In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Planck's law describes the
spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...
of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
emitted by a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
at a given
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, when there is no net flow of
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
or
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
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
Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
, which by then had been accurately measured, diverged significantly at higher
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
from that predicted by existing theories. In 1900, German physicist
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
heuristically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged
oscillator
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
in a cavity that contained black-body radiation could only change its
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
in a minimal increment, , that was proportional to the frequency of its associated
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. This resolved the problem of the
ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
predicted by
classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. This discovery was a pioneering insight of
modern physics
Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity and general ...
and is of fundamental importance to
quantum theory
Quantum theory may refer to:
Science
*Quantum mechanics, a major field of physics
*Old quantum theory, predating modern quantum mechanics
* Quantum field theory, an area of quantum mechanics that includes:
** Quantum electrodynamics
** Quantum ...
.
The law
![Black body](https://upload.wikimedia.org/wikipedia/commons/1/19/Black_body.svg)
Every
physical body
In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space. The boundary must be defined and identified by t ...
spontaneously and continuously emits
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
and the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Internat ...
of a body, , describes the spectral emissive power per unit area, per unit solid angle, 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 this, the spectral radiance of a body for
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
at
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
is given by
where is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in the medium, whether material or vacuum.
The spectral radiance can also be expressed per unit
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
instead of per unit frequency.
By choosing an appropriate system of
unit of measure
A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multip ...
(i.e. natural
Planck units), the law can be simplified to become:
Equating the integral of spectral radiance in per unit wavelength to that of per unit frequency
where the second integral integrates from
to
because integrating forward in frequency space is integrating backwards in wavelength space. (Wavelength increases as frequency decreases so if
then
). Because this equation holds for any limits
Using
, we see that
showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths. The law may also 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
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
of are , while those of are .
In the limit of low frequencies (i.e. long wavelengths), Planck's law tends to the
Rayleigh–Jeans law
In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is:
B_ (T) = \ ...
, while in the limit of high frequencies (i.e. small wavelengths) it tends to the
Wien approximation
Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
.
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
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
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
.
As an energy distribution, it is one of a family of thermal equilibrium distributions which include the
Bose–Einstein distribution Bose–Einstein may refer to:
* Bose–Einstein condensate
** Bose–Einstein condensation (network theory)
* Bose–Einstein correlations
* Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describ ...
, the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
and the
Maxwell–Boltzmann distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used ...
.
Black-body radiation
![EffectiveTemperature 300dpi e](https://upload.wikimedia.org/wikipedia/commons/0/0d/EffectiveTemperature_300dpi_e.png)
A black-body is an idealised object which absorbs and emits all radiation frequencies. Near
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
, the emitted radiation is closely described by Planck's law and because of its dependence on
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, 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
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
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
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nanometer, nm (with a corresponding frequency around 30 Hertz, PHz) to 400 nm (750 Hertz, THz), shorter than that of visible light, but longer than ...
and even
x-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
s. 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
Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
.
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
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
of the interface (the ratio of the actual
radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
to the theoretical Planck radiance), 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
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used ...
is the unique maximum
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
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
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
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 is the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Internat ...
as a function of temperature and frequency. It has units of
W·
m−2·
sr−1·
Hz−1 in the
SI system
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
. 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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of over those three quantities, and is given by the
Stefan–Boltzmann law
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
. 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.
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
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...
, per spectral unit (frequency, wavelength, wavenumber or their angular equivalents). Since the radiance is
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
(i.e. independent of direction), the power emitted at an angle to the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
is proportional to the projected area, and therefore to the cosine of that angle as per
Lambert's cosine law
In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle ''θ'' between the direction ...
, 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, . 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 the sub-section ]Percentiles
In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage falls ...
of the section Properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy and ...
.
Spectral energy density form
Planck's law can also be written in terms of the spectral energy density
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or .
Often only the ''useful'' or extract ...
() by multiplying by :
These distributions have units of energy per volume per spectral unit.
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
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether ...
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
The General Conference on Weights and Measures (GCWM; french: Conférence générale des poids et mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established i ...
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
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
, to the ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
, 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
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
, and the Wien displacement law
Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck ...
. 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
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
. 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 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 Bose–Einstein may refer to:
* Bose–Einstein condensate
** Bose–Einstein condensation (network theory)
* Bose–Einstein correlations
* Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describ ...
, the energy distribution describing non-interactive bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
in thermodynamic equilibrium. In the case of massless bosons such as photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alway ...
and gluons
A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind qu ...
, the chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
is zero and the Bose–Einstein distribution reduces to the Planck distribution. There is another fundamental equilibrium energy distribution: the Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
, which describes fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, 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
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used ...
.
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
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
and radiative heat transfer
Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
. 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 heat it radiates 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
Carbon black (subtypes are acetylene black, channel black, furnace black, lamp black and thermal black) is a material produced by the Combustion#Incomplete, incomplete combustion of coal and coal tar, vegetable matter, or petroleum products, inc ...
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 surrounds 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 surrounds 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 at a temperature , lying in a radiation field within a large cavity with walls of material labeled at a temperature . The body emits its own thermal radiation. At a particular frequency , the radiation emitted from a particular cross-section through the centre of in one sense in a direction normal to that cross-section may be denoted , characteristically for the material of . At that frequency , the radiative power from the walls into that cross-section in the opposite sense in that direction may be denoted , for the wall temperature . For the material of , defining the absorptivity as the fraction of that incident radiation absorbed by , that incident energy is absorbed at a rate .
