Experiment: The Blackbody Spectrum Measured

Contents

Stefan: the fourth power, from Tyndall's data (1879)
Lummer and Kurlbaum: a black body one can build (1898)
Lummer and Pringsheim: Wien's law breaks (1899)
Rubens and Kurlbaum: residual rays in the far infrared (1900)
Radiometric realization of the kelvin (2019)
What the measurements settle

Black-Body Radiation and Planck's Hypothesis makes a prediction of an unusual kind. It does not say what some particular substance will do; it says that the light leaving a small hole in any isothermal cavity is governed by one universal function of frequency and temperature, that thermodynamics fixes the integral of that function and its behaviour under a change of temperature but not the function itself, and that the function is Planck's law — which contains a constant of nature absent from classical physics. That chapter derives the law. It does not show it being measured, and the omission matters more here than anywhere else in this treatise, because in this one case the measurement came first. Planck did not predict the spectrum and wait for the laboratory; the laboratory handed him a spectrum that no classical theory could produce, and he wrote down a formula to fit it. This chapter is that measurement.

Five experiments are reported, in the order in which they were made. Stefan reads a fourth-power law out of somebody else's thermopile deflections [Stefan:1879]; Lummer and Kurlbaum build the first object whose emission may honestly be called Kirchhoff's universal function [Lummer:1898]; Lummer and Pringsheim take its spectrum far enough into the infrared to kill the accepted distribution law [Lummer:1899]; Rubens and Kurlbaum, by fixing the wavelength and sweeping the temperature instead, produce the single result that Planck was told about on 7 October 1900 and answered the same evening [Rubens:1900] [Planck:1900a]. The fifth is modern: under the SI of 2019 the same cavity, viewed by a calibrated radiometer, is a primary realization of the kelvin [BIPM:2019] [CCT:2019], so that the constant Planck read off the Reichsanstalt data now defines the units in which the measurement is expressed. Together they are the reason Part VIII — The Transition to Quantum Physics exists.

Derivation pending.

Experiment: The Blackbody Spectrum Measured: what is pending is the tabulation of the measured radiance against wavelength and temperature for each experiment, in SI with its uncertainty budget, together with the residuals against the Wien and the Planck distributions. The phenomena stated below each carry their derivation inline.

Stefan: the fourth power, from Tyndall's data (1879)

The first quantitative law of thermal radiation was obtained without an experiment. Stefan had no apparatus in this matter; he had Tyndall's published numbers for the total emission of an incandescent platinum wire [Tyndall:1865], and he asked what power of the absolute temperature would reproduce them [Stefan:1879]. The experiment of record is therefore Tyndall's and the measurement of record is Stefan's re-analysis of it, a distinction this section keeps because it is what makes the honest assessment of the result possible.

Apparatus

[Reserved: Tyndall's arrangement — a platinum wire heated electrically to incandescence in air, its temperature judged from its colour against a scale of known incandescent hues, and its total radiant emission collected and measured with a thermopile and galvanometer. To be given: the wire dimensions and its surface condition, the geometry that fixes what fraction of the emitted power reaches the pile, the thermopile's linearity, and above all the provenance and uncertainty of the two temperature values, which dominate everything that follows.]

Procedure

[Reserved: raise the wire through a sequence of temperatures and record the galvanometer deflection at each. The quantity used is a ratio of deflections between a pair of temperatures, which cancels the absolute calibration of the thermopile and leaves only its linearity to be established — an economy worth stating explicitly, since it is the reason a crude instrument could settle an exponent. Stefan's own step is arithmetical: convert both temperatures to the absolute scale and ask what power of $T$ reproduces the measured ratio.]

Observations and data

[Reserved: the deflection at each temperature and the derived emission ratios, with Stefan's parallel re-analysis of the cooling measurements of Dulong and Petit, which he treated in the same paper and by the same method. To be tabulated when written: temperature pairs in kelvin, measured emission ratio, exponent implied by each pair, and the propagated uncertainty of that exponent.]

Phenomenon 73.1 (Total emission rises as the fourth power of the absolute temperature).

