Blackbody Radiation

Blackbody radiation is the electromagnetic radiation emitted by an ideal body that absorbs all incident radiation, at every wavelength and from every direction. In thermodynamic equilibrium, its spectrum depends only on its absolute temperature, not on its composition. This universal spectrum provides a reference for thermal emission and radiation measurement. Explaining it was a decisive step in the development of quantum mechanics. (nvlpubs.nist.gov)

The blackbody idealization

An ideal blackbody neither reflects nor transmits incident radiation. “Black” describes its absorption properties, not necessarily its appearance: a sufficiently hot blackbody emits visible light. A practical approximation is a small opening in an opaque cavity maintained at uniform temperature. Radiation entering the opening undergoes repeated interactions with the walls, making escape unlikely; radiation emerging from it closely approximates the equilibrium spectrum. (nvlpubs.nist.gov)

The connection between absorption and emission is expressed by Kirchhoff’s law of thermal radiation. At thermal equilibrium, a material’s spectral directional emissivity equals its absorptivity under the corresponding conditions. Consequently, a perfect absorber is also a perfect thermal emitter. Equilibrium does not mean that emission stops: emission and absorption continue while their energy transfers balance. These principles connect radiation with thermodynamics. (nvlpubs.nist.gov)

Planck’s radiation law

Planck’s law gives the blackbody spectrum. In vacuum, the spectral radiance per unit frequency is

Here $\nu$ is frequency, $h$ is the Planck constant, $c$ is the speed of light, and $k_{\mathrm B}$ is the Boltzmann constant. Temperature $T$ is measured in kelvins. Spectral radiance specifies radiant power per projected area, per solid angle, and per frequency interval; $B_\nu$ has units of $\mathrm{W\,m^{-2}\,sr^{-1}\,Hz^{-1}}$. (nvlpubs.nist.gov)

The equivalent expression per unit vacuum wavelength is

These expressions describe the same radiation using different spectral coordinates. They are related by $B_\lambda=B_\nu|d\nu/d\lambda|$, with $\nu=c/\lambda$. The conversion factor matters: the maximum of $B_\lambda$ does not correspond simply to $c$ divided by the frequency maximizing $B_\nu$. A “peak wavelength” must therefore specify the spectral representation. (nvlpubs.nist.gov)

Total emission and spectral peak

Integrating the spectrum over wavelength and the outward hemisphere gives the Stefan–Boltzmann law:

where $M$ is radiant power emitted per unit surface area and $\sigma\approx5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$. Doubling the absolute temperature increases total emission sixteenfold. For a blackbody sphere of radius $R$, the total emitted power is $4\pi R^2\sigma T^4$. This is emitted power, not net heat loss, which also depends on radiation absorbed from the surroundings. (asd.gsfc.nasa.gov)

Wien’s displacement law locates the maximum of the wavelength-based spectrum:

Hotter bodies thus peak at shorter wavelengths. Calculated examples are approximately $9.7\,\mu\mathrm m$ at $300\,\mathrm K$, in the infrared, and $0.50\,\mu\mathrm m$ at $5800\,\mathrm K$, in the visible range. The peak is not the only emitted wavelength: blackbody radiation forms a broad, continuous spectrum across the electromagnetic spectrum. (nvlpubs.nist.gov)

Classical limits and quantum interpretation

At low frequencies, where $h\nu\ll k_{\mathrm B}T$, Planck’s law approaches the Rayleigh–Jeans law,

Extending this classical expression to arbitrarily high frequencies predicts divergent total emission, the ultraviolet catastrophe. At high frequencies, the correct spectrum instead falls exponentially. The Wien approximation describes this limit by neglecting the minus one in Planck’s denominator. (nvlpubs.nist.gov)

In 1900, Max Planck obtained the successful radiation formula and introduced discrete energy elements proportional to frequency. His oscillator-based derivation associated allowed energies with multiples of $h\nu$, establishing the importance of energy quantization. This should be distinguished from Albert Einstein’s subsequent light-quantum interpretation: Planck’s original work did not itself establish the particle nature of light. (nobelprize.org)

In modern statistical mechanics, the spectrum follows from counting electromagnetic modes and assigning their photons the equilibrium occupations given by Bose–Einstein statistics. The mean thermal energy of a mode is $h\nu/[\exp(h\nu/k_{\mathrm B}T)-1]$; high-frequency modes are sparsely occupied. (arxiv.org)

Real emitters and applications

Real surfaces generally depart from the ideal spectrum. Their emitted spectral radiance is expressed as $\epsilon_\lambda B_\lambda(T)$, with emissivity potentially depending on wavelength, direction, and temperature. A graybody assumes wavelength-independent emissivity. Noncontact temperature measurements must account for these properties; otherwise, a measured radiance temperature need not equal the material’s thermodynamic temperature. Laboratory blackbody sources provide calibration references for such measurements. (nvlpubs.nist.gov)

In astronomy, thermal spectra provide approximate temperature information about stars and other emitters. A particularly close natural example is the cosmic microwave background. The COBE satellite’s FIRAS instrument measured a spectrum closely matching a blackbody at approximately $2.725\,\mathrm K$, supporting its interpretation as radiation left from a hot early universe. (asd.gsfc.nasa.gov)