The rate of accumulation of energy in one sense into the cross-section of the body can then be expressed
Kirchhoff's seminal insight, mentioned just above, was that, at thermodynamic equilibrium at temperature , there exists a unique universal radiative distribution, nowadays denoted , that is independent of the chemical characteristics of the materials and , 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 , the cavity radiation from the walls has that unique universal value, so that . Further, one may define the emissivity of the material of the body just so that at thermodynamic equilibrium at temperature , one has .
When thermal equilibrium prevails at temperature , the rate of accumulation of energy vanishes so that . It follows that in thermodynamic equilibrium, when ,
Kirchhoff pointed out that it follows that in thermodynamic equilibrium, when ,
Introducing the special notation for the absorptivity of material at thermodynamic equilibrium at temperature (justified by a discovery of Einstein, as indicated below), one further has the equality
at thermodynamic equilibrium.
The equality of absorptivity and emissivity here demonstrated is specific for thermodynamic equilibrium at temperature 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 , 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 .
Black body
In physics, one considers an ideal black body, here labeled , defined as one that completely absorbs all of the electromagnetic radiation falling upon it at every frequency (hence the term "black"). According to Kirchhoff's law of thermal radiation, this entails that, for every frequency , at thermodynamic equilibrium at temperature , one has , 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 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 In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Internat ...
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 from a given infinitesimal element of area of the actual emitting surface of the black body, detected from a given direction that makes an angle with the normal to the actual emitting surface at , into an element of solid angle of detection centred on the direction indicated by , in an element of frequency bandwidth , can be represented as
where denotes the flux, per unit area per unit frequency per unit solid angle, that area would show if it were measured in its normal direction .
The factor 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 . 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 and so
Thus Lambert's cosine law expresses the independence of direction of the spectral radiance of the surface of a black body in thermodynamic equilibrium.
Stefan–Boltzmann law
The total power emitted per unit area at the surface of a black body () may be found by integrating the black body spectral flux found from Lambert's law over all frequencies, and over the solid angles corresponding to a hemisphere () above the surface.
The infinitesimal solid angle can be expressed in spherical polar coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
:
So that:
where is known as the Stefan–Boltzmann constant
The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter ''σ'' (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths inc ...
.
Radiative transfer
The equation of radiative transfer describes the way in which radiation is affected as it travels through a material medium. For the special case in which the material medium is in thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
in the neighborhood of a point in the medium, Planck's law is of special importance.
For simplicity, we can consider the linear steady state, without scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
. The equation of radiative transfer states that for a beam of light going through a small distance , energy is conserved: The change in the (spectral) radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
of that beam () is equal to the amount removed by the material medium plus the amount gained from the material medium. If the radiation field is in equilibrium with the material medium, these two contributions will be equal. The material medium will have a certain emission coefficient
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an electron making a transition from a high energy state to a lower energy state. The photon energy of t ...
and absorption coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
.
The absorption coefficient is the fractional change in the intensity of the light beam as it travels the distance , and has units of length−1. It is composed of two parts, the decrease due to absorption and the increase due to stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
. Stimulated emission is emission by the material body which is caused by and is proportional to the incoming radiation. It is included in the absorption term because, like absorption, it is proportional to the intensity of the incoming radiation. Since the amount of absorption will generally vary linearly as the density of the material, we may define a "mass absorption coefficient" which is a property of the material itself. The change in intensity of a light beam due to absorption as it traverses a small distance will then be
The "mass emission coefficient" is equal to the radiance per unit volume of a small volume element divided by its mass (since, as for the mass absorption coefficient, the emission is proportional to the emitting mass) and has units of power⋅solid angle−1⋅frequency−1⋅density−1. Like the mass absorption coefficient, it too is a property of the material itself. The change in a light beam as it traverses a small distance will then be
The equation of radiative transfer will then be the sum of these two contributions:
If the radiation field is in equilibrium with the material medium, then the radiation will be homogeneous (independent of position) so that and:
which is another statement of Kirchhoff's law, relating two material properties of the medium, and which yields the radiative transfer equation at a point around which the medium is in thermodynamic equilibrium:
Einstein coefficients
The principle of detailed balance The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...
states that, at thermodynamic equilibrium, each elementary process is equilibrated by its reverse process.
In 1916, Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
applied this principle on an atomic level to the case of an atom radiating and absorbing radiation due to transitions between two particular energy levels, giving a deeper insight into the equation of radiative transfer and Kirchhoff's law for this type of radiation. If level 1 is the lower energy level with energy , and level 2 is the upper energy level with energy , then the frequency of the radiation radiated or absorbed will be determined by Bohr's frequency condition:
If and are the number densities of the atom in states 1 and 2 respectively, then the rate of change of these densities in time will be due to three processes:
; Spontaneous emission
:
; Stimulated emission
:
; Photo-absorption
:
where is the spectral energy density of the radiation field. The three parameters , and , known as the Einstein coefficients, are associated with the photon frequency produced by the transition between two energy levels (states). As a result, each line in a spectra has its own set of associated coefficients. When the atoms and the radiation field are in equilibrium, the radiance will be given by Planck's law and, by the principle of detailed balance, the sum of these rates must be zero:
Since the atoms are also in equilibrium, the populations of the two levels are related by the Boltzmann factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, su ...
