Thelinear attenuation coefficient,attenuation coefficient, ornarrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam oflight,sound,particles, or otherenergy ormatter.^[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 (derived)SI unit of attenuation coefficient is thereciprocal metre (m⁻¹).Extinction coefficient is another term for this quantity,^[1] often used inmeteorology andclimatology.^[3] Theattenuation length is the reciprocal of the attenuation coefficient.^[4]

The attenuation coefficient describes the extent to which theradiant flux of a beam is reduced as it passes through a specific material. It is used in the context of:

• X-rays orgamma rays, where it is denotedμ and measured in cm⁻¹;
• neutrons andnuclear reactors, where it is calledmacroscopic cross section (although actually it is not a section dimensionally speaking), denotedΣ and measured in m⁻¹;
• ultrasound attenuation, where it is denotedα and measured indB⋅cm⁻¹⋅MHz⁻¹;^[5][6]
• acoustics for characterizingparticle size distribution, where it is denotedα and measured in m⁻¹.

The attenuation coefficient is called the "extinction coefficient" or sometimesabsorption coefficient in the context ofsolar andinfrared radiative transfer in theatmosphere.^[4]: 423

A small attenuation coefficient indicates that the material in question is relativelytransparent, while a larger value indicates greater degrees ofopacity. 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 of light as it moves through a thin layer of a homogenous material is proportional to the layer thickness,${\displaystyle d}$ and the initial intensity,${\displaystyle I_0}$. The resulting intensity is given by$${\displaystyle I(d)=I_0{e}^{-\alpha d}}$$where${\displaystyle \alpha }$ is the attenuation coefficient. This formula is known as the Beer-Lambert law.^[7]This attenuation coefficient measures theexponential decay of intensity, that is, the value of downwarde-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⁻¹ means that after passing through 1 metre, the radiation will be reduced by a factor ofe, and for material with a coefficient of 2 m⁻¹, it will be reduced twice bye, ore². Other measures may use a different factor thane, such as thedecadic attenuation coefficient below. Thebroad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable toradiation shielding.Themass attenuation coefficient is the attenuation coefficient normalized by the density of the material.

Theattenuation coefficient of a volume, denotedμ, is defined as^[8]

${\displaystyle \mu =-\frac{1}{{\mathrm{\Phi }}_{\mathrm{e}}}\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{e}}}{\mathrm{d}z},}$

where

• Φ_e is theradiant flux;
• z is the path length of the beam.

Note that for an attenuation coefficient which does not vary withz, this equation is solved along a line from${\displaystyle z}$=0 to${\displaystyle z}$ as:

${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}}={\mathrm{\Phi }}_{\mathrm{e}0}{e}^{-\mu z}}$

where${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}0}}$ is the incoming radiation flux at${\displaystyle z}$=0 and${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}}}$ is the radiation flux at${\displaystyle z}$.

Thespectral hemispherical attenuation coefficient in frequency andspectral hemispherical attenuation coefficient in wavelength of a volume, denotedμ_ν andμ_λ respectively, are defined as:^[8]

${\displaystyle {\mu }_{\nu }=-\frac{1}{{\mathrm{\Phi }}_{\mathrm{e},\nu }}\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{e},\nu }}{\mathrm{d}z},}$

${\displaystyle {\mu }_{\lambda }=-\frac{1}{{\mathrm{\Phi }}_{\mathrm{e},\lambda }}\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{e},\lambda }}{\mathrm{d}z},}$

where

• Φ_e,ν is thespectral radiant flux in frequency;
• Φ_e,λ is thespectral radiant flux in wavelength.

Thedirectional attenuation coefficient of a volume, denotedμ_Ω, is defined as^[8]

${\displaystyle {\mu }_{\mathrm{\Omega }}=-\frac{1}{L_{\mathrm{e},\mathrm{\Omega }}}\frac{\mathrm{d}{L}_{\mathrm{e},\mathrm{\Omega }}}{\mathrm{d}z},}$

whereL_e,Ω is theradiance.

