2.5.2 Thermal radiation
In a field of thermal radiation, the radiant power
δ2φ falling on an infinitesimal area
δA, in the range of directions contained by an infinitesimal cone
of solid angle δΩ, is given by
δ2φ =
LδAδΩ cos θ, where
θ is the angle between the cone and the normal to δA.
The quantity L is known as the radiance of the field at
δA in the direction of the cone. Radiance has the property that,
in a transparent medium of constant refractive index, the radiance at any point
along a ray path in the direction of the ray path is
constant†.
The total radiant power δφ
falling on δA from one side is given by
| |
δ = δA |
 |
L cos θ dΩ = E
δA ...(1) |
where the integration is carried out over the
appropriate hemisphere on δA. The quantity E which, unlike
L, usually depends on the orientation of δA, is known as
irradiance. If δA is an element of a radiating surface, the
equivalent quantity in terms of the radiation passing out through
δA is known as exitance. The radiation emitted by the whole
of a radiation source in a given direction is described by the quantity
I, known as radiant intensity, defined by the equation
δ = 1δΩ, where
δ is the power radiated in a cone of
directions of solid angle δΩ about the specified
direction.
If the radiation has a continuous spectrum, the power in
the frequency interval δν about ν or the wavelength interval
δλ about
λ is proportional to the
interval, and is therefore denoted by a spectral power density (ν) = ∂/∂ν or
(λ)
=∂/∂λ. These and derived quantities such
as L(v) are commonly distinguished from ordinary functions of
ν or λ by writing the ν
or λ as a subscript. If
vacuum wavelengths are used, then νλ = c and λ = (c/λ2)v, etc.
Black-body radiation
The thermal radiation inside a closed cavity, with
opaque walls at a uniform temperature T, is known as full or black-body
radiation. It has a continuous spectrum and is uniform and isotropic, i.e. the
spectral radiance Lv within the cavity is independent
of position and direction. Moreover, Lv is determined solely
by T.‡
This radiation may be observed by making a hole in the
cavity wall small enough not to reduce appreciably the radiance of the emerging
radiation. A means of testing whether this condition is satisfied is provided
by the extended form of Kirchhoff’s law, which says that the factor
(known as emissivity) by which the radiance in a particular direction is
reduced, equals the absorption factor for radiation incident on the hole in the
opposite direction. A radiation source of this kind with emissivity sensibly
equal to unity is known as a black-body radiator.
Spectral distribution of black-body radiation
According to Planck’s formula
| |
Lv = |
2h |
v3(ekv/kT −
1)−1 |
| c2 |
|
...(2) |
| |
|
The spectral radiance is usually expressed in terms of
Lλ: |
| |
|
 |
...(3) |
| |
|
where c'1 =
2hc2 = 1.1911 × 10−16 W m2
sr−1 |
| |
|
and c2
= hc/k = 1.43877 × 10−2 m K
|
| |
|
For Eλ we have, since
Eλ =
πLλ for isotropic radiation,
|
| |
|
 |
| |
|
where c1 =
πc'1 = 3.74177 x
10−16 W m2 |
...(4) |
† More generally, if the radiation passes
without loss through a medium of varying refractive index n, then
L/n2 is constant.
‡
Strictly speaking, Lv is proportional to
n2, where n is the refractive index inside the cavity, and
the subsequent formulae refer to an evacuated cavity.
The constants c1 and c2 are known as
the 1st and 2nd radiation constants; more accurate values are given in section
1.2.3.
| The total radiance over the
spectrum, |
L = ∫ |
 |
∞ |
Lλ dλ |
, is given by L =
σ' T4, where |
| 0 |
is given by L = σ'
T4, where
| |
σ' = |
2π4k4 |
= 1.80498 ×
10−8 W m−2 sr−1
K−4 |
...(5) |
| 15c2h3 |
Likewise the total exitance is given by E =
σT4, where
σ
= πσ' = 5.6705 ×
10−8 W m−2
K−4 …(6)
σ is known as the Stefan–Boltzmann
radiation constant.