:
where and are the multiplicities of the respective energy levels. Combining the above two equations with the requirement that they be valid at any temperature yields two relationships between the Einstein coefficients:
so that knowledge of one coefficient will yield the other two. For the case of isotropic absorption and emission, the emission coefficient () and absorption coefficient () defined in the radiative transfer section above, can be expressed in terms of the Einstein coefficients. The relationships between the Einstein coefficients will yield the expression of Kirchhoff's law expressed in the ''Radiative transfer'' section above, namely that
These coefficients apply to both atoms and molecules.
Properties
Peaks
The distributions , , and peak at a photon energy of
where is the Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
and is Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
.
The distributions and however, peak at a different energy[
The reason for this is that, as mentioned above, one cannot go from (for example) to simply by substituting by . In addition, one must also multiply the result of the substitution by
This factor shifts the peak of the distribution to higher energies. These peaks are the '']mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
'' energy of a photon, when binned using equal-size bins of frequency or wavelength, respectively. Meanwhile, the ''average'' energy of a photon from a blackbody is
where is the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. Dividing by this energy expression gives the wavelength of the peak. For this one can use = .
The spectral radiance at these peaks is given by:
:
with and
with
Approximations
In the limit of low frequencies (i.e. long wavelengths), Planck's law becomes the Rayleigh–Jeans law
In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is:
B_ (T) = \ ...
or
The radiance increases as the square of the frequency, illustrating the ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
. In the limit of high frequencies (i.e. small wavelengths) Planck's law tends to the Wien approximation
Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
:
or
Both approximations were known to Planck before he developed his law. He was led by these two approximations to develop a law which incorporated both limits, which ultimately became Planck's law.
Percentiles
Wien's displacement law
Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
in its stronger form states that the shape of Planck's law is independent of temperature. It is therefore possible to list the percentile points of the total radiation as well as the peaks for wavelength and frequency, in a form which gives the wavelength when divided by temperature . The second row of the following table lists the corresponding values of , that is, those values of for which the wavelength is micrometers
The micrometre (American and British English spelling differences#-re, -er, international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American and British English spelling differences# ...
at the radiance percentile point given by the corresponding entry in the first row.
That is, 0.01% of the radiation is at a wavelength below µm, 20% below , etc. The wavelength and frequency peaks are in bold and occur at 25.0% and 64.6% respectively. The 41.8% point is the wavelength-frequency-neutral peak (i.e. the peak in power per unit change in logarithm of wavelength or frequency). These are the points at which the respective Planck-law functions , and , respectively, divided by attain their maxima. The much smaller gap in ratio of wavelengths between 0.1% and 0.01% (1110 is 22% more than 910) than between 99.9% and 99.99% (113374 is 120% more than 51613) reflects the exponential decay of energy at short wavelengths (left end) and polynomial decay at long.
Which peak to use depends on the application. The conventional choice is the wavelength peak at 25.0% given by Wien's displacement law
Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
in its weak form. For some purposes the median or 50% point dividing the total radiation into two-halves may be more suitable. The latter is closer to the frequency peak than to the wavelength peak because the radiance drops exponentially at short wavelengths and only polynomially at long. The neutral peak occurs at a shorter wavelength than the median for the same reason.
For the Sun, is 5778 K, allowing the percentile points of the Sun's radiation, in nanometers, to be tabulated as follows when modeled as a black body radiator, to which the Sun is a fair approximation. For comparison a planet modeled as a black body radiating at a nominal 288 K (15 °C) as a representative value of the Earth's highly variable temperature has wavelengths more than twenty times that of the Sun, tabulated in the third row in micrometers (thousands of nanometers).
That is, only 1% of the Sun's radiation is at wavelengths shorter than 251 nm, and only 1% at longer than 3961 nm. Expressed in micrometers this puts 98% of the Sun's radiation in the range from 0.251 to 3.961 µm. The corresponding 98% of energy radiated from a 288 K planet is from 5.03 to 79.5 µm, well above the range of solar radiation (or below if expressed in terms of frequencies instead of wavelengths ).
A consequence of this more-than-order-of-magnitude difference in wavelength between solar and planetary radiation is that filters designed to pass one and block the other are easy to construct. For example, windows fabricated of ordinary glass or transparent plastic pass at least 80% of the incoming 5778 K solar radiation, which is below 1.2 µm in wavelength, while blocking over 99% of the outgoing 288 K thermal radiation from 5 µm upwards, wavelengths at which most kinds of glass and plastic of construction-grade thickness are effectively opaque.
The Sun's radiation is that arriving at the top of the atmosphere (TOA). As can be read from the table, radiation below 400 nm, or ultraviolet
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nanometer, nm (with a corresponding frequency around 30 Hertz, PHz) to 400 nm (750 Hertz, THz), shorter than that of visible light, but longer than ...
, is about 12%, while that above 700 nm, or infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
, starts at about the 49% point and so accounts for 51% of the total. Hence only 37% of the TOA insolation is visible to the human eye. The atmosphere shifts these percentages substantially in favor of visible light as it absorbs most of the ultraviolet and significant amounts of infrared.
