Black body

A widely used model of a black surface is a small hole in a cavity with walls that are opaque to radiation.[8] Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large. The hole is not quite a perfect black surface — in particular, if the wavelength of the incident radiation is greater than the diameter of the hole, part will be reflected. Similarly, even in perfect thermal equilibrium, the radiation inside a finite-sized cavity will not have an ideal Planck spectrum for wavelengths comparable to or larger than the size of the cavity.[9]

Suppose the cavity is held at a fixed temperature T and the radiation trapped inside the enclosure is at thermal equilibrium with the enclosure. The hole in the enclosure will allow some radiation to escape. If the hole is small, radiation passing in and out of the hole has negligible effect upon the equilibrium of the radiation inside the cavity. This escaping radiation will approximate black-body radiation that exhibits a distribution in energy characteristic of the temperature T and does not depend upon the properties of the cavity or the hole, at least for wavelengths smaller than the size of the hole.[9] See the figure in the Introduction for the spectrum as a function of the frequency of the radiation, which is related to the energy of the radiation by the equation E=hf, with E = energy, h = Planck's constant, f = frequency.

At any given time the radiation in the cavity may not be in thermal equilibrium, but the second law of thermodynamics states that if left undisturbed it will eventually reach equilibrium,[10] although the time it takes to do so may be very long.[11] Typically, equilibrium is reached by continual absorption and emission of radiation by material in the cavity or its walls.[12][13][14][15] Radiation entering the cavity will be "thermalized" by this mechanism: the energy will be redistributed until the ensemble of photons achieves a Planck distribution. The time taken for thermalization is much faster with condensed matter present than with rarefied matter such as a dilute gas. At temperatures below billions of Kelvin, direct photon–photon interactions[16] are usually negligible compared to interactions with matter.[17] Photons are an example of an interacting boson gas,[18] and as described by the H-theorem,[19] under very general conditions any interacting boson gas will approach thermal equilibrium.

A body's behavior with regard to thermal radiation is characterized by its transmission τ, absorption α, and reflection ρ.

The boundary of a body forms an interface with its surroundings, and this interface may be rough or smooth. A nonreflecting interface separating regions with different refractive indices must be rough, because the laws of reflection and refraction governed by the Fresnel equations for a smooth interface require a reflected ray when the refractive indices of the material and its surroundings differ.[20] A few idealized types of behavior are given particular names:

An opaque body is one that transmits none of the radiation that reaches it, although some may be reflected.[21][22] That is, τ=0 and α+ρ=1

A transparent body is one that transmits all the radiation that reaches it. That is, τ=1 and α=ρ=0.

A grey body is one where α, ρ and τ are uniform for all wavelengths. This term also is used to mean a body for which α is temperature and wavelength independent.

A white body is one for which all incident radiation is reflected uniformly in all directions: τ=0, α=0, and ρ=1.

For a black body, τ=0, α=1, and ρ=0. Planck offers a theoretical model for perfectly black bodies, which he noted do not exist in nature: besides their opaque interior, they have interfaces that are perfectly transmitting and non-reflective.[23]

Kirchhoff in 1860 introduced the theoretical concept of a perfect black body with a completely absorbing surface layer of infinitely small thickness, but Planck noted some severe restrictions upon this idea. Planck noted three requirements upon a black body: the body must (i) allow radiation to enter but not reflect; (ii) possess a minimum thickness adequate to absorb the incident radiation and prevent its re-emission; (iii) satisfy severe limitations upon scattering to prevent radiation from entering and bouncing back out. As a consequence, Kirchhoff's perfect black bodies that absorb all the radiation that falls on them cannot be realized in an infinitely thin surface layer, and impose conditions upon scattering of the light within the black body that are difficult to satisfy.[24][25]

The concept of black-body was formulated initially for size of it to be much less than the wave length. To say about diffraction, the method of geometrical optics is valid then. To apply the Plank law for emitting of black body one has to regard the restriction:

${\displaystyle \lambda \ll \ L}$,

where ${\displaystyle L}$ is the characteristic size of object. In 20-th age, the series of attempts was taken to find approach that is valid for any wave length - similar to ideal reflecting surface. In that case, an according mathematical condition has to be formulated on surface of black-body.[26] As to be known, impedance matching is effective for the only angle of incidence. In 1911, Macdonald H.M. proposed nearly self-evident approach.[27] He used two well formulated problems in electrodynamics- that of reflection from ideal metal:

${\displaystyle \mathbf {E} _{\tau }=0}$,

and that of reflection from ideal magnetic:

${\displaystyle \mathbf {H} _{\tau }=0}$.

