Planck's law

In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature, when there is no net flow of matter or energy between the body and its environment.
At the end of the 19th century, physicists were unable to explain why the observed spectrum of black-body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, German physicist Max Planck heuristically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment,, that was proportional to the frequency of its associated electromagnetic wave. While Planck originally regarded the hypothesis of dividing energy into increments as a mathematical artifice, introduced merely to get the correct answer, other physicists including Albert Einstein built on his work, and Planck's insight is now recognized to be of fundamental importance to quantum theory.

Definition

Every physical body spontaneously and continuously emits electromagnetic radiation and the spectral radiance of a body,, describes the spectral emissive power per unit area, per unit solid angle and per unit frequency for particular radiation frequencies. The relationship given by Planck's radiation law, given below, shows that with increasing temperature, the total radiated energy of a body increases and the peak of the emitted spectrum shifts to shorter wavelengths. According to Planck's distribution law, the spectral energy density at given temperature is given by:
Alternatively, the law can be expressed for the spectral radiance of a body for frequency at absolute temperature
given as:
where is the Boltzmann constant, is the Planck constant, and is the speed of light in the medium, whether material or vacuum.
The SI units of the spectral radiance are. The spectral radiance are.
The cgs units of spectral radiance are.
The terms and are related to each other by a factor of, since is independent of direction and radiation travels at speed.
The spectral radiance can also be expressed per unit wavelength instead of per unit frequency. In addition, the law may be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation.
In the limit of low frequencies, Planck's law tends to the Rayleigh–Jeans law, while in the limit of high frequencies it tends to the Wien approximation.
Max Planck developed the law in 1900 with only empirically determined constants, and later showed that, expressed as an energy distribution, it is the unique stable distribution for radiation in thermodynamic equilibrium. As an energy distribution, it is one of a family of thermal equilibrium distributions which include the Bose–Einstein distribution, the Fermi–Dirac distribution and the Maxwell–Boltzmann distribution.

Black-body radiation

A black-body is an idealised object which absorbs and emits all radiation frequencies. Near thermodynamic equilibrium, the emitted radiation is closely described by Planck's law and because of its dependence on temperature, Planck radiation is said to be thermal radiation, such that the higher the temperature of a body the more radiation it emits at every wavelength.
Planck radiation has a maximum intensity at a wavelength that depends on the temperature of the body. For example, at room temperature, a body emits thermal radiation that is mostly infrared and invisible. At higher temperatures the amount of infrared radiation increases and can be felt as heat, and more visible radiation is emitted so the body glows visibly red. At higher temperatures, the body is bright yellow or blue-white and emits significant amounts of short wavelength radiation, including ultraviolet and even x-rays. The surface of the Sun emits large amounts of both infrared and ultraviolet radiation; its emission is peaked in the visible spectrum. This shift due to temperature is called Wien's displacement law.
Planck radiation is the greatest amount of radiation that any body at thermal equilibrium can emit from its surface, whatever its chemical composition or surface structure. The passage of radiation across an interface between media can be characterized by the emissivity of the interface, usually denoted by the symbol. It is in general dependent on chemical composition and physical structure, on temperature, on the wavelength, on the angle of passage, and on the polarization. The emissivity of a natural interface is always between and 1.
A body that interfaces with another medium which both has and absorbs all the radiation incident upon it is said to be a black body. The surface of a black body can be modelled by a small hole in the wall of a large enclosure which is maintained at a uniform temperature with opaque walls that, at every wavelength, are not perfectly reflective. At equilibrium, the radiation inside this enclosure is described by Planck's law, as is the radiation leaving the small hole.
Just as the Maxwell–Boltzmann distribution is the unique maximum entropy energy distribution for a gas of material particles at thermal equilibrium, so is Planck's distribution for a gas of photons. By contrast to a material gas where the masses and number of particles play a role, the spectral radiance, pressure and energy density of a photon gas at thermal equilibrium are entirely determined by the temperature.
If the photon gas is not Planckian, the second law of thermodynamics guarantees that interactions will cause the photon energy distribution to change and approach the Planck distribution. In such an approach to thermodynamic equilibrium, photons are created or annihilated in the right numbers and with the right energies to fill the cavity with a Planck distribution until they reach the equilibrium temperature. It is as if the gas is a mixture of sub-gases, one for every band of wavelengths, and each sub-gas eventually attains the common temperature.
The quantity is the spectral radiance as a function of temperature and frequency. It has units of W·m⁻²·sr⁻¹·Hz⁻¹ in the SI system. An infinitesimal amount of power is radiated in the direction described by the angle from the surface normal from infinitesimal surface area into infinitesimal solid angle in an infinitesimal frequency band of width centered on frequency. The total power radiated into any solid angle is the integral of over those three quantities, and is given by the Stefan–Boltzmann law. The spectral radiance of Planckian radiation from a black body has the same value for every direction and angle of polarization, and so the black body is said to be a Lambertian radiator.

Different forms

Planck's law can be encountered in several forms depending on the conventions and preferences of different scientific fields. The various forms of the law for spectral radiance are summarized in the table below. Forms on the left are most often encountered in experimental fields, while those on the right are most often encountered in theoretical fields.
In the fractional bandwidth formulation, and the integration is with respect to
Planck's law can also be written in terms of the spectral energy density by multiplying by :
These distributions represent the spectral radiance of blackbodies—the power emitted from the emitting surface, per unit projected area of emitting surface, per unit solid angle, per spectral unit. Since the radiance is isotropic, the power emitted at an angle to the normal is proportional to the projected area, and therefore to the cosine of that angle as per Lambert's cosine law, and is unpolarized.

Correspondence between spectral variable forms

Different spectral variables require different corresponding forms of expression of the law. In general, one may not convert between the various forms of Planck's law simply by substituting one variable for another, because this would not take into account that the different forms have different units. Wavelength and frequency units are reciprocal.
Corresponding forms of expression are related because they express one and the same physical fact: for a particular physical spectral increment, a corresponding particular physical energy increment is radiated.
This is so whether it is expressed in terms of an increment of frequency,, or, correspondingly, of wavelength,, or of fractional bandwidth, or. Introduction of a minus sign can indicate that an increment of frequency corresponds with decrement of wavelength.
In order to convert the corresponding forms so that they express the same quantity in the same units we multiply by the spectral increment. Then, for a particular spectral increment, the particular physical energy increment may be written
which leads to
Also,, so that. Substitution gives the correspondence between the frequency and wavelength forms, with their different dimensions and units.
Consequently,
Evidently, the location of the peak of the spectral distribution for Planck's law depends on the choice of spectral variable. Nevertheless, in a manner of speaking, this formula means that the shape of the spectral distribution is independent of temperature, according to Wien's displacement law, as detailed below in § Properties §§ Percentiles.
The fractional bandwidth form is related to the other forms by

First and second radiation constants

In the above variants of Planck's law, the wavelength and wavenumber variants use the terms and which comprise physical constants only. Consequently, these terms can be considered as physical constants themselves, and are therefore referred to as the first radiation constant and the second radiation constant with
and
Using the radiation constants, the wavelength variant of Planck's law can be simplified to
and the wavenumber variant can be simplified correspondingly.
is used here instead of because it is the SI symbol for spectral radiance. The in refers to that. This reference is necessary because Planck's law can be reformulated to give spectral radiant exitance rather than spectral radiance, in which case replaces, with
so that Planck's law for spectral radiant exitance can be written as
As measuring techniques have improved, the General Conference on Weights and Measures has revised its estimate of ; see for details.