Girish Mahajan (Editor)

Attenuation coefficient

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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. 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, but still used in meteorology and climatology

Contents

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;
  • 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.

    Hemispherical attenuation coefficient

    Hemispherical attenuation coefficient of a volume, denoted μ, is defined as

    μ = − 1 Φ e d Φ e 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

    μ ν = − 1 Φ e , ν d Φ e , ν d z , μ λ = − 1 Φ e , λ d Φ e , λ d z ,

    where

  • Φe,ν is the spectral radiant flux in frequency;
  • Φe,λ is the spectral radiant flux in wavelength.

    • Directional attenuation coefficient

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

    μ Ω = − 1 L e , Ω d L e , Ω d z ,

    where Le,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 μΩ,ν and μΩ,λ respectively, are defined as

    μ Ω , ν = − 1 L e , Ω , ν d L e , Ω , ν d z , μ Ω , λ = − 1 L e , Ω , λ d L e , Ω , λ d z ,

    where

  • Le,Ω,ν is the spectral radiance in frequency;
  • Le,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 μa, and scattering coefficient of a volume, denoted μs, are defined the same way as for attenuation coefficient.

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

    μ = μ a + μ s , μ ν = μ a , ν + μ s , ν , μ λ = μ a , λ + μ s , λ , μ Ω = μ a , Ω + μ s , Ω , μ Ω , ν = μ a , Ω , ν + μ s , Ω , ν , μ Ω , λ = μ a , Ω , λ + μ s , Ω , λ .

    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

    μ ρ m , μ a ρ m , μ s ρ 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

    μ 10 = μ 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 − ∫ 0 ℓ μ ( z ) d z = 10 − ∫ 0 ℓ μ 10 ( z ) d z ,

    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

    T = e − μ ℓ = 10 − μ 10 ℓ .

    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

    μ ( z ) = ∑ i = 1 N μ i ( z ) = ∑ i = 1 N σ i n i ( z ) , μ 10 ( z ) = ∑ i = 1 N μ 10 , i ( z ) = ∑ i = 1 N ε i c i ( z ) ,

    where

  • σi is the attenuation cross section of the attenuating specie i in the material sample;
  • ni is the number density of the attenuating specie i in the material sample;
  • εi is the molar attenuation coefficient of the attenuating specie i in the material sample;
  • ci is the amount concentration of the attenuating specie 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

    ε i = N A ln ⁡ 10 σ i ,

    and number density and amount concentration by

    c i = n i 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.

    References

    Attenuation coefficient Wikipedia
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