Blackbody Radiation & Planck's Desperate Act

Everything warmer than absolute zero glows. The atoms in any material jiggle with thermal motion, their charged parts oscillate as they do, and oscillating charges launch the electromagnetic waves of Chapter 2. The result is thermal radiation: light emitted simply because an object is warm. You are glowing right now, at wavelengths near 10 micrometres in the infrared, invisible to your eyes. How much an ordinary object emits, and at which wavelengths, depends on its material and surface finish, which makes a general theory look hopeless. In 1859 Gustav Kirchhoff found the way through: in thermal equilibrium, meaning a single steady temperature with no net exchange of energy, a surface that absorbs a wavelength well must emit it equally well.

So consider the best absorber imaginable. In 1860 Kirchhoff named it a black body, now usually written blackbody: an idealized object that absorbs all electromagnetic radiation falling on it, at every wavelength and angle. In equilibrium it out-glows any other body at the same temperature, and its spectrum depends on that temperature alone; material and shape drop out. A cavity with a tiny hole comes remarkably close, because light entering the hole is absorbed before it can bounce back out; Otto Lummer and Ferdinand Kurlbaum built an electrically heated one in Berlin in 1898. A glowing stove element is a decent blackbody, the Sun's surface a good one, and the cosmic microwave background, the afterglow of the hot early universe detected by Arno Penzias and Robert Wilson in 1964, the best ever measured. Black describes absorption, not appearance: a hot blackbody can be dazzling.

Plot a blackbody's output against wavelength and you get a lopsided hump. The height of the curve is the spectral radiance: the power emitted per square metre of surface, per unit of solid angle (a measure of a cone of directions), per metre of wavelength. Heat the body and two things happen at once: the curve swells at every wavelength, and its peak slides towards shorter wavelengths. The slide is Wien's displacement law, derived by Wilhelm Wien in 1893: λₘₐₓ·T = b ≈ 2.898×10⁻³ m·K, with T the absolute temperature, so peak wavelength and temperature are inversely proportional. Your skin, near 306 K, peaks around 9.5 μm, which is why thermal cameras look between 8 and 14 μm. The Sun, at 5,772 K, peaks near 502 nm, and the cosmic microwave background, at 2.725 K, near 1.06 mm, squarely in the microwave band.

The swelling is the Stefan-Boltzmann law. Josef Stefan extracted it from measurements in 1879, and his former student Ludwig Boltzmann derived it from thermodynamics and Maxwell's theory in 1884: the total power radiated is P = εσA·T⁴. Here A is the surface area, σ = 5.670×10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant, and ε, the emissivity, measures how closely a real surface approaches the ideal, from 0 for a perfect mirror to exactly 1 for a blackbody. The fourth power is ferocious: double the absolute temperature and the power rises sixteenfold. A surface at 798 K, just hot enough to glow visibly, radiates about 23 kilowatts per square metre; each square metre of the Sun's surface radiates about 63 million watts.

Heat an iron poker in a dark room and it stays invisible until about 798 K, or 525 degrees Celsius, the Draper point, measured by John William Draper in 1847. Even then the peak sits near 3.6 μm in the infrared; you see only the faint short-wavelength tail of the curve spilling into the red end of the visible band, which runs from about 380 to 750 nm. Heat it further and the curve adds orange and then yellow, until the visible band is filled fairly evenly and the eye reports white. That settles a stubborn myth: white-hot is hotter than red-hot. Red is merely the first glow of a body barely hot enough to shine; lighting catalogues even invert the physics when they call reddish light warm and bluish light cool. The same curve doomed the incandescent bulb: a tungsten filament near 2,800 K peaks around 1.0 μm, and a blackbody at that temperature puts only about 8 per cent of its output into the visible band.

