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{{Short description|Light or sound absorption in a substance}}
{{for|"attenuation coefficient" as it applies to electromagnetic theory and telecommunications|Attenuation constant}}

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]], [[particle]]s, or other [[energy]] or [[matter]].<ref name=GoldBook>{{GoldBookRef|title=Attenuation coefficient|file=A00516|accessdate=2015-03-15}}</ref> A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss.<ref>{{Cite book|last=Serway|first=Raymond|title=Modern Physics|last2=Moses|first2=Clement|last3=Moyer|first3=Curt|publisher=Brooks/Cole|year=2005|isbn=978-0-534-49339-4|location=California, USA|pages=529}}</ref> The (derived) [[International System of Units|SI unit]] of attenuation coefficient is the [[reciprocal metre]] (m<sup>−1</sup>). '''Extinction coefficient''' is another term for this quantity,<ref name=GoldBook/> often used in [[meteorology]] and [[climatology]].<ref name="AMSGlossary">{{cite web|url=https://glossary.ametsoc.org/wiki/Extinction_coefficient|title=2nd Edition of the Glossary of Meteorology|publisher=[[American Meteorological Society]]|access-date=2015-11-03}}</ref> Most commonly, the quantity measures the [[exponential decay]] of intensity, that is, the value of downward [[e-folding|''e''-folding]] distance of the original intensity as the energy of the intensity passes through a unit (''e.g.'' one meter) thickness of material, so that an attenuation coefficient of 1 m<sup>−1</sup> means that after passing through 1 metre, the radiation will be reduced by a factor of ''[[e (mathematical constant)|e]]'', and for material with a coefficient of 2 m<sup>−1</sup>, it will be reduced twice by ''e'', or ''e''<sup>2</sup>. Other measures may use a different factor than ''e'', such as the ''decadic attenuation coefficient'' below. The '''broad-beam attenuation coefficient''' counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to [[radiation shielding]].
The ''[[mass attenuation coefficient]]'' is the attenuation coefficient normalized by the density of the material.

==Overview==
The attenuation coefficient describes the extent to which the [[radiant flux]] of a beam is reduced as it passes through a specific material. It is used in the context of:
*[[X-ray]]s or [[gamma ray]]s, where it is denoted ''μ'' and measured in cm<sup>−1</sup>;
*[[neutron]]s and [[nuclear reactor]]s, where it is called ''macroscopic cross section'' (although actually it is not a section dimensionally speaking), denoted ''Σ'' and measured in m<sup>−1</sup>;
*[[ultrasound attenuation]], where it is denoted ''α'' and measured in [[decibel|dB]]⋅cm<sup>−1</sup>⋅MHz<sup>−1</sup>;<ref name="ReferenceA">ISO 20998-1:2006 "Measurement and characterization of particles by acoustic methods"</ref><ref name="autogenerated2002">Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, 2002</ref>
*[[acoustics]] for characterizing [[particle size distribution]], where it is denoted ''α'' and measured in m<sup>−1</sup>.

The attenuation coefficient is called the "extinction coefficient" in the context of
*[[visible spectrum|solar]] and [[infrared]] radiative transfer in the [[atmosphere]], albeit usually denoted with another symbol (given the standard use of {{nowrap|1=''μ'' = cos ''θ''}} for slant paths);

A small attenuation coefficient indicates that the material in question is relatively [[transparency and translucency|transparent]], while a larger value indicates greater degrees of [[Opacity (optics)|opacity]]. The attenuation coefficient is dependent upon the type of material and the energy of the radiation. Generally, for electromagnetic radiation, the higher the energy of the incident photons and the less dense the material in question, the lower the corresponding attenuation coefficient will be.

==Mathematical definitions==

===Attenuation coefficient===
The '''attenuation coefficient''' of a volume, denoted ''μ'', is defined as<ref name="ISO_9288-1989">{{cite web|url=https://www.iso.org/standard/16943.html|title=Thermal insulation — Heat transfer by radiation — Physical quantities and definitions|work=ISO 9288:1989|publisher=[[International Organization for Standardization|ISO]] catalogue|year=1989|access-date=2015-03-15}}</ref>
:<math>\mu = -\frac{1}{\Phi_\mathrm{e}} \frac{\mathrm{d}\Phi_\mathrm{e}}{\mathrm{d}z},</math>
where
*Φ<sub>e</sub> is the [[radiant flux]];
*''z'' is the path length of the beam.

===Spectral hemispherical attenuation coefficient===
The '''spectral hemispherical attenuation coefficient in frequency''' and '''spectral hemispherical attenuation coefficient in wavelength''' of a volume, denoted ''μ''<sub>ν</sub> and ''μ''<sub>λ</sub> respectively, are defined as:<ref name="ISO_9288-1989"/>
:<math>\mu_\nu = -\frac{1}{\Phi_{\mathrm{e},\nu}} \frac{\mathrm{d}\Phi_{\mathrm{e},\nu}}{\mathrm{d}z},</math>
:<math>\mu_\lambda = -\frac{1}{\Phi_{\mathrm{e},\lambda}} \frac{\mathrm{d}\Phi_{\mathrm{e},\lambda}}{\mathrm{d}z},</math>
where
*Φ<sub>e,ν</sub> is the [[radiant flux|spectral radiant flux in frequency]];
*Φ<sub>e,λ</sub> is the [[radiant flux|spectral radiant flux in wavelength]].