The total power radiated per unit area by a hot body, integrated over all frequencies, rises much faster than in proportion to its temperature. Stefan found that Tyndall's platinum wire, taken from $798\,\mathrm{K}$ to $1473\,\mathrm{K}$, radiated $11.7$ times as much power at the higher temperature as at the lower, and identified this as the fourth power of the ratio of the absolute temperatures [Stefan:1879]. The thermodynamic derivation of the same exponent, which requires the radiation pressure and no data at all, came five years later [Boltzmann:1884].

Derivation. Take the law in the form $M=\varepsilon\sigma T^{4}$ of Black-Body Radiation and Planck's Hypothesis and suppose the emissivity $\varepsilon$ of the wire to be the same at both temperatures. The absolute calibration then cancels from a ratio,

\begin{equation}\tag{73.1} \frac{M_{2}}{M_{1}}=\left(\frac{T_{2}}{T_{1}}\right)^{4}\ec \end{equation}

and with $T_{1}=798\,\mathrm{K}$ and $T_{2}=1473\,\mathrm{K}$ the temperature ratio is $1.8459$, whose fourth power is $11.61$. Against Tyndall's $11.7$ that is agreement to better than one percent, which is how the law was found.

It is also better agreement than the data deserve, and the derivation should say so. Solving Equation (73.1) for the exponent instead of assuming it gives

\begin{equation}\tag{73.2} n=\frac{\ln\left(M_{2}/M_{1}\right)} {\ln\left(T_{2}/T_{1}\right)}=4.01\ec \end{equation}

but the sensitivity of that number to the inputs is severe. From Equation (73.2), a fractional error $\delta$ in the measured emission ratio moves $n$ by $\delta/\ln(T_{2}/T_{1}) \simeq1.6\,\delta$, so one percent on the ratio is worth $0.016$ on the exponent and is negligible. An error in the hotter temperature is not: it enters as

\begin{equation}\tag{73.3} \delta n=\frac{n}{\ln\left(T_{2}/T_{1}\right)}\, \frac{\delta T_{2}}{T_{2}}\ec \end{equation}

which for an error of $100\,\mathrm{K}$ in $T_{2}$ alone gives $\delta n\approx0.44$. Tyndall judged his temperature from the colour of the wire, so an error of that size is optimistic rather than pessimistic, and the datum by itself is consistent with any exponent from roughly $3.5$ to $4.5$. Two further defects work in the same direction: platinum is not a black body, and its emissivity rises with temperature, so the assumption that $\varepsilon$ cancels from Equation (73.1) is false in the direction that inflates the apparent exponent. The fourth power is right; it was not established here, and the sections that follow are the reason it is now believed.

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Interpretation

[Reserved: what the analysis does and does not establish. It establishes that the emission of a hot solid rises far faster than linearly with temperature and is consistent with a fourth power; it does not establish the exponent to better than a few tenths, nor does it concern a black body at all. The section should state plainly that the law became secure only when Boltzmann derived it from the second law together with the radiation pressure [Boltzmann:1884], and when the Reichsanstalt cavity of Section 73.2 made it testable on a radiator that really was black. Forward link: the Stefan–Boltzmann constant is today not measured but computed from the defining constants [BIPM:2019] [Mohr:2025], so this experiment has become a test of the defining constants rather than a determination of anything.]

Primary references

[Stefan:1879]; the thermodynamic derivation of the same law is [Boltzmann:1884]. Tyndall's measurements [Tyndall:1865] are used here as Stefan reports them and the original was not consulted.

Lummer and Kurlbaum: a black body one can build (1898)

Kirchhoff's theorem tells a laboratory what to build — a small hole in a large isothermal enclosure — and says nothing about how well any actual hole will do [Kirchhoff:1860a]. Until that gap was closed there was no object in the world whose spectrum could be compared with the universal function, and every measurement of thermal radiation was a measurement of a material. The electrically heated cavity radiator of Lummer and Kurlbaum is where that changes [Lummer:1898]: it is the first artefact for which the departure from blackness can be estimated rather than hoped for, and it is the direct ancestor of every radiometric standard in use today.

Apparatus

[Reserved: a cylindrical cavity heated electrically along its whole length rather than from one end, so that the wall temperature is uniform; internal diaphragms and a blackened, roughened inner surface, both of which raise the number of reflections a ray must survive before it can leave; a small aperture at one end and a water-cooled front plate to define it; thermocouples distributed along the wall to establish isothermality rather than assume it. Detection is by the surface bolometer Lummer and Kurlbaum had developed for the purpose, with a thermopile as an independent check. To be given: cavity length and bore, aperture diameter, the resulting ratio of aperture area to total interior area, the wall temperature range and its uniformity, and the bolometer's time constant and noise floor.]

