# Attenuation coefficient

For "attenuation coefficient" as it applies to electromagnetic theory and telecommunications see propagation constant. For the "mass attenuation coefficient", see the article mass attenuation coefficient.

Attenuation coefficient or narrow beam attenuation coefficient of the volume of a material characterizes how easily it 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 still used in meteorology and climatology [2]

## 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

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]

$\mu = -\frac{1}{\Phi_\mathrm{e}} \frac{\mathrm{d}\Phi_\mathrm{e}}{\mathrm{d}z},$

where

• Φe 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 μν and μλ respectively, are defined as[5]

$\mu_\nu = -\frac{1}{\Phi_{\mathrm{e},\nu}} \frac{\mathrm{d}\Phi_{\mathrm{e},\nu}}{\mathrm{d}z},$
$\mu_\lambda = -\frac{1}{\Phi_{\mathrm{e},\lambda}} \frac{\mathrm{d}\Phi_{\mathrm{e},\lambda}}{\mathrm{d}z},$

where

### Directional attenuation coefficient

Directional attenuation coefficient of a volume, denoted μΩ, is defined as[5]

$\mu_\Omega = -\frac{1}{L_{\mathrm{e},\Omega}} \frac{\mathrm{d}L_{\mathrm{e},\Omega}}{\mathrm{d}z},$

### Spectral directional attenuation coefficient

Spectral directional attenuation coefficient in frequency and spectral directional attenuation coefficient in wavelength of a volume, denoted μΩ,ν and μΩ,λ respectively, are defined as[5]

$\mu_{\Omega,\nu} = -\frac{1}{L_{\mathrm{e},\Omega,\nu}} \frac{\mathrm{d}L_{\mathrm{e},\Omega,\nu}}{\mathrm{d}z},$
$\mu_{\Omega,\lambda} = -\frac{1}{L_{\mathrm{e},\Omega,\lambda}} \frac{\mathrm{d}L_{\mathrm{e},\Omega,\lambda}}{\mathrm{d}z},$

where

## 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 μa, and scattering coefficient of a volume, denoted μs, 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]

$\mu = \mu_\mathrm{a} + \mu_\mathrm{s},$
$\mu_\nu = \mu_{\mathrm{a},\nu} + \mu_{\mathrm{s},\nu},$
$\mu_\lambda = \mu_{\mathrm{a},\lambda} + \mu_{\mathrm{s},\lambda},$
$\mu_\Omega = \mu_{\mathrm{a},\Omega} + \mu_{\mathrm{s},\Omega},$
$\mu_{\Omega,\nu} = \mu_{\mathrm{a},\Omega,\nu} + \mu_{\mathrm{s},\Omega,\nu},$
$\mu_{\Omega,\lambda} = \mu_{\mathrm{a},\Omega,\lambda} + \mu_{\mathrm{s},\Omega,\lambda}.$

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 always larger than the absorption coefficient, although 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]

$\frac{\mu}{\rho_m},\quad \frac{\mu_\mathrm{a}}{\rho_m},\quad \frac{\mu_\mathrm{s}}{\rho_m},$

where ρm is the mass density.

## Napierian and decadic attenuation coefficients

Decadic attenuation coefficient or decadic narrow beam attenuation coefficient, denoted μ10, is defined as

$\mu_{10} = \frac{\mu}{\ln 10}.$

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:

$T = e^{-\int_0^\ell \mu(z)\mathrm{d}z} = 10^{-\int_0^\ell \mu_{10}(z)\mathrm{d}z},$

where

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

In case of uniform attenuation, these relations become

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

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

$\mu(z) = \sum_{i = 1}^N \mu_i(z) = \sum_{i = 1}^N \sigma_i n_i(z),$
$\mu_{10}(z) = \sum_{i = 1}^N \mu_{10,i}(z) = \sum_{i = 1}^N \varepsilon_i c_i(z),$

where

by definition of attenuation cross section and molar attenuation coefficient.

Attenuation cross section and molar attenuation coefficient are related by

$\varepsilon_i = \frac{\mathrm{N_A}}{\ln{10}}\,\sigma_i,$

and number density and amount concentration by

$c_i = \frac{n_i}{\mathrm{N_A}},$

where NA 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. It is from these equations that engineers decide how much protection is needed for "safety" from potentially harmful radiation.

Attenuation coefficient is also inversely related to mean free path. Moreover, it is very closely related to the attenuation cross section.

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy per unit volume.
Radiant flux Φe[nb 2] watt W or J/s ML2T−3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]
or
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−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".
or
Le,Ω,λ[nb 4]
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
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".
Irradiance Ee[nb 2] watt per square metre W/m2 MT−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
or
Ee,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Irradiance of a surface per unit frequency or wavelength. The terms spectral flux density or more confusingly "spectral intensity" are also used. Non-SI units of spectral irradiance include Jansky = 10−26 W⋅m−2⋅Hz−1 and solar flux unit (1SFU = 10−22 W⋅m−2⋅Hz−1).
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
or
Je,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
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".
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−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 Me,ν[nb 3]
or
Me,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
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".
Radiant exposure He joule per square metre J/m2 MT−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 He,ν[nb 3]
or
He,λ[nb 4]
joule per square metre per hertz
or
joule per square metre, per metre
J⋅m−2⋅Hz−1
or
J/m3
MT−1
or
ML−1T−2
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".
Hemispherical emissivity ε 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
or
ελ
1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
or
εΩ,λ
1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance 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 Aν
or
Aλ
1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
or
AΩ,λ
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 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
or
Rλ
1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
or
RΩ,λ
1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
or
Tλ
1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
or
TΩ,λ
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 μν
or
μλ
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 μΩ 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 μΩ,ν
or
μΩ,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.
1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. 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.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).