Physics · Atomic and quantum physics · Grade 11

Thermal radiation and Planck

Why does heated iron glow first red, then yellow, then white?

The situation

A knife-maker from Chust puts a piece of steel into the hearth and works the bellows. The steel first glows dark red, barely visible in the dark, then cherry, then orange, then yellow-white — the master knows the temperature by its colour, without a thermometer. Beside the hearth, the hand is warmed by rays the eye cannot see. Why does the colour of a hot body depend on its temperature, and how did this question change physics at the start of the 20th century?

Topic

Thermal radiation and Planck's hypothesis. Blacksmith's workshop: heated iron from red to white

Key ideas

Planck's formula
Wien's law
Stefan–Boltzmann law

Formulas

B_{λ} (T) = \frac{2 h c^{2}}{λ^{5}} \frac{1}{e^{h c / λ k T} - 1}

Planck's formula: the spectral radiance of an absolute black body.

λ_{m a x} T = b = 2.898 \cdot 10^{- 3} m \cdot K

Wien's displacement law: for steel at 1000 °C, $λ_{m a x} \approx 2.28$ µm — the infrared region.

M = ε σ T^{4}, σ = 5.670 \cdot 10^{- 8} V t / (m^{2} \cdot K^{4})

The Stefan–Boltzmann law ( $ε = 1$ — an absolute black body).

B_{λ}^{R J} = \frac{2 c k T}{λ^{4}}

The Rayleigh–Jeans formula: it follows from Planck's formula when $λ k T ≫ h c$ ; at short wavelengths — the “ultraviolet catastrophe”.

ε (λ, T) = α (λ, T), u = \frac{4 σ T^{4}}{c}, p = \frac{u}{3}

Kirchhoff's law; energy density and pressure of a photon gas.

Theory

Thermal radiation and the absolute black body

Every body above absolute zero has charges oscillating in thermal motion, and they emit electromagnetic waves — thermal radiation; its spectrum is continuous. The absorptivity $α$ is the fraction of the incident radiation that is absorbed, and the emissivity $ε$ is the ratio of a body's radiation to that of an ideal radiator at the same temperature. An absolute black body absorbs all waves completely ( $α = 1$ ). Its best model is a small hole in the wall of a cavity: light that enters is reflected many times and is almost completely absorbed. The radiation in an equilibrium cavity depends only on the temperature, not on the wall material — which is why black-body radiation is universal. The furnace in “Laboratory” mode is exactly such a black-body model.

Kirchhoff's law and pyrometers

Kirchhoff's law: at each wavelength $ε (λ, T) = α (λ, T)$ — a good absorber is a good emitter. Soot and the hole of a cavity, which look blackest at room temperature, glow brightest when heated, while polished metal reflects light and radiates little. White glaze reflects visible light (looks white, glows weakly) but is almost black in the infrared ( $ε \approx 0.9$ ) — a thermocouple registers it close to soot. In 1804 John Leslie did such an experiment with a cube of hot water with different coatings. An optical pyrometer measures the brightness of a glowing body at 650 nm; if the correct $ε$ is not entered, it shows a “brightness temperature” lower than the true temperature.

Wien's and Stefan–Boltzmann laws

As the temperature rises, the spectral maximum shifts towards short wavelengths: $λ_{m a x} = b / T$ . For steel at 1000 °C, $λ_{m a x} \approx 2.28$ µm — that is infrared! Visible light is only the short-wavelength “tail” of the spectrum, so red appears first; if heating continues, the “tail” widens towards yellow and blue, and the colour becomes orange, yellow, white. The Sun ( $T \approx 5 772$ K) has its maximum at ≈ 502 nm. The total emission is $M = σ T^{4}$ : if the temperature doubles, the power rises 16 times. Only ≈ 10 % of the radiation of an incandescent lamp (≈ 2 800 K) falls in the visible region — the rest is infrared; beside the hearth it is exactly these rays that warm the hand.

The ultraviolet catastrophe and Planck's hypothesis

According to classical physics every standing wave (mode) in the cavity has an average energy $k T$ , and the number of modes grows as $λ^{- 4}$ at short wavelengths. As a result the Rayleigh–Jeans formula agrees with experiment at long wavelengths, but at short wavelengths it tends to infinity — the total energy comes out infinite (the ultraviolet catastrophe). Switch on the Rayleigh–Jeans curve on the graph. In 1900 Max Planck assumed that an oscillator can take up energy only in portions $h ν$ ( $E = n h ν$ ). Then high-frequency modes “freeze out”: the average energy is $h ν / (e^{h ν / k T} - 1)$ , and Planck's formula agrees with experiment over the whole spectrum; $h = 6.626 \cdot 10^{- 34}$ J·s. In 1905 Einstein concluded that light itself consists of photons (the photoelectric effect, photon momentum and the Compton effect are in other experiments).

Photon gas and fluctuations

Equilibrium radiation can be treated as a photon gas: photons obey Bose–Einstein statistics, their chemical potential is zero, and the mode occupation is $\overline{n} = 1 / (e^{h ν / k T} - 1)$ . From this follow the energy density $u = 4 σ T^{4} / c$ , the pressure $p = u / 3$ , the entropy density $s = 4 u / (3 T)$ and the photon concentration $n \approx 2.03 \cdot 10^{7} T^{3}$ m⁻³. The number of photons fluctuates: $⟨ Δ N^{2} ⟩ = \overline{N} (1 + \overline{n})$ ; in visible light $\overline{n} ≪ 1$ , so the signal-to-noise ratio is $\approx \sqrt{N}$ . For this reason every radiation meter has a sensitivity limit: to measure a weak signal, longer collection (averaging) is needed — try this in laboratory mode.

Stars and the limits of the model

The colour of a star gives its surface temperature (Wien's law), and from its luminosity and temperature its radius is found: $L = 4 π R^{2} σ T^{4}$ . Red Betelgeuse is cool (≈ 3 600 K) but, being gigantic, is about 100 thousand times brighter than the Sun; the white dwarf Sirius B is hot (≈ 25 000 K) but is about the size of the Earth. Interstellar dust absorbs and scatters blue light more, so distant stars look reddened — this must be taken into account when determining temperature. Model: stars are taken as black bodies; the emissivity of surfaces is a two-band approximate model (waves shorter / longer than 2.5 µm); the temperature distribution along the iron and the glow colour on screen are approximate.

Did you know?

Max Planck announced the quantum hypothesis on 14 December 1900 to the German Physical Society — this day is often called the birthday of quantum physics. Planck received the Nobel Prize in Physics for 1918.
The cosmic background (relic) radiation is almost ideal black-body radiation with a temperature of ≈ 2.725 K; the COBE satellite showed in 1990 that it matches the Planck spectrum extremely precisely. Every cubic centimetre of space contains about 410 relic photons.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Radioactivity, half-life — Grade 11
Nuclear reactions — Grade 11
Orbitals of the hydrogen atom — Grade 11
Photon momentum and the Compton effect — Grade 11
De Broglie waves — Grade 11
Rutherford's experiment — Grade 11
The atomic nucleus and binding — Grade 11
Thermonuclear fusion — Grade 11