Derivation
Consider a cube of side with conducting walls filled with electromagnetic radiation in thermal equilibrium at temperature . If there is a small hole in one of the walls, the radiation emitted from the hole will be characteristic of a perfect black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
. We will first calculate the spectral energy density within the cavity and then determine the spectral radiance of the emitted radiation.
At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
, one finds that the fields are superpositions of periodic functions. The three wavelengths , , and , in the three directions orthogonal to the walls can be:
where the are positive integers. For each set of integers there are two linearly independent solutions (known as modes). According to quantum theory, the energy levels of a mode are given by:
The quantum number can be interpreted as the number of photons in the mode. The two modes for each set of correspond to the two polarization states of the photon which has a spin of 1. For the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect
In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pr ...
. In the following we will calculate the internal energy of the box at absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
.
According to statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the equilibrium probability distribution over the energy levels of a particular mode is given by:
Here
The denominator , is the partition function of a single mode and makes properly normalized:
Here we have implicitly defined
which is the energy of a single photon. As explained here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here
Television
* Here TV (form ...
, the average energy in a mode can be expressed in terms of the partition function:
This formula, apart from the first vacuum energy term, is a special case of the general formula for particles obeying Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic e ...
. Since there is no restriction on the total number of photons, the chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
is zero.
If we measure the energy relative to the ground state, the total energy in the box follows by summing over all allowed single photon states. This can be done exactly in the thermodynamic limit as approaches infinity. In this limit, becomes continuous and we can then integrate over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between and as , where is the density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
(which is evaluated below), then we can write:
To calculate the density of states we rewrite equation () as follows:
where is the norm of the vector :
For every vector with integer components larger than or equal to zero, there are two photon states. This means that the number of photon states in a certain region of -space is twice the volume of that region. An energy range of corresponds to shell of thickness in -space. Because the components of have to be positive, this shell spans an octant of a sphere. The number of photon states , in an energy range , is thus given by:
Inserting this in Eq. () gives:
From this equation one can derive the spectral energy density as a function of frequency and as a function of wavelength :
where
And:
where
This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
s. In this case, however, it is possible to calculate the integral in closed form using only elementary functions. Substituting
in Eq. (), makes the integration variable dimensionless giving:
where is a Bose–Einstein integral
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natu ...
given by:
The total electromagnetic energy inside the box is thus given by:
where is the volume of the box.
The combination has the value .
This is not the Stefan–Boltzmann law
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
(which provides the total energy ''radiated'' by a black body per unit surface area per unit time), but it can be written more compactly using the Stefan–Boltzmann constant
The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter ''σ'' (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths inc ...
, giving
The constant is sometimes called the radiation constant.
Since the radiation is the same in all directions, and propagates at the speed of light (), the spectral radiance of radiation exiting the small hole is
which yields
It can be converted to an expression for in wavelength units by substituting by and evaluating
Dimensional analysis shows that the unit of steradians, shown in the denominator of the right hand side of the equation above, is generated in and carried through the derivation but does not appear in any of the dimensions for any element on the left-hand-side of the equation.
This derivation is based on .
History
Balfour Stewart
In 1858, Balfour Stewart
Balfour Stewart (1 November 182819 December 1887) was a Scottish physicist and meteorologist.
His studies in the field of radiant heat led to him receiving the Rumford Medal of the Royal Society in 1868. In 1859 he was appointed director of K ...
described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature. Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of John Leslie. He wrote "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power."
Stewart measured radiated power with a thermo-pile and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Helmholtz reciprocity principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium.
Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid. According to historian D. M. Siegel: "He was not a practitioner of the more sophisticated techniques of nineteenth-century mathematical physics; he did not even make use of the functional notation in dealing with spectral distributions." He made no mention of thermodynamics in this paper, though he did refer to conservation of ''vis viva''. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He concluded that his experiments showed that, in the interior of an enclosure in thermal equilibrium, the radiant heat, reflected and emitted combined, leaving any part of the surface, regardless of its substance, was the same as would have left that same portion of the surface if it had been composed of lamp-black. He did not mention the possibility of ideally perfectly reflective walls; in particular he noted that highly polished real physical metals absorb very slightly.
Gustav Kirchhoff
In 1859, not knowing of Stewart's work, Gustav Robert Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
He coin ...
reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.
Kirchhoff then went on to consider bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature .
Here is used a notation different from Kirchhoff's. Here, the emitting power denotes a dimensioned quantity, the total radiation emitted by a body labeled by index at temperature . The total absorption ratio of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because is dimensionless. Also here the wavelength-specific emitting power of the body at temperature is denoted by and the wavelength-specific absorption ratio by . Again, the ratio of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power.
In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers. His theoretical proof was and still is considered by some writers to be invalid. His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio has one and the same value for all bodies, that is for all values of index . In this report there was no mention of black bodies.
In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio , has one and the same value common to all bodies, that is, for every value of the material index . Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.
But more importantly, it relied on a new theoretical postulate of "perfectly black bodies", which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.
Kirchhoff's proof considered an arbitrary non-ideal body labeled as well as various perfect black bodies labeled . It required that the bodies be kept in a cavity in thermal equilibrium at temperature . His proof intended to show that the ratio was independent of the nature of the non-ideal body, however partly transparent or partly reflective it was.
His proof first argued that for wavelength and at temperature , at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power , with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio is again just , with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio was equal to , which may now be denoted , a continuous function, dependent only on at fixed temperature , and an increasing function of at fixed wavelength , at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.)
