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Chapter: 2 General physics
    Section: 2.5 Radiation and optics
        SubSection: 2.5.2 Thermal radiation

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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 = AδΩ 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 = Ω, 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 = spacer  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

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