A filament glows red at approx. 700 ° C and orange to yellow at 2500 ° C.
Max Planck at the first Solvay conference (1911) with his radiation law in the background on the blackboard

The Planck's radiation law, are for each temperature the distribution of the electromagnetic energy of the thermal radiation of a black body , depending on the wavelength or the frequency of the radiation of the body.

Max Planck found the radiation law in 1900 and noticed that a derivation within the framework of classical physics is not possible. Rather, it turned out to be necessary to introduce a new postulate according to which the energy exchange between oscillators and the electromagnetic field does not take place continuously, but in the form of tiny energy packets (later referred to as quanta ). Planck's derivation of the radiation law is therefore considered to be the hour of birth of quantum physics .

## Basics and meaning

According to kirchhoff's law of radiation , the absorption capacity and the emissivity for thermal radiation are proportional to each other for every body for every wavelength . A black body (or black body ) is a hypothetical body that completely absorbs radiation of any wavelength and intensity. Since its absorption capacity assumes the greatest possible value for each wavelength, its emissivity also assumes the maximum possible value for all wavelengths. A real (or also real ) body cannot emit more thermal radiation at any wavelength than a black body, which therefore represents an ideal source of thermal radiation. Since the spectrum of the black body (also called black body spectrum and Planck spectrum ) does not depend on any other parameter than temperature , it is a useful reference model for numerous purposes .

In addition to the considerable practical importance of the blackbody, the discovery of Planck's law of radiation in 1900 is also considered to be the birth of quantum physics , since in order to explain the formula , which was initially found empirically , Planck had to assume that light (or electromagnetic radiation in general) is not continuous , but only discrete in quanta (today one speaks of photons ) is absorbed and released.

Furthermore, Planck's law of radiation united and confirmed laws that had already been found before its discovery, partly empirically and partly on the basis of thermodynamic considerations:

• the Stefan-Boltzmann law , which indicates the radiated power of a blackbody (proportional to T 4 ).
• the Rayleigh-Jeans law , which describes the spectral energy distribution for long wavelengths.
• the Wien's law of radiation , which represents the spectral energy distribution for small wavelengths.
• the Wien's displacement law , the Wilhelm Wien (1864-1928) 1893 formulated, and that establishes the relationship between the emission maximum of a black body, and its temperature.

## Derivation and history

As a simplified example, consider a cube-shaped cavity of side length and volume that contains cavity electromagnetic radiation in thermal equilibrium. Only standing waves can form in equilibrium; the permitted waves can run in any direction, but must meet the condition that an integral number of half waves fit between two opposing cavity surfaces. The reason for this is as follows: Since the electromagnetic waves cannot exist within the walls of the cavity, the electric and magnetic field strengths there are zero. This means that the nodes of the waves must be on the surfaces of the inner walls. So only certain discrete oscillation states are allowed; the entire cavity radiation is composed of these standing waves. ${\ displaystyle L}$${\ displaystyle V}$

### The density of states

The number of allowed oscillation states increases at higher frequencies because there are more possibilities for waves with a smaller wavelength to fit into the cavity in such a way that the integer conditions for their components in -, - and - direction are met. The number of these permitted oscillation states in the frequency interval between and and per volume is called the density of states and is calculated as follows ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle \ nu}$${\ displaystyle \ nu + \ mathrm {d} \ nu}$${\ displaystyle g (\ nu) \, \ mathrm {d} \ nu}$

${\ displaystyle g (\ nu) \, \ mathrm {d} \ nu \, = \, {\ frac {8 \ pi} {c ^ {3}}} \, \ nu ^ {2} \, \ mathrm {d} \ nu}$.

