The Black Body: The Perfect Absorber That Launched Quantum Physics

A black body is an idealised object that absorbs all the electromagnetic radiation that falls on it, whatever its frequency and whatever the angle at which it arrives. Nothing is reflected and nothing passes through. A black body at room temperature would look perfectly black, because it sends none of the light that hits it back to your eyes.

That definition sounds like a curiosity, but black bodies sit at the crossroads of thermodynamics, electromagnetism, astronomy and quantum mechanics. The radiation a black body gives off when it is warm, called black-body radiation, has a spectrum that depends only on its temperature, not on what it is made of or what shape it is. Explaining that spectrum defeated nineteenth-century physics and led Max Planck, in 1900, to the idea that energy comes in discrete packets, the starting point of quantum theory. Today black-body physics is used to measure the temperatures of stars, calibrate infrared instruments, design solar collectors and test the Big Bang theory.

This article explains what a black body is, the laws that describe its radiation, how scientists approximate one in the laboratory, where nature provides near-perfect examples, and why the concept still matters.

Absorbing, reflecting and transmitting

When radiation strikes any object, three things can happen to it. It can be absorbed, reflected, or transmitted through. Physicists describe these with three fractions, which must add up to one:

• α (alpha), the absorptivity: the fraction absorbed
• ρ (rho), the reflectivity: the fraction reflected
• τ (tau), the transmissivity: the fraction transmitted

So α + ρ + τ = 1. The Wikipedia article on black bodies uses these to define a family of ideal objects:

A white body reflects everything; a transparent body lets everything through; a black body absorbs everything. Real materials are somewhere in between, and their fractions usually vary with wavelength. Snow, for example, is highly reflective in visible light but behaves very differently in the infrared.

Emission: why good absorbers are good emitters

Every object above absolute zero gives off thermal radiation. The question is how much, and at which wavelengths.

In 1859 and 1860 the German physicist Gustav Kirchhoff showed that, for a body in thermal equilibrium, emission and absorption are tightly linked. According to the Wikipedia article on Kirchhoff's law of thermal radiation, the law states that the emissivity of a body equals its absorptivity at each wavelength: ε(λ) = α(λ). A good absorber at a particular wavelength is a good emitter at that wavelength; a poor absorber, such as a shiny reflector, is a poor emitter.

Kirchhoff's deeper insight was that the ratio of a body's emissive power to its absorptivity is the same for all bodies at a given temperature, and equals the emission of a perfect black body. In other words, there exists a single universal function, depending only on wavelength and temperature, that describes the maximum radiation any body can emit. The Kirchhoff's law article notes that this implies no body can emit more thermal radiation than a black body at the same temperature. The black body is therefore both the perfect absorber and the perfect emitter.

The Wikipedia black body article adds two more properties of an ideal black body in equilibrium. It is an ideal emitter: at every frequency, it emits at least as much thermal energy as any other body at the same temperature. And it is a diffuse emitter: its radiation is the same in every direction per unit of area perpendicular to that direction.

Emissivity

Real objects emit only a fraction of what a black body would. That fraction is the emissivity, ε. A black body has ε = 1. A "gray body" is an idealisation with an emissivity below 1 that does not change with frequency. According to the black body article, some authors treat sources with an emissivity above about 0.99 as black bodies for practical purposes.

Kirchhoff found the universal function existed but could not say what it was. Finding its mathematical form took about four decades.

Who first thought of it?

The Wikipedia article traces the idea back to Isaac Newton, who raised it in Query 6 of his 1704 book Opticks. Kirchhoff introduced the term "perfectly black" bodies in 1860. His definition involved absorbing layers of infinitely small thickness. Max Planck later pointed out, in his 1914 book The Theory of Heat Radiation, that such an infinitely thin perfect absorber cannot actually be made and that Kirchhoff's ideal imposes difficult conditions on how radiation scatters inside the material. Modern definitions drop the infinitely-thin requirement and simply describe a black body as a perfect absorber of all incident radiation.

Making a black body: the cavity with a hole

No real material absorbs every wavelength perfectly. But there is an elegant trick that gets extremely close.

Take a hollow box with opaque walls and drill a small hole in it. Any light that enters the hole bounces around inside, and at each bounce some of it is absorbed by the walls. For it to escape, it would have to find its way back out of the tiny hole, which becomes extremely unlikely after many bounces. To an outside observer, the hole absorbs nearly everything that enters it. The hole, not the box, is the black body.