The functions Lv for different
T are of the form f(T)g(v/T),
so that if normalized to a given maximum value and plotted against v/T,
they reduce to a single curve. The maximum of this curve occurs at
| |
vm |
= 5.8787 x 1010 Hz
K−1 |
| T |
This result is more conveniently expressed in terms of the wave number
σ used by spectroscopists, σ = 1/λ:
σm/T
= 1.9609 K−1 cm−1
(σ should not be confused with σ denoting the
Stefan–Boltzmann constant.)
Likewise the distribution functions in terms of λ reduce to a single function of
λT with a maximum
at
λmT = 2.8979 ×
10−3 m K
The following table gives the fraction f of the
total radiation for temperature T which lies between zero wavelength and
that corresponding to various values of λT. It may be used in
conjunction with formulae (5) and (6) above to obtain the radiance or exitance
in extended spectral bands.
Example: for a black-body at
1000 K the exitance in the band 0.6–0.7 μm (λT = 600–700 μm K)
is
(1.84 × 10−6
− 9.29 × 10−8) x 5.67 x 10−8
× 10004 W m−2
Fraction of total radiation from 0 to
λT
|
λT /(μm
K) |
f |
λT /(μm
K) |
f |
λT /(μm
K) |
f |
|
600 |
9.29 × 10−8 |
2 500 |
0.1613 |
12 000 |
0.945 05 |
|
700 |
1.84 × 10−6 |
3 000 |
0.2732 |
14 000 |
0.962 85 |
|
800 |
1.64 × 10−5 |
3 500 |
0.3829 |
16 000 |
0.973 77 |
|
900 |
8.70 × 10−5 |
4 000 |
0.4808 |
18 000 |
0.980 81 |
|
1 000 |
3.21 × 10−4 |
4 500 |
0.5643 |
20 000 |
0.985 56 |
|
1 200 |
2.13 × 10−3 |
5 000 |
0.6337 |
30 000 |
0.995 29 |
|
1 400 |
7.79 × 10−3 |
6 000 |
0.7378 |
40 000 |
0.997 92 |
|
1 600 |
1.97 × 10−2 |
7 000 |
0.8081 |
50 000 |
0.998 91 |
|
1 800 |
3.93 × 10−2 |
8 000 |
0.8562 |
75 000 |
0.999 67 |
|
2 000 |
6.67 × 10−2 |
10 000 |
0.9142 |
100 000 |
0.999 86 |
E.J.Gillham
Colour-temperature
The colour of an illuminant often resembles that of a black-body
radiator at a particular temperature and can conveniently be specified in terms
of this temperature. If an exact colour match can be obtained, the temperature
is known as the colour-temperature. If only an approximate match can be
obtained, then the term correlated colour-temperature is used. This can
be given a precise meaning by representing the colours of the illuminant and of
the black-body radiation as points on the CIE (u', v')
Uniform Chromaticity Scale diagram. The correlated colour-temperature is then
defined as the black-body temperature for which the two points are closest
together. For correlated colour-temperatures of various illuminants, see
section 2.5.3.
With some illuminants, such as the tungsten-filament lamp, both the
colour and the relative spectral power distribution resemble that of a
black-body radiator at a suitable temperature. In such cases, the term
‘distribution temperature’ is used for the temperature which
gives the best spectral fit, i.e. the temperature which, by adjustment of
a and T, minimises the integral:
| |
|
[1 − St(λ)/aSb(λ, T)]2 dλ |
where λ is the
wavelength, St(λ) is the relative spectral
distribution of the radiation being considered and
Sb(λ,
T) is the relative spectral distribution of a Planckian radiator at
temperature T. The wavelength range used for the integration must be
stated, with a range of 400 to 750 nm being preferred, and the use of the term
is usually restricted to incandescent radiators, such as tungsten filament
lamps.
Colour and distribution temperatures are usually expressed in kelvins
(K). The reciprocal of the temperature in megakelvins may also be used, the
unit being known as the mired: 1 mired = 10−6
K−1.
T.M.Goodman
|
