2. Thermal radiation

Grey and coloured emitters     (1/2)

Planck's radiation law describes the radiation of ideal black bodies. Their degrees of absorption α and emission ε equals 1. It is applicable to grey bodies as well considering a degree of emission of ε<1 as a factor:

u f,grau =ε 8π f 2 c 3 hf exp{ hf / kT }−1
u λ,grau =ε 8πhc λ 5 hf exp{ hc / λkT }−1

Instead of ε, it is possible to use α since their values are identical corresponding to Kirchhoff's law.

The term “grey emitter” indicates that the degree of absorption over the total spectral range is below 1. The non-absorbed light is reflected of passes through the body. The body appears grey in reflexion or transparency at white illumination.

Just as black bodies, grey bodies are idealisations which do not occur in reality. Even if ε seems to be nearly constant in certain spectral ranges, this does not represent the situation in general. Every body shows areas in which it is black besides others in which it is grey or white. These are coloured emitters. Examples:

  • glass is white in the visible (in the sense of non-absorbent), but in the ultraviolet and infrared it s black
  • black tissue is black in the visible but grey or white in the infrared
  • the sunlight spectrum shows absorption lines, the Fraunhofer lines
  • trace gases in the atmosphere show absorption lines which cause the greenhouse effect
  • discharges in gases emit spectral lines; they are black bodies in the range of their spectral lines, since radiation of the same wavelength is emitted and absorbed. At other wavelengths, they are white since they neither emit nor absorb radiation.

Equations ↓

An object is a grey emitter in a certain spectral range if it emits not enough radiation lacking by the factor ε. It appears cooler than the measured temperature T. The temperature corresponding to the black emitter is the emission temperature Trad. It is always below the thermometric temperature.

This is valid for the integral of the radiation over all wavelengths i.e. the Stephan-Boltzmann law as well. The specific radiation of a black body becomes:

M=εσ T 4 =σ T rad 4

For the correlation between the emission temperature and the real temperature, the relation mentioned in the Section about Kichhoff's radiation law must be considered:

T rad = ε 1/4 T
Knowing the degree of emission or absorption it is possible to calculate the real temperature of an object from the measured emission temperature by the help of this equation.

Likewise, it is possible to convert the measured emission temperatures of grey bodies into real temperatures by knowing their degree of emission or absorption. This is particularly important for the correction of temperature images in what grey objects are represented.