### The ultraviolet disaster

Now one understands each of these oscillation states per frequency interval as a harmonic oscillator of the frequency . If all oscillators oscillate in thermal equilibrium at the temperature , then according to the uniform distribution law of classical thermodynamics one would expect that each of these oscillators carries the kinetic energy and the potential energy , i.e. the total energy . Where is the Boltzmann constant . The energy density of the cavity radiation in the frequency interval between and would therefore be the product of the density of states of the permitted oscillation states and the average energy for each classical oscillation state , i.e. ${\ displaystyle \ nu}$${\ displaystyle T}$${\ displaystyle kT / 2}$${\ displaystyle kT / 2}$${\ displaystyle kT}$${\ displaystyle k}$${\ displaystyle \ nu}$${\ displaystyle \ nu + \ mathrm {d} \ nu}$${\ displaystyle g (\ nu) \, \ mathrm {d} \ nu}$${\ displaystyle kT}$

${\ displaystyle U _ {\ nu} ^ {RJ} (\ nu, T) \, \ mathrm {d} \ nu \, = \, {\ frac {8 \ pi} {c ^ {3}}} \, kT \, \ nu ^ {2} \, \ mathrm {d} \ nu}$.

This is the Rayleigh-Jeans law of radiation . It reflects the actually measured energy density at low frequencies, but incorrectly predicts an energy density that always increases quadratically with higher frequencies, so that the cavity would have to contain an infinite energy integrated over all frequencies ( ultraviolet catastrophe ). The problem is: Each existing oscillation state only carries the energy on average , but according to classical analysis an infinite number of such oscillation states are excited, which is not the case (due to the quantization) and therefore only incorrectly leads to an infinite energy density in the cavity. ${\ displaystyle kT}$

### The empirical solution

In his derivation of the radiation law, Planck did not rely on Rayleigh's approach, rather he started from entropy and added various additional terms to the equations on a trial basis, which, according to the physics knowledge of the time, were incomprehensible - but did not contradict them. An additional term that led to a formula that described the spectral curves already measured very well was particularly simple (1900). This formula remained purely empirical - but it correctly described the known experimental measurements over the entire frequency spectrum. But Planck was not satisfied with that. He succeeded in replacing the radiation constants and from the Viennese formula with natural constants, only one factor ("help") remained. ${\ displaystyle C}$${\ displaystyle c}$${\ displaystyle h}$

### The quantum hypothesis

Based on the improved empirical radiation formula, Planck came to an epoch-making result within a few months. Quantum physics was born. Against his own conviction, Planck had to admit that he could only derive the curve confirmed by the experiment if the energy output was not continuous, but only in multiples of the smallest units at each frequency. These units have the size , whereby there is a new fundamental natural constant, which was soon referred to as Planck's quantum of action . This is the quantum hypothesis introduced by Planck . ${\ displaystyle h \ nu}$${\ displaystyle h}$

According to this, a minimum energy is required for an oscillator of the frequency to be excited at all. Oscillators, the minimum energies of which are significantly higher than the average thermally available energy , can hardly or not at all be excited, they remain frozen . Those whose minimum energy is only slightly above can be excited with a certain probability, so that a certain fraction of them with their vibrational states contribute to the total cavity radiation. Only oscillation states with low minimum energy, i.e. lower frequencies, can safely absorb the offered thermal energy and are excited according to the classic value. ${\ displaystyle h \ nu}$${\ displaystyle \ nu}$${\ displaystyle kT}$${\ displaystyle kT}$${\ displaystyle h \ nu}$

### Quantized vibrational states

Statistical thermodynamics shows that through the application of the quantum hypothesis and Bose-Einstein statistics, an oscillation state of frequency carries the following energy on average: ${\ displaystyle \ nu}$

${\ displaystyle E (\ nu, T) \, = {\ frac {h \ nu} {e ^ {\ left ({\ frac {h \ nu} {kT}} \ right)} - ​​1}}}$

According to geometric criteria, higher-frequency electromagnetic oscillation states could well exist in the cavity. However, the above connection means that these oscillation states cannot be excited by the available energy because their excitation threshold is too high. These states therefore do not contribute to the energy density in the cavity.