===Directional attenuation coefficient===
The '''directional attenuation coefficient''' of a volume, denoted ''μ''<sub>Ω</sub>, is defined as<ref name="ISO_9288-1989"/>
:<math>\mu_\Omega = -\frac{1}{L_{\mathrm{e},\Omega}} \frac{\mathrm{d}L_{\mathrm{e},\Omega}}{\mathrm{d}z},</math>
where ''L''<sub>e,Ω</sub> is the [[radiance]].

===Spectral directional attenuation coefficient===
The '''spectral directional attenuation coefficient in frequency''' and '''spectral directional attenuation coefficient in wavelength''' of a volume, denoted ''μ''<sub>Ω,ν</sub> and ''μ''<sub>Ω,λ</sub> respectively, are defined as<ref name="ISO_9288-1989"/>
:<math>\begin{align}
      \mu_{\Omega,\nu} &= -\frac{1}{L_{\mathrm{e},\Omega,\nu}} \frac{\mathrm{d}L_{\mathrm{e},\Omega,\nu}}{\mathrm{d}z}, \\
  \mu_{\Omega,\lambda} &= -\frac{1}{L_{\mathrm{e},\Omega,\lambda}} \frac{\mathrm{d}L_{\mathrm{e},\Omega,\lambda}}{\mathrm{d}z},
\end{align}</math>
where
*''L''<sub>e,Ω,ν</sub> is the [[radiance|spectral radiance in frequency]];
*''L''<sub>e,Ω,λ</sub> is the [[radiance|spectral radiance in wavelength]].

==Absorption and scattering coefficients==
{{further|Optical extinction coefficient}}

When a narrow ([[collimated light|collimated]]) beam passes through a volume, the beam will lose intensity due to two processes: [[absorption (electromagnetic radiation)|absorption]] and [[scattering]]. Absorption indicates energy that is lost from the beam, while scattering indicates light that is redirected in a (random) direction, and hence is no longer in the beam, but still present, resulting in diffuse light.

The '''absorption coefficient''' of a volume, denoted ''μ''<sub>a</sub>, and the '''scattering coefficient''' of a volume, denoted ''μ''<sub>s</sub>, are defined the same way as the attenuation coefficient.<ref name="ISO_9288-1989"/>

The attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficients:<ref name="ISO_9288-1989"/>
:<math>\begin{align}
                   \mu &= \mu_\mathrm{a} + \mu_\mathrm{s}, \\
               \mu_\nu &= \mu_{\mathrm{a},\nu} + \mu_{\mathrm{s},\nu}, \\
           \mu_\lambda &= \mu_{\mathrm{a},\lambda} + \mu_{\mathrm{s},\lambda}, \\
            \mu_\Omega &= \mu_{\mathrm{a},\Omega} + \mu_{\mathrm{s},\Omega}, \\
      \mu_{\Omega,\nu} &= \mu_{\mathrm{a},\Omega,\nu} + \mu_{\mathrm{s},\Omega,\nu}, \\
  \mu_{\Omega,\lambda} &= \mu_{\mathrm{a},\Omega,\lambda} + \mu_{\mathrm{s},\Omega,\lambda}.
\end{align}</math>

Just looking at the narrow beam itself, the two processes cannot be distinguished. However, if a detector is set up to measure beam leaving in different directions, or conversely using a non-narrow beam, one can measure how much of the lost radiant flux was scattered, and how much was absorbed.

In this context, the "absorption coefficient" measures how quickly the beam would lose radiant flux due to the absorption ''alone'', while "attenuation coefficient" measures the ''total'' loss of narrow-beam intensity, including scattering as well. "Narrow-beam attenuation coefficient" always unambiguously refers to the latter. The attenuation coefficient is at least as large as the absorption coefficient; they are equal in the idealized case of no scattering.

==Mass attenuation, absorption, and scattering coefficients==
{{main|Mass attenuation coefficient}}
The '''mass attenuation coefficient''', '''mass absorption coefficient''', and '''mass scattering coefficient''' are defined as<ref name="ISO_9288-1989"/>
:<math>\frac{\mu}{\rho_m},\quad \frac{\mu_\mathrm{a}}{\rho_m},\quad \frac{\mu_\mathrm{s}}{\rho_m},</math>
where ''ρ''<sub>''m''</sub> is the [[mass density]].

