The curve that would not come down

Heat a body and it glows. Heat it more and the glow shifts from red toward white, and the total power leaving it rises steeply. Both facts were measured to high precision in the 1890s, and neither had an explanation.

The shape of that curve is the subject. It is not a complicated shape and it has no free parameters: fix the temperature and the whole curve is determined, with no reference to what the body is made of. That last fact is the strange one, and it is what makes the problem worth a name. A lump of soot and a hollow ceramic cavity at the same temperature emit identically. The material has dropped out.

Why the material drops out #

The reason is a thought experiment about equilibrium rather than a fact about surfaces.

Consider a sealed cavity whose walls are held at one temperature. Radiation bounces around inside it, absorbed and re-emitted by the walls, until as much of every wavelength is being created as destroyed. The radiation inside has then reached thermal equilibrium with the walls, and its spectrum cannot depend on what the walls are made of — because if two cavities of different materials held different spectra at the same temperature, joining them through a filter that passed one wavelength would let energy flow from one to the other with no temperature difference to drive it, and that is the one thing the second law forbids.

So the spectrum inside a cavity is a universal function of temperature and wavelength. A small hole in such a cavity samples it, and that hole is what “a black body” means in practice. The measurements the theorists were trying to explain were made on furnaces with holes in them, not on lumps of coal.

What classical physics actually predicted #

The classical calculation has two halves and both of them are right.

The first half is counting. The radiation in the cavity is a set of standing waves, exactly as a string fixed at both ends carries only the modes that fit, and in three dimensions the counting is straightforward: the number of modes with wavelength near λ, per unit volume, goes as 1/λ⁴. Short wavelengths are more numerous because there are more ways to fit a small wave into a box than a large one, and the count grows without limit as the wavelength shrinks.

The second half is sharing. Equipartition says that in thermal equilibrium every degree of freedom carries the same average energy, kT/2 per quadratic term, so kT per mode of an oscillator. It is one of the most successful results in nineteenth-century physics: it predicts the specific heats of gases, the mean speed of a molecule, the equilibrium of a Brownian particle. There was no reason to doubt it.

Multiply the two and the Rayleigh–Jeans law comes out: the energy per unit wavelength goes as kT/λ⁴. And that expression diverges.

The consequence is not a small discrepancy. It is that any object at any temperature above absolute zero should radiate infinite power, instantaneously, at the short-wavelength end. Opening an oven door should sterilise the room with gamma rays. The name that stuck — the ultraviolet catastrophe — was coined by Ehrenfest in 1911, more than a decade after the problem was solved, and it slightly overstates how alarmed anyone was at the time. What the theorists knew was that the formula worked at long wavelengths and failed at short ones, which felt like a missing correction rather than a crisis.

Where the two curves part company #

The place where they separate is the informative part, and it is set by one dimensionless number: hc/λkT, the ratio of the energy a single quantum of that wavelength would carry to the thermal energy available.

Where that ratio is small — long wavelengths, high temperatures — the classical answer is right, and not approximately right but right to whatever precision is measured. Where the ratio is large, the classical answer is wrong by a factor that grows exponentially. The crossover is not a sharp edge; it is where the ratio passes about one, which at room temperature is a wavelength of around 50 micrometres, and at the surface of the Sun about 500 nanometres.

That is worth stating plainly, because it is the site’s standing obligation and this is the essay where it points backwards. Equipartition is not wrong. It has a domain, and every classical application of it lives inside that domain. A nitrogen molecule’s translational degrees of freedom are spaced so finely that hc/λkT is astronomically small; the classical treatment of a gas at room temperature is not an approximation that happens to be good, it is the correct limit of the quantum answer. What broke was applying the same result to modes whose quanta are large compared with kT, and nothing in classical physics contained a warning that such modes exist.

Planck’s move, and what it cost #

In October 1900 Planck produced a formula by interpolation — an expression that reduced to Wien’s empirical short-wavelength law at one end and to Rayleigh’s at the other. It fitted the data immediately and he had no derivation for it. He spent the next two months finding one, and the finding is the beginning of quantum physics.

The derivation requires abandoning the second half of the classical argument. Instead of letting a mode of frequency f take any energy, Planck required its energy to be a whole multiple of hf, where h is a constant to be fitted. Given that, the average energy of a mode in thermal equilibrium is no longer kT. It is

$$\langle E\rangle = \frac{hf}{e^{hf/kT}-1},$$

which equals kT when hf ≪ kT — recovering equipartition exactly — and collapses toward zero when hf ≫ kT.

That collapse is the whole solution. A mode whose smallest possible excitation costs more than the thermal energy typically available is not excited at all. It is not that such modes carry a little energy; they are, overwhelmingly, empty. The infinite reservoir of short-wavelength modes is still there and still counted; it simply cannot be filled.

Planck’s assumption imposes a ladder on each mode of the cavity, evenly spaced by $hf$. An oscillator can then absorb that much or nothing, and never a third of it — which is the whole of the move. Nothing about light was assumed; the quantisation was applied to the oscillators in the walls, and Planck regarded it as a mathematical device for six years before Einstein took it literally.