Procedure

[Reserved: bring the radiator to a chosen temperature and hold it there; verify isothermality by traversing a thermocouple along the wall and by comparing the emission through the aperture before and after reversing the heating current. Then the two tests that make the claim of blackness an experimental one rather than a design intention: vary the aperture diameter and show that the radiance emerging is unchanged (a black aperture has no memory of its size, a grey one does), and compare two cavities built of different wall materials at the same temperature and show that they agree. Finally, measure the total emission as a function of temperature and compare with the fourth-power law of Section 73.1, this time on a radiator that is actually black.]

Observations and data

[Reserved: total radiance against wall temperature over the working range, with the fitted exponent and the fitted Stefan–Boltzmann constant; the invariance of the radiance under change of aperture and of wall material, quoted as a bound rather than as an equality; the residual temperature non-uniformity along the wall and what it contributes to the uncertainty. All in SI with uncertainties.]

Phenomenon 73.2 (The aperture of an isothermal cavity is black to a computable accuracy).

An electrically heated isothermal enclosure viewed through a small aperture emits, to within a small and calculable correction, the universal function of [Kirchhoff:1860a], and its emission is insensitive to the material of the walls: a mediocre absorber makes an excellent cavity provided the aperture is small enough. This is what Lummer and Kurlbaum realized in hardware [Lummer:1898], and it is the reason every spectrum reported in the sections that follow is a measurement of a function of nature rather than of a surface.

Derivation. Let the enclosure be isothermal, let its wall have emissivity $\varepsilon$ — equal to its absorptivity by Black-Body Radiation and Planck's Hypothesis — and let

\begin{equation}\tag{73.4} f:=\frac{A_{\text{hole}}}{A_{\text{total}}} \end{equation}

be the fraction of the interior area occupied by the aperture. Compute the absorptance of the aperture, which by Kirchhoff's theorem is also its emissivity. Send unit flux in through the hole and assume, as an isothermal cavity with a rough wall makes reasonable, that after each diffuse reflection the flux is spread uniformly over the whole interior. The first encounter with the wall absorbs $\varepsilon$ and reflects $1-\varepsilon$; of what is reflected, the fraction $f$ finds the hole and escapes and the fraction $1-f$ meets the wall again. The survival factor per round trip is therefore $\left(1-\varepsilon\right)\left(1-f\right)$, and summing the geometric series of absorbed contributions,

\[ \varepsilon_{\text{eff}} =\varepsilon\sum_{k\geq0} \left[\left(1-\varepsilon\right)\left(1-f\right)\right]^{k} =\frac{\varepsilon} {1-\left(1-\varepsilon\right)\left(1-f\right)}\ec \]

and expanding the denominator, $1-(1-\varepsilon)(1-f) =\varepsilon+f-\varepsilon f$, gives

\begin{equation}\tag{73.5} \varepsilon_{\text{eff}} =\frac{\varepsilon}{\varepsilon+\left(1-\varepsilon\right)f}\ep \end{equation}

The design rule follows by inspection. The wall material enters only through the product $\left(1-\varepsilon\right)f$, so it is the aperture fraction and not the blackness of the paint that carries the accuracy: a wall of $\varepsilon=0.5$ — barely better than polished — with an aperture fraction $f=10^{-3}$ gives $\varepsilon_{\text{eff}}=0.999$, while the same wall with the lid off would be half a black body. Diaphragms and a roughened surface work by making the effective $f$ seen by an escaping ray smaller still.

Two honesties are owed here. First, Equation (73.5) is the modern integrating-cavity estimate and not Lummer and Kurlbaum's own formula; they argued the case by construction and by the experimental invariances of the procedure above, which is the stronger argument since it does not assume perfect diffusion. Second, the formula assumes isothermality exactly, and a real cavity is graded in temperature along its length; that error is not covered by Equation (73.5) and has to be bounded separately, which is why the thermocouple traverse is part of the apparatus and not of the diagnostics.