The product of the density of states of the permitted oscillation states and the mean energy per quantized oscillation state then already gives the Planck energy density ${\ displaystyle g (\ nu) \, \ mathrm {d} \ nu}$${\ displaystyle E (\ nu, T) \,}$

${\ displaystyle U _ {\ nu} ^ {o} (\ nu, T) \, \ mathrm {d} \ nu = \, {\ frac {8 \ pi h \ nu ^ {3}} {c ^ {3 }}} {\ frac {1} {e ^ {\ left ({\ frac {h \ nu} {kT}} \ right)} - ​​1}} \, \ mathrm {d} \ nu}$.

Because the mean energy decreases more strongly at high frequencies than the density of states increases, the spectral energy density - as its product - decreases again towards higher frequencies - after it has passed through a maximum - and the total energy density remains finite. Using his quantum thesis, Planck explained why the ultraviolet catastrophe predicted by classical thermodynamics does not actually take place.

In astronomy , especially in astrophysics and in the physics of cosmic background radiation , the spectral radiance is often used instead of the energy density

${\ displaystyle B _ {\ nu} (T) = \, {\ frac {2h \ nu ^ {3}} {c ^ {2}}} {\ frac {1} {e ^ {h \ nu / kT} -1}}}$

used, which indicates the energy flux density per solid angle and differs from the energy density by the factor . ${\ displaystyle c / 4 \ pi}$

## meaning

Planck radiation spectra for different temperatures
Planck radiation spectra for different temperatures in double-logarithmic plot

The first picture on the right shows Planck's radiation spectra of a black body for various temperatures between 300 K and 1000 K in a linear representation. The typical shape can be seen with a clearly pronounced radiation maximum, a steep drop towards short wavelengths and a longer drop towards larger wavelengths. The position of the radiation maximum shifts , as required by Wien's law of displacement , with increasing temperature to shorter wavelengths. At the same time the taking according Stefan-Boltzmann law , the total emissivity (radiation power of the area ) with the fourth power of the absolute temperature to ${\ displaystyle P}$${\ displaystyle A}$${\ displaystyle T}$

${\ displaystyle P = \ sigma AT ^ {4}}$

with the Stefan-Boltzmann constant . ${\ displaystyle \ textstyle \ sigma \ approx 5 {,} 67 \ cdot 10 ^ {- 8} \ mathrm {\ frac {W} {m ^ {2} K ^ {4}}}}$

This disproportionate increase in radiation intensity with increasing temperature explains the increasing importance of heat radiation compared to the heat given off via convection with increasing temperature. At the same time, this relationship makes it difficult to display radiation curves over a larger temperature range in a diagram.

The second picture therefore uses a logarithmic subdivision for both axes. Spectra for temperatures between 100 K and 10,000 K are shown here.

The curve for 300 K is highlighted in red, which corresponds to typical ambient temperatures. The maximum of this curve is at 10 μm; In the range around this wavelength, the middle infrared (MIR), the radiation exchange from objects takes place at room temperature. Infrared thermometers for low temperatures and thermographic cameras work in this area.

The curve for 3000 K corresponds to the typical radiation spectrum of an incandescent lamp . Part of the emitted radiation is already emitted in the schematically indicated visible spectral range . However, the radiation maximum is still in the near infrared (NIR).

The curve for 5777 K, the effective temperature of the sun , is highlighted in yellow . Their radiation maximum lies in the middle of the visible spectral range. Fortunately, most of the UV radiation emitted by the sun is filtered out by the ozone layer in the earth's atmosphere .

Planck's law of radiation is represented in different formula variants, which use quantities for intensities , flux densities and spectral distributions that are appropriate for the facts under consideration. All forms of the different radiation sizes are just different forms of the one law.

## Frequently used formulas and units

There are numerous different variants for the mathematical representation of the law, depending on whether the law is to be formulated as a function of the frequency or the wavelength, whether the intensity of the radiation is to be considered in a certain direction or the radiation in the entire half-space, whether beam sizes are to be considered , Energy densities or photon numbers are to be described.