The colour sequence turns every star into a thermometer light-years away. Betelgeuse, the orange-red shoulder of Orion, has a surface temperature of about 3,600 to 3,800 K, depending on the study, and peaks near 780 nm, at the edge of the infrared. The Sun, at 5,772 K, looks white from space. Sirius, at about 9,800 K, peaks near 295 nm in the ultraviolet and shines blue-white, and Rigel, at about 12,000 K, is bluer still. These are effective temperatures: the temperature of a blackbody of the same size that would radiate the same total power. Astronomers estimate them for millions of stars simply by comparing each star's brightness through blue and yellow filters.

Now for an explanation. Radiation inside a hot cavity bounces between the walls, and, just as on the guitar string of Chapter 2, only waves that fit survive as standing waves. Each allowed standing-wave pattern is called a mode, and the cavity's radiation is the sum of all its modes. Counting them is pure geometry. In three dimensions, the number of distinct patterns with frequencies below f grows like the volume of a sphere whose radius is proportional to f, that is, as f³. The number of modes in a narrow frequency window therefore grows as f². Including light's two independent polarizations, the count is 8πf²/c³ modes per unit volume per unit frequency. Double the frequency and four times as many modes crowd into each window, and there is no highest-frequency mode at all.

The energy per mode comes from classical statistical mechanics, the theory Maxwell and Boltzmann built to explain heat as random motion. Its equipartition theorem states that in thermal equilibrium every independent quadratic energy term, kinetic or potential, holds on average kT/2, where k = 1.381×10⁻²³ joules per kelvin is Boltzmann's constant. An oscillator stores energy both in motion and in stretch, so it averages kT, and every cavity mode is an oscillator. At room temperature kT is about 4.1×10⁻²¹ joules, or 0.026 electron-volts (eV), where one electron-volt is 1.602×10⁻¹⁹ joules. Multiply the mode count by kT and you have the Rayleigh-Jeans law. Lord Rayleigh argued in June 1900 that long-wavelength radiation should grow as T/λ⁴, and in 1905 he and James Jeans completed the classical formula.

Follow the Rayleigh-Jeans law towards short wavelengths and disaster unfolds. The modes keep multiplying and each still holds kT, so the predicted spectrum climbs without limit through the ultraviolet, X-ray and gamma-ray bands instead of falling to zero, and the total energy, summed over infinitely many modes, is infinite. Taken literally, radiation would drain energy from the walls into ever shorter wavelengths forever. In 1911 Paul Ehrenfest named this the ultraviolet catastrophe; Rayleigh, Jeans and Albert Einstein had each recognized the divergence by 1905. Measure how wrong it is for a body at the Sun's temperature: at 10 μm the classical prediction is only 14 per cent too high, at 500 nm it is 29 times too high, and at 100 nm about 2.7 billion times too high.

Notice the shape of the failure. Classical physics is excellent where wavelengths are long and frequencies low, and fails ever more absurdly as frequency rises. This was no slip in arithmetic: Maxwell's electromagnetism, the mode count and Boltzmann's statistics each had a long record of success, although equipartition had already stumbled over the heat capacities of gases. One assumption had to give, and the culprit was hidden inside equipartition: that an oscillator can hold any amount of energy whatsoever, however small.

Max Planck, born in Kiel in 1858 and a professor at the University of Berlin, was a master of thermodynamics and a conservative scientist who had long distrusted Boltzmann's statistical methods. In 1899 he championed Wien's formula. Then measurements in Berlin, first by Otto Lummer and Ernst Pringsheim and then, deeper in the infrared, by Rubens and Kurlbaum, showed it failing at long wavelengths. On Sunday 7 October 1900 Rubens visited Planck and told him that at long wavelengths the radiation grew in proportion to temperature. That evening, by the familiar account, Planck found a formula blending both behaviours. He presented it to the German Physical Society on 19 October, and Rubens soon confirmed that it fitted his measurements at every wavelength.

