Blackbody Radiation and the Quantum Hypothesis

A black body is an ideal object that absorbs all radiation falling on it and emits a characteristic continuous spectrum that depends only on its temperature; classical physics could not explain this spectrum (the “ultraviolet catastrophe”), and Max Planck solved the problem in 1900 by proposing that energy is exchanged in discrete quanta, E = hf, where h = 6.626 × 10⁻³⁴ J·s. In brief: the total power radiated per unit area is σT⁴ (Stefan-Boltzmann law), and the wavelength of the peak of the spectrum is λ_max = 2.898 × 10⁻³ m·K ÷ T (Wien’s law). Hotter objects glow brighter and with a shorter peak wavelength. Planck’s quantum hypothesis was the beginning of quantum theory.

A hot piece of metal glows red, then orange, then white as its temperature rises. The shape of the emitted spectrum posed one of the biggest problems of physics at the end of the 19th century. This guide describes black-body radiation, the failure of classical theory, Planck’s solution, compares the classical and quantum predictions in a table, works through numerical examples and corrects common mistakes.

What is a black body?

A black body absorbs all radiation incident on it, at all wavelengths, and is also the best possible emitter at a given temperature. A close approximation is a small hole in a closed cavity with walls at a uniform temperature: radiation entering through the hole is reflected many times and absorbed, and the radiation leaving is in equilibrium with the walls. The spectrum of this radiation depends only on the temperature, not on the material. The Sun, a star, a heated iron bar and the cosmic microwave background are approximately black-body radiators. The cosmic microwave background, at 2.725 K, is the most perfect black-body spectrum known.

What do experiments show?

• Stefan-Boltzmann law: the power emitted per unit area is j = σT⁴, with σ = 5.67 × 10⁻⁸ W/(m²·K⁴). Doubling the absolute temperature increases the emitted power 16 times.
• Wien’s displacement law: the wavelength at which the intensity is highest is λ_max = b ÷ T, with b = 2.898 × 10⁻³ m·K. As the temperature rises, the peak moves to shorter wavelengths: a heated object turns from red to orange, yellow and white, and very hot stars are blue-white.
• The spectrum is a smooth curve that rises to a maximum and then falls at short wavelengths, going to zero for both very short and very long wavelengths.

What was the ultraviolet catastrophe?

Classical physics (the Rayleigh-Jeans law, 1900 to 1905) treated the radiation in the cavity as standing electromagnetic waves, each with an average energy kT (equipartition). The number of modes increases as the wavelength decreases (proportionally to 1/λ⁴ in intensity), so the predicted intensity grows without limit at short wavelengths, and the total emitted energy would be infinite. This was in obvious conflict with experiments, in which the intensity falls to zero at short wavelengths; Paul Ehrenfest called it the “ultraviolet catastrophe”. Wien’s law (1896) fitted the short-wavelength data but not the long-wavelength data. The classical theories were correct at long wavelengths and wrong at short wavelengths.

How did Planck solve it?

In December 1900, Max Planck found a formula that fitted all the data, and gave it a theoretical basis with an assumption he considered a “mathematical trick”: the oscillators in the walls of the cavity can only emit or absorb energy in discrete packets (quanta), E = nhf (n = 1, 2, 3, …), with h a new constant of nature. The average energy of a mode of frequency f is then hf ÷ (e^hf/kT − 1), which tends to kT at low frequencies (agreeing with classical theory), but decreases exponentially at high frequencies, where the energy quantum hf is much larger than kT and such modes are rarely excited. The result is Planck’s law, which agrees with the experimental spectrum at all wavelengths. In 1905, Einstein extended the idea, proposing that light itself consists of quanta (photons) of energy hf, which explained the photoelectric effect.

Classical and quantum predictions compared

Worked example: stars and photon energies

1. The Sun. Surface temperature 5,772 K: λ_max = 2.898 × 10⁻³ ÷ 5,772 = 5.02 × 10⁻⁷ m = 502 nm (green). The power emitted per unit area: σT⁴ = 5.67 × 10⁻⁸ × (5,772)⁴ = 6.29 × 10⁷ W/m².
2. Other stars. A red star at 3,500 K peaks at 828 nm (in the infrared, so it looks red). A blue-white star at 10,000 K peaks at 290 nm (in the ultraviolet, so it looks blue-white).
3. Cosmic background. At 2.725 K: λ_max = 2.898 × 10⁻³ ÷ 2.725 = 1.06 mm, in the microwave region.
4. Photon energy. At the Sun’s peak (500 nm), E = hc ÷ λ = 1,240 eV·nm ÷ 500 nm = 2.48 eV = 3.97 × 10⁻¹⁹ J. At the temperature of the Sun, kT = 8.617 × 10⁻⁵ × 5,772 = 0.50 eV, so a visible photon has about 5 kT.
5. Increase of power. If the temperature of a furnace rises from 1,000 K to 1,500 K, the emitted power per unit area rises by (1,500 ÷ 1,000)⁴ = 5.06 times.
6. Number of quanta. A 100 W lamp emitting 5 percent as visible light at 500 nm gives 5.0 W ÷ 3.97 × 10⁻¹⁹ J = 1.3 × 10¹⁹ photons per second.

How did quantum theory begin?

Gustav Kirchhoff introduced the concept of the black body in 1859 to 1860. Josef Stefan found the T⁴ law in 1879 and Ludwig Boltzmann derived it in 1884; Wilhelm Wien published the displacement law in 1893. Precise measurements by Heinrich Rubens and Ferdinand Kurlbaum in Berlin in 1900 showed deviations from Wien’s formula at long wavelengths. Planck presented his law on 14 December 1900, a date regarded as the birth of quantum theory, and received the Nobel Prize in Physics in 1918. The COBE satellite measured the spectrum of the cosmic microwave background in 1990 and found it to be a perfect black body at 2.725 K, for which John Mather and George Smoot shared the 2006 Nobel Prize in Physics.

Which mistakes are common?

Mistake 1: Thinking that a black body must look black

A black body is defined by perfect absorption, and glows when hot (the Sun is almost a black body).

Mistake 2: Using Celsius temperatures in Wien’s or Stefan’s laws

Use kelvins.

Mistake 3: Believing that Planck’s constant is only important for small objects

h is the same everywhere. Quantum effects are not noticeable for large objects because hf is tiny compared with the macroscopic energies involved.

Mistake 4: Thinking that the peak wavelength is where most of the energy lies

The spectrum is broad. For the Sun, a large part of the energy is in the infrared and ultraviolet as well.