Planck's law

Planck's radiation law, describes the distribution of the electromagnetic energy of the thermal radiation field of a black body, depending on the frequency of the radiation.

In the derivation of the radiation law by Max Planck in 1900 showed that a description in the framework of classical physics is not possible. Rather, it proved necessary to introduce a new postulate, which states that the energy exchange between the oscillators and the electromagnetic radiation field is not continuous but in the form of tiny packets of energy (later called quanta) takes place. Therefore, Planck's derivation of the radiation law is now regarded as the birth of quantum physics.

Fundamentals and importance

According to the Kirchhoff's law of radiation, the absorbance and emissivity for thermal radiation for each body for each wavelength is proportional to each other. A blackbody is a hypothetical body, the radiation hitting any wavelength and intensity completely absorbed him. Since its absorbance takes the largest possible value for each wavelength, and its emissivity assumes the maximum possible value for all wavelengths. A real body can on no more wavelength thermal radiation emit as a black body, which therefore represents an ideal thermal radiation source. Since the spectrum of a black body also depends on any other parameters than the temperature, in particular the material properties, it represents a useful reference source for a variety of purposes

In addition to the considerable practical importance of the black body, the discovery of Planck's radiation law in 1900 is considered the same as the birth of quantum physics as Planck to explain the formula first found empirically had to assume that light (or electromagnetic radiation in general) is not continuous but only discrete quantum (today one speaks of photons) is absorbed and released.

Furthermore, united and confirmed Planck's radiation law, laws which partly empirical, partly due to thermodynamic considerations had been found before his discovery:

• The Stefan- Boltzmann law, which indicates the power emitted by a blackbody (proportional to T4).
• The Rayleigh-Jeans law, which describes the spectral energy distribution for long wavelengths.
• Wien's radiation law, which reproduces the spectral energy distribution for short wavelengths.
• Wien's displacement law, Wilhelm Wien (1864-1928) formulated in 1893, and which establishes the relationship between maximum emission of a black body and its temperature.

Derivation and History

Consider the simplified example of a cube-shaped cavity of the side length L and the volume V, which contains electromagnetic radiation cavity is in thermal equilibrium. In equilibrium, only standing waves may form; the permitted wave may run in any direction, however, must meet the condition that between two opposite surfaces of the cavity in each case fits an integer number of half-waves. The reason is as follows: Since the electromagnetic waves can not exist within the walls of the cavity, there is the electric and magnetic field strength is zero. So that the nodes of the waves must be located on the surfaces of the inner walls. There are therefore permitted only certain discrete vibrational states; the whole body radiation is made up of these standing waves.

The density of

The number of allowed states of vibration increases with higher frequencies, because there are more ways for waves with a lower wavelength, so to fit in the cavity that the integrality are met for their components in the x-, y -and z- direction. The number of allowed vibrational states in the frequency interval between ν and ν dv per unit volume is called density of states and is calculated as

The ultraviolet catastrophe

Now it summarizes each of these vibrational states per frequency interval as a harmonic oscillator of frequency ν on. If all oscillators oscillate in thermal equilibrium at temperature T, then would be expected from the equipartition theorem of classical thermodynamics that each of these oscillators contributes on average, the kinetic energy of kT / 2 and the potential energy of kT / 2, for a total energy kT. The energy density of the black body radiation in the frequency interval between ν and ν dv would therefore be the product of the density of the states and allowable oscillation of the average energy per classical vibrational state kT, so

This is the law of radiation Rayleigh - Jeans. There are the actual measured energy density at low frequencies well again, but says an erroneously higher frequencies always square growing energy density required, so that the cavity integrated over all frequencies would contain an infinite energy ( ultraviolet catastrophe ). The problem is: Although each existing vibrational state contributes on average only the energy kT, but there are infinitely inspired by classic view many of these vibrational states, which, however (due to quantization) does not apply and falsely leads to infinite energy density in the cavity.

The empirical approach

Planck relied on for its derivation of the radiation law not on the Rayleigh approach, but he went by the entropy and added into the equations tentatively various additional terms, which indeed were incomprehensible by the then knowledge of physics - but they may not even disagreed. Especially was simply an additional term, which resulted in a formula that described the already measured spectral curves very well ( 1900). That left this formula pure empiricism - but she described the known experimental measurements over the entire frequency spectrum correctly. But Planck was thus not satisfied. He managed the radiation constants C and c to replace from the Wien's formula by natural constants, only one factor h ( "help " ) remained.

