly divided into physical and chemical interactions. 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. Chemical interactions in contrast change the polarizability and thus absorption.
In solids, attenuation is usually an addition of absorption coefficient
(creation of electron-hole pairs) or scattering (for example
Rayleigh scattering
Rayleigh scattering ( ) is the scattering or deflection of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scat ...
if the scattering centers are much smaller than the incident wavelength). Also note that for some systems we can put
(1 over inelastic mean free path) in place of
Applications
In plasma physics
The BBL extinction law also arises as a solution to the
BGK equation.
Chemical analysis by spectrophotometry
The Beer–Lambert law can be applied to the analysis of a mixture by
spectrophotometry
Spectrophotometry is a branch of electromagnetic spectroscopy concerned with the quantitative measurement of the reflection or transmission properties of a material as a function of wavelength. Spectrophotometry uses photometers, known as spe ...
, without the need for extensive pre-processing of the sample. An example is the determination of
bilirubin
Bilirubin (BR) (adopted from German, originally bili—bile—plus ruber—red—from Latin) is a red-orange compound that occurs in the normcomponent of the straw-yellow color in urine. Another breakdown product, stercobilin, causes the brown ...
in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient is known. Measurements of decadic attenuation coefficient are made at one wavelength that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration is then given by
For a more complicated example, consider a mixture in solution containing two species at amount concentrations and . The decadic attenuation coefficient at any wavelength is, given by
Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations and as long as the molar attenuation coefficients of the two components, and are known at both wavelengths. This two system equation can be solved using
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
. In practice it is better to use
linear least squares to determine the two amount concentrations from measurements made at more than two wavelengths.
Mixtures containing more than two components can be analyzed in the same way, using a minimum of wavelengths for a mixture containing components. So, in general:
where
is the absorbance at wavelength
,
is the molar absorptivity of component
at
,
is the concentration of component
, and
is the path length.
The law is used widely in
infra-red spectroscopy and
near-infrared spectroscopy for analysis of
polymer degradation and
oxidation
Redox ( , , reduction–oxidation or oxidation–reduction) is a type of chemical reaction in which the oxidation states of the reactants change. Oxidation is the loss of electrons or an increase in the oxidation state, while reduction is ...
(also in biological tissue) as well as to measure the
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
of various compounds in different
food samples. The
carbonyl group
In organic chemistry, a carbonyl group is a functional group with the formula , composed of a carbon atom double-bonded to an oxygen atom, and it is divalent at the C atom. It is common to several classes of organic compounds (such as aldehydes ...
attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the
polymer
A polymer () is a chemical substance, substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeat unit, repeating subunits derived from one or more species of monomers. Due to their br ...
calculated.
In-atmosphere astronomy
The Bouguer–Lambert law may be applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The optical depth for a slant path is , where refers to a vertical path, is called the
relative airmass, and for a plane-parallel atmosphere it is determined as where is the
zenith angle corresponding to the given path. The Bouguer-Lambert law for the atmosphere is usually written
where each is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
* refers to
aerosols
An aerosol is a suspension of fine solid particles or liquid droplets in air or another gas. Aerosols can be generated from natural or human causes. The term ''aerosol'' commonly refers to the mixture of particulates in air, and not to t ...
(that absorb and scatter);
* are uniformly mixed gases (mainly
carbon dioxide
Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
(CO
2) and molecular
oxygen
Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
(O
2) which only absorb);
* is
nitrogen dioxide
Nitrogen dioxide is a chemical compound with the formula . One of several nitrogen oxides, nitrogen dioxide is a reddish-brown gas. It is a paramagnetic, bent molecule with C2v point group symmetry. Industrially, is an intermediate in the s ...
, mainly due to urban pollution (absorption only);
* are effects due to
Raman scattering
In chemistry and physics, Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrationa ...
in the atmosphere;
* is
water vapour
Water vapor, water vapour, or aqueous vapor is the gaseous phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Water vapor ...
absorption;
* is
ozone
Ozone () (or trioxygen) is an Inorganic compound, inorganic molecule with the chemical formula . It is a pale blue gas with a distinctively pungent smell. It is an allotrope of oxygen that is much less stable than the diatomic allotrope , break ...
(absorption only);
* is
Rayleigh scattering
Rayleigh scattering ( ) is the scattering or deflection of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scat ...
from molecular
oxygen
Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
() and
nitrogen
Nitrogen is a chemical element; it has Symbol (chemistry), symbol N and atomic number 7. Nitrogen is a Nonmetal (chemistry), nonmetal and the lightest member of pnictogen, group 15 of the periodic table, often called the Pnictogen, pnictogens. ...
() (responsible for the blue color of the sky);
* the selection of the attenuators which have to be considered depends on the wavelength range and can include various other compounds. This can include
tetraoxygen,
HONO,
formaldehyde
Formaldehyde ( , ) (systematic name methanal) is an organic compound with the chemical formula and structure , more precisely . The compound is a pungent, colourless gas that polymerises spontaneously into paraformaldehyde. It is stored as ...
,
glyoxal, a series of halogen radicals and others.
is the ''optical mass'' or ''
airmass factor'', a term approximately equal (for small and moderate values of ) to where is the observed object's
zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve , the aerosol
optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.
See also
*
Applied spectroscopy
*
Atomic absorption spectroscopy
*
Absorption spectroscopy
Absorption spectroscopy is spectroscopy that involves techniques that measure the absorption of electromagnetic radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons ...
*
Cavity ring-down spectroscopy
*
Clausius–Mossotti relation
*
Infra-red spectroscopy
*
Job plot
*
Laser absorption spectrometry
*
Lorentz–Lorenz relation
*
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
*
Polymer degradation
*
Scientific laws named after people
*
Quantification of nucleic acids
*
Tunable diode laser absorption spectroscopy
*
Transmittance#Beer–Lambert law
References
External links
Beer–Lambert Law Calculator
{{DEFAULTSORT:Beer-Lambert Law
Eponymous laws of physics
Scattering, absorption and radiative transfer (optics)
Spectroscopy
Electromagnetic radiation
Visibility