# 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.[1] A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the medium, and a small attenuation coefficient means that the medium is relatively transparent to the beam. The SI unit of attenuation coefficient is the reciprocal metre (m^{−1}). **Extinction coefficient** is an old term for this quantity[1] but is still used in meteorology and climatology.[2] Most commonly, the quantity measures the number of downward *e*-foldings of the original intensity that will be had as the energy 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 **broad-beam attenuation coefficient** counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding.

*For "attenuation coefficient" as it applies to electromagnetic theory and telecommunications see Attenuation constant. For the "mass attenuation coefficient", see Mass attenuation coefficient.*

## Overview

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-rays or gamma rays, where it is denoted
*μ*and measured in cm^{−1}; - neutrons and nuclear reactors, where it is called
*macroscopic cross section*(although actually it is not a section dimensionally speaking), denoted*Σ*and measured in m^{−1}; - ultrasound attenuation, where it is denoted
*α*and measured in dB⋅cm^{−1}⋅MHz^{−1};[3][4] - acoustics for characterizing particle size distribution, where it is denoted
*α*and measured in m^{−1}.

The attenuation coefficient is called the "extinction coefficient" in the context of

- solar and infrared radiative transfer in the atmosphere, albeit usually denoted with another symbol (given the standard use of
*μ*= cos*θ*for slant paths);

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.

## Mathematical definitions

### Hemispherical attenuation coefficient

**Hemispherical attenuation coefficient** of a volume, denoted *μ*, is defined as[5]

where

- Φ
_{e}is the radiant flux; *z*is the path length of the beam.

### Spectral hemispherical attenuation coefficient

**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[5]

where

- Φ
_{e,ν}is the spectral radiant flux in frequency; - Φ
_{e,λ}is the spectral radiant flux in wavelength.

### Directional attenuation coefficient

**Directional attenuation coefficient** of a volume, denoted *μ*_{Ω}, is defined as[5]

where *L*_{e,Ω} is the radiance.

### Spectral directional attenuation coefficient

**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[5]

where

*L*_{e,Ω,ν}is the spectral radiance in frequency;*L*_{e,Ω,λ}is the spectral radiance in wavelength.

## Absorption and scattering coefficients

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

**Absorption coefficient** of a volume, denoted *μ*_{a}, and **scattering coefficient** of a volume, denoted *μ*_{s}, are defined the same way as for attenuation coefficient.[5]

Attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficient:[5]

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

**Mass attenuation coefficient**, **mass absorption coefficient**, and **mass scattering coefficient** are defined as[5]

where *ρ*_{m} is the mass density.

## Napierian and decadic attenuation coefficients

**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 *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^{−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

*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

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 densities and the amount concentrations of its *N* attenuating species as

where

*σ*_{i}is the attenuation cross section of the attenuating species*i*in the material sample;*n*_{i}is the number density of the attenuating species*i*in the material sample;*ε*_{i}is the molar attenuation coefficient of the attenuating species*i*in the material sample;*c*_{i}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

and number density and amount concentration by

where *N*_{A} 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 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.

## SI radiometry units

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol[nb 1] | Name | Symbol | Symbol | ||||

Radiant energy | Q_{e}[nb 2] |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}[nb 2] |
watt | W = J/s | M⋅L^{2}⋅T^{−3} |
Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||

Spectral flux | Φ_{e,ν}[nb 3] |
watt per hertz | W/Hz | M⋅L^{2}⋅T^{−2} |
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm^{−1}. | |||

Φ_{e,λ}[nb 4] |
watt per metre | W/m | M⋅L⋅T^{−3} | |||||

Radiant intensity | I_{e,Ω}[nb 5] |
watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. | |||

Spectral intensity | I_{e,Ω,ν}[nb 3] |
watt per steradian per hertz | W⋅sr^{−1}⋅Hz^{−1} |
M⋅L^{2}⋅T^{−2} |
Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅nm^{−1}. This is a directional quantity. | |||

I_{e,Ω,λ}[nb 4] |
watt per steradian per metre | W⋅sr^{−1}⋅m^{−1} |
M⋅L⋅T^{−3} | |||||

Radiance | L_{e,Ω}[nb 5] |
watt per steradian per square metre | W⋅sr^{−1}⋅m^{−2} |
M⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity". | |||

Spectral radiance | L_{e,Ω,ν}[nb 3] |
watt per steradian per square metre per hertz | W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1} |
M⋅T^{−2} |
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. This is a directional quantity. This is sometimes also confusingly called "spectral intensity". | |||

