Radiant energy density

In radiometry, radiant energy density is the radiant energy per unit volume.<ref name=GoldBook>IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Radiant energy density". doi:10.1351/goldbook.goldbook.R05040</ref> The SI unit of radiant energy density is the joule per cubic metre (J/m³).

Mathematical definition

Radiant energy density, denoted w_e ("e" for "energetic", to avoid confusion with photometric quantities), is defined as<ref name="Rusňák">Karel Rusňák. Přenos energie elektromagnetickým vlněním. Department of Physics, Faculty of Applied Sciences, University of West Bohemia. 2005-11. Visited 2013-10-06</ref>

<math>w_\mathrm{e} = \frac{\partial Q_\mathrm{e}}{\partial V},</math>

where

• ∂ is the partial derivative symbol;
• Q_e is the radiant energy;
• V is the volume.

Relation to other radiometric quantities

Because radiation always transmits the energy,<ref name="Rusňák" /> it is useful to wonder what the speed of the transmission is. If all the radiation at given location propagates in the same direction, then the radiant flux through a unit area perpendicular to the propagation direction is given by the irradiance:<ref name="Rusňák" />

<math>E_\mathrm{e} = c w_\mathrm{e},</math>

where c is the radiation propagation speed.

Contrarily if the radiation intensity is equal in all directions, like in a cavity in a thermodynamic equilibrium, then the energy transmission is best described by radiance:<ref>Max Planck. The Theory of Heat Radiation. Equation 21. 1914.</ref>

<math>L_\mathrm{e} = \frac{c}{4\pi} w_\mathrm{e}.</math>

Radiant exitance through a small opening from such a cavity is:<ref>Max Planck. The Theory of Heat Radiation. Equation 7. 1914.</ref>

<math>M_\mathrm{e} = \pi L_\mathrm{e} = \frac{c}{4} w_\mathrm{e}.</math>

These relations can be used for example in the black-body radiation equation's derivation.

SI radiometry units

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol<ref group="nb" name="note-suffix-e">Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.</ref>	Name	Symbol
Radiant energy	Qe<ref group="nb" name="note-alternative-symbol-radiometric">Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.</ref>	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Φe<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	watt	W = J/s	M⋅L2⋅T−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in Astronomy.
Spectral flux	Φe,ν<ref group="nb" name="note-suffix-nu">Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)</ref>	watt per hertz	W/Hz	M⋅L2⋅T −2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
	Φe,λ<ref group="nb" name="note-suffix-lambda">Spectral quantities given per unit wavelength are denoted with suffix "λ".</ref>	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Ie,Ω<ref group="nb" name="note-suffix-omega">Directional quantities are denoted with suffix "Ω".</ref>	watt per steradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν<ref group="nb" name="note-suffix-nu"></ref>	watt per steradian per hertz	W⋅sr−1⋅Hz−1	M⋅L2⋅T−2	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
	Ie,Ω,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Le,Ω<ref group="nb" name="note-suffix-omega"></ref>	watt per steradian per square metre	W⋅sr−1⋅m−2	M⋅T−3	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance Specific intensity	Le,Ω,ν<ref group="nb" name="note-suffix-nu"></ref>	watt per steradian per square metre per hertz	W⋅sr−1⋅m−2⋅Hz−1	M⋅T−2	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
	Le,Ω,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Irradiance Flux density	Ee<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	watt per square metre	W/m2	M⋅T−3	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance Spectral flux density	Ee,ν<ref group="nb" name="note-suffix-nu"></ref>	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
	Ee,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Je<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	watt per square metre	W/m2	M⋅T−3	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity	Je,ν<ref group="nb" name="note-suffix-nu"></ref>	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
	Je,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	watt per square metre	W/m2	M⋅T−3	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance	Me,ν<ref group="nb" name="note-suffix-nu"></ref>	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
	Me,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	He	joule per square metre	J/m2	M⋅T−2	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure	He,ν<ref group="nb" name="note-suffix-nu"></ref>	joule per square metre per hertz	J⋅m−2⋅Hz−1	M⋅T−1	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
	He,λ<ref group="nb" name="note-suffix-lambda"></ref>	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: SI Radiometry Photometry