The linear attenuation coefficient, attenuation coefficient, or narrowbeam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter.^{[1]} 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.^{[2]} The SI unit of attenuation coefficient is the reciprocal metre (m^{−1}). Extinction coefficient is another term for this quantity,^{[1]} often used in meteorology and climatology.^{[3]} Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward efolding 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^{−1} means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m^{−1}, it will be reduced twice by e, or e^{2}. Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broadbeam attenuation coefficient counts forwardscattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding.
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:
The attenuation coefficient is called the "extinction coefficient" in the context of
A small attenuation coefficient indicates that the material in question is relatively transparent, while a larger value indicates greater degrees of 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.
The attenuation coefficient of a volume, denoted μ, is defined as^{[6]}
where
The spectral hemispherical attenuation coefficient in frequency and spectral hemispherical attenuation coefficient in wavelength of a volume, denoted μ_{ν} and μ_{λ} respectively, are defined as:^{[6]}
where
The directional attenuation coefficient of a volume, denoted μ_{Ω}, is defined as^{[6]}
where L_{e,Ω} is the radiance.
The spectral directional attenuation coefficient in frequency and spectral directional attenuation coefficient in wavelength of a volume, denoted μ_{Ω,ν} and μ_{Ω,λ} respectively, are defined as^{[6]}
where
When a narrow (collimated) beam passes through a volume, the beam will lose intensity due to two processes: 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_{a}, and the scattering coefficient of a volume, denoted μ_{s}, are defined the same way as the attenuation coefficient.^{[6]}
The attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficients:^{[6]}
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 nonnarrow 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 narrowbeam intensity, including scattering as well. "Narrowbeam 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.
The mass attenuation coefficient, mass absorption coefficient, and mass scattering coefficient are defined as^{[6]}
where ρ_{m} is the mass density.
Engineering applications often express attenuation in the logarithmic units of decibels, 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:
The decadic attenuation coefficient or decadic narrow beam attenuation coefficient, denoted μ_{10}, is defined as
Just as the usual attenuation coefficient measures the number of efold reductions that occur over a unit length of material, this coefficient measures how many 10fold reductions occur: a decadic coefficient of 1 m^{−1} 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 in the Beer–Lambert law for a material sample, in which the two attenuation coefficients take part:
where
In case of uniform attenuation, these relations become
Cases of nonuniform 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 densities and the amount concentrations of its N attenuating species as
where
by definition of attenuation cross section and molar attenuation coefficient.
Attenuation cross section and molar attenuation coefficient are related by
and number density and amount concentration by
where N_{A} is the Avogadro constant.
The halfvalue 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 halfvalue layer is about 69% (ln 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.
Quantity  SI units  Notes  

Name  Sym.  
Hemispherical emissivity  ε  —  Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface. 
Spectral hemispherical emissivity  ε_{ν} ε_{λ} 
—  Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. 
Directional emissivity  ε_{Ω}  —  Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. 
Spectral directional emissivity  ε_{Ω,ν} ε_{Ω,λ} 
—  Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface. 
Hemispherical absorptance  A  —  Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance". 
Spectral hemispherical absorptance  A_{ν} A_{λ} 
—  Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance". 
Directional absorptance  A_{Ω}  —  Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance". 
Spectral directional absorptance  A_{Ω,ν} A_{Ω,λ} 
—  Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance". 
Hemispherical reflectance  R  —  Radiant flux reflected by a surface, divided by that received by that surface. 
Spectral hemispherical reflectance  R_{ν} R_{λ} 
—  Spectral flux reflected by a surface, divided by that received by that surface. 
Directional reflectance  R_{Ω}  —  Radiance reflected by a surface, divided by that received by that surface. 
Spectral directional reflectance  R_{Ω,ν} R_{Ω,λ} 
—  Spectral radiance reflected by a surface, divided by that received by that surface. 
Hemispherical transmittance  T  —  Radiant flux transmitted by a surface, divided by that received by that surface. 
Spectral hemispherical transmittance  T_{ν} T_{λ} 
—  Spectral flux transmitted by a surface, divided by that received by that surface. 
Directional transmittance  T_{Ω}  —  Radiance transmitted by a surface, divided by that received by that surface. 
Spectral directional transmittance  T_{Ω,ν} T_{Ω,λ} 
—  Spectral radiance transmitted by a surface, divided by that received by that surface. 
Hemispherical attenuation coefficient
μ
m−1
Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient
μνμλ
m−1
Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient
μΩ
m−1
Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient
μΩ,νμΩ,λ
m−1
Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
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