Thus Kirchhoff's law of thermal radiation can be stated: ''For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature ', for every wavelength ', the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by '.'' (For our notation , Kirchhoff's original notation was simply .)
Kirchhoff announced that the determination of the function was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. That function has occasionally been called 'Kirchhoff's (emission, universal) function', though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later. Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so. Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with "Carnot's principle", which is a form of the second law.
According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "blackbody" radiation that Robert Kirchhoff had first defined in 1859–1860."
Empirical and theoretical ingredients for the scientific induction of Planck's law
In 1860, Kirchhoff predicted experimental difficulties for the empirical determination of the function that described the dependence of the black-body spectrum as a function only of temperature and wavelength. And so it turned out. It took some forty years of development of improved methods of measurement of electromagnetic radiation to get a reliable result.
In 1865, John Tyndall
John Tyndall FRS (; 2 August 1820 – 4 December 1893) was a prominent 19th-century Irish physicist. His scientific fame arose in the 1850s from his study of diamagnetism. Later he made discoveries in the realms of infrared radiation and the p ...
described radiation from electrically heated filaments and from carbon arcs as visible and invisible. Tyndall spectrally decomposed the radiation by use of a rock salt prism, which passed heat as well as visible rays, and measured the radiation intensity by means of a thermopile.
In 1880, André-Prosper-Paul Crova published a diagram of the three-dimensional appearance of the graph of the strength of thermal radiation as a function of wavelength and temperature. He determined the spectral variable by use of prisms. He analyzed the surface through what he called "isothermal" curves, sections for a single temperature, with a spectral variable on the abscissa and a power variable on the ordinate. He put smooth curves through his experimental data points. They had one peak at a spectral value characteristic for the temperature, and fell either side of it towards the horizontal axis. Such spectral sections are widely shown even today.
In a series of papers from 1881 to 1886, Langley reported measurements of the spectrum of heat radiation, using diffraction gratings and prisms, and the most sensitive detectors that he could make. He reported that there was a peak intensity that increased with temperature, that the shape of the spectrum was not symmetrical about the peak, that there was a strong fall-off of intensity when the wavelength was shorter than an approximate cut-off value for each temperature, that the approximate cut-off wavelength decreased with increasing temperature, and that the wavelength of the peak intensity decreased with temperature, so that the intensity increased strongly with temperature for short wavelengths that were longer than the approximate cut-off for the temperature.
Having read Langley, in 1888, Russian physicist V.A. Michelson published a consideration of the idea that the unknown Kirchhoff radiation function could be explained physically and stated mathematically in terms of "complete irregularity of the vibrations of ... atoms". At this time, Planck was not studying radiation closely, and believed in neither atoms nor statistical physics. Michelson produced a formula for the spectrum for temperature:
where denotes specific radiative intensity at wavelength and temperature , and where and are empirical constants.
In 1898, Otto Lummer
Otto Richard Lummer (17 July 1860 – 5 July 1925) was a German physicist and researcher. He was born in the city of Gera, Germany. With Leon Arons, Lummer helped to design and build the Arons–Lummer mercury-vapor lamp. Lummer primarily worked ...
and Ferdinand Kurlbaum
Ferdinand Kurlbaum (October 4, 1857 in Burg bei Magdeburg – July 29, 1927 in Berlin) was a German people, German physicist.
He was a son of a judicial officer. Until 23 he made his Abitur. He studied mathematics and physics in Heidelberg a ...
published an account of their cavity radiation source. Their design has been used largely unchanged for radiation measurements to the present day. It was a platinum box, divided by diaphragms, with its interior blackened with iron oxide. It was an important ingredient for the progressively improved measurements that led to the discovery of Planck's law. A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.
The importance of the Lummer and Kurlbaum cavity radiation source was that it was an experimentally accessible source of black-body radiation, as distinct from radiation from a simply exposed incandescent solid body, which had been the nearest available experimental approximation to black-body radiation over a suitable range of temperatures. The simply exposed incandescent solid bodies, that had been used before, emitted radiation with departures from the black-body spectrum that made it impossible to find the true black-body spectrum from experiments.
Planck's views before the empirical facts led him to find his eventual law
Planck first turned his attention to the problem of black-body radiation in 1897.[, p. 460.]
Theoretical and empirical progress enabled Lummer and Pringsheim to write in 1899 that available experimental evidence was approximately consistent with the specific intensity law where and denote empirically measurable constants, and where and denote wavelength and temperature respectively. For theoretical reasons, Planck at that time accepted this formulation, which has an effective cut-off of short wavelengths.
Gustav Kirchhoff was Max Planck's teacher and surmised that there was a universal law for blackbody radiation and this was called "Kirchhoff's challenge". Planck, a theorist, believed that Wilhelm Wien had discovered this law and Planck expanded on Wien's work presenting it in 1899 to the meeting of the German Physical Society. Experimentalists Otto Lummer
Otto Richard Lummer (17 July 1860 – 5 July 1925) was a German physicist and researcher. He was born in the city of Gera, Germany. With Leon Arons, Lummer helped to design and build the Arons–Lummer mercury-vapor lamp. Lummer primarily worked ...