What it cost is worth being honest about. Planck did not believe he had discovered that energy is quantised; he described the step as “an act of desperation” and spent a decade trying to derive the same formula without it. The quantisation in his 1900 paper applies to the oscillators in the walls, not to the radiation, and the modern reading — that the electromagnetic field itself comes in quanta — is Einstein’s, five years later and about a different experiment.

The average that does the work #

The expression for ⟨E⟩ above is not an extra assumption. It is what the Boltzmann factor gives once the ladder is imposed, and the derivation is three lines of a geometric series.

A mode in contact with a reservoir at temperature T occupies its nth rung with probability proportional to e^(−nhf/kT) — the same exponential weighting that gives a gas its distribution of speeds and the atmosphere its scale height. Summing nhf against those weights and dividing by their total gives hf/(e^(hf/kT) − 1) exactly. No approximation is made anywhere in it.

The exponential weighting was not new, and that is worth saying because it is what made the move seem modest. The same factor governs the distribution of molecular speeds, where raising the temperature does not lift every speed equally but reweights a fixed set of possibilities. Planck applied a familiar weighting to an unfamiliar set of possibilities, and the unfamiliar part was that the set was discrete.

The important consequence is the ratio between two neighbouring rungs, e^(−hf/kT). For a mode in the far infrared at room temperature that ratio is 0.99, the rungs are so close together that the ladder is indistinguishable from a ramp, and the classical average returns. For a mode in the ultraviolet the ratio is about 10⁻²⁵: the chance of finding that mode on even its first rung is negligible, so its average energy is not kT but effectively nothing.

So the resolution is statistical rather than mechanical. Nothing prevents a short-wavelength mode from being excited; it is simply that at thermal energies, almost none of them ever are. The catastrophe was a prediction about an average, and it was wrong because the average was taken over the wrong set.

Reading a constant of nature off a curve #

The constant h is not an adjustable shape parameter. Fixing it once fixes every blackbody curve at every temperature, and the value that fits the furnace measurements of 1900 is within half a per cent of the modern one.

That is the strongest possible form of evidence, and it becomes stronger when the same constant turns up somewhere unrelated.

The constant can be read off a completely different experiment. The energy of electrons knocked out of a metal by light, plotted against the light’s frequency, gives straight lines whose slope is $h$ — measured with no cavity, no thermal equilibrium and no fitting. That two such different measurements give the same number is what turned a device for fitting a curve into a constant of nature.

Two laws that fall out on the way #

Both of the empirical regularities that had been measured before the theory arrived are consequences of the formula, and both are visible on the drawing.

Wien’s displacement law. The peak of the curve sits at a wavelength inversely proportional to temperature: λ_max T = 2.898 × 10⁻³ m K. The figure marks each peak and prints its wavelength, and the products come out at that constant to four figures. This is the fact that makes a colour into a thermometer, and it is why a kiln’s temperature can be judged by eye.

Stefan’s law. The total area under the curve goes as T⁴. Less obviously, the height of the peak goes as T⁵ — the curve gets taller by five powers and narrower by one, and the product is the fourth power.

The fourth power has a consequence that is easy to underestimate. Doubling the absolute temperature of a filament multiplies its radiated power by sixteen, which is why a lamp’s efficiency changes so violently with the voltage across it, and why a star twice as hot as the Sun and the same size is sixteen times as luminous.

The evening the formula was written #

Planck’s interpolation was produced in one evening, and it was produced because an experimentalist came to tea.

The measurements everybody was trying to fit came from the Physikalisch-Technische Reichsanstalt in Berlin — a national laboratory founded in 1887 partly at the urging of the electrical industry, which needed standards for lamps and for the efficiency of light sources. Its cavity-radiation work through the 1890s was the best in the world, and it was done for the most practical of reasons: to know how much of a filament’s output was light and how much was heat.

The hard part was the long wavelengths, where Wien’s empirical formula was suspected of failing. Getting to fifty micrometres in 1900 required a technique with no filters and no gratings: bounce the beam repeatedly off crystals of rock salt or potassium bromide, each of which reflects strongly only in a narrow band near its own lattice vibration and passes almost everything else. After five or six reflections, what is left is a narrow band tens of micrometres out. The residual rays, Reststrahlen, were an instrument built out of a material property, and they opened the region where the theory was in doubt.

Rubens and Kurlbaum measured there and found that the energy at long wavelengths was proportional to the temperature — not to Wien’s exponential, which falls far too fast.

On 7 October 1900 Rubens and his wife visited the Plancks and Rubens mentioned the result in conversation. Planck already had a thermodynamic argument for the short-wavelength end; being told that the long end went as $T$ gave him the other limit, and the two together determined an interpolation between them almost uniquely. He worked it out that evening and sent Rubens a postcard with the formula on it. He presented it to the Physical Society on 19 October, twelve days later, with no derivation at all.

The derivation took the following two months, and it is the one that required the quanta.

