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. 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 is still used in meteorology and climatology. Most commonly, the quantity measures the value of downward "e"-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 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.
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
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.
Attenuation coefficient.
Attenuation coefficient of a volume, denoted "μ", is defined as
where
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
where
Directional attenuation coefficient.
Directional attenuation coefficient of a volume, denoted "μ"Ω, is defined as
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
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 "μ"a, and scattering coefficient of a volume, denoted "μ"s, are defined the same way as for attenuation coefficient.
Attenuation coefficient of a volume is the sum of absorption coefficient and scattering coefficient:
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
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
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
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.