, Ferdinand Kurlbaum
Ferdinand Kurlbaum (October 4, 1857 in Burg bei Magdeburg – July 29, 1927 in Berlin) was a German people, German physicist.
He was a son of a judicial officer. Until 23 he made his Abitur. He studied mathematics and physics in Heidelberg a ...
, Ernst Pringsheim Sr., and Heinrich Rubens
Heinrich Rubens (30 March 1865, Wiesbaden, Nassau, Germany – 17 July 1922, Berlin, Germany) was a German physicist. He is known for his measurements of the energy of black-body radiation which led Max Planck to the discovery of his radiation l ...
did experiments that appeared to support Wien's law especially at higher frequency short wavelengths which Planck so wholly endorsed at the German Physical Society that it began to be called the Wien-Planck Law. However, by September 1900, the experimentalists had proven beyond a doubt that the Wien-Planck law failed at the longer wavelengths. They would present their data on October 19. Planck was informed by his friend Rubens and quickly created a formula within a few days. In June of that same year, Lord Raleigh had created a formula that would work for short lower frequency wavelengths based on the widely accepted theory of equipartition. So Planck submitted a formula combining both Raleigh's Law (or a similar equipartition theory) and Wien's law which would be weighted to one or the other law depending on wavelength to match the experimental data. However, although this equation worked, Planck himself said unless he could explain the formula derived from a "lucky intuition" into one of "true meaning" in physics, it did not have true significance. Planck explained that thereafter followed the hardest work of his life. Planck did not believe in atoms, nor did he think the second law of thermodynamics should be statistical because probability does not provide an absolute answer, and Boltzmann's entropy law rested on the hypothesis of atoms and was statistical. But Planck was unable to find a way to reconcile his Blackbody equation with continuous laws such as Maxwell's wave equations. So in what Planck called "an act of desperation", he turned to Boltzmann's atomic law of entropy as it was the only one that made his equation work. Therefore, he used the Boltzmann constant ''k'' and his new constant ''h'' to explain the blackbody radiation law which became widely known through his published paper.
Finding the empirical law
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
produced his law on 19 October 1900 as an improvement upon the Wien approximation
Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
, published in 1896 by Wilhelm Wien
Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody ...
, which fit the experimental data at short wavelengths (high frequencies) but deviated from it at long wavelengths (low frequencies). In June 1900, based on heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
theoretical considerations, Rayleigh had suggested a formula that he proposed might be checked experimentally. The suggestion was that the Stewart–Kirchhoff universal function might be of the form . This was not the celebrated Rayleigh–Jeans formula , which did not emerge until 1905, though it did reduce to the latter for long wavelengths, which are the relevant ones here. According to Klein, one may speculate that it is likely that Planck had seen this suggestion though he did not mention it in his papers of 1900 and 1901. Planck would have been aware of various other proposed formulas which had been offered. On 7 October 1900, Rubens told Planck that in the complementary domain (long wavelength, low frequency), and only there, Rayleigh's 1900 formula fitted the observed data well.
For long wavelengths, Rayleigh's 1900 heuristic formula approximately meant that energy was proportional to temperature, . It is known that and this leads to and thence to for long wavelengths. But for short wavelengths, the Wien formula leads to and thence to for short wavelengths. Planck perhaps patched together these two heuristic formulas, for long and for short wavelengths, to produce a formula
This led Planck to the formula
where Planck used the symbols and to denote empirical fitting constants.
Planck sent this result to Rubens, who compared it with his and Kurlbaum's observational data and found that it fitted for all wavelengths remarkably well. On 19 October 1900, Rubens and Kurlbaum briefly reported the fit to the data, and Planck added a short presentation to give a theoretical sketch to account for his formula. Within a week, Rubens and Kurlbaum gave a fuller report of their measurements confirming Planck's law. Their technique for spectral resolution of the longer wavelength radiation was called the residual ray method. The rays were repeatedly reflected from polished crystal surfaces, and the rays that made it all the way through the process were 'residual', and were of wavelengths preferentially reflected by crystals of suitably specific materials.
Trying to find a physical explanation of the law
Once Planck had discovered the empirically fitting function, he constructed a physical derivation of this law. His thinking revolved around entropy rather than being directly about temperature. Planck considered a cavity with perfectly reflective walls; inside the cavity, there are finitely many distinct but identically constituted resonant oscillatory bodies of definite magnitude, with several such oscillators at each of finitely many characteristic frequencies. These hypothetical oscillators were for Planck purely imaginary theoretical investigative probes, and he said of them that such oscillators do not need to "really exist somewhere in nature, provided their existence and their properties are consistent with the laws of thermodynamics and electrodynamics.". Planck did not attribute any definite physical significance to his hypothesis of resonant oscillators but rather proposed it as a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths. He tentatively mentioned the possible connection of such oscillators with atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s. In a sense, the oscillators corresponded to Planck's speck of carbon; the size of the speck could be small regardless of the size of the cavity, provided the speck effectively transduced energy between radiative wavelength modes.
