In physics, Planck's law (also Planck radiation law) describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, 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, E, 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, Bν, 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 (energy per unit volume per unit frequency) at given temperature is given by:
u ν ( ν , T ) = 8 π h ν 3 c 3 1 exp ( h ν k B T ) − 1 . {\displaystyle u_{\nu }(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}{\frac {1}{\exp \left({\frac {h\nu }{k_{\mathrm {B} }T}}\right)-1}}.}
Alternatively, the law can be expressed for the spectral radiance of a body for frequency ν at absolute temperature T given as:
B ν ( ν , T ) = 2 h ν 3 c 2 1 exp ( h ν k B T ) − 1 {\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{\exp \left({\frac {h\nu }{k_{\mathrm {B} }T}}\right)-1}}}
where kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light in the medium, whether material or vacuum. The spectral radiance can also be expressed per unit wavelength λ {\displaystyle \lambda } instead of per unit frequency:
B λ ( λ , T ) = 2 h c 2 λ 5 1 exp ( h c λ k B T ) − 1 {\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{\exp \left({\frac {hc}{\lambda k_{\mathrm {B} }T}}\right)-1}}}
However this form is not related to B ν ( ν , T ) {\displaystyle B_{\nu }(\nu ,T)} by the substitution λ = c / ν {\displaystyle \lambda =c/\nu } . These are different functions because the spectral radiance is defined in terms of equal amounts of radiation for each increment in the independent variable ( ν {\displaystyle \nu } or λ {\displaystyle \lambda } ) and these increments are not the same in the two forms. 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. The SI units of the spectral radiance Bν are W·sr−1·m−2·Hz−1. The spectral radiance Bλ are W·sr−1·m−3. The cgs units of spectral radiance Bν are erg·s−1·sr−1·cm−2·Hz−1. The terms B and u are related to each other by a factor of 4π/c, since B is independent of direction and radiation travels at speed c. In the limit of low frequencies (i.e. long wavelengths), Planck's law tends to the Rayleigh–Jeans law, while in the limit of high frequencies (i.e. small wavelengths) 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 (~300 K), 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 (~6000 K) 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 (the ratio of the actual radiance to the theoretical Planck radiance), usually denoted by the symbol ε. 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 ε = 0 and 1. A body that interfaces with another medium which both has ε = 1 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 (between photons and other particles or even, at sufficiently high temperatures, between the photons themselves) 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 Bν(ν, T) 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 Bν(ν, T) cos θ dA dΩ dν is radiated in the direction described by the angle θ from the surface normal from infinitesimal surface area dA into infinitesimal solid angle dΩ in an infinitesimal frequency band of width dν centered on frequency ν. The total power radiated into any solid angle is the integral of Bν(ν, T) 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, x = h ν k B T = h c λ k B T {\textstyle x={\frac {h\nu }{k_{\mathrm {B} }T}}={\frac {hc}{\lambda k_{\mathrm {B} }T}}} , and the integration is with respect to d ( ln x ) = d ( ln ν ) = d ν ν = − d λ λ = − d ( ln λ ) {\textstyle \mathrm {d} (\ln x)=\mathrm {d} (\ln \nu )={\frac {\mathrm {d} \nu }{\nu }}=-{\frac {\mathrm {d} \lambda }{\lambda }}=-\mathrm {d} (\ln \lambda )} . Planck's law can also be written in terms of the spectral energy density (u) by multiplying B by 4π/c: u i ( T ) = 4 π c B i ( T ) . {\displaystyle u_{i}(T)={\frac {4\pi }{c}}B_{i}(T).}
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 (frequency, wavelength, wavenumber or their angular equivalents, or fractional frequency or wavelength). Since the radiance is isotropic (i.e. independent of direction), 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 The frequency and wavelength forms of the law are different functions. The correspondence involves differential relations and the wavelength and frequency units are reciprocal. Both forms express an amount of energy per step of the independent variable, but the step sizes differ. Therefore, the two forms of Planck's law are not related by substituting one variable for another because this would not take into account that the different forms have different units. 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