The quantum hypothesis

Based on the improved empirical radiation formula came Planck within a few months to a epoch-making result, it was the birth of quantum physics: he had himself to admit to his own belief that the energy output is not continuous, but only in multiples of the smallest "h" - units, which was then called in his honor later than Planck's constant. After this introduced by Planck 's quantum hypothesis, an oscillator frequency can only take integer multiples of the energy hv ν instead of arbitrary amounts of energy; In particular, it requires a minimum energy hv, so as to be at all stimulated. Vibrational states, the minimum energy hv are significantly higher than the thermal energy kT provided, can not be excited, they remain frozen. Those vibrational states whose minimum energy is only slightly above kT, can be excited with a certain probability, so that they contribute a certain fraction of the total body radiation. Only vibrational states with lower minimum energy hv, so lower frequencies, the offered thermal energy can completely absorb and be excited (on average) with certainty.

Quantized vibrational states

Statistical thermodynamics show that by the application of quantum hypothesis and Bose -Einstein statistics, a vibration condition of the frequency ν in the middle carries following power:

After geometric criteria high-frequency electromagnetic vibration states may well exist in the cavity. The above connection now says, however, that these vibrational states can not be excited by the energy available because their excitation threshold is too high. These states do not contribute to the energy density in the cavity.

The radiation law

The product of the density of states of the allowed vibrational states and the mean energy per quantized vibrational state then gives already the Planckian energy density

Because the average energy stronger at high frequencies decreases as increases the density of states takes the spectral energy density - as their product - to higher frequencies again - after passing through a maximum - and the total energy density remains finite. So Planck explained by his quantum theory why the predicted by classical thermodynamics ultraviolet catastrophe in reality does not take place.

Importance

The first picture shows Planck's radiation spectra of a black radiator at various temperatures between 300 K and 1000 K in a linear representation. One recognizes the typical shape out with a very pronounced maximum radiation, a steep drop to the short wavelength side and a longer leaking waste to large wavelengths. The location of the maximum radiation shifts, as required by the Wien's displacement law, as the temperature increases to shorter wavelengths. At the same time according to the Stefan- Boltzmann law, the total specific emission ( radiation power P of area A ) with the fourth power of the absolute temperature T to:

With - Stefan- Boltzmann constant:

This disproportionate increase in the radiation intensity with increasing temperature explains the increasing importance with increasing temperature of the heat radiation compared to the votes on convection heat. At the same time it makes this relationship difficult to represent radiation curves over a wider temperature range in a chart.

The second picture, therefore, uses a logarithmic division for both axes. Shown here are spectra for temperatures between 100 K and 10,000 K.

Highlighted red curve is for 300 K, which corresponds to typical ambient temperatures. The maximum of this curve is 10 microns; in the range of this wavelength, the mid-infrared (MIR), which is the exchange of radiation of objects takes place at room temperature. Infrared thermometer for low temperatures and thermal imaging cameras work in this area.

The curve for 3000 K corresponds to the typical radiation spectrum of an incandescent lamp. Now already part of the emitted radiation is emitted in the visible spectral range, indicated schematically. The maximum radiation is still in the Near Infrared ( NIR).

Highlighted in yellow is the curve for 5777 K, the effective temperature of the sun. Your radiation maximum is located in the visible spectral range. The thermally emitted by the sun's ultraviolet radiation is filtered out, fortunately, for the most part by the ozone layer of the Earth's atmosphere.

The Planck radiation law is represented in various formula types, the sizes of intensities, flux densities and spectral use, which are appropriate for the observed facts. All forms of radiation differing sizes are only different forms of a law.

Frequently used formulas and units

The mathematical representation of the law, there are numerous different types, depending on whether the law in dependence on the frequency or wavelength to be formulated, if the intensity of the radiation in a certain direction, or the radiation to be viewed in the entire half-space, if beam sizes, energy density or photon numbers are to be described.

Frequently used the formula for the spectral emittance of a black body of the absolute temperature. For they

In the frequency representation:

And in the wave -length representation:

The radiation power emitted by the surface element in the frequency range and in the entire half-space. Further, h is the Planck constant, c is the speed of light and k is the Boltzmann constant.

When converting between frequency and wavelength representation is important to note that because

Applies

With the help of the two radiation constants and allows the spectral emittance also written in the form: