Thermal or Black-Body Radiation |

**In General** > s.a. radiation [including pressure];
physical constants [Stefan-Boltzmann constant].

* __Idea__: Radiation emitted
by a perfectly absorbing and emitting body at a fixed temperature *T*;
It describes (maximum) heat transfer from an object in the far field; The calculation
of its spectrum was "a loose thread that when tugged (by Planck) would eventually
unravel the entire fabric of what had passed for reality."

* __Spectrum__: It is given by the Planck distribution

*E*(*f*, *T*) d*f* = (2π*h*/*c*^{2})
(*f*^{ 3} / e^{hf/kT }−1)
d*f* .

* __Stefan-Boltzmann law__: For a
macroscopic black body at temperature *T*, the total energy emitted per unit
surface and unit time is *E* = *σT*^{ 4},
with *σ* the Stefan-Boltzmann constant.

* __Wien's law__: For a macroscopic
black body at temperature *T*, *λ*_{max}
= (0.0029 m/K) *T*^{ −1}.

* __Deviations from the general relations__:
When the size of the radiating body approaches the typical wavelength of the emitted radiation,
the spectrum and radiation rate need to be calculated using a more general and detailed theory
rather than following Planck's law, based on fluctuational electrodynamics and the
fluctuation-dissipation theorem, and using the objects shape and absorption characteristics.

> __Online resources__:
see Wikipedia's Planck law page.

**Phenomenology and Related Effects** > s.a. temperature [and Lorentz transformations].

* __Energy-level shifts and induced forces__:
Blackbody radiation around hot objects induces ac Stark shifts of the energy levels of nearby
atoms and molecules; These shifts are roughly proportional to the fourth power of the temperature
and induce a force decaying with the third power of the distance from the object.

@ __General references__: Silva e Costa PLA(04) [as seen by moving observer];
Balasanyan & Mkrtchian a0907 [drag on a relativistic moving mirror];
Sonnleitner et al PRL(13)
+ news pw(13)jul [net attractive force between tiny objects].

@ __Radiative heat transfer__:
Francoeur & Mengüç JQSRT(08) [and fluctuational electrodynamics];
Golyk et al EPL(13)-a1210 [between curved objects, small-distance expansion];
> s.a. Heat Flow.

**References** > s.a. light
and thermodynamic systems [thermal light].

@ __General__: Planck 06; Agnese et al NCB(99) [simple];
Chang & Guan qp/04 [in a compact space];
Boyer FP(07) [and relativity and discrete charge];
Kramm & Herbert a0801 [using dimensional analysis];
Kramm & Mölders a0901;
Smerlak EJP(11) [more consistent derivation];
Marr & Wilkin AJP(12)may [Planck's law in introductory physics];
Ford & O'Connell PRE(13)-a1310 [Lorentz transformation of blackbody radiation].

@ __Quantum aspects__: Boya et al qp/00 [multiphoton states];
Greffet et al Nat(02)mar,
Greffet & Henkel CP(07) [coherent emission];
Johansen JOB(04)qp.

@ __History__: Boya RACZ(03)phy/04 [Planck 1900];
Varró FNL(06)qp [Einstein's fluctuation formula];
Carvalho Martins a1308-ln [and quantum theory];
Persson AJP(18)dec
[presentation of the history of Planck's blackbody radiation equation];
Boyer AJP(18)jul [in classical electrodynamics].

@ __Coupling to atoms__: Fröhlich & Merkli CMP(04)mp [return to equilibrium].

@ __Classical derivation__: Marshall NC(65);
Boyer PRD(84) [from equivalence principle and zero-point radiation].

@ __Relativistic__: Lee & Cleaver a1507 [inertial and non-inertial reference frames].

@ __Quantum-gravity motivated corrections__:
Kim et al PRD(07)-a0705 [in *κ*-Minkowski spacetime];
Mania & Maziashvili PLB(11)-a0911 [minimum-length corrections];
Husain et al PRD(13)-a1305 [and evidence for dimensional reduction];
Ramos & Boschi PhyA(14)-a1404 [with compact extra dimensions].

@ __Nanoscale objects__: Wuttke & Rauschenbeutel PRL(13)-a1209
+ news pw(12)sep [and fluctuational electrodynamics];
> s.a. QED phenomenology.

@ __Other variations__: García-García PRA(08)-a0709 [finite-size corrections];
Moreira & Ribeiro a1512
[massive-photon generalization of the Stefan-Boltzmann law];
> s.a. GUP phenomenology.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 12 mar 2019