B λ ( λ , T ) d λ = − B ν ( ν ( λ ) , T ) d ν , {\displaystyle B_{\lambda }(\lambda ,T)\,d\lambda =-B_{\nu }(\nu (\lambda ),T)\,d\nu ,}
which leads to
B λ ( λ , T ) = − d ν d λ B ν ( ν ( λ ) , T ) . {\displaystyle B_{\lambda }(\lambda ,T)=-{\frac {d\nu }{d\lambda }}B_{\nu }(\nu (\lambda ),T).} The minus sign indicates that an increment of frequency corresponds with decrement of wavelength. Since ν(λ) = c/λ, the ratio dν/dλ = − c/λ2. Substitution gives the correspondence between the frequency and wavelength forms, with their different dimensions and units:
B λ ( T ) B ν ( T ) = c λ 2 = ν 2 c . {\displaystyle {\frac {B_{\lambda }(T)}{B_{\nu }(T)}}={\frac {c}{\lambda ^{2}}}={\frac {\nu ^{2}}{c}}.}
The location of the peak of the spectral distribution for Planck's law depends on the choice of spectral variable. The fractional bandwidth form is related to the other forms by
B ln x = ν B ν = λ B λ {\displaystyle B_{\ln x}=\nu B_{\nu }=\lambda B_{\lambda }} .
First and second radiation constants In the above variants of Planck's law, the wavelength and wavenumber variants use the terms 2hc2 and hc/kB 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 c1L and the second radiation constant c2 with
and
Using the radiation constants, the wavelength variant of Planck's law can be simplified to
L ( λ , T ) = c 1 L λ 5 1 exp ( c 2 λ T ) − 1 {\displaystyle L(\lambda ,T)={\frac {c_{1L}}{\lambda ^{5}}}{\frac {1}{\exp \left({\frac {c_{2}}{\lambda T}}\right)-1}}}
and the wavenumber variant can be simplified correspondingly. L is used here instead of B because it is the SI symbol for spectral radiance. The L in c1L refers to that. This reference is necessary because Planck's law can be reformulated to give spectral radiant exitance M(λ, T) rather than spectral radiance L(λ, T), in which case c1 replaces c1L, with
so that Planck's law for spectral radiant exitance can be written as
M ( λ , T ) = c 1 λ 5 1 exp ( c 2 λ T ) − 1 {\displaystyle M(\lambda ,T)={\frac {c_{1}}{\lambda ^{5}}}{\frac {1}{\exp \left({\frac {c_{2}}{\lambda T}}\right)-1}}}
As measuring techniques have improved, the General Conference on Weights and Measures has revised its estimate of c2; see Planckian locus § International Temperature Scale for details.
Physics
Planck's law describes the unique and characteristic spectral distribution for electromagnetic radiation in thermodynamic equilibrium, when there is no net flow of matter or energy. Its physics is most easily understood by considering the radiation in a cavity with rigid opaque walls. Motion of the walls can affect the radiation. If the walls are not opaque, then the thermodynamic equilibrium is not isolated. It is of interest to explain how the thermodynamic equilibrium is attained. There are two main cases: (a) when the approach to thermodynamic equilibrium is in the presence of matter, when the walls of the cavity are imperfectly reflective for every wavelength or when the walls are perfectly reflective while the cavity contains a small black body (this was the main case considered by Planck); or (b) when the approach to equilibrium is in the absence of matter, when the walls are perfectly reflective for all wavelengths and the cavity contains no matter. For matter not enclosed in such a cavity, thermal radiation can be approximately explained by appropriate use of Planck's law. Classical physics led, via the equipartition theorem, to the ultraviolet catastrophe, a prediction that the total blackbody radiation intensity was infinite. If supplemented by the classically unjustifiable assumption that for some reason the radiation is finite, classical thermodynamics provides an account of some aspects of the Planck distribution, such as the Stefan–Boltzmann law, and the Wien displacement law. For the case of the presence of matter, quantum mechanics provides a good account, as found below in the section headed Einstein coefficients. This was the case considered by Einstein, and is nowadays used for quantum optics. For the case of the absence of matter, quantum field theory is necessary, because non-relativistic quantum mechanics with fixed particle numbers does not provide a sufficient account.
Photons Quantum theoretical explanation of Planck's law views the radiation as a gas of massless, uncharged, bosonic particles, namely photons, in thermodynamic equilibrium. Photons are viewed as the carriers of the electromagnetic interaction between electrically charged elementary particles. Photon numbers are not conserved. Photons are created or annihilated in the right numbers and with the right energies to fill the cavity with photons described by the Planck distribution. For a photon gas in thermodynamic equilibrium, the internal energy density is entirely determined by the temperature; moreover, the pressure is entirely determined by the internal energy density. This is unlike the case of thermodynamic equilibrium for material gases, for which the internal energy is determined not only by the temperature, but also, independently, by the respective numbers of the different molecules, and independently again, by the specific characteristics of the different molecules. For different material gases at given temperature, the pressure and internal energy density can vary independently, because different molecules can carry independently different excitation energies. Planck's law arises as a limit of the Bose–Einstein distribution, the energy distribution describing non-interactive bosons in thermodynamic equilibrium. In the case of massless bosons such as photons and gluons, the chemical potential is zero and the Bose–Einstein distribution reduces to the Planck distribution. There is another fundamental equilibrium energy distribution: the Fermi–Dirac distribution, which describes fermions, such as electrons, in thermal equilibrium. The two distributions differ because multiple bosons can occupy the same quantum state, while multiple fermions cannot. At low densities, the number of available quantum states per particle is large, and this difference becomes irrelevant. In the low density limit, the Bose–Einstein and the Fermi–Dirac distribution each reduce to the Maxwell–Boltzmann distribution.
Kirchhoff's law of thermal radiation
Kirchhoff's law of thermal radiation is a succinct and brief account of a complicated physical situation. The following is an introductory sketch of that situation, and is very far from being a rigorous physical argument. The purpose here is only to summarize the main physical factors in the situation, and the main conclusions.
Spectral dependence of thermal radiation There is a difference between conductive heat transfer and radiative heat transfer. Radiative heat transfer can be filtered to pass only a definite band of radiative frequencies. It is generally known that the hotter a body becomes, the more thermal radiation it emits at every frequency. In a cavity in an opaque body with rigid walls that are not perfectly reflective at any frequency, in thermodynamic equilibrium, there is only one temperature, and it must be shared in common by the radiation of every frequency. One may imagine two such cavities, each in its own isolated radiative and thermodynamic equilibrium. One may imagine an optical device that allows radiative heat transfer between the two cavities, filtered to pass only a definite band of radiative frequencies. If the values of the spectral radiances of the radiations in the cavities differ in that frequency band, heat may be expected to pass from the hotter to the colder. One might propose to use such a filtered transfer of heat in such a band to drive a heat engine. If the two bodies are at the same temperature, the second law of thermodynamics does not allow the heat engine to work. It may be inferred that for a temperature common to the two bodies, the values of the spectral radiances in the pass-band must also be common. This must hold for every frequency band. This became clear to Balfour Stewart and later to Kirchhoff. Balfour Stewart found experimentally that of all surfaces, one of lamp-black emitted the greatest amount of thermal radiation for every quality of radiation, judged by various filters. Thinking theoretically, Kirchhoff went a little further and pointed out that this implied that the spectral radiance, as a function of radiative frequency, of any such cavity in thermodynamic equilibrium must be a unique universal function of temperature. He postulated an ideal black body that interfaced with its surroundings in just such a way as to absorb all the radiation that falls on it. By the Helmholtz reciprocity principle, radiation from the interior of such a body would pass unimpeded directly to its surroundings without reflection at the interface. In thermodynamic equilibrium, the thermal radiation emitted from such a body would have that unique universal spectral radiance as a function of temperature. This insight is the root of Kirchhoff's law of thermal radiation.