Half-sum of solutions is the field around the black-body in Macdonald's model. The approach is clear in the scope of the geometrical optics. Two reflected rays have equal amplitudes of opposite signs and cancel each other. Therefore, the convex surface does not reflect rays at all. At the same time, surface having convex and concave parts of that allows double reflections. The second reflections have equal amplitudes of two rays what does not accord to black-body concept. Consequently, Macdonald's model is reasonable for the convex surface only. Diagrams of scattered fields around black ball for Macdonald's model are calculated on the base of Maxwell's equations in monography.[26]

Sommerfeld proposed to consider black flat screen as surface of continued space what is analogous to procedure in the theory of complex analysis. Therefore, the problem is got to be spacious instead of surface one. [28]

The idea to continue physical space was developed later. In 1978, Sergei P. Efimov from Bauman Moscow State Technical University found that Macdonald's model is equivalent to that with symmetrical adjunct space. [29] The spaces are connected formally on the surface of black-body. Actually, two problems are considered outside of the surface. One is with charges and currents, other is without that. Boundary conditions on the surface equate tangential components of electric and magnetic fields of two problems to each other with changing sign of the magnetic component. (In scalar problem, potentials are equal on the surface when their derivatives are differed by sign.) In such a way, electric field in physical space is equal to half-sum of solutions of two problems for ideal reflecting surfaces:

${\displaystyle \mathbf {E} ={\frac {(\,\mathbf {E} _{+}+\mathbf {E} _{-})}{2}}\qquad }$ (in physical space),

where ${\displaystyle \mathbf {E} _{+}}$ is field from problem for ideal metal and ${\displaystyle \mathbf {E} _{-}}$ is the field from problem for ideal magnetic. In the adjunct space, where no charges and currents, the sought electric field is equal to the difference of the same fields:

${\displaystyle \mathbf {E} ={\frac {(\,\mathbf {E} _{+}-\mathbf {E} _{-})}{2}}\qquad }$ (in adjunct space).

The concept of adjunct space proves that Macdonald's model is physically correct for all frequencies. The causality holds in the approach and considerations of scattering of wave packs is acceptable. From symmetry of physical anf adjunct spaces follows two electrodynamical theorems:

• In state of the heat equilibrium, heat fluxes from surface in physical and adjunct spaces are equal to each other.
• Scattered field from thin black disc is equal to that from hole in flat thin screen (Babinet's principle).

Macdonald's model and Efimov's consideration are valid for equations of acoustics, to equations of hydrodynamics and for diffusion equation. It should be noticed that half-sum of two subsidiary solutions as a solution of the problem is valid for linear equations only. It is clear that theoretical model needs a way for realizations.[30][31]

The concept of adjunct space can be applied to the black hole in theory of gravitation. The famous Schwarzschild metric looks mathematically simple:

${\displaystyle \mathbf {-} \,{\frac {\left(1-{\frac {r_{s}}{4R}}\right)^{2}}{\left(1+{\frac {r_{s}}{4R}}\right)^{2}}}dt^{2}+\left(1+{\frac {r_{s}}{4R}}\right)^{4}(dx^{2}+dy^{2}+dz^{2})}$,

where ${\displaystyle \mathbf {R} =(x,y,z)}$ is radius-vector, ${\displaystyle r_{s}}$ is the Schwarzschild radius i.e. radius of black-hole. From point of view of concept based on the adjunct space , it is useful to apply the following transformation of physical space:[29]

${\displaystyle {\boldsymbol {\rho }}={\frac {a^{2}\mathbf {R} }{R^{2}}}}$,

where ${\displaystyle a}$ is radius of sphere that adjunct space is attached to. As to be known, it is inversion in sphere. Radius ${\displaystyle a}$ is taken to give the Schwarzschild metrics again:

${\displaystyle a={\frac {r^{s}}{4}}}$.