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Interpretation

[Reserved: the significance is metrological before it is theoretical. Kirchhoff's universal function ceases to be an idealization and becomes an object with a serial number, so that a spectrum measured in Berlin can be compared with a formula and with a spectrum measured anywhere else. The cavity radiator became the primary radiometric standard of the Physikalisch-Technische Reichsanstalt and the ancestor of the fixed-point blackbodies of Section 73.5; the traceability chain and its uncertainty budget belong to Measurement, SI Units, and the Theory of Errors. Worth recording plainly: the Reichsanstalt built this instrument to serve the German lighting industry, and the overthrow of classical physics was a by-product of a standards laboratory doing standards work.]

Primary references

[Lummer:1898]; the theorem the instrument realizes is [Kirchhoff:1860a].

Lummer and Pringsheim: Wien's law breaks (1899)

With a black body available, the universal function could at last be measured rather than guessed. Wien's distribution [Wien:1896] was the accepted answer and fitted everything then in evidence; what Lummer and Pringsheim did was to push the measurement into the region where it was not in evidence, at long wavelength and high temperature, and there it failed [Lummer:1899] [Lummer:1900]. This is the point at which the accepted law of thermal radiation died, and it died to data rather than to an argument: no successor to Wien's formula existed for another year.

Apparatus

[Reserved: the cavity radiator of Section 73.2 as source; a spectrometer whose prism and windows are transparent well into the infrared, where glass and quartz are opaque, hence fluorite and rock-salt optics; a linear bolometer at the exit slit, with its galvanometer; and the calibration that the whole result rests on — the dispersion curve of the prism, which converts an angular position into a wavelength and an angular slit width into a bandwidth. To be given: prism material and apex angle, slit widths and the resolving power they imply, the bolometer strip dimensions, and the measured transmission of the optical train as a function of wavelength.]

Procedure

[Reserved: hold the radiator at a chosen temperature and scan the bolometer through the spectrum, recording deflection against angle. Reduce each scan to a spectral radiance: correct for the dispersion of the prism, so that equal angular steps become unequal wavelength intervals; for the transmission of prism, windows and mirrors; and for the spectral response of the bolometer. Repeat at each temperature over the working range. The comparison with theory is then made by fitting the two constants of Wien's distribution to the short-wavelength part of each isotherm and examining the residuals at long wavelength — a procedure that must be fixed in advance, since a fit over the whole range would distribute the discrepancy and hide it.]

Observations and data

[Reserved: isotherms of spectral radiance against wavelength from about $300\,\mathrm{K}$ to $1650\,\mathrm{K}$, reaching about $8\,\mu\mathrm{m}$ [Lummer:1899], with the residuals against the fitted Wien distribution. The residuals are the observation: they are of one sign, they grow with the product of wavelength and temperature, and they exceed the estimated uncertainty of the measurement by a wide margin at the long-wavelength end. To be tabulated: wavelength, temperature, measured spectral radiance and its uncertainty, and the ratio of measurement to Wien's formula and to Planck's.]

Phenomenon 73.3 (The measured spectrum exceeds Wien's distribution at long wavelength).

Measured with a black body of known temperature and a spectrometer transparent into the infrared, the spectral radiance of cavity radiation is systematically greater than Wien's distribution [Wien:1896] predicts, and the excess is not random: it has one sign, it is negligible in the visible and near infrared where the formula had been tested, and it grows steadily with the product of wavelength and temperature. Lummer and Pringsheim found it out to about $8\,\mu\mathrm{m}$ over temperatures from about $300\,\mathrm{K}$ to $1650\,\mathrm{K}$, far beyond any plausible error of their measurement [Lummer:1899].

Derivation. The excess is exactly what Planck's law of Black-Body Radiation and Planck's Hypothesis requires, and its dependence on a single variable is the content of the phenomenon. Write the dimensionless variable

\begin{equation}\tag{73.6} x:=\frac{hc}{\lambda k_{\text{B}}T}=\frac{c_{2}}{\lambda T}\ec\qquad c_{2}:=\frac{hc}{k_{\text{B}}}=1.438777\times 10^{-2}\,\mathrm{m}\,\mathrm{K}\ec \end{equation}

the second radiation constant, whose value is exact in the present SI because $h$, $c$ and $k_{\text{B}}$ are all defining constants [BIPM:2019] [Mohr:2025]. In terms of $x$ the Planck spectral radiance carries the factor $1/(\ee^{x}-1)$ and Wien's distribution the factor $\ee^{-x}$, with the same prefactor, so their ratio is

\begin{equation}\tag{73.7} \frac{L_{\lambda}^{\text{Planck}}}{L_{\lambda}^{\text{Wien}}} =\frac{\ee^{x}}{\ee^{x}-1} =\frac{1}{1-\ee^{-x}}\ep \end{equation}

