# Emissivity

Every body with a temperature above absolute zero emits thermal radiation . The emissivity of a body indicates how much radiation it emits compared to an ideal heat radiator, a black body . This value is always between 0 (no absorption) and 1 (100% absorption).

## Black body as a radiation reference Two views of a Leslie cube , all sides of the cube are at the same temperature but with different degrees of emission. Image taken with a thermal imaging camera and in comparison in the visible area

A black body is a hypothetical idealized body that completely absorbs any electromagnetic radiation it hits at any frequency . According to Kirchhoff's law of radiation , the absorption and emissivity of a body are always proportional. Since the black body has the greatest possible absorption capacity at every frequency (namely 100%), it must also emit the strongest physically possible thermal radiation output at every frequency that is possible at the given temperature . In other words: If it stands next to another body of the same name with a lower emissivity, it gives off its energy faster and also shines brighter than the other body.

Since it radiates equally maximally in every direction, the radiation it emits is equally strong in all directions; it radiates completely diffusely . In addition, the intensity and frequency distribution of the radiation emitted by a black body do not depend on its material nature or on its history, but only on its temperature; they are described by Planck's law of radiation .

The universal character of the thermal radiation emitted by a black body and the fact that no real body can radiate more strongly than a black body at any frequency suggest that the emissivity of a real body should be set to the maximum possible value given by the black body Respectively. The ratio of the radiation intensity emitted by a body to the radiation intensity of a black body at the same temperature is called the body's emissivity. The emissivity can have values ​​between 0 and 1. Depending on whether the frequency and directional distribution of the radiation are to be taken into account, four different emissivities can be specified.

The emissivity of a body must be known so that its temperature can be determined from the intensity of the heat radiation emitted using a pyrometer or a thermal imaging camera .

## Emissivities

### Directed spectral emissivity

The spectral radiance (unit: W · m −2 · Hz −1 · sr −1 ) of a body of temperature indicates which radiant power the body at the frequency in the direction given by the polar angle and the azimuth angle per area, per frequency width and emits per solid angle . The spectral radiance of a black body is independent of direction and is given by Planck's law of radiation. ${\ displaystyle L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T)}$ ${\ displaystyle T}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ varphi}$ ${\ displaystyle L _ {\ Omega \ nu} ^ {o} (\ nu, T)}$ The directed spectral emissivity of a body is the ratio of the spectral radiance emitted by a surface element of the body at the frequency in the given angle and direction to the spectral radiance emitted by a blackbody of the same temperature at the same frequency in the same direction : ${\ displaystyle \ nu}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ varphi}$ ${\ displaystyle L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T)}$ ${\ displaystyle L _ {\ Omega \ nu} ^ {o} (\ nu, T)}$ ${\ displaystyle \ varepsilon _ {\ nu} ^ {\ prime} (\ beta, \ varphi, \ nu, T) = {\ frac {L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T )} {L _ {\ Omega \ nu} ^ {o} (\ nu, T)}}}$ .

### Hemispheric spectral emissivity

The spectral specific radiation (unit: W · m −2 · Hz −1 ) of a body of temperature indicates which radiation power the body emits at the frequency in the entire half-space per unit area and per unit frequency interval. The spectral specific radiation of a black body is given by Planck's law of radiation. ${\ displaystyle \, M _ {\ nu} (\ nu, T)}$ ${\ displaystyle T}$ ${\ displaystyle \ nu}$ ${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T)}$ The hemispherical spectral emissivity of a body is the ratio of the spectral specific radiation emitted by a surface element of the body at the frequency in the half space to the spectral specific radiation emitted by a black body of the same temperature at the same frequency in the half space : ${\ displaystyle \ nu}$ ${\ displaystyle \, M _ {\ nu} (\ nu, T)}$ ${\ displaystyle M _ {\ nu} ^ {o} (\ nu, T)}$ ${\ displaystyle \ varepsilon _ {\ nu} (\ nu, T)}$ ${\ displaystyle = {\ frac {M _ {\ nu} (\ nu, T)} {M _ {\ nu} ^ {o} (\ nu, T)}}}$ ${\ displaystyle = {\ frac {\ int L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T) \ cos (\ beta) \, \ mathrm {d} \ Omega} {\ int L_ { \ Omega \ nu} ^ {o} (\ nu, T) \ cos (\ beta) \, \ mathrm {d} \ Omega}}}$ ${\ displaystyle = {\ frac {1} {\ pi}} \, \ int \ varepsilon _ {\ nu} ^ {\ prime} (\ beta, \ varphi, \ nu, T) \, \ cos (\ beta ) \, \ mathrm {d} \ Omega}$ .

### Directed total emissivity

The total radiance or radiance (unit: W m −2 sr −1 ) of a body of temperature indicates which radiant power the body emits at all frequencies in the direction given by the polar angle and the azimuth angle per unit area and per solid angle unit. The radiance of a black body is independent of direction and is given by Planck's law of radiation. ${\ displaystyle L _ {\ Omega} (\ beta, \ varphi, T)}$ ${\ displaystyle T}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ varphi}$ ${\ displaystyle L _ {\ Omega} ^ {o} (T)}$ The directed total emissivity of a body is the ratio of the radiance emitted by a surface element of the body at all frequencies in the given angle and direction to the radiance emitted by a blackbody of the same temperature at all frequencies in the same direction : ${\ displaystyle \ beta}$ ${\ displaystyle \ varphi}$ ${\ displaystyle L _ {\ Omega} (\ beta, \ varphi, T)}$ ${\ displaystyle L _ {\ Omega} ^ {o} (T)}$ ${\ displaystyle \ varepsilon ^ {\ prime} (\ beta, \ varphi, T)}$ ${\ displaystyle = {\ frac {L _ {\ Omega} (\ beta, \ varphi, T)} {L _ {\ Omega} ^ {o} (T)}}}$ ${\ displaystyle = {\ frac {\ int L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T) \, \ mathrm {d} \ nu} {\ int L _ {\ Omega \ nu} ^ {o} (\ nu, T) \, \ mathrm {d} \ nu}}}$ ${\ displaystyle = {\ frac {\ pi} {\ sigma T ^ {4}}} \, \ int \ varepsilon _ {\ nu} ^ {\ prime} (\ beta, \ varphi, \ nu, T) L_ {\ Omega \ nu} ^ {o} (\ nu, T) \, \ mathrm {d} \ nu}$ .

