Samiksha Jaiswal (Editor)

Transmittance

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Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.

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Internal transmittance refers to energy loss by absorption, whereas (total) transmittance is that due to absorption, scattering, reflection, etc.

Hemispherical transmittance

Hemispherical transmittance of a surface, denoted T, is defined as

T = Φ e t Φ e i ,

where

  • Φet is the radiant flux transmitted by that surface;
  • Φei is the radiant flux received by that surface.

    • Spectral hemispherical transmittance

    Spectral hemispherical transmittance in frequency and spectral hemispherical transmittance in wavelength of a surface, denoted Tν and Tλ respectively, are defined as

    T ν = Φ e , ν t Φ e , ν i , T λ = Φ e , λ t Φ e , λ i ,

    where

  • Φe,νt is the spectral radiant flux in frequency transmitted by that surface;
  • Φe,νi is the spectral radiant flux in frequency received by that surface;
  • Φe,λt is the spectral radiant flux in wavelength transmitted by that surface;
  • Φe,λi is the spectral radiant flux in wavelength received by that surface.

    • Directional transmittance

    Directional transmittance of a surface, denoted TΩ, is defined as

    T Ω = L e , Ω t L e , Ω i ,

    where

  • Le,Ωt is the radiance transmitted by that surface;
  • Le,Ωi is the radiance received by that surface.

    • Spectral directional transmittance

    Spectral directional transmittance in frequency and spectral directional transmittance in wavelength of a surface, denoted Tν,Ω and Tλ,Ω respectively, are defined as

    T ν , Ω = L e , Ω , ν t L e , Ω , ν i , T λ , Ω = L e , Ω , λ t L e , Ω , λ i ,

    where

  • Le,Ω,νt is the spectral radiance in frequency transmitted by that surface;
  • Le,Ω,νi is the spectral radiance received by that surface;
  • Le,Ω,λt is the spectral radiance in wavelength transmitted by that surface;
  • Le,Ω,λi is the spectral radiance in wavelength received by that surface.

    • Beer–Lambert law

    By definition, transmittance is related to optical depth and to absorbance as

    T = e − τ = 10 − A ,

    where

  • τ is the optical depth;
  • A is the absorbance.

    • The Beer–Lambert law states that, for N attenuating species in the material sample,

    T = e − ∑ i = 1 N σ i ∫ 0 ℓ n i ( z ) d z = 10 − ∑ i = 1 N ε i ∫ 0 ℓ c i ( z ) d z ,

    or equivalently that

    τ = ∑ i = 1 N τ i = ∑ i = 1 N σ i ∫ 0 ℓ n i ( z ) d z , A = ∑ i = 1 N A i = ∑ i = 1 N ε i ∫ 0 ℓ c i ( z ) d 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;
  • ℓ is the path length of the beam of light through the material sample.

    • 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.

    In case of uniform attenuation, these relations become

    T = e − ∑ i = 1 N σ i n i ℓ = 10 − ∑ i = 1 N ε i c i ℓ ,

    or equivalently

    τ = ∑ i = 1 N σ i n i ℓ , A = ∑ i = 1 N ε i c i ℓ .

    Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

    References

    Transmittance Wikipedia
    (Text) CC BY-SA