This is greater than unity for every finite $x$, so the discrepancy can only have one sign, and it is a decreasing function of $x$ — that is, an increasing function of $\lambda T$ — so it is invisible in the visible and unmistakable in the infrared. That it depends on $\lambda$ and $T$ only through their product is Wien's own displacement scaling of Black-Body Radiation and Planck's Hypothesis, which both formulas respect; it means the whole family of isotherms collapses onto one curve, and a residual that does not so collapse would indicate an instrumental error rather than a failure of the law.

Table 73.1 evaluates Equation (73.7). The numbers are computed here from Planck's law, not taken from any measurement, and they say what an experimenter had to reach before Wien's formula could be convicted: below $\lambda T\approx3000\,\mu\mathrm{m}\,\mathrm{K}$ the excess is under one percent and no instrument of 1899 could have seen it, which is why measurements confined to the visible and near infrared continued to support Wien's law while these did not.

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The factor by which Planck's law exceeds Wien's distribution, from Equation (73.7), as a function of the product of wavelength and temperature. Computed from the formula; not a measurement. The short-wavelength end is the region in which Wien's law had been tested and found good.

$\lambda T$	excess of Planck over Wien
$3000\,\mu\mathrm{m}\,\mathrm{K}$	$0.8\,\mathrm{\%}$
$6000\,\mu\mathrm{m}\,\mathrm{K}$	$10\,\mathrm{\%}$
$9000\,\mu\mathrm{m}\,\mathrm{K}$	$25\,\mathrm{\%}$
$12000\,\mu\mathrm{m}\,\mathrm{K}$	$43\,\mathrm{\%}$

Interpretation

[Reserved: the accepted distribution law is falsified, and nothing replaces it. The section should resist the retrospective tidying that makes this look like a step towards Planck: for a year there was simply a measured curve with no formula, and the candidate interpolations then in circulation were fitted to it rather than derived. Also the methodological point, which Epistemology and the Scientific Method takes up: a law tested only where its predictions are insensitive to the thing at issue has not been tested, and the entire agreement of Wien's formula in the visible was of that kind — Table 73.1 shows the discrepancy there was below the noise. Honest caveat: the size of the deviations was disputed at the time, and settling the dispute needed the different and far more decisive measurement of Section 73.4.]

Primary references

[Lummer:1899] [Lummer:1900]; the distribution under test is [Wien:1896] and the scaling law both formulas obey is [Wien:1893].

Rubens and Kurlbaum: residual rays in the far infrared (1900)

This is the decisive measurement, and it is decisive because of an inversion of method. Every spectral measurement before it fixed the temperature and scanned the wavelength, so that the result depended on a dispersion calibration and a chain of transmission corrections. Rubens and Kurlbaum fixed the wavelength — at values far beyond anything a prism could reach — and swept the temperature [Rubens:1900]. What they obtained was a curve of radiance against temperature at one wavelength, needing no spectral calibration at all, and on that curve Wien's law and its successors make qualitatively different predictions. Rubens described the result to Planck privately on 7 October 1900; the formula that fits it was presented to the Deutsche Physikalische Gesellschaft twelve days later [Planck:1900a] and derived on 14 December [Planck:1900b].

Apparatus

[Reserved: the cavity radiator of Section 73.2 as source, with the furnace extended downward in temperature by cooled baths; the residual-ray (Reststrahlen) train, in which the beam is reflected in succession from several polished crystal plates — fluorite for $24\,\mu\mathrm{m}$, rock salt for $51.2\,\mu\mathrm{m}$ — each of which reflects strongly only in the narrow band near its lattice resonance and weakly elsewhere; and a bolometer or thermopile to measure what survives. To be given: the number of successive reflections, the measured reflectance of each plate in and out of band, the residual short-wavelength contamination of the emerging beam, and the detector's sensitivity and noise floor, which set the lowest temperature reachable.]