### Hemispheric total emissivity

The specific radiation (unit: W · m −2 ) of a body of temperature indicates which radiation power the body emits per unit area at all frequencies into the half-space. The specific radiation of a black body is given by the Stefan-Boltzmann law . ${\ displaystyle \, M (T)}$ ${\ displaystyle T}$ ${\ displaystyle \, M ^ {o} (T)}$ The total hemispherical emissivity of a body is the ratio of the specific radiation emitted by a surface element of the body at all frequencies in the half-space to the radiance emitted by a black body of the same temperature at all frequencies in the half-space : ${\ displaystyle \, M (T)}$ ${\ displaystyle \, M ^ {o} (T)}$ ${\ displaystyle \ varepsilon (T)}$ ${\ displaystyle = {\ frac {M (T)} {M ^ {o} (T)}}}$ ${\ displaystyle = {\ frac {\ int \ int L _ {\ Omega \ nu} (\ beta, \ varphi, \ nu, T) \, \ cos (\ beta) \, \ mathrm {d} \ nu \, \ mathrm {d} \ Omega} {\ int \ int L _ {\ Omega \ nu} ^ {o} (\ nu, T) \, \ cos (\ beta) \, \ mathrm {d} \ nu \, \ mathrm {d} \ Omega}}}$ ${\ displaystyle = {\ frac {1} {\ sigma T ^ {4}}} \, \ int \ varepsilon _ {\ nu} (\ nu, T) \, M _ {\ nu} ^ {o} (\ nu, T) \, \ mathrm {d} \ nu}$ ${\ displaystyle = {\ frac {1} {\ pi}} \, \ int \ varepsilon ^ {\ prime} (\ beta, \ varphi, T) \, \ cos (\ beta) \, \ mathrm {d} \ Omega}$ .

All beam sizes and emissivities can of course also be formulated as a function of wavelength instead of frequency.

## properties

All four emissivities described are material properties of the body under consideration (in the case of the absorption degrees defined in the same way , this only applies to the directed spectral absorption degree). The directional spectral emissivity describes the direction and frequency dependence of the emitted radiation by comparison with the radiation emitted by a black body. The hemispherical spectral emissivity only describes the frequency dependency, the directed total emissivity only the directional dependence and the hemispherical total emissivity only the total radiated power emitted. For many materials, only the latter is known.

A body whose directed spectral emissivity does not depend on the direction is a Lambert radiator ; it emits completely diffuse radiation. A body whose directional spectral emissivity does not depend on the frequency is a gray body . In both cases, there are often considerable simplifications for radiation calculations, so that real bodies are often - as far as possible - approximately viewed as diffuse emitters and gray bodies.

According to kirchhoff's law of radiation , the directed spectral emissivity is equal to the directed spectral absorption for every body. For the other degrees of emission and absorption, the same applies only under additional conditions.

## Tables

Basically, the information on the emissivity in the many tables to be found should be treated with caution. Due to the many possible variations, which are seldom all specified, there can be larger differences.

Examples of emissivities of non-metallic surfaces.
material Temperature / ° C ${\ displaystyle \ vartheta}$ Total emissivity in the direction of the surface normal ${\ displaystyle \ varepsilon _ {n}}$ Total hemispherical emissivity ${\ displaystyle \ varepsilon}$ Beech wood 70 0.94 0.91
Ice, smooth, thickness> 4 mm −9.6 0.965 0.918
Enamel paint , white 20th 0.91
coal 150 0.81
Paper, white, matte 95 0.92 0.89
Tire surface, rough 0 0.985
sand 20th 0.76
Sheet glass, 6 mm thick −60 ... 0 0.910
60 0.913
120 0.919
Soda-lime glass 9.85 0.837
Water, thickness> 0.1 mm 10 ... 50 0.965 0.91
Examples of emissivities of metal surfaces
material Temperature / ° C ${\ displaystyle \ vartheta}$ Total emissivity in the direction of the surface normal ${\ displaystyle \ varepsilon _ {n}}$ Total hemispherical emissivity ${\ displaystyle \ varepsilon}$ Iron polished −73 ... 727 0.04 ... 0.19 0.06 ... 0.25
-, oxidized −73 ... 727 0.32 ... 0.60
-, sanded smooth 25th 0.24
-, etched blank 150 0.128 0.158
-, cast skin 100 0.80
-, rusty 25th 0.61
Gold , polished 227 ... 627 0.020 ... 0.035
-, oxidized −173 ... 827   0.013 ... 0.070
Copper polished 327 ... 727 0.012 ... 0.019
-, oxidized 130 0.76 0.725
-, heavily oxidized 25th 0.78
327 0.83
427 0.89
527 ... 727 0.91 ... 0.92
aluminum 0.04
platinum 0.05

## Terminology

The term emissivity is often used synonymously for emissivity , whereby emissivity has a broader meaning. In the older literature in particular , emissivity is also used in the sense of a physical radiation quantity, such as B. the spectral radiance .