Attenuation coefficient

The attenuation coefficient of a volume, denoted μ, is defined as^[6]

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

where

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

Spectral hemispherical attenuation coefficient

The 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:^[6]

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

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

where

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

Directional attenuation coefficient

The directional attenuation coefficient of a volume, denoted μ_Ω, is defined as^[6]

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

where L_e,Ω is the radiance.

Spectral directional attenuation coefficient

The 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^[6]

${\displaystyle {\begin{aligned}\mu _{\Omega ,\nu }&=-{\frac {1}{L_{\mathrm {e} ,\Omega ,\nu }}}{\frac {\mathrm {d} L_{\mathrm {e} ,\Omega ,\nu }}{\mathrm {d} z}},\\\mu _{\Omega ,\lambda }&=-{\frac {1}{L_{\mathrm {e} ,\Omega ,\lambda }}}{\frac {\mathrm {d} L_{\mathrm {e} ,\Omega ,\lambda }}{\mathrm {d} z}},\end{aligned}}}$

where

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

Decibels

Engineering applications often express attenuation in the logarithmic units of decibels, 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: ${\displaystyle I=I_{o}10^{-(dB/10)}.}$

Naperian attenuation

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

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

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:

${\displaystyle T=e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z}=10^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z},}$

where

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

In case of uniform attenuation, these relations become

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

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

${\displaystyle {\begin{aligned}\mu (z)&=\sum _{i=1}^{N}\mu _{i}(z)=\sum _{i=1}^{N}\sigma _{i}n_{i}(z),\\\mu _{10}(z)&=\sum _{i=1}^{N}\mu _{10,i}(z)=\sum _{i=1}^{N}\varepsilon _{i}c_{i}(z),\end{aligned}}}$

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

${\displaystyle \varepsilon _{i}={\frac {N_{\text{A}}}{\ln {10}}}\,\sigma _{i},}$

and number density and amount concentration by

${\displaystyle c_{i}={\frac {n_{i}}{N_{\text{A}}}},}$

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.