==Napierian and decadic attenuation coefficients==

===Decibels===
{{main|decibel}}

Engineering applications often express attenuation in the [[Logarithmic_scale#Logarithmic_units|logarithmic units]] of [[decibel]]s, or "dB", where 10 dB represents attenuation by a factor of 10. The units for attenuation coefficient are thus dB/m (or, in general, dB per unit distance). Note that in logarithmic units such as dB, the attenuation is a linear function of distance, rather than exponential.  This has the advantage that the result of multiple attenuation layers can be found by simply adding up the dB loss for each individual passage. However, if intensity is desired, the logarithms must be converted back into linear units by using an exponential: <math>I = I_o 10^{-(dB/10)}.</math>

===Naperian attenuation===
The '''decadic attenuation coefficient''' or '''decadic narrow beam attenuation coefficient''', denoted ''μ''<sub>10</sub>, is defined as
:<math>\mu_{10} = \frac{\mu}{\ln 10}.</math>

Just as the usual attenuation coefficient measures the number of ''e''-fold reductions that occur over a unit length of material, this coefficient measures how many 10-fold reductions occur: a decadic coefficient of 1 m<sup>−1</sup> means 1 m of material reduces the radiation once by a factor of 10.

''μ'' is sometimes called '''Napierian attenuation coefficient''' or '''Napierian narrow beam attenuation coefficient''' rather than just simply "attenuation coefficient". The terms "decadic" and "Napierian" come from the base used for the [[exponential function|exponential]] in the [[Beer–Lambert law]] for a material sample, in which the two attenuation coefficients take part:
:<math>T = e^{-\int_0^\ell \mu(z)\mathrm{d}z} = 10^{-\int_0^\ell \mu_{10}(z)\mathrm{d}z},</math>
where
*''T'' is the [[transmittance]] of the material sample;
*''ℓ'' is the path length of the beam of light through the material sample.

In case of ''uniform'' attenuation, these relations become
:<math>T = e^{-\mu\ell} = 10^{-\mu_{10}\ell}.</math>
Cases of ''non-uniform'' attenuation occur in [[atmospheric science]] applications and [[radiation shielding]] theory for instance.

The (Napierian) attenuation coefficient and the decadic attenuation coefficient of a material sample are related to the [[number density|number densities]] and the [[amount concentration]]s of its ''N'' attenuating species as
:<math>\begin{align}
       \mu(z) &= \sum_{i = 1}^N \mu_i(z) = \sum_{i = 1}^N \sigma_i n_i(z), \\
  \mu_{10}(z) &= \sum_{i = 1}^N \mu_{10,i}(z) = \sum_{i = 1}^N \varepsilon_i c_i(z),
\end{align}</math>
where
*''σ''<sub>''i''</sub> is the attenuation [[cross section (physics)|cross section]] of the attenuating species ''i'' in the material sample;
*''n''<sub>''i''</sub> is the [[number density]] of the attenuating species ''i'' in the material sample;
*''ε''<sub>''i''</sub> is the [[molar attenuation coefficient]] of the attenuating species ''i'' in the material sample;
*''c''<sub>''i''</sub> is the [[amount concentration]] of the attenuating species ''i'' in the material sample,
by definition of attenuation cross section and molar attenuation coefficient.

Attenuation cross section and molar attenuation coefficient are related by
:<math>\varepsilon_i = \frac{N_\text{A}}{\ln{10}}\,\sigma_i,</math>
and number density and amount concentration by
:<math>c_i = \frac{n_i}{N_\text{A}},</math>
where ''N''<sub>A</sub> is the [[Avogadro constant]].

The [[half-value layer]] (HVL) is the thickness of a layer of material required to reduce the radiant flux of the transmitted radiation to half its incident magnitude. The half-value layer is about 69% (ln&nbsp;2) of the [[penetration depth]]. Engineers use these equations predict how much shielding thickness is required to attenuate radiation to acceptable or regulatory limits.

Attenuation coefficient is also inversely related to [[mean free path]]. Moreover, it is very closely related to the attenuation [[cross section (physics)|cross section]].

==Other radiometric coefficients==
{{Radiometry coefficients}}

==See also==
*[[Absorption (electromagnetic radiation)]]
*[[Absorption cross section]]
*[[Absorption spectrum]]
*[[Acoustic attenuation]]
*[[Attenuation]]
*[[Attenuation length]]
*[[Beer–Lambert law]]
*[[Cargo scanning]]
*[[Compton edge]]
*[[Compton scattering]]
*[[Computation of radiowave attenuation in the atmosphere]]
*[[Cross section (physics)]]
*[[Grey atmosphere]]
*[[High-energy X-rays]]
*[[Mass attenuation coefficient]]
*[[Mean free path]]
*[[Propagation constant]]
*[[Radiation length]]
*[[Scattering theory]]
*[[Transmittance]]

==References==
{{Reflist}}

==External links==
*[http://www.sengpielaudio.com/calculator-RT60Coeff.htm Absorption Coefficients α of Building Materials and Finishes]
*[https://www.engineeringtoolbox.com/accoustic-sound-absorption-d_68.html Sound Absorption Coefficients for Some Common Materials]
*[https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest]
*{{GoldBookRef|title=Absorption coefficient|file=A00037|accessdate=2019-06-05}}<!--(versus Attenuation coefficient...)-->

[[Category:Physical quantities]]
[[Category:Radiometry]]
[[Category:Acoustics]]