Therefore, the black hole can be considered as the connection of two symmetrical spaces on the surface of ball of radius ${\displaystyle a}$. Adjunct space has the same Schwarzschild metrics. In that case, well known singularity ${\displaystyle R=0}$ is disappeared.

Non-reflecting chamber has absolutely absorbing walls. Regarding physical picture, adjunct space now is simply the surrounding space as far as the walls are missed. Therefore, adjunct spaces for the totally convex surface and concave one are identical. The adjunct space is continued along normal directed into side of convexity of surface. Details are described in the paper. [29] The equivalent electrodynamical problem can be formulated on the base of boundary condition. It analogous to the impedance matching. Nevertheless, the boundary condition binds tangential components of electric and magnet fields not in the point but on all surface. The condition is based on Stratton - Chu formula.[32][33]

To demonstrate approach, it is useful to deduce boundary condition for scalar problem when Helmholtz equation is valid. Fields on the surface are bound by Green's function in two points - ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {y} }$:

${\displaystyle G(\mathbf {x,y} )={\frac {\exp(\mathbf {\mid x-y\mid } )}{4\pi \mathbf {\mid x-y\mid } }}.}$

Let be charges (or radiation sources) are placed in non-reflecting chamber i.e. in free space. Green's formula defines field ${\displaystyle u(\mathbf {x} )}$ in adjunct space by boundary values on the surface of non-reflecting chamber. Upon sending argument ${\displaystyle \mathbf {x} }$ on the surface, formula gives boundary condition:

${\displaystyle {\frac {u(\mathbf {x} )}{2}}=\iint \limits _{S,\,y\neq x}\left[{\mathbf {-} \,G(\mathbf {x,y} ){\frac {\partial u(\mathbf {y} )}{\partial n_{y}}}+u(\mathbf {y} ){\frac {\partial G(\mathbf {x,y)} }{\partial n_{y}}}}\right]\,dS_{y}.}$

The surface integral is calculated in the sense of Hadamard regularization. Normal ${\displaystyle n_{y}}$ is directed outside of chamber i.e. in side of convexity of surface. Vector ${\displaystyle \mathbf {x} }$ lays on the surface.

Boundary condition for convex black-body (for example ball) differes by sign of first item in the integral as far as derivative changes sign on the surface for going from physical space to adjunct one. Normal is directed outside of black-body.

At last, boundary condition for arbitrary surface, containing convex and concave parts, conserves its form under requirement that first sign is (+) for convex part and sign (-) for concave that.

In 1898, Otto Lummer and Ferdinand Kurlbaum published an account of their cavity radiation source.[34] Their design has been used largely unchanged for radiation measurements to the present day. It was a hole in the wall of 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.[35][36] A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.[37] See also Hohlraum.

There is interest in blackbody-like materials for camouflage and radar-absorbent materials for radar invisibility.[38][39] They also have application as solar energy collectors, and infrared thermal detectors. As a perfect emitter of radiation, a hot material with black body behavior would create an efficient infrared heater, particularly in space or in a vacuum where convective heating is unavailable.[40] They are also useful in telescopes and cameras as anti-reflection surfaces to reduce stray light, and to gather information about objects in high-contrast areas (for example, observation of planets in orbit around their stars), where blackbody-like materials absorb light that comes from the wrong sources.

It has long been known that a lamp-black coating will make a body nearly black. An improvement on lamp-black is found in manufactured carbon nanotubes. Nano-porous materials can achieve refractive indices nearly that of vacuum, in one case obtaining average reflectance of 0.045%.[5][41] In 2009, a team of Japanese scientists created a material called nanoblack which is close to an ideal black body, based on vertically aligned single-walled carbon nanotubes. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.[40]

Other examples of nearly perfect black materials are super black, prepared by chemically etching a nickel–phosphorus alloy,[42] vertically aligned carbon nanotube arrays and flower carbon nanostructures;[43] all absorb 99.9% of light or more.