A formula that fits is not yet physics. To derive it, Planck took what he later called an act of desperation: he adopted Boltzmann's method, in which entropy counts the number of ways energy can be shared among many oscillators. Counting requires countable pieces, so Planck divided the energy of the hypothetical charged oscillators he pictured in the cavity walls, which he called resonators, into finite energy elements. To agree with his formula and with Wien's displacement law, each element had to be proportional to frequency, equal to hf. Read as a physical rule, which is how his successors came to understand it, this means an oscillator of frequency f can hold only 0, hf, 2hf, 3hf and so on: E = n·hf, with n a whole number. He presented the derivation to the Society on 14 December 1900, often called the birthday of quantum theory.

Planck immediately put a number on his new constant. From the Stefan-Boltzmann and Wien constants as measured at the time, he obtained h = 6.55×10⁻³⁴ J·s, about 1 per cent below today's exact value of 6.62607015×10⁻³⁴ J·s, and k = 1.346×10⁻²³ joules per kelvin, within 3 per cent of the modern figure. He was the first to evaluate this constant, which now carries Boltzmann's name although Boltzmann never introduced it, and from it he calculated Avogadro's number and the electron's charge. The worksheet's mad scientist challenge lets you repeat his calculation.

The key is a rule from Boltzmann's statistics. In thermal equilibrium at temperature T, a state with energy E occurs with relative probability e^(−E/kT), the Boltzmann factor: states costing a few kT are common, and states costing a hundred kT absurdly rare. Classically a mode can hold any energy, however small, so every mode averages kT. With Planck's rule the smallest energy above zero is one packet, hf, and everything hinges on comparing hf with kT. When hf is much smaller than kT, the steps are so fine that the ladder behaves like a ramp, the mode averages almost exactly kT, and Planck's law reproduces Rayleigh-Jeans and the infrared data. When hf is much larger than kT, even one packet is a huge thermal expense, the chance of holding one is about e^(−hf/kT), and the mode almost always sits empty.

Averaging over the ladder gives the mean energy per mode, ⟨E⟩ = hf/(e^(hf/kT) − 1), which slides smoothly from kT down to essentially nothing. There are still infinitely many modes, but the high-frequency ones are frozen out exponentially faster than their number grows, so the total is finite and the curve turns over. At room temperature a green-light packet costs about 96 times kT, and its Boltzmann factor, about 2×10⁻⁴², is why your coffee mug does not glow green; at the Sun's surface the ratio is about 5, and visible modes are well stocked. Planck's law also contains all four earlier results: at long wavelengths it becomes the Rayleigh-Jeans law, at short ones Wien's 1896 formula; its peak gives λₘₐₓ·T = hc/(4.965·k) = 2.898×10⁻³ m·K, and its total gives σT⁴ with σ = 2π⁵k⁴/(15·c²·h³).

It is often said that Planck discovered that light is made of particles. He did not. He quantized the energy of his wall oscillators as a counting device and treated the radiation as Maxwell's continuous waves. Writing to the American physicist Robert Williams Wood in 1931, he called the step a purely formal assumption, made because he had to obtain a positive result whatever the cost. Historians still debate how literally he meant it; Thomas Kuhn argued in 1978 that Planck did not regard the discontinuity as physical until years later. Others took the packets more seriously than their inventor. In 1905 Einstein proposed that light itself consists of quanta of energy hf, the subject of the next chapter, and Planck resisted: as late as 1911 and 1912 he devised a theory in which oscillators absorbed energy continuously and emitted it only in quanta.

In 1913, recommending Einstein for the Prussian Academy of Sciences, Planck and three colleagues wrote that Einstein “may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta.” Planck nevertheless received the 1918 Nobel Prize in Physics, awarded in 1919, for his discovery of energy quanta, and since 20 May 2019 his constant has defined the kilogram: the SI fixes h at exactly 6.62607015×10⁻³⁴ J·s. Planck wanted to preserve classical physics, not overthrow it, but he followed the data and the mathematics where they led and published what he found. A revolution does not require an eager revolutionary, only an honest one.