Procedure

[Reserved: form the residual-ray beam and verify its purity, then hold the wavelength fixed and step the source through its temperature range, recording the detector deflection at each step. Because the wavelength never changes, the dispersion curve, the transmission corrections and the detector's spectral response all drop out of the comparison, and only the linearity of the detector and the accuracy of the source temperature remain — this is the whole methodological advantage over Section 73.3. The candidate formulas of the day are then fitted to the measured curve and their residuals compared over the whole temperature range, not over a favourable part of it.]

Observations and data

[Reserved: radiance against source temperature at $24\,\mu\mathrm{m}$ and at $51.2\,\mu\mathrm{m}$, over a range extending from below room temperature to the upper limit of the furnace [Rubens:1900]. The observation is the shape of that curve: it rises without any sign of the ceiling Wien's distribution requires, and over the upper part of the range it is close to a straight line through the origin. To be tabulated: source temperature, measured radiance and its uncertainty at each wavelength, together with the residuals against each of the formulas Rubens and Kurlbaum tested — Wien's and the interpolations of Thiesen, of Rayleigh and of Planck.]

Phenomenon 73.4 (At fixed long wavelength the radiance grows without bound, nearly in proportion to the temperature).

Far beyond the maximum of the spectrum the radiance of a black body does not saturate as the source is made hotter; it goes on rising, and over the upper part of the accessible range it rises very nearly in proportion to the absolute temperature. Rubens and Kurlbaum isolated $24\,\mu\mathrm{m}$ and $51.2\,\mu\mathrm{m}$ by repeated reflection from fluorite and rock salt and found this behaviour at both wavelengths [Rubens:1900]. Wien's distribution [Wien:1896] requires the opposite: at fixed wavelength it approaches a finite ceiling as $T\to\infty$. The disagreement is not a matter of percentages, and it is what Planck was told about on 7 October 1900.

Derivation. Compare the two formulas of Black-Body Radiation and Planck's Hypothesis at fixed $\lambda$ as functions of $T$, using the variable $x=c_{2}/\lambda T$ of Equation (73.6). Wien's distribution carries the factor $\ee^{-x}$; as $T$ grows, $x\to0$ and $\ee^{-x}\to1$, so

\begin{equation}\tag{73.8} L_{\lambda}^{\text{Wien}}\longrightarrow \frac{2hc^{2}}{\lambda^{5}}\qquad\left(T\to\infty\right)\ec \end{equation}

a ceiling fixed by the wavelength alone, which no temperature can exceed. Planck's law carries $1/(\ee^{x}-1)$ instead, and for $x\ll1$ the expansion $\ee^{x}-1\simeq x$ gives

\begin{equation}\tag{73.9} L_{\lambda}^{\text{Planck}}\simeq \frac{2hc^{2}}{\lambda^{5}}\,\frac{1}{x} =\frac{2ck_{\text{B}}T}{\lambda^{4}}\ec \end{equation}

in which $h$ has cancelled: the radiance is proportional to $T$, grows without bound, and is the equipartition result of Kinetic Theory of Gases. One formula predicts a plateau and the other a rising straight line, so a single wavelength and a wide temperature range decide between them without any spectral calibration whatever — which is the point of the method.

The wavelengths chosen put the experiment squarely in the regime where Equation (73.9) applies. The crossover, $x=1$, falls at $T=c_{2}/\lambda$, which is close to $600\,\mathrm{K}$ at $24\,\mu\mathrm{m}$ and to $281\,\mathrm{K}$ at $51.2\,\mu\mathrm{m}$: above those temperatures the linear behaviour sets in, and at the longer wavelength every furnace temperature lies above it. The size of the resulting discrepancy is brutal. At $51.2\,\mu\mathrm{m}$ and a source at $1500\,\mathrm{K}$ one has $x=0.187$, and by Equation (73.7) the true radiance exceeds Wien's by the factor $1/(1-\ee^{-x})=5.9$ — a factor of six, not a discrepancy of percentages, and beyond the reach of any excuse about calibration.

