# Physics:Transmittance

__: Effectiveness of a material in transmitting radiant energy__

**Short description**In optical physics, **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.^{[2]}

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

## Mathematical definitions

### Hemispherical transmittance

**Hemispherical transmittance** of a surface, denoted *T*, is defined as^{[3]}

- [math]\displaystyle{ T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}}, }[/math]

where

- Φ
_{e}^{t}is the radiant flux*transmitted*by that surface; - Φ
_{e}^{i}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 *T*_{ν} and *T*_{λ} respectively, are defined as^{[3]}

- [math]\displaystyle{ T_\nu = \frac{\Phi_{\mathrm{e},\nu}^\mathrm{t}}{\Phi_{\mathrm{e},\nu}^\mathrm{i}}, }[/math]
- [math]\displaystyle{ T_\lambda = \frac{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}{\Phi_{\mathrm{e},\lambda}^\mathrm{i}}, }[/math]

where

- Φ
_{e,ν}^{t}is the spectral radiant flux in frequency*transmitted*by that surface; - Φ
_{e,ν}^{i}is the spectral radiant flux in frequency received by that surface; - Φ
_{e,λ}^{t}is the spectral radiant flux in wavelength*transmitted*by that surface; - Φ
_{e,λ}^{i}is the spectral radiant flux in wavelength received by that surface.

### Directional transmittance

**Directional transmittance** of a surface, denoted *T*_{Ω}, is defined as^{[3]}

- [math]\displaystyle{ T_\Omega = \frac{L_{\mathrm{e},\Omega}^\mathrm{t}}{L_{\mathrm{e},\Omega}^\mathrm{i}}, }[/math]

where

*L*_{e,Ω}^{t}is the radiance*transmitted*by that surface;*L*_{e,Ω}^{i}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 *T*_{ν,Ω} and *T*_{λ,Ω} respectively, are defined as^{[3]}

- [math]\displaystyle{ T_{\nu,\Omega} = \frac{L_{\mathrm{e},\Omega,\nu}^\mathrm{t}}{L_{\mathrm{e},\Omega,\nu}^\mathrm{i}}, }[/math]
- [math]\displaystyle{ T_{\lambda,\Omega} = \frac{L_{\mathrm{e},\Omega,\lambda}^\mathrm{t}}{L_{\mathrm{e},\Omega,\lambda}^\mathrm{i}}, }[/math]

where

*L*_{e,Ω,ν}^{t}is the spectral radiance in frequency*transmitted*by that surface;*L*_{e,Ω,ν}^{i}is the spectral radiance received by that surface;*L*_{e,Ω,λ}^{t}is the spectral radiance in wavelength*transmitted*by that surface;*L*_{e,Ω,λ}^{i}is the spectral radiance in wavelength received by that surface.

## Beer–Lambert law

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

- [math]\displaystyle{ T = e^{-\tau} = 10^{-A}, }[/math]

where

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

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

- [math]\displaystyle{ T = e^{-\sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\mathrm{d}z} = 10^{-\sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\mathrm{d}z}, }[/math]

or equivalently that

- [math]\displaystyle{ \tau = \sum_{i = 1}^N \tau_i = \sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\,\mathrm{d}z, }[/math]
- [math]\displaystyle{ A = \sum_{i = 1}^N A_i = \sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrm{d}z, }[/math]

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;*ℓ*is the path length of the beam of light through the material sample.

Attenuation cross section and molar attenuation coefficient are related by

- [math]\displaystyle{ \varepsilon_i = \frac{\mathrm{N_A}}{\ln{10}}\,\sigma_i, }[/math]

and number density and amount concentration by

- [math]\displaystyle{ c_i = \frac{n_i}{\mathrm{N_A}}, }[/math]

where N_{A} is the Avogadro constant.

In case of *uniform* attenuation, these relations become^{[4]}

- [math]\displaystyle{ T = e^{-\sum_{i = 1}^N \sigma_i n_i\ell} = 10^{-\sum_{i = 1}^N \varepsilon_i c_i\ell}, }[/math]

or equivalently

- [math]\displaystyle{ \tau = \sum_{i = 1}^N \sigma_i n_i\ell, }[/math]
- [math]\displaystyle{ A = \sum_{i = 1}^N \varepsilon_i c_i\ell. }[/math]

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

## Other radiometric coefficients

## See also

## References

- ↑ "Electronic warfare and radar systems engineering handbook". http://ewhdbks.mugu.navy.mil/EO-IR.htm#transmission.
- ↑ IUPAC,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484 - ↑
^{3.0}^{3.1}^{3.2}^{3.3}"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions".*ISO 9288:1989*. ISO catalogue. 1989. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943. - ↑ IUPAC,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626

Original source: https://en.wikipedia.org/wiki/Transmittance.
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