The formula for the spectral specific radiation of a blackbody of the absolute temperature is often used . For them applies ${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T)}$${\ displaystyle T}$

in the frequency display:

 ${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T) \, \ mathrm {d} A \, \ mathrm {d} \ nu \, = {\ frac {2 \ pi h \ nu ^ { 3}} {c ^ {2}}} {\ frac {1} {e ^ {\ left ({\ frac {h \ nu} {kT}} \ right)} - ​​1}} \ mathrm {d} A \, \ mathrm {d} \ nu}$ SI unit of : W m −2 Hz −1${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T)}$

and in the wavelength display:

 ${\ displaystyle M _ {\ lambda} ^ {o} (\ lambda, T) \, \ mathrm {d} A \, \ mathrm {d} \ lambda \, = {\ frac {2 \ pi hc ^ {2} } {\ lambda ^ {5}}} {\ frac {1} {e ^ {\ left ({\ frac {hc} {\ lambda kT}} \ right)} - ​​1}} \ mathrm {d} A \ , \ mathrm {d} \ lambda}$ SI unit of : W m −2 m −1 . ${\ displaystyle M _ {\ lambda} ^ {o} (\ lambda, T)}$

${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T) \, \ mathrm {d} A \, \ mathrm {d} \ nu}$is the radiant power that is radiated by the surface element in the frequency range between and into the entire half-space. Next are the Planck constant , the speed of light and the Boltzmann constant . ${\ displaystyle \ mathrm {d} A}$${\ displaystyle \ nu}$${\ displaystyle \ nu + \ mathrm {d} \ nu}$${\ displaystyle h}$${\ displaystyle c}$${\ displaystyle k}$

When converting between frequency and wavelength representation, it should be noted that because of

${\ displaystyle \ lambda = {\ frac {c} {\ nu}}}$

applies

${\ displaystyle | \ mathrm {d} \ lambda | = {\ frac {c} {\ nu ^ {2}}} | \ mathrm {d} \ nu | \ quad {\ text {and}} \ quad | \ mathrm {d} \ nu | = {\ frac {c} {\ lambda ^ {2}}} | \ mathrm {d} \ lambda |}$.

With the help of the two radiation constants and , the specific spectral radiation can also be written in the form: ${\ displaystyle c_ {1} = 2 \ pi hc ^ {2} \,}$${\ displaystyle c_ {2} = {\ tfrac {hc} {k}}}$

${\ displaystyle M _ {\ lambda} ^ {o} (\ lambda, T) \, \ mathrm {d} A \, \ mathrm {d} \ lambda \, = {\ frac {c_ {1}} {\ lambda ^ {5}}} {\ frac {1} {e ^ {\ left ({\ frac {c_ {2}} {\ lambda T}} \ right)} - ​​1}} \ mathrm {d} A \, \ mathrm {d} \ lambda}$

## literature

• Hans Dieter Baehr, Karl Stephan : Heat and mass transfer. 4th edition. Springer, Berlin 2004, ISBN 3-540-40130-X (Chapter 5: Thermal radiation).
• Dieter Hoffmann: 100 years of quantum physics: Black bodies in the laboratory . Experimental preliminary work for Planck's quantum hypothesis. In: Physical sheets . tape 56 , no. 12 , December 1, 2000, pp. 43-47 , doi : 10.1002 / phbl.20000561215 ( wiley.com [PDF; 765 kB ]).
• Gerd Wedler: Textbook of physical chemistry. 4th edition. Wiley-VCH, Weinheim 1997, ISBN 3-527-29481-3 , pp. 111-114 and pp. 775-779.
• Thomas Engel, Philip Reid: Physical chemistry. Pearson, Munich 2006, ISBN 3-8273-7200-3 , pp. 330-332.