L_{e,Ω,λ}[nb 4] |
watt per steradian per square metre, per metre | W⋅sr^{−1}⋅m^{−3} |
M⋅L^{−1}⋅T^{−3} | |||||

Irradiance Flux density |
E_{e}[nb 2] |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral irradiance Spectral flux density |
E_{e,ν}[nb 3] |
watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} |
M⋅T^{−2} |
Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10^{−26} W⋅m^{−2}⋅Hz^{−1}) and solar flux unit (1 sfu = 10^{−22} W⋅m^{−2}⋅Hz^{−1} = 10^{4} Jy). | |||

E_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiosity | J_{e}[nb 2] |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral radiosity | J_{e,ν}[nb 3] |
watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} |
M⋅T^{−2} |
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. This is sometimes also confusingly called "spectral intensity". | |||

J_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exitance | M_{e}[nb 2] |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". | |||

Spectral exitance | M_{e,ν}[nb 3] |
watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} |
M⋅T^{−2} |
Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". | |||

M_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exposure | H_{e} |
joule per square metre | J/m^{2} |
M⋅T^{−2} |
Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence". | |||

Spectral exposure | H_{e,ν}[nb 3] |
joule per square metre per hertz | J⋅m^{−2}⋅Hz^{−1} |
M⋅T^{−1} |
Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m^{−2}⋅nm^{−1}. This is sometimes also called "spectral fluence". | |||

H_{e,λ}[nb 4] |
joule per square metre, per metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} | |||||

Hemispherical emissivity | ε |
N/A | 1 |
Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||

Spectral hemispherical emissivity | ε_{ν}orε_{λ} |
N/A | 1 |
Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||

Directional emissivity | ε_{Ω} |
N/A | 1 |
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}orε_{Ω,λ} |
N/A | 1 |
Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface. | ||||

Hemispherical absorptance | A |
N/A | 1 |
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}orA_{λ} |
N/A | 1 |
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} |
N/A | 1 |
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}orA_{Ω,λ} |
N/A | 1 |
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 |
N/A | 1 |
Radiant flux reflected by a surface, divided by that received by that surface. | ||||

Spectral hemispherical reflectance | R_{ν}orR_{λ} |
N/A | 1 |
Spectral flux reflected by a surface, divided by that received by that surface. | ||||

Directional reflectance | R_{Ω} |
N/A | 1 |
Radiance reflected by a surface, divided by that received by that surface. | ||||

Spectral directional reflectance | R_{Ω,ν}orR_{Ω,λ} |
N/A | 1 |
Spectral radiance reflected by a surface, divided by that received by that surface. | ||||

Hemispherical transmittance | T |
N/A | 1 |
Radiant flux transmitted by a surface, divided by that received by that surface. | ||||

Spectral hemispherical transmittance | T_{ν}orT_{λ} |
N/A | 1 |
Spectral flux transmitted by a surface, divided by that received by that surface. | ||||

Directional transmittance | T_{Ω} |
N/A | 1 |
Radiance transmitted by a surface, divided by that received by that surface. | ||||

Spectral directional transmittance | T_{Ω,ν}orT_{Ω,λ} |
N/A | 1 |
Spectral radiance transmitted by a surface, divided by that received by that surface. | ||||

Hemispherical attenuation coefficient | μ |
reciprocal metre | m^{−1} |
L^{−1} |
Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Spectral hemispherical attenuation coefficient | μ_{ν}orμ_{λ} |
reciprocal metre | m^{−1} |
L^{−1} |
Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Directional attenuation coefficient | μ_{Ω} |
reciprocal metre | m^{−1} |
L^{−1} |
Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

Spectral directional attenuation coefficient | μ_{Ω,ν}orμ_{Ω,λ} |
reciprocal metre | m^{−1} |
L^{−1} |
Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||

See also: SI · Radiometry · Photometry |

- Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
- Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- Directional quantities are denoted with suffix "Ω" (Greek).

## 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

- IUPAC,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Attenuation coefficient". doi:10.1351/goldbook.A00516 - "2nd Edition of the Glossary of Meteorology". American Meteorological Society. Retrieved 2015-11-03.
- ISO 20998-1:2006 "Measurement and characterization of particles by acoustic methods"
- Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, 2002
- "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions".
*ISO 9288:1989*. ISO catalogue. 1989. Retrieved 2015-03-15.

## External links

- 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*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Absorption coefficient". doi:10.1351/goldbook.A00037