The same hole also emits. If the walls are held at a uniform temperature, the radiation inside comes into equilibrium with them, and what leaks out of the hole is very nearly ideal black-body radiation. This is often called cavity radiation.

The black body article notes the limits of the trick. It works best when the cavity is large compared with the hole. Imperfections appear for wavelengths comparable to or larger than the hole's diameter or the cavity's size. Equilibrium is reached through repeated absorption and emission by the walls or by matter inside; radiation alone does not equilibrate easily, because photons almost never interact with one another except at temperatures in the billions of kelvin.

In 1898 Otto Lummer and Ferdinand Kurlbaum described such a cavity source: a hole in a platinum box whose inside was blackened with iron oxide. A 1901 version used oxides of chromium, nickel and cobalt. Sources like these allowed the precise measurements of black-body spectra that would soon overturn classical physics.

The black-body spectrum

If you measure how much radiation a black body emits at each wavelength, you get a characteristic hump-shaped curve. At low temperature, the hump sits in the infrared and is small. As temperature rises, the whole curve grows dramatically taller and its peak shifts towards shorter wavelengths. This is why a heated poker first glows dull red, then orange, then yellow-white as it gets hotter.

Three laws describe the curve.

Wien's displacement law: where the peak sits

In 1893 Wilhelm Wien showed, using a thermodynamic argument, that the wavelength at which the emission peaks is inversely proportional to temperature. According to the Wikipedia article on Wien's displacement law:

λ_peak = b / T, with b ≈ 2898 micrometre-kelvins (2.897771955 × 10⁻³ m·K)

Some examples from that article:

• The Sun, with a surface temperature of about 5778 kelvin, has its emission peak per nanometre near 500 nanometres, in the green part of the spectrum. (Its light looks white to us because it spans the whole visible range.)
• Human skin, at about 300 kelvin, peaks at roughly 10 micrometres, in the far infrared. This is the range sensed by passive infrared cameras and by the heat-sensing pits of pit vipers.
• An incandescent bulb that is dimmed has a cooler filament, so its peak moves to longer wavelengths and the light looks redder.

A useful subtlety: where the peak falls depends on whether you plot the spectrum per unit wavelength or per unit frequency. The Wien article notes that for the Sun, the peak per unit frequency is about 883 nanometres, in the near infrared. Neither is "wrong"; they answer slightly different questions.

The Stefan–Boltzmann law: how much in total

The total power radiated by a black body rises with the fourth power of its absolute temperature. According to the Wikipedia article on the Stefan–Boltzmann law, John Tyndall measured infrared emission from a heated platinum filament in 1864; Josef Stefan deduced the fourth-power relationship from Tyndall's data in 1877 and published it in 1879; and Ludwig Boltzmann derived it theoretically in 1884. The law is:

M = σT⁴

where M is the power radiated per square metre of surface (the radiant exitance), T is the absolute temperature in kelvin, and σ, the Stefan–Boltzmann constant, is about 5.670374419 × 10⁻⁸ watts per square metre per kelvin to the fourth power. Since the 2019 redefinition of the SI units, σ has an exact value derived from the fixed values of the Planck constant, the Boltzmann constant and the speed of light.

For real surfaces, multiply by the emissivity: M = εσT⁴.

The fourth power is dramatic. Doubling the temperature multiplies the radiated power by 2⁴ = 16. The Wikipedia black body article gives the figure for a 5780-kelvin black body (about the Sun's effective temperature): a radiant exitance of 63.3 megawatts per square metre, with a peak at 501 nanometres. You can check this yourself: 5.67 × 10⁻⁸ × 5780⁴ ≈ 6.33 × 10⁷ W/m².

The law also lets astronomers estimate sizes. Since total output depends on temperature and area, measuring an object's brightness and temperature reveals its area. The black body article notes that astronomers used this reasoning to conclude that X-ray bursts came from neutron stars about 10 kilometres in radius, not from black holes.

Planck's law: the full curve

Wien's law gives the peak, and the Stefan–Boltzmann law gives the area under the curve, but neither gives the shape. That is Planck's law. In wavelength form, as given in the Wikipedia article on Planck's law, the spectral radiance is:

B_λ(λ, T) = (2hc² / λ⁵) × 1 / (e^(hc / λk_BT) − 1)

where h is the Planck constant, c the speed of light, and k_B the Boltzmann constant. There is an equivalent form per unit frequency:

B_ν(ν, T) = (2hν³ / c²) × 1 / (e^(hν / k_BT) − 1)

The Planck article cautions that the two forms are not interchangeable by just substituting λ = c/ν, because one gives energy per unit wavelength and the other per unit frequency, which is also why their peaks fall in different places.