Partly following a heuristic method of calculation pioneered by Boltzmann for gas molecules, Planck considered the possible ways of distributing electromagnetic energy over the different modes of his hypothetical charged material oscillators. This acceptance of the probabilistic approach, following Boltzmann, for Planck was a radical change from his former position, which till then had deliberately opposed such thinking proposed by Boltzmann. In Planck's words, "I considered the uantum hypothesisa purely formal assumption, and I did not give it much thought except for this: that I had obtained a positive result under any circumstances and at whatever cost." Heuristically, Boltzmann had distributed the energy in arbitrary merely mathematical quanta , which he had proceeded to make tend to zero in magnitude, because the finite magnitude had served only to allow definite counting for the sake of mathematical calculation of probabilities, and had no physical significance. Referring to a new universal constant of nature, , Planck supposed that, in the several oscillators of each of the finitely many characteristic frequencies, the total energy was distributed to each in an integer multiple of a definite physical unit of energy, , not arbitrary as in Boltzmann's method, but now for Planck, in a new departure, characteristic of the respective characteristic frequency. His new universal constant of nature, , is now known as the Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
.
Planck explained further that the respective definite unit, , of energy should be proportional to the respective characteristic oscillation frequency of the hypothetical oscillator, and in 1901 he expressed this with the constant of proportionality :
Planck did not propose that light propagating in free space is quantized. The idea of quantization of the free electromagnetic field was developed later, and eventually incorporated into what we now know as quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.
In 1906, Planck acknowledged that his imaginary resonators, having linear dynamics, did not provide a physical explanation for energy transduction between frequencies. Present-day physics explains the transduction between frequencies in the presence of atoms by their quantum excitability, following Einstein. Planck believed that in a cavity with perfectly reflecting walls and with no matter present, the electromagnetic field cannot exchange energy between frequency components. This is because of the linearity
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
of Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. Present-day quantum field theory predicts that, in the absence of matter, the electromagnetic field obeys nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
equations and in that sense does self-interact.[, Chapter 13] Such interaction in the absence of matter has not yet been directly measured because it would require very high intensities and very sensitive and low-noise detectors, which are still in the process of being constructed. Planck believed that a field with no interactions neither obeys nor violates the classical principle of equipartition of energy, and instead remains exactly as it was when introduced, rather than evolving into a black body field. Thus, the linearity of his mechanical assumptions precluded Planck from having a mechanical explanation of the maximization of the entropy of the thermodynamic equilibrium thermal radiation field. This is why he had to resort to Boltzmann's probabilistic arguments.
Some recent proposals in the possible physical explanation of the Planck constant suggest that, following de Broglie's spirit of wave-particle duality, if, regarding the radiation as a wave packet, the Planck constant is determined by the physical properties of the vacuum and a critical amount of disturbance in the electromagnetic field.
Planck's law may be regarded as fulfilling the prediction of Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
He coine ...
that his law of thermal radiation was of the highest importance. In his mature presentation of his own law, Planck offered a thorough and detailed theoretical proof for Kirchhoff's law, theoretical proof of which until then had been sometimes debated, partly because it was said to rely on unphysical theoretical objects, such as Kirchhoff's perfectly absorbing infinitely thin black surface.
Subsequent events
It was not until five years after Planck made his heuristic assumption of abstract elements of energy or of action that Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
conceived of really existing quanta of light in 1905 as a revolutionary explanation of black-body radiation, of photoluminescence, of the photoelectric effect
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid st ...
, and of the ionization of gases by ultraviolet light. In 1905, "Einstein believed that Planck's theory could not be made to agree with the idea of light quanta, a mistake he corrected in 1906." Contrary to Planck's beliefs of the time, Einstein proposed a model and formula whereby light was emitted, absorbed, and propagated in free space in energy quanta localized in points of space. As an introduction to his reasoning, Einstein recapitulated Planck's model of hypothetical resonant material electric oscillators as sources and sinks of radiation, but then he offered a new argument, disconnected from that model, but partly based on a thermodynamic argument of Wien, in which Planck's formula played no role. Einstein gave the energy content of such quanta in the form . Thus Einstein was contradicting the undulatory theory of light held by Planck. In 1910, criticizing a manuscript sent to him by Planck, knowing that Planck was a steady supporter of Einstein's theory of special relativity, Einstein wrote to Planck: "To me it seems absurd to have energy continuously distributed in space without assuming an aether."
According to Thomas Kuhn
Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American philosopher of science whose 1962 book ''The Structure of Scientific Revolutions'' was influential in both academic and popular circles, introducing the term '' paradigm ...
, it was not till 1908 that Planck more or less accepted part of Einstein's arguments for physical as distinct from abstract mathematical discreteness in thermal radiation physics. Still in 1908, considering Einstein's proposal of quantal propagation, Planck opined that such a revolutionary step was perhaps unnecessary. Until then, Planck had been consistent in thinking that discreteness of action quanta was to be found neither in his resonant oscillators nor in the propagation of thermal radiation. Kuhn wrote that, in Planck's earlier papers and in his 1906 monograph, there is no "mention of discontinuity, orof talk of a restriction on oscillator energy, or of
Or or OR may refer to:
Arts and entertainment Film and television
* "O.R.", a 1974 episode of M*A*S*H
* Or (My Treasure), a 2004 movie from Israel (''Or'' means "light" in Hebrew)
Music
* ''Or'' (album), a 2002 album by Golden Boy with Miss ...
any formula like ." Kuhn pointed out that his study of Planck's papers of 1900 and 1901, and of his monograph of 1906, had led him to "heretical" conclusions, contrary to the widespread assumptions of others who saw Planck's writing only from the perspective of later, anachronistic, viewpoints. Kuhn's conclusions, finding a period till 1908, when Planck consistently held his 'first theory', have been accepted by other historians.
