Condensed matter physics
Blackbody radiation and Planck’s law
A blackbody in thermal equilibrium emits a spectrum determined only by temperature. Planck’s law combines electromagnetic mode density with Bose–Einstein photon statistics, resolves the ultraviolet catastrophe, and yields the Stefan–Boltzmann and Wien laws.
An ideal blackbody absorbs all incident radiation; in equilibrium its universal emission spectrum depends on temperature T, not on cavity material. Each electromagnetic mode is a bosonic photon mode with zero chemical potential.
Definition: Planck’s law per frequency
u(ν,T)dν is radiation energy per unit volume in a frequency interval dν. The formula is spectral energy density; the wavelength spectrum has a different form and is not obtained by merely substituting ν=c/λ.
Quantized modes and limiting laws
The mode density in dν is 8πν²/c³; the mean energy per photon mode is hν/[exp(hν/kBT)−1]. Their product gives Planck’s spectrum. At low frequencies it reduces to the classical Rayleigh–Jeans limit; at high frequencies it has Wien’s exponential suppression.
Integrating the spectrum gives energy density proportional to T⁴, related to the Stefan–Boltzmann law for emitted power per area. Wien’s displacement law says the peak wavelength is inversely proportional to T; the peak location depends on which spectral quantity is used.
Example: Worked example
A blackbody has λmax=580 nm. Using b≈2.90×10⁻³ m·K, estimate its temperature.
Solution
T=b/λmax≈(2.90×10⁻³)/(580×10⁻⁹)≈5.0×10³ K.
Quick check
Planck’s spectrum resolves the ultraviolet catastrophe by assigning discrete energies n hν to oscillators exchanging energy with the field. The thermal mean energy of photon modes yields the Bose–Einstein denominator with zero chemical potential; 8πν²/c³ counts the two polarizations and mode density. Integrating the spectrum gives the Stefan–Boltzmann law, while its wavelength peak follows Wien’s displacement law. Converting frequency to wavelength requires the Jacobian |dν/dλ|=c/λ².
An observable consequence is the color shift of a hot object: its spectrum moves toward shorter wavelengths and total power rises rapidly. Real objects are only approximate blackbodies; emissivity depends on wavelength, direction, and surface. In thermal equilibrium, Kirchhoff’s law relates emissivity to absorptivity at the same wavelength and direction. Distinguishing spectral energy density from spectral radiance prevents confusion about geometric factors.
What is the chemical potential of equilibrium blackbody photons?
Which sign is in the denominator of the corresponding distribution?
References
- Max Planck (1914). The Theory of Heat Radiation
- Charles Kittel, Herbert Kroemer (1980). Thermal Physics