Integrating Planck's law over all wavelengths gives the Stefan–Boltzmann law, and finding where it peaks gives Wien's displacement law. Planck's formula contains both.

The crisis that Planck solved

By the late nineteenth century, physicists had excellent cavity sources and good measurements, but their theories failed.

Classical physics, applying the equipartition theorem (the rule that energy is shared equally among all possible modes of motion), predicted the Rayleigh–Jeans law. At long wavelengths it matched experiment. At short wavelengths it predicted that the emission should keep rising without limit, so that a cavity would contain infinite energy. In 1911 Paul Ehrenfest coined the name "ultraviolet catastrophe" for this absurd result, according to the Planck's law article.

Wien had proposed a different formula, now called the Wien approximation, that worked well at short wavelengths. According to the Planck's law article, Lummer and Pringsheim wrote in 1899 that available data roughly matched such an empirical law. But in late 1900, Heinrich Rubens and Ferdinand Kurlbaum showed that Wien's law failed at long wavelengths. Rubens told Planck, who constructed a formula to fit the data. Planck presented it on 19 October 1900, and Rubens and Kurlbaum reported the same day that it fitted their measurements.

To justify his formula, Planck assumed that the energy of the oscillators in the cavity walls could change only in steps proportional to frequency, with energy E = hν. He regarded this at first as a mathematical device. In 1905 Albert Einstein proposed that light itself really consists of energy quanta, which explained both black-body radiation and the photoelectric effect. Planck's constant h became one of the fundamental constants of nature, and quantum mechanics grew from these beginnings.

Planck's law also contains the two older formulas as limits: at low frequencies it reduces to the Rayleigh–Jeans law, and at high frequencies to the Wien approximation.

Near-perfect black bodies in the laboratory

Practical uses demand surfaces, not just holes in boxes. Near-black materials are used in camouflage, radar-absorbing materials, solar collectors and infrared detectors.

The black body article describes several generations of such materials:

• Carbon black coatings make surfaces nearly black.
• Carbon nanotubes do better. In 2009 a Japanese team led by Mizuno reported, in the Proceedings of the National Academy of Sciences, a material nicknamed "nanoblack" based on vertically aligned single-walled carbon nanotubes that absorbed 98 to 99 percent of light from the ultraviolet to the far infrared.
• Nano-porous materials have reached an average reflectance of 0.045 percent.
• Super-black coatings, Vantablack and flower-like carbon nanostructures absorb 99.9 percent or more. Vantablack, produced in 2014, is among the blackest substances known.

These materials work on the same principle as the cavity: a forest of tiny tubes or pores traps light in countless small cavities until it is absorbed.

Black bodies in nature

Stars

Stars are not perfect black bodies, but their outer layers, the photospheres, emit spectra close enough that astronomers model them as such. A star's effective temperature is the temperature of a black body that would emit the same energy flux per square metre. According to the black body article, the Sun's effective temperature is 5780 kelvin, while its photosphere ranges from about 5000 kelvin at the outer boundary to about 9500 kelvin at the inner boundary with the convection zone, a layer roughly 500 kilometres deep.

Astronomers also use colour indices, which compare a star's brightness through different filters, to estimate temperature. The black body article gives the Sun's B−V colour index as +0.648 ± 0.006.

Planets and Earth's temperature

The Stefan–Boltzmann law gives a quick estimate of a planet's temperature. Balance the sunlight absorbed with the thermal radiation emitted. The Stefan–Boltzmann law article reports that for a body at Earth's distance with no albedo (reflectivity) and no atmosphere, this gives about 279 kelvin. With Earth's albedo of about 0.3, the estimate drops to about 255 kelvin (−18 °C). Earth's actual average surface temperature is about 288 kelvin, and the 33-degree difference is due to the greenhouse effect of its atmosphere. This simple calculation is a standard first step in climate science.

The cosmic microwave background

The most perfect black body ever measured is not an object but the universe itself. The black body article explains that about one second after its formation, the universe was a near-ideal black body at a temperature above 10¹⁰ kelvin. As it expanded and cooled, that radiation stretched to longer wavelengths. Today it fills space as the cosmic microwave background (CMB), with a nearly perfect Planck spectrum at about 2.7 kelvin and variations of only about one part in 100,000. The black body article quotes a source calling it "the most perfect black body ever measured in nature."