Two qualifications keep the statement honest. The proportionality to $T$ is asymptotic, not exact: the next term of the expansion gives $L_{\lambda}\simeq(2ck_{\text{B}}T/\lambda^{4})(1-x/2+\dots)$, so the measured curve bends below the straight line at the cold end, by about $9\,\mathrm{\%}$ at $x=0.187$, and the linear region has to be identified rather than assumed. And Equation (73.9) is a limit of Planck's law, not evidence for it: what the measurement establishes directly is that Wien's exponential is wrong at long wavelength. That a single formula reproduces both this limit and the visible spectrum of Section 73.3 is the additional claim, and it is what Rubens and Kurlbaum tested by fitting the competing interpolations to the same curve.

The purity of the residual-ray beam is what makes any of this measurable, and it is worth one line of arithmetic. A source at $1500\,\mathrm{K}$ emits overwhelmingly more power near its maximum, at a few micrometres, than at $51.2\,\mu\mathrm{m}$, so a leak of short-wave radiation would swamp the signal. If a crystal plate reflects a fraction $R_{\text{in}}$ in its residual-ray band and $R_{\text{out}}$ outside it, then $n$ successive reflections change the ratio of in-band to out-of-band flux by

\begin{equation}\tag{73.10} \left(\frac{R_{\text{in}}}{R_{\text{out}}}\right)^{n}\ec \end{equation}

a geometric gain in purity for a merely linear loss of signal: a reflectance contrast of ten, which a strong lattice band easily provides, gives four orders of magnitude of suppression after four reflections. This is why a filter that no one can manufacture is replaced by a mirror used four times.

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Interpretation

[Reserved: the causal chain, which is the clearest instance in physics of a measurement dictating a formula and should be set out step by step with its dates. Rubens communicated the far-infrared result to Planck on 7 October 1900; Planck, who knew that the short-wavelength limit had to be Wien's, now knew that the long-wavelength limit had to be linear in $T$, and constructed the simplest expression for the second derivative of the entropy with respect to the energy that reduces to both — presented on 19 October [Planck:1900a]. That step is interpolation and Planck said so; the physics enters only with the combinatorial derivation of 14 December [Planck:1900b], in which the energy elements $h\nu$ appear. Fitting the new law to the Reichsanstalt data gave the first values of $h$ and of $k_{\text{B}}$ [Planck:1901], and through the latter the first good values of the Avogadro constant and of the elementary charge — a bonus of the measurement that is often forgotten and that did more than anything else to make the formula credible before the hypothesis behind it was.]

Primary references

[Rubens:1900]; the formula it produced within the fortnight is [Planck:1900a], its derivation [Planck:1900b] and its considered exposition [Planck:1901]. The distribution it falsified is [Wien:1896], and the classical limit it exhibits is that of [Rayleigh:1900] [Jeans:1905].

Radiometric realization of the kelvin (2019)

The last experiment in this chapter is not a test of Planck's law but a use of it, and the change of status is the point. Since the revision of the SI that took effect in 2019 the kelvin is defined by fixing the numerical value of the Boltzmann constant, and $h$ and $c$ are fixed as well [BIPM:2019] [Fischer:2018]. Planck's law therefore contains no measured constant and no property of any substance: an absolute measurement of the spectral radiance leaving a cavity yields the thermodynamic temperature of its walls directly. The instrument of Section 73.2, which was built to make the universal function observable, has become a primary realization of a base unit.

Apparatus

[Reserved: a cavity radiator of the modern kind — either held at a fixed point of a pure metal or continuously variable — with an effective emissivity established by the argument of Phenomenon 73.2 and by direct reflectance measurement; a filter radiometer defining a narrow spectral band and a known solid angle, whose absolute spectral responsivity is the quantity everything rests on; and the chain that establishes that responsivity, running through a trap detector to an absolute cryogenic radiometer [Quinn:1985], which equates optical power to electrical power dissipated in the same receiver and so ties the measurement to the watt and thence to the SI base units. To be given: the band centre and width, the aperture geometry and its dimensional metrology, the emissivity budget of the cavity, and the responsivity uncertainty.]

Procedure

[Reserved: calibrate the filter radiometer against the cryogenic radiometer; measure the spectral radiance of the cavity in the chosen band, with the geometry known dimensionally; invert Planck's law for $T$ as in Equation (73.12) below. Cross-checks that make the result a realization rather than a reading: repeat in a second band, where the sensitivity factor of Equation (73.13) differs, and compare with an independent primary thermometer of a different physical kind. The comparison against the defined temperature scale in use, and the difference between a thermodynamic temperature and a scale value, belong to Measurement, SI Units, and the Theory of Errors.]