Relation between absorptivity and emissivity One may imagine a small homogeneous spherical material body labeled X at a temperature TX, lying in a radiation field within a large cavity with walls of material labeled Y at a temperature TY. The body X emits its own thermal radiation. At a particular frequency ν, the radiation emitted from a particular cross-section through the centre of X in one sense in a direction normal to that cross-section may be denoted Iν,X(TX), characteristically for the material of X. At that frequency ν, the radiative power from the walls into that cross-section in the opposite sense in that direction may be denoted Iν,Y(TY), for the wall temperature TY. For the material of X, defining the absorptivity αν,X,Y(TX, TY) as the fraction of that incident radiation absorbed by X, that incident energy is absorbed at a rate αν,X,Y(TX, TY) Iν,Y(TY). The rate q(ν,TX,TY) of accumulation of energy in one sense into the cross-section of the body can then be expressed
q ( ν , T X , T Y ) = α ν , X , Y ( T X , T Y ) I ν , Y ( T Y ) − I ν , X ( T X ) . {\displaystyle q(\nu ,T_{X},T_{Y})=\alpha _{\nu ,X,Y}(T_{X},T_{Y})I_{\nu ,Y}(T_{Y})-I_{\nu ,X}(T_{X}).}
Kirchhoff's seminal insight, mentioned just above, was that, at thermodynamic equilibrium at temperature T, there exists a unique universal radiative distribution, nowadays denoted Bν(T), that is independent of the chemical characteristics of the materials X and Y, that leads to a very valuable understanding of the radiative exchange equilibrium of any body at all, as follows. When there is thermodynamic equilibrium at temperature T, the cavity radiation from the walls has that unique universal value, so that Iν,Y(TY) = Bν(T). Further, one may define the emissivity εν,X(TX) of the material of the body X just so that at thermodynamic equilibrium at temperature TX = T, one has Iν,X(TX) = Iν,X(T) = εν,X(T) Bν(T). When thermal equilibrium prevails at temperature T = TX = TY, the rate of accumulation of energy vanishes so that q(ν,TX,TY) = 0. It follows that in thermodynamic equilibrium, when T = TX = TY,
0 = α ν , X , Y ( T , T ) B ν ( T ) − ϵ ν , X ( T ) B ν ( T ) . {\displaystyle 0=\alpha _{\nu ,X,Y}(T,T)B_{\nu }(T)-\epsilon _{\nu ,X}(T)B_{\nu }(T).}
Kirchhoff pointed out that it follows that in thermodynamic equilibrium, when T = TX = TY,
α ν , X , Y ( T , T ) = ϵ ν , X ( T ) . {\displaystyle \alpha _{\nu ,X,Y}(T,T)=\epsilon _{\nu ,X}(T).}
Introducing the special notation αν,X(T) for the absorptivity of material X at thermodynamic equilibrium at temperature T (justified by a discovery of Einstein, as indicated below), one further has the equality
α ν , X ( T ) = ϵ ν , X ( T ) {\displaystyle \alpha _{\nu ,X}(T)=\epsilon _{\nu ,X}(T)}
at thermodynamic equilibrium. The equality of absorptivity and emissivity here demonstrated is specific for thermodynamic equilibrium at temperature T and is in general not to be expected to hold when conditions of thermodynamic equilibrium do not hold. The emissivity and absorptivity are each separately properties of the molecules of the material but they depend differently upon the distributions of states of molecular excitation on the occasion, because of a phenomenon known as "stimulated emission", that was discovered by Einstein. On occasions when the material is in thermodynamic equilibrium or in a state known as local thermodynamic equilibrium, the emissivity and absorptivity become equal. Very strong incident radiation or other factors can disrupt thermodynamic equilibrium or local thermodynamic equilibrium. Local thermodynamic equilibrium in a gas means that molecular collisions far outweigh light emission and absorption in determining the distributions of states of molecular excitation. Kirchhoff pointed out that he did not know the precise character of Bν(T), but he thought it important that it should be found out. Four decades after Kirchhoff's insight of the general principles of its existence and character, Planck's contribution was to determine the precise mathematical expression of that equilibrium distribution Bν(T).
