Blackbody radiation

Blackbody radiation is the light emitted by an object that absorbs every wavelength falling on it, held at a uniform temperature $T$. Its spectrum depends on temperature alone, which makes it the reference against which thermal sources, infrared detectors and radiation thermometers are calibrated. A body at room temperature (300 K) radiates 459 W/m², almost all of it in the thermal infrared with a peak near 9.7 µm; the Sun, close to a blackbody at 5772 K, peaks near 0.50 µm; a tungsten lamp at 2856 K peaks near 1.01 µm and puts only 10.5 % of its power into the visible band.

Planck's law

The spectral radiance of a blackbody, per unit wavelength, is

$$L_\lambda(T) = \frac{2hc^2}{\lambda^5} \,\frac{1}{e^{hc/\lambda k T} - 1},$$

in W/(m²·sr·m). Planck derived it in 1900 by assuming the oscillators exchanging energy with the field could do so only in units of $h\nu$, the first appearance of the quantum; the photon energy $hc/\lambda$ compared with $kT$ controls the exponential cutoff on the short-wavelength side.

Two results follow by calculus. Wien's displacement law gives the wavelength of peak spectral radiance,

$$\lambda_{\max} = \frac{b}{T}, \quad b = 2897.77\ \mu\mathrm{m\,K},$$

and integrating over all wavelengths and the hemisphere gives the Stefan-Boltzmann law for the total exitance,

$$M = \sigma T^4, \quad \sigma = 5.670 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}.$$

For the Sun at 5772 K, $\lambda_{\max}$ = 2897.77/5772 = 0.502 µm and $M$ = 6.29 × 10⁷ W/m². For human skin at 310 K the peak is at 9.35 µm and $M$ = 524 W/m²; with an emissivity of 0.98 and surroundings at 293 K, the net loss is about 104 W per square metre of skin.

Where the peak sits

The location of the peak depends on the variable used. Plotted per unit frequency, the spectrum peaks at 58.79 GHz per kelvin, 339 THz for the Sun, which corresponds to 0.88 µm rather than 0.50 µm. Counted in photons per unit wavelength, the solar peak moves to 0.64 µm. These are the same spectrum under different changes of variable, so a quoted "peak wavelength" needs its convention stated.

Real surfaces and emissivity

Real materials emit $\varepsilon(\lambda, T)$ times the blackbody value, where the emissivity $\varepsilon$ lies between 0 and 1 and, by Kirchhoff's law, equals the absorptance at the same wavelength and direction. Polished metals have $\varepsilon$ of a few percent in the infrared; skin, water, most paints and oxidized surfaces are above 0.9 in the 8–14 µm band. A laboratory blackbody approximates $\varepsilon$ = 1 with a cavity: a heated enclosure with a small aperture, whose internal walls are rough and dark so that light entering the aperture is absorbed after many reflections. The integrating sphere uses the same geometry with a highly reflective coating.

Measurement and use

Cavity blackbodies with calibrated temperature serve as radiance standards for infrared cameras and radiometers and for thermal detectors such as the thermopile and bolometer. A detector's responsivity or noise-equivalent power is often specified against a 500 K blackbody for this reason. Thermal imagers work in the 8–14 µm window of the infrared bands, which holds 38 % of the emission of a 310 K body and 0.8 % of a 2856 K one. The sensitivity is steep: at 300 K a 1 K temperature change alters total exitance by 1.3 %, and the change at a fixed short wavelength is larger still.

In the visible, a blackbody temperature defines color temperature, and CIE illuminant A is specified as a 2856 K Planckian radiator. Only 10.5 % of its emission falls between 380 and 780 nm, against 46 % for a 5772 K spectrum; the conversion between these radiometric fractions and luminous quantities is covered under radiometry vs photometry.

Pitfalls

Emissivity dominates error budgets in radiation thermometry: an unknown $\varepsilon$ of 0.9 read as 1.0 gives a temperature reading about 8 K low for a 300 K target in a total-radiation measurement, before reflected background is considered. Reflected radiation from warmer surroundings adds to the emitted signal on low-emissivity targets. At room temperature, the instrument's own housing and optics radiate strongly in the same band as the target, which is why infrared detectors sit behind cold shields and why background subtraction or chopping is standard.

Common questions

Is the Sun a blackbody?

Approximately. Its effective temperature of 5772 K is the blackbody temperature that gives the same total exitance, but absorption lines and the temperature structure of the solar atmosphere make the real spectrum depart noticeably from Planck's law, most strongly in the ultraviolet.

Does a hot object emit light at all wavelengths?

In principle yes, but the exponential factor makes short-wavelength emission negligible at modest temperatures. A 310 K body puts about 5 × 10⁻²² of its power into the visible band, so it is invisible without an infrared camera; visible glow becomes noticeable in the dark only above roughly 800 K.

References: M. Planck, The Theory of Heat Radiation (1914; Dover reprint 1991); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics 3rd ed. (Wiley, 2019); R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983); E. Tiesinga et al., CODATA recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys. 93, 025010 (2021).