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]}
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]}
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]}
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]}
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
where
- τ is the optical depth;
- A is the absorbance.
The Beer–Lambert law states that, for N attenuating species in the material sample,
or equivalently that
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
and number density and amount concentration by
where N_{A} is the Avogadro constant.
In case of uniform attenuation, these relations become^{[4]}
or equivalently
Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.
SI radiometry units
Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|
Name | Symbol^{[nb 1]} | Name | Symbol | Symbol | ||||
Radiant energy | Q_{e}^{[nb 2]} | joule | J | M⋅L^{2}⋅T^{−2} | Energy of electromagnetic radiation. | |||
Radiant energy density | w_{e} | joule per cubic metre | J/m^{3} | M⋅L^{−1}⋅T^{−2} | Radiant energy per unit volume. | |||
Radiant flux | Φ_{e}^{[nb 2]} | watt | W = J/s | M⋅L^{2}⋅T^{−3} | Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||
Spectral flux | Φ_{e,ν}^{[nb 3]} | watt per hertz | W/Hz | M⋅L^{2}⋅T^{−2} | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm^{−1}. | |||
Φ_{e,λ}^{[nb 4]} | watt per metre | W/m | M⋅L⋅T^{−3} | |||||
Radiant intensity | I_{e,Ω}^{[nb 5]} | watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} | Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. | |||
Spectral intensity | I_{e,Ω,ν}^{[nb 3]} | watt per steradian per hertz | W⋅sr^{−1}⋅Hz^{−1} | M⋅L^{2}⋅T^{−2} | Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅nm^{−1}. This is a directional quantity. | |||
I_{e,Ω,λ}^{[nb 4]} | watt per steradian per metre | W⋅sr^{−1}⋅m^{−1} | M⋅L⋅T^{−3} | |||||
Radiance | L_{e,Ω}^{[nb 5]} | watt per steradian per square metre | W⋅sr^{−1}⋅m^{−2} | M⋅T^{−3} | Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity". | |||
Spectral radiance | L_{e,Ω,ν}^{[nb 3]} | watt per steradian per square metre per hertz | W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1} | M⋅T^{−2} | Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. This is a directional quantity. This is sometimes also confusingly called "spectral intensity". | |||
L_{e,Ω,λ}^{[nb 4]} | watt per steradian per square metre, per metre | W⋅sr^{−1}⋅m^{−3} | M⋅L^{−1}⋅T^{−3} | |||||
Irradiance Flux density |
E_{e}^{[nb 2]} | watt per square metre | W/m^{2} | M⋅T^{−3} | Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". | |||
Spectral irradiance Spectral flux density |
E_{e,ν}^{[nb 3]} | watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} | M⋅T^{−2} | Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10^{−26} W⋅m^{−2}⋅Hz^{−1}) and solar flux unit (1 sfu = 10^{−22} W⋅m^{−2}⋅Hz^{−1} = 10^{4} Jy). | |||
E_{e,λ}^{[nb 4]} | watt per square metre, per metre | W/m^{3} | M⋅L^{−1}⋅T^{−3} | |||||
Radiosity | J_{e}^{[nb 2]} | watt per square metre | W/m^{2} | M⋅T^{−3} | Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity". | |||
Spectral radiosity | J_{e,ν}^{[nb 3]} | watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} | M⋅T^{−2} | Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. This is sometimes also confusingly called "spectral intensity". | |||
J_{e,λ}^{[nb 4]} | watt per square metre, per metre | W/m^{3} | M⋅L^{−1}⋅T^{−3} | |||||
Radiant exitance | M_{e}^{[nb 2]} | watt per square metre | W/m^{2} | M⋅T^{−3} | Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". | |||
Spectral exitance | M_{e,ν}^{[nb 3]} | watt per square metre per hertz | W⋅m^{−2}⋅Hz^{−1} | M⋅T^{−2} | Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m^{−2}⋅nm^{−1}. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". | |||
M_{e,λ}^{[nb 4]} | watt per square metre, per metre | W/m^{3} | M⋅L^{−1}⋅T^{−3} | |||||
Radiant exposure | H_{e} | joule per square metre | J/m^{2} | M⋅T^{−2} | Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence". | |||
Spectral exposure | H_{e,ν}^{[nb 3]} | joule per square metre per hertz | J⋅m^{−2}⋅Hz^{−1} | M⋅T^{−1} | Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m^{−2}⋅nm^{−1}. This is sometimes also called "spectral fluence". | |||
H_{e,λ}^{[nb 4]} | joule per square metre, per metre | J/m^{3} | M⋅L^{−1}⋅T^{−2} | |||||
Hemispherical emissivity | ε | N/A | 1 | Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||
Spectral hemispherical emissivity | ε_{ν} or ε_{λ} |
N/A | 1 | Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||
Directional emissivity | ε_{Ω} | N/A | 1 | Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. | ||||
Spectral directional emissivity | ε_{Ω,ν} or ε_{Ω,λ} |
N/A | 1 | Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface. | ||||
Hemispherical absorptance | A | N/A | 1 | Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance". | ||||
Spectral hemispherical absorptance | A_{ν} or A_{λ} |
N/A | 1 | Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance". | ||||
Directional absorptance | A_{Ω} | N/A | 1 | Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance". | ||||
Spectral directional absorptance | A_{Ω,ν} or A_{Ω,λ} |
N/A | 1 | Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance". | ||||
Hemispherical reflectance | R | N/A | 1 | Radiant flux reflected by a surface, divided by that received by that surface. | ||||
Spectral hemispherical reflectance | R_{ν} or R_{λ} |
N/A | 1 | Spectral flux reflected by a surface, divided by that received by that surface. | ||||
Directional reflectance | R_{Ω} | N/A | 1 | Radiance reflected by a surface, divided by that received by that surface. | ||||
Spectral directional reflectance | R_{Ω,ν} or R_{Ω,λ} |
N/A | 1 | Spectral radiance reflected by a surface, divided by that received by that surface. | ||||
Hemispherical transmittance | T | N/A | 1 | Radiant flux transmitted by a surface, divided by that received by that surface. | ||||
Spectral hemispherical transmittance | T_{ν} or T_{λ} |
N/A | 1 | Spectral flux transmitted by a surface, divided by that received by that surface. | ||||
Directional transmittance | T_{Ω} | N/A | 1 | Radiance transmitted by a surface, divided by that received by that surface. | ||||
Spectral directional transmittance | T_{Ω,ν} or T_{Ω,λ} |
N/A | 1 | Spectral radiance transmitted by a surface, divided by that received by that surface. | ||||
Hemispherical attenuation coefficient | μ | reciprocal metre | m^{−1} | L^{−1} | Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||
Spectral hemispherical attenuation coefficient | μ_{ν} or μ_{λ} |
reciprocal metre | m^{−1} | L^{−1} | Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||
Directional attenuation coefficient | μ_{Ω} | reciprocal metre | m^{−1} | L^{−1} | Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||
Spectral directional attenuation coefficient | μ_{Ω,ν} or μ_{Ω,λ} |
reciprocal metre | m^{−1} | L^{−1} | Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. | |||
See also: SI · Radiometry · Photometry |
- ^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- ^ ^{a} ^{b} Directional quantities are denoted with suffix "Ω" (Greek).
See also
References
- ^ "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001.CS1 maint: unfit URL (link)
- ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484
- ^ ^{a} ^{b} ^{c} ^{d} "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
- ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626