Black body
In physics, one considers an ideal black body, here labeled B, defined as one that completely absorbs all of the electromagnetic radiation falling upon it at every frequency ν (hence the term "black"). According to Kirchhoff's law of thermal radiation, this entails that, for every frequency ν, at thermodynamic equilibrium at temperature T, one has αν,B(T) = εν,B(T) = 1, so that the thermal radiation from a black body is always equal to the full amount specified by Planck's law. No physical body can emit thermal radiation that exceeds that of a black body, since if it were in equilibrium with a radiation field, it would be emitting more energy than was incident upon it. Though perfectly black materials do not exist, in practice a black surface can be accurately approximated. As to its material interior, a body of condensed matter, liquid, solid, or plasma, with a definite interface with its surroundings, is completely black to radiation if it is completely opaque. That means that it absorbs all of the radiation that penetrates the interface of the body with its surroundings, and enters the body. This is not too difficult to achieve in practice. On the other hand, a perfectly black interface is not found in nature. A perfectly black interface reflects no radiation, but transmits all that falls on it, from either side. The best practical way to make an effectively black interface is to simulate an 'interface' by a small hole in the wall of a large cavity in a completely opaque rigid body of material that does not reflect perfectly at any frequency, with its walls at a controlled temperature. Beyond these requirements, the component material of the walls is unrestricted. Radiation entering the hole has almost no possibility of escaping the cavity without being absorbed by multiple impacts with its walls.
Lambert's cosine law
As explained by Planck, a radiating body has an interior consisting of matter, and an interface with its contiguous neighbouring material medium, which is usually the medium from within which the radiation from the surface of the body is observed. The interface is not composed of physical matter but is a theoretical conception, a mathematical two-dimensional surface, a joint property of the two contiguous media, strictly speaking belonging to neither separately. Such an interface can neither absorb nor emit, because it is not composed of physical matter; but it is the site of reflection and transmission of radiation, because it is a surface of discontinuity of optical properties. The reflection and transmission of radiation at the interface obey the Stokes–Helmholtz reciprocity principle. At any point in the interior of a black body located inside a cavity in thermodynamic equilibrium at temperature T the radiation is homogeneous, isotropic and unpolarized. A black body absorbs all and reflects none of the electromagnetic radiation incident upon it. According to the Helmholtz reciprocity principle, radiation from the interior of a black body is not reflected at its surface, but is fully transmitted to its exterior. Because of the isotropy of the radiation in the body's interior, the spectral radiance of radiation transmitted from its interior to its exterior through its surface is independent of direction. This is expressed by saying that radiation from the surface of a black body in thermodynamic equilibrium obeys Lambert's cosine law. This means that the spectral flux dΦ(dA, θ, dΩ, dν) from a given infinitesimal element of area dA of the actual emitting surface of the black body, detected from a given direction that makes an angle θ with the normal to the actual emitting surface at dA, into an element of solid angle of detection dΩ centred on the direction indicated by θ, in an element of frequency bandwidth dν, can be represented as
d Φ ( d A , θ , d Ω , d ν ) d Ω = L 0 ( d A , d ν ) d A d ν cos θ {\displaystyle {\frac {d\Phi (dA,\theta ,d\Omega ,d\nu )}{d\Omega }}=L^{0}(dA,d\nu )\,dA\,d\nu \,\cos \theta }
where L0(dA, dν) denotes the flux, per unit area per unit frequency per unit solid angle, that area dA would show if it were measured in its normal direction θ = 0. The factor cos θ is present because the area to which the spectral radiance refers directly is the projection, of the actual emitting surface area, onto a plane perpendicular to the direction indicated by θ . This is the reason for the name cosine law. Taking into account the independence of direction of the spectral radiance of radiation from the surface of a black body in thermodynamic equilibrium, one has L0(dA, dν) = Bν(T) and so
d Φ ( d A , θ , d Ω , d ν ) d Ω = B ν ( T ) d A d ν cos θ . {\displaystyle {\frac {d\