Wien's displacement law
Named after Wilhelm Wien, Wien's displacement law specifies at which wavelength or frequency, a black body emits the most radiation power or the greatest photon rate, depending on its temperature.
- 3.1 wavelength representation
- 3.2 frequency representation
General
The heat emitted by a black body radiation is a mixture of electromagnetic wave from a wide range of wavelengths. The distribution of the irradiance on the different wavelengths can be described by the Planck's law of radiation. It has a clear maximum, whose position can be calculated with the Wien's displacement law.
In particular, the higher the temperature of a body is, the more in shorter wavelengths, the maximum of the distribution. Thus, for example, are steel at room temperature invisible infrared radiation ( " heat radiation" ) from, warm glowing steel lights dark red, hot liquid steel glows almost white.
The most common formulation of the displacement law:
- : Wavelength at which the intensity per wavelength interval is maximum
- : Absolute temperature of the radiating surface
The frequency of these waves is not wavelength, the frequency at which the intensity of each frequency interval is at maximum. Two other maximum points result if, instead of intensity considered the photon rate. There is no " objective" maximum.
Other legislation affecting the black body radiation Planck's law of radiation are, the Stefan- Boltzmann law, Wien's radiation law and the Rayleigh-Jeans law. Approximation, these laws are often the output of non-black radiators heat radiation.
Maximum radiation power
Wavelength representation
The spectral emittance of a black body of temperature is described by Planck's radiation law and is in the wavelength representation:
Sought is the wavelength at which this function takes the maximum. Zero the derivative with respect to supplies:
The substitution of the expression simplifies to:
The numerical solution yields
And the back-substitution leads to Wien's displacement law in the wavelength representation:
The wavelength of maximum radiation power is thus displaced to a temperature change simply inversely proportional to the absolute temperature of the black body: to Double the temperature of the radiator, the largest radiation power occurs at half the wavelength.
The constant is also referred to as Wien's displacement constant. Its value is in accordance with current measurement accuracy:
The spectral emittance of the maximum is proportional to:
Frequency representation
In the frequency representation of the spectral emittance is given by
Zero the derivative with respect to frequency yields:
The substitution simplifies to the expression.
The numerical solution yields
And back-substitution leads to Wien's displacement law in frequency representation:
The frequency of maximum radiation power that is shifted in proportion to the absolute temperature of the radiator. The recommended value of Wien's constant in the frequency representation is:
The spectral emittance of the maximum is proportional to:
Different maxima in both representations
Note that because of the non-linear conversion between the wavelength and frequency intervals not the following equation:
But the following equation is valid:
For example, the points in the adjacent diagram, follow a Planck distribution ( for T = 600 K). Marked are also frequency intervals of constant width (both 10 THz) and wavelength intervals of constant width (each 1 micron ). For each interval, the number of points included therein is shown. As can be seen immediately, the frequency interval 30-40 THz contains more points than any other frequency interval (13 ), while the wavelength interval 4-5 microns contains more points than any other wavelength interval (10). Is expected, however, the frequency at which the maximum occurs in the wavelength corresponding to, one does not obtain that wavelength at which the maximum occurs when the wavelength intervals are considered.
Plotting the graph once against a linear frequency axis and once against a linear wavelength axis, so is visually recognizable as crowd together the points each in different areas of the axis. There is no " objective" maximum.
Maximum photon rate
Wavelength representation
The spectral emittance expressed by the emission rate of photons is given in the wavelength representation by
Zero the derivative with respect to supplies:
The substitution simplifies to the expression.
The numerical solution yields
And back-substitution leads to Wien's displacement law for the photon rate in the wavelength representation:
The spectral photon rate of the maximum is proportional to.
Frequency representation
In the frequency representation, the spectral emittance is expressed by the radiation rate of the photons is given by
The substitution simplifies to the expression.
The numerical solution yields
And back-substitution leads to Wien's displacement law for the photon rate in the frequency representation:
The spectral photon rate of the maximum is proportional to.
Application Examples
Assuming, for the sun lambda max ≈ 500 nm and considers them approximately as a black body, it follows after the Wien's displacement law surface temperature to about 5800 K. The temperature determined in this way is called Wien's temperature. They compare well with the calculated using the Stefan- Boltzmann law effective temperature of 5777 K. The difference stems from the fact that the two calculations underlying assumption that the sun is a black body, is indeed a good approximation, but not perfectly satisfied.
Glutfarben give information about the temperature of hot (over 500 ° C) materials.
Other examples are the radiating surface and the greenhouse gases. At temperatures in the range of 0 ° C, the maximum radiation in the infrared range of 10 microns. In the greenhouse gases is a case that they are only partial ( selective) black body.
History
Based on the experimental studies of Josef Stefan and the thermodynamic derivation by Ludwig Boltzmann was known that from a black body with the absolute temperature T thermally emitted radiation power increases with the fourth power of the temperature ( Main article: Stefan- Boltzmann law ). However, the distribution of the radiation energy to the various emitted wavelengths was still unknown.
Wien 's displacement law could be derived, which produced a correlation between the wave length distributions at various temperatures due to thermodynamic considerations:
"Well, the change in radiation sets with temperature according to the Boltzmann and me given theory consists of an increase in the Gesammtenergie in proportion to the fourth power of the absolute temperature and a change in the wavelength of each d.lambda enclosed between λ and λ quantum of energy in the sense that the Relevant wavelength is inversely proportional to the absolute temperature changes. If we imagine the energy at a temperature plotted as a function of the wavelength, so this curve would remain unchanged at different temperature when the standard of drawing would be changed so that the ordinates reduced in proportion 1/θ4 and the abscissae in the ratio θ increases would. "
Thus the real wavelength distribution of the radiation was still unknown, but it was an additional condition found that they had to be subject to a temperature change. With the help of some additional assumptions Vienna could derive a radiation law, which on exposure to temperature changes, in fact behaves as required by the law of displacement. The comparison with the experiment, however, showed that this Wien law provides in the long wavelength range to low values.
Max Planck was finally by a clever interpolation between the Rayleigh-Jeans law (correct for large wavelengths ) and the Wien's radiation law (correct for small wavelengths) Planck's radiation law derived which reproduces the emitted radiation in all wavelengths correctly.
Nowadays, the general form of Wien's displacement law is no longer relevant because the Planck's radiation law, the spectral shift with temperature change concretely describes. Only the temperature-induced shift of the radiation maximum, whose legality is already derivable from the Wien's displacement law, has survived under the name of vienna cal displacement law.