Two things about that sequence are worth keeping. The formula came from data rather than from theory, and the data came from a laboratory set up to serve the lighting industry — so the founding equation of quantum physics was written to fit measurements made because somebody wanted better lamps. And the crucial experimental step was an instrumental trick with rock salt, which is the kind of thing that never appears in an account of the theory and without which there would have been nothing to interpolate between.

The peak that is in three places #

The essay’s warning that the axis is a choice has a well-known casualty, and it is worth spending a section on because the casualty is repeated constantly.

The claim is that the eye is tuned to the Sun: solar radiation peaks at about 500 nanometres, human sensitivity peaks at 555, and the closeness is offered as evidence of adaptation.

Every number in that argument is shaky. The Sun’s spectrum peaks at 502 nanometres if it is plotted per unit wavelength, and at 883 nanometres if it is plotted per unit frequency — the same physical radiation, the same temperature, a factor of 1.76 between the two answers. Neither is more correct; they are answers to different questions, and the difference is entirely the Jacobian of the change of variable.

There is a third peak, and it is arguably the relevant one. An eye does not respond to energy; it responds to photons, one absorption at a time. Counting photons per unit wavelength rather than joules moves the maximum to about 635 nanometres, well to the red of both the sensitivity peak and the energy peak.

So the Sun’s output peaks at 502, 635 or 883 nanometres according to what is being counted and against what it is plotted, and the eye’s 555 lies between them without matching any. The coincidence being appealed to is a choice of axis.

There are better reasons why vision occupies the band it does, and they have little to do with the peak. The atmosphere is transparent from about 300 to 700 nanometres and much less so on either side, so the band is set by what arrives rather than by what leaves. Water is transparent over a narrower window in the blue-green, which matters if vision began in the sea. And the strongest constraint is chemical: driving a photochemical change in a molecule requires a quantum above roughly 1.7 electronvolts, and quanta above about 3.3 begin to break ordinary covalent bonds rather than isomerise them. Those two energies bracket almost exactly the band an eye uses.

The band is therefore set by a window in the air, a window in water and a window between two chemical thresholds — three constraints that happen to overlap — rather than by the position of a maximum that depends on how the graph was drawn.

What the picture cannot show #

Three things.

The first is that the axis is a choice. The curve plotted against wavelength and the curve plotted against frequency are not the same curve rescaled — they peak at different places, because the conversion carries a factor of c/λ² that reshapes the distribution. Wien’s law in frequency gives a peak at 2.82 kT/h, and converting that to a wavelength does not give the peak of the wavelength curve. Both are correct answers to different questions, and quoting “the peak of the blackbody spectrum” without saying which axis is a common way to be wrong by 76 per cent.

The second is the equilibrium. Every curve here describes radiation that has come into balance with a wall at one temperature. A flame, a fluorescent tube, an aurora and a laser are all sources of light and none of them is anywhere near this condition; their spectra are not blackbody curves and are not even approximately so. Assigning a temperature to such a source describes the blackbody it most resembles, and can be misleading about everything else it does.

The third is that the figure shows a smooth continuum, and the physics underneath it is not smooth. The curve is an average over an enormous number of discrete events, each of which delivers exactly one quantum. Nothing about the drawing hints at that, and the hint would have been the whole story.

The five years the idea was ignored #

The gap between Planck’s paper and the acceptance of quantisation is the part of the history worth keeping, because it shows what the argument was actually short of.

Planck’s formula was accepted immediately — it fitted. His derivation was not, and reasonably so: it introduced a discrete energy scale into a subject with no other evidence for one, in order to fix a single anomaly, and the man who introduced it said openly that he hoped to remove it again. A hypothesis invented to fix exactly one problem and doing no other work is a weak hypothesis, whatever it explains.

What changed that was not a better derivation. It was the constant turning up elsewhere. Einstein used the same h on the photoelectric effect in 1905 and on the specific heat of solids in 1907, where the same silencing of high-frequency modes explains why diamond’s heat capacity falls far below the classical prediction at room temperature. Three unrelated phenomena, one constant, one mechanism. By 1911 the first Solvay conference was devoted to it.

The general shape is worth naming: an assumption made to save one calculation is a repair, and the same assumption saving three unrelated calculations is a discovery. Nothing about the assumption changed in between.

Where the ladder goes next #

The rungs from here: the derivation of the mode count itself, which is a lattice-counting argument and the same one that gives a string its harmonic series; the Bose–Einstein statistics that Planck’s counting turns out to have assumed without saying so; the cosmic microwave background, which is the most perfect blackbody ever measured and is the radiation left over from a cavity the size of the universe; the specific heat of solids, where the same silencing of stiff modes explains a curve that fooled everyone for fifty years; and the connection to the discreteness of atomic spectra, which is quantisation observed directly rather than inferred from a fit.

The claim to carry forward is the one that makes this the field’s first rung. The classical calculation was not sloppy, and the fix is not a correction term. Counting the modes was right; sharing the energy equally between them was right in every case anyone had tested; and the two together produce nonsense, because energy is not infinitely divisible and nothing in the physics of 1899 could have suggested it.