Thespectral directional attenuation coefficient in frequency andspectral directional attenuation coefficient in wavelength of a volume, denotedμ_Ω,ν andμ_Ω,λ respectively, are defined as^[8]

where

• L_e,Ω,ν is thespectral radiance in frequency;
• L_e,Ω,λ is thespectral radiance in wavelength.

When a narrow (collimated) beam passes through a volume, the beam will lose intensity due to two processes:absorption andscattering. 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.

Theabsorption coefficient of a volume, denotedμ_a, and thescattering coefficient of a volume, denotedμ_s, are defined the same way as the attenuation coefficient.^[8]

The attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficients:^[8]

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 absorptionalone, while "attenuation coefficient" measures thetotal 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.

The absorption coefficient may be expressed in terms of a number density of absorbing centersn and an absorbing cross section areaσ.^[9] For a slab of areaA and thicknessdz, the total number of absorbing centers contained isn A dz. Assuming that dz is so small that there will be no overlap of the cross section areas, the total area available for absorption will ben A σ dz and the fraction of radiation absorbed is thenn σ dz. The absorption coefficient is thusμ = n σ

Themass attenuation coefficient,mass absorption coefficient, andmass scattering coefficient are defined as^[8]

${\displaystyle \frac{\mu }{{\rho }_m},\frac{{\mu }_{\mathrm{a}}}{{\rho }_m},\frac{{\mu }_{\mathrm{s}}}{{\rho }_m},}$

whereρ_m is themass density.

Engineering applications often express attenuation in thelogarithmic units ofdecibels, 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:${\displaystyle I=I_o{10}^{-(dB/10)}.}$

Thedecadic attenuation coefficient ordecadic narrow beam attenuation coefficient, denotedμ₁₀, is defined as

${\displaystyle {\mu }_{10}=\frac{\mu }{\ln 10}.}$

Just as the usual attenuation coefficient measures the number ofe-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⁻¹ means 1 m of material reduces the radiation once by a factor of 10.

μ is sometimes calledNapierian attenuation coefficient orNapierian narrow beam attenuation coefficient rather than just simply "attenuation coefficient". The terms "decadic" and "Napierian" come from the base used for theexponential in theBeer–Lambert law for a material sample, in which the two attenuation coefficients take part:

${\displaystyle T=e^{-{\int }_0^{\ell }\mu (z)\mathrm{d}z}={10}^{-{\int }_0^{\ell }{\mu }_{10}(z)\mathrm{d}z},}$

where

• T is thetransmittance of the material sample;
• ℓ is the path length of the beam of light through the material sample.

In case ofuniform attenuation, these relations become

${\displaystyle T=e^{-\mu \ell }={10}^{-{\mu }_{10}\ell }.}$

Cases ofnon-uniform attenuation occur inatmospheric science applications andradiation shielding theory for instance.

The (Napierian) attenuation coefficient and the decadic attenuation coefficient of a material sample are related to thenumber densities and theamount concentrations of itsN attenuating species as

where

• σ_i is the attenuationcross section of the attenuating speciesi in the material sample;
• n_i is thenumber density of the attenuating speciesi in the material sample;
• ε_i is themolar attenuation coefficient of the attenuating speciesi in the material sample;
• c_i is theamount concentration of the attenuating speciesi in the material sample,

by definition of attenuation cross section and molar attenuation coefficient.

Attenuation cross section and molar attenuation coefficient are related by

${\displaystyle {\epsilon }_i=\frac{N_{\text{A}}}{\ln 10}{\sigma }_i,}$

and number density and amount concentration by

${\displaystyle c_i=\frac{n_i}{N_{\text{A}}},}$

whereN_A is theAvogadro constant.

Thehalf-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 2) of thepenetration 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 tomean free path. Moreover, it is very closely related to the attenuationcross section.

• 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

• Absorption Coefficients α of Building Materials and Finishes
• Sound Absorption Coefficients for Some Common Materials
• 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
• IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Absorption coefficient".doi:10.1351/goldbook.A00037

• Physical quantities
• Radiometry
• Acoustics

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