An idealized view of the cross-section of a star. The photosphere contains photons of light nearly in thermal equilibrium, and some escape into space as near-black-body radiation.

A star or planet often is modeled as a black body, and electromagnetic radiation emitted from these bodies as black-body radiation. The figure shows a highly schematic cross-section to illustrate the idea. The photosphere of the star, where the emitted light is generated, is idealized as a layer within which the photons of light interact with the material in the photosphere and achieve a common temperature T that is maintained over a long period of time. Some photons escape and are emitted into space, but the energy they carry away is replaced by energy from within the star, so that the temperature of the photosphere is nearly steady. Changes in the core lead to changes in the supply of energy to the photosphere, but such changes are slow on the time scale of interest here. Assuming these circumstances can be realized, the outer layer of the star is somewhat analogous to the example of an enclosure with a small hole in it, with the hole replaced by the limited transmission into space at the outside of the photosphere. With all these assumptions in place, the star emits black-body radiation at the temperature of the photosphere.[44]

Effective temperature of a black body compared with the B-V and U-B color index of main sequence and super giant stars in what is called a color-color diagram.[45]

Using this model the effective temperature of stars is estimated, defined as the temperature of a black body that yields the same surface flux of energy as the star. If a star were a black body, the same effective temperature would result from any region of the spectrum. For example, comparisons in the B (blue) or V (visible) range lead to the so-called B-V color index, which increases the redder the star,[46] with the Sun having an index of +0.648 ± 0.006.[47] Combining the U (ultraviolet) and the B indices leads to the U-B index, which becomes more negative the hotter the star and the more the UV radiation. Assuming the Sun is a type G2 V star, its U-B index is +0.12.[48] The two indices for two types of most common star sequences are compared in the figure (diagram) with the effective surface temperature of the stars if they were perfect black bodies. There is a rough correlation. For example, for a given B-V index measurement, the curves of both most common sequences of star (the main sequence and the supergiants) lie below the corresponding black-body U-B index that includes the ultraviolet spectrum, showing that both groupings of star emit less ultraviolet light than a black body with the same B-V index. It is perhaps surprising that they fit a black body curve as well as they do, considering that stars have greatly different temperatures at different depths.[49] For example, the Sun has an effective temperature of 5780 K,[50] which can be compared to the temperature of its photosphere (the region generating the light), which ranges from about 5000 K at its outer boundary with the chromosphere to about 9500 K at its inner boundary with the convection zone approximately 500 km (310 mi) deep.[51]

A black hole is a region of spacetime from which nothing escapes. Around a black hole there is a mathematically defined surface called an event horizon that marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, making it almost an ideal black body[52] (radiation with a wavelength equal to or larger than the diameter of the hole may not be absorbed, so black holes are not perfect black bodies).[53] Physicists believe that to an outside observer, black holes have a non-zero temperature and emit radiation with a nearly perfect black-body spectrum, ultimately evaporating.[54] The mechanism for this emission is related to vacuum fluctuations in which a virtual pair of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted.[55] The energy distribution of emission is described by Planck's law with a temperature T:

${\displaystyle T={\frac {\hbar c^{3}}{8\pi Gk_{B}M}}\ ,}$

where c is the speed of light, ℏ is the reduced Planck constant, k_B is Boltzmann's constant, G is the gravitational constant and M is the mass of the black hole.[56] These predictions have not yet been tested either observationally or experimentally.[57]

The big bang theory is based upon the cosmological principle, which states that on large scales the Universe is homogeneous and isotropic. According to theory, the Universe approximately a second after its formation was a near-ideal black body in thermal equilibrium at a temperature above 10¹⁰ K. The temperature decreased as the Universe expanded and the matter and radiation in it cooled. The cosmic microwave background radiation observed today is "the most perfect black body ever measured in nature".[58] It has a nearly ideal Planck spectrum at a temperature of about 2.7 K. It departs from the perfect isotropy of true black-body radiation by an observed anisotropy that varies with angle on the sky only to about one part in 100,000.