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1
M a t h e m a t i c a l d e f i n i t i o n s
T o g g l e M a t h e m a t i c a l d e f i n i t i o n s s u b s e c t i o n
1 . 1
H e m i s p h e r i c a l t r a n s m i t t a n c e
1 . 2
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1 . 3
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1 . 4
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1 . 5
L u m i n o u s t r a n s m i t t a n c e
2
B e e r – L a m b e r t l a w
3
O t h e r r a d i o m e t r i c c o e f f i c i e n t s
4
S e e a l s o
5
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
T r a n s m i t t a n c e
1 8 l a n g u a g e s
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Effectiveness of a material in transmitting radiant energy
This article is about transmission through a
volume . For transmission through a
surface , see
Surface transmittance .
Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region[1] ). Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.
Transmittance of ruby in optical and near-IR spectra. Note the two broad blue and green absorption bands and one narrow absorption band on the wavelength of 694 nm, which is the wavelength of the ruby laser .
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 [ edit ]
Hemispherical transmittance [ edit ]
Hemispherical transmittance of a surface, denoted T , is defined as[3]
T
=
Φ
e
t
Φ
e
i
,
{\displaystyle T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}},}
where
Φe t is the radiant flux transmitted by that surface;
Φe i is the radiant flux received by that surface.
Spectral hemispherical transmittance [ edit ]
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]
T
ν
=
Φ
e
,
ν
t
Φ
e
,
ν
i
,
{\displaystyle T_{\nu }={\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}},}
T
λ
=
Φ
e
,
λ
t
Φ
e
,
λ
i
,
{\displaystyle T_{\lambda }={\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}},}
where
Directional transmittance [ edit ]
Directional transmittance of a surface, denoted T Ω , is defined as[3]
T
Ω
=
L
e
,
Ω
t
L
e
,
Ω
i
,
{\displaystyle T_{\Omega }={\frac {L_{\mathrm {e} ,\Omega }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega }^{\mathrm {i} }}},}
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 [ edit ]
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]
T
ν
,
Ω
=
L
e
,
Ω
,
ν
t
L
e
,
Ω
,
ν
i
,
{\displaystyle T_{\nu ,\Omega }={\frac {L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {i} }}},}
T
λ
,
Ω
=
L
e
,
Ω
,
λ
t
L
e
,
Ω
,
λ
i
,
{\displaystyle T_{\lambda ,\Omega }={\frac {L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {i} }}},}
where
Luminous transmittance [ edit ]
In the field of photometry (optics) , the luminous transmittance of a filter is a measure of the amount of luminous flux or intensity transmitted by an optical filter. It is generally defined in terms of a standard illuminant (e.g. Illuminant A, Iluminant C, or Illuminant E). The luminous transmittance with respect to the standard illuminant is defined as:
T
l
u
m
=
∫
0
∞
I
(
λ
)
T
(
λ
)
V
(
λ
)
d
λ
∫
0
∞
I
(
λ
)
V
(
λ
)
d
λ
{\displaystyle T_{lum}={\frac {\int _{0}^{\infty }I(\lambda )T(\lambda )V(\lambda )d\lambda }{\int _{0}^{\infty }I(\lambda )V(\lambda )d\lambda }}}
where:
I
(
λ
)
{\displaystyle I(\lambda )}
is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).
T
(
λ
)
{\displaystyle T(\lambda )}
is the spectral transmittance of the filter
V
(
λ
)
{\displaystyle V(\lambda )}
is the luminous efficiency function
The luminous transmittance is independent of the magnitude of the flux or intensity of the standard illuminant used to measure it, and is a dimensionless quantity.
Beer–Lambert law [ edit ]
By definition, internal transmittance is related to optical depth and to absorbance as
T
=
e
−
τ
=
10
−
A
,
{\displaystyle T=e^{-\tau }=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
,
{\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},}
or equivalently that
τ
=
∑
i
=
1
N
τ
i
=
∑
i
=
1
N
σ
i
∫
0
ℓ
n
i
(
z
)
d
z
,
{\displaystyle \tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z )\,\mathrm {d} z,}
A
=
∑
i
=
1
N
A
i
=
∑
i
=
1
N
ε
i
∫
0
ℓ
c
i
(
z
)
d
z
,
{\displaystyle A=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z )\,\mathrm {d} z,}
where
Attenuation cross section and molar attenuation coefficient are related by
ε
i
=
N
A
ln
10
σ
i
,
{\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},}
and number density and amount concentration by
c
i
=
n
i
N
A
,
{\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}
where NA is the Avogadro constant .
In case of uniform attenuation, these relations become[4]
T
=
e
−
∑
i
=
1
N
σ
i
n
i
ℓ
=
10
−
∑
i
=
1
N
ε
i
c
i
ℓ
,
{\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}n_{i}\ell }=10^{-\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell },}
or equivalently
τ
=
∑
i
=
1
N
σ
i
n
i
ℓ
,
{\displaystyle \tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell ,}
A
=
∑
i
=
1
N
ε
i
c
i
ℓ
.
{\displaystyle A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell .}
Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.
Other radiometric coefficients [ edit ]
e
Quantity
SI units
Notes
Name
Sym.
Hemispherical emissivity
ε
—
Radiant exitance of a surface , divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity
εν ελ
—
Spectral exitance of a surface , divided by that of a black body at the same temperature as that surface.
Directional emissivity
ε Ω
—
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 ε Ω,λ
—
Spectral radiance emitted by a surface , divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance
A
—
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 A λ
—
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
—
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 A Ω,λ
—
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
—
Radiant flux reflected by a surface , divided by that received by that surface.
Spectral hemispherical reflectance
R ν R λ
—
Spectral flux reflected by a surface , divided by that received by that surface.
Directional reflectance
R Ω
—
Radiance reflected by a surface , divided by that received by that surface.
Spectral directional reflectance
R Ω,ν R Ω,λ
—
Spectral radiance reflected by a surface , divided by that received by that surface.
Hemispherical transmittance
T
—
Radiant flux transmitted by a surface , divided by that received by that surface.
Spectral hemispherical transmittance
T ν T λ
—
Spectral flux transmitted by a surface , divided by that received by that surface.
Directional transmittance
T Ω
—
Radiance transmitted by a surface , divided by that received by that surface.
Spectral directional transmittance
T Ω,ν T Ω,λ
—
Spectral radiance transmitted by a surface , divided by that received by that surface.
Hemispherical attenuation coefficient
μ
m −1
Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient
μν μλ
m −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
m −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 μ Ω,λ
m −1
Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also [ edit ]
References [ edit ]
^ 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
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Transmittance&oldid=1215034764 "
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