In the second edition of his monograph, in 1912, Planck sustained his dissent from Einstein's proposal of light quanta. He proposed in some detail that absorption of light by his virtual material resonators might be continuous, occurring at a constant rate in equilibrium, as distinct from quantal absorption. Only emission was quantal. This has at times been called Planck's "second theory".
It was not till 1919 that Planck in the third edition of his monograph more or less accepted his 'third theory', that both emission and absorption of light were quantal.
The colourful term "ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
" was given by Paul Ehrenfest
Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition a ...
in 1911 to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
of classical statistical mechanics is (mistakenly) applied to black-body radiation. But this had not been part of Planck's thinking, because he had not tried to apply the doctrine of equipartition: when he made his discovery in 1900, he had not noticed any sort of "catastrophe". It was first noted by Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
in 1900, and then in 1901 by Sir James Jeans
Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician.
Early life
Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans was ...
; and later, in 1905, by Einstein when he wanted to support the idea that light propagates as discrete packets, later called 'photons', and by Rayleigh and by Jeans.
In 1913, Bohr gave another formula with a further different physical meaning to the quantity . In contrast to Planck's and Einstein's formulas, Bohr's formula referred explicitly and categorically to energy levels of atoms. Bohr's formula was where and denote the energy levels of quantum states of an atom, with quantum numbers and . The symbol denotes the frequency of a quantum of radiation that can be emitted or absorbed as the atom passes between those two quantum states. In contrast to Planck's model, the frequency has no immediate relation to frequencies that might describe those quantum states themselves.
Later, in 1924, Satyendra Nath Bose
Satyendra Nath Bose (; 1 January 1894 – 4 February 1974) was a Bengali mathematician and physicist specializing in theoretical physics. He is best known for his work on quantum mechanics in the early 1920s, in developing the foundation for ...
developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law. The actual word 'photon' was invented still later, by G.N. Lewis in 1926, who mistakenly believed that photons were conserved, contrary to Bose–Einstein statistics; nevertheless the word 'photon' was adopted to express the Einstein postulate of the packet nature of light propagation. In an electromagnetic field isolated in a vacuum in a vessel with perfectly reflective walls, such as was considered by Planck, indeed the photons would be conserved according to Einstein's 1905 model, but Lewis was referring to a field of photons considered as a system closed with respect to ponderable matter but open to exchange of electromagnetic energy with a surrounding system of ponderable matter, and he mistakenly imagined that still the photons were conserved, being stored inside atoms.
Ultimately, Planck's law of black-body radiation contributed to Einstein's concept of quanta of light carrying linear momentum, which became the fundamental basis for the development of quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.
The above-mentioned linearity of Planck's mechanical assumptions, not allowing for energetic interactions between frequency components, was superseded in 1925 by Heisenberg's original quantum mechanics. In his paper submitted on 29 July 1925, Heisenberg's theory accounted for Bohr's above-mentioned formula of 1913. It admitted non-linear oscillators as models of atomic quantum states, allowing energetic interaction between their own multiple internal discrete Fourier frequency components, on the occasions of emission or absorption of quanta of radiation. The frequency of a quantum of radiation was that of a definite coupling between internal atomic meta-stable oscillatory quantum states. At that time, Heisenberg knew nothing of matrix algebra, but Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
read the manuscript of Heisenberg's paper and recognized the matrix character of Heisenberg's theory. Then Born and Jordan
Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...
published an explicitly matrix theory of quantum mechanics, based on, but in form distinctly different from, Heisenberg's original quantum mechanics; it is the Born and Jordan matrix theory that is today called matrix mechanics. Heisenberg's explanation of the Planck oscillators, as non-linear effects apparent as Fourier modes of transient processes of emission or absorption of radiation, showed why Planck's oscillators, viewed as enduring physical objects such as might be envisaged by classical physics, did not give an adequate explanation of the phenomena.
Nowadays, as a statement of the energy of a light quantum, often one finds the formula , where , and denotes angular frequency, and less often the equivalent formula . This statement about a really existing and propagating light quantum, based on Einstein's, has a physical meaning different from that of Planck's above statement about the abstract energy units to be distributed amongst his hypothetical resonant material oscillators.
An article by Helge Kragh published in ''Physics World
''Physics World'' is the membership magazine of the Institute of Physics, one of the largest physical societies in the world. It is an international monthly magazine covering all areas of physics, pure and applied, and is aimed at physicists in ...
'' gives an account of this history.
See also
* Emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
* Radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
* Sakuma–Hattori equation The Sakuma–Hattori equation is a mathematical model for predicting the amount of thermal radiation, radiometric flux or radiometric power emitted from a perfect blackbody or received by a thermal radiation detector.
History
The Sakuma–Hattori ...
* Wien's displacement law
Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
References
Bibliography
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* Translated as "Quantum-theoretical Re-interpretation of kinematic and mechanical relations" in
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* a translation of ''Frühgeschichte der Quantentheorie (1899–1913)'', Physik Verlag, Mosbach/Baden, 1969.
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External links
Summary of Radiation
– interactive simulation to play with Planck's law
{{DEFAULTSORT:Planck's Law
Statistical mechanics
Foundational quantum physics
Max Planck
1900 in science
1900 in Germany