According to the Wikipedia article on the cosmic microwave background, Arno Penzias and Robert Wilson made their first clear measurement of the background on 20 May 1964 and were awarded the 1978 Nobel Prize in Physics. NASA's Cosmic Background Explorer (COBE) satellite, launched on 18 November 1989, carried an instrument called FIRAS that measured its spectrum. The CMB article gives its temperature as 2.72548 ± 0.00057 kelvin, and records that the COBE team received the 2006 Nobel Prize in Physics.

Black holes

A black hole might seem the ultimate black body, since nothing escapes it. The black body article notes that it is not quite perfect, because radiation with wavelengths equal to or larger than the black hole's size may not be absorbed. Stephen Hawking's theory predicts that black holes nonetheless emit thermal radiation with a temperature:

T = ħc³ / (8πGk_BM)

where ħ is the reduced Planck constant, G the gravitational constant, and M the black hole's mass. Larger black holes are colder. The black body article notes that these predictions have not been tested by observation.

Radiative cooling

The Stefan–Boltzmann law also tells you how fast hot things lose heat by radiation. A body at constant temperature must absorb or generate as much power as it radiates. If it doesn't, it cools. Combining M = εσT⁴ with the object's heat capacity gives an estimate of its cooling rate. The black body article cautions that this is an approximation that ignores many complications, such as conduction, convection and the radiation the object receives from its surroundings.

A worked example: a 1-square-metre black panel at 300 kelvin (27 °C) radiates σ × 300⁴ ≈ 459 watts. If its surroundings are also at 300 kelvin, it absorbs the same amount back and its net loss is zero. If it faces a clear night sky whose effective temperature is much colder, it loses more than it gains and cools, which is why frost can form on car roofs on clear nights even when the air is above freezing.

Why the black body still matters

• Temperature measurement. Infrared thermometers and thermal cameras convert emitted radiation into temperature using black-body laws and the target's emissivity. Laboratory cavity sources serve as reference standards for calibrating such instruments.
• Lighting. The "colour temperature" quoted for a light source refers to the black body whose colour it matches.
• Astronomy. Black-body fits give star temperatures and sizes.
• Climate. Earth's energy balance begins with the Stefan–Boltzmann law.
• Cosmology. The CMB's black-body spectrum is one of the strongest pieces of evidence for the hot Big Bang.
• Solar energy. Collectors use near-black surfaces to absorb as much sunlight as possible.

Frequently asked questions

Is a black body always black? Only when it is cold. A black body absorbs all incident light, so at room temperature it looks black. When heated, it glows, and at high enough temperatures it can be very bright. The Sun is a reasonable approximation of a black body.

What is the difference between a black body and a gray body? A black body has an emissivity of exactly 1. A gray body has an emissivity below 1 that is the same at all wavelengths.

Why does a hot object change colour as it heats up? Wien's displacement law: the peak of the emission moves to shorter wavelengths as temperature rises, shifting from infrared to red, then to orange, yellow and white.

What was the ultraviolet catastrophe? The prediction of classical physics that a black body should emit unlimited energy at short wavelengths. Paul Ehrenfest named it in 1911. Planck's 1900 law, based on energy quanta, resolved it.

What is the most perfect black body known? The cosmic microwave background, with a nearly perfect Planck spectrum at about 2.725 kelvin.

What is the blackest material? Carbon nanotube-based coatings such as Vantablack and similar super-black materials absorb 99.9 percent or more of visible light.

Summary

A black body absorbs all radiation that falls on it and, by Kirchhoff's law, emits the maximum possible thermal radiation at every wavelength for its temperature. Its spectrum is universal: it depends only on temperature, peaks at a wavelength given by Wien's law, carries total power given by the Stefan–Boltzmann law, and has the exact shape given by Planck's law. Explaining that shape forced Planck, in 1900, to introduce energy quanta. Cavities with small holes, carbon nanotube coatings, stars, planets and the cosmic microwave background all approximate black bodies to varying degrees, making the concept one of the most practically useful idealisations in physics.

Sources and further reading

• Wikipedia, "Black body": https://en.wikipedia.org/wiki/Black_body
• Wikipedia, "Planck's law": https://en.wikipedia.org/wiki/Planck%27s_law
• Wikipedia, "Stefan–Boltzmann law": https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law
• Wikipedia, "Wien's displacement law": https://en.wikipedia.org/wiki/Wien%27s_displacement_law
• Wikipedia, "Kirchhoff's law of thermal radiation": https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation
• Wikipedia, "Cosmic microwave background": https://en.wikipedia.org/wiki/Cosmic_microwave_background
• NASA Science, COBE mission page: https://science.nasa.gov/mission/cobe/