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].

@ __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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 14
jun 2016