Observations and data

[Reserved: measured spectral radiance in each band, the inferred thermodynamic temperature, and the uncertainty budget — which is dominated by the radiometric scale and by the cavity emissivity, and which shrinks relative to the temperature by the factor computed in the derivation below. To be tabulated when written: band, measured radiance, sensitivity factor, inferred temperature and combined standard uncertainty, in SI throughout.]

Phenomenon 73.5 (Thermodynamic temperature is readable from an absolute spectral radiance).

The spectral radiance emerging from a black cavity determines the thermodynamic temperature of its walls through Planck's law alone. In the SI that took effect in 2019 the constants entering that law — $h$, $c$ and $k_{\text{B}}$ — are all fixed by definition [BIPM:2019] [Mohr:2025], so no material property and no reference temperature intervene: the measurement is a primary realization of the kelvin [CCT:2019]. The same relation that was extracted from the Reichsanstalt spectra now supplies the unit in which those spectra would be reported.

Derivation. Write Planck's law of Black-Body Radiation and Planck's Hypothesis as a spectral radiance,

\begin{equation}\tag{73.11} L_{\lambda}\left(\lambda,T\right) =\frac{2hc^{2}}{\lambda^{5}}\, \frac{1}{\ee^{x}-1}\ec\qquad x=\frac{c_{2}}{\lambda T}\ec \end{equation}

and solve for the temperature. Rearranging Equation (73.11) gives $\ee^{x}-1=2hc^{2}/(\lambda^{5}L_{\lambda})$, hence $x$ by a logarithm and hence

\begin{equation}\tag{73.12} T=\frac{c_{2}}{\lambda}\left[ \ln\left(1+\frac{2hc^{2}}{\lambda^{5}L_{\lambda}}\right) \right]^{-1}\ep \end{equation}

Everything on the right of Equation (73.12) is either a defining constant or a measured quantity: the wavelength, which is metrology of the filter, and the absolute spectral radiance, which is metrology of the radiometer. No thermometer appears.

The design of such a measurement turns on how an error in $L_{\lambda}$ propagates into $T$. Differentiating Equation (73.11) at fixed $\lambda$, and using $\dd x/x=-\dd T/T$,

\[ \frac{\dd L_{\lambda}}{L_{\lambda}} =-\frac{\ee^{x}}{\ee^{x}-1}\,\dd x =\frac{x\ee^{x}}{\ee^{x}-1}\,\frac{\dd T}{T}\ec \]

so that

\begin{equation}\tag{73.13} \frac{\dd T}{T}=\frac{1-\ee^{-x}}{x}\, \frac{\dd L_{\lambda}}{L_{\lambda}}\ep \end{equation}

The factor in front is close to unity when $x\ll1$ and close to $1/x$ when $x\gg1$, so radiometric thermometry is done at short wavelengths, on the steep Wien side of the spectrum, where a coarse measurement of radiance buys a fine measurement of temperature. At $\lambda=650\,\mathrm{nm}$ and $T=1235\,\mathrm{K}$ — representative of a metal fixed point — Equation (73.6) gives $x=17.9$, so the sensitivity factor is $0.056$ and an error of $1\,\mathrm{\%}$ in the measured radiance appears as $0.056$ of a percent in temperature, about $0.7\,\mathrm{K}$. The same one percent at $\lambda=51.2\,\mu\mathrm{m}$, where $x$ is small, would appear almost undiminished. The far infrared that decided the physics is the worst possible place to realize the unit, and for exactly the reason that made it decisive: there the spectrum has forgotten $h$.

One consequence should be stated plainly, because it inverts the logic of every earlier section. Since $h$, $c$ and $k_{\text{B}}$ are now defined rather than measured, Equation (73.12) cannot be used to test Planck's law — it presupposes it. The law is tested instead by comparing this realization against primary thermometers of a different physical kind, and, on a far more exacting scale, by the astrophysical spectrum of Experiment: The Cosmic Microwave Background.

∎

Interpretation

[Reserved: what has and has not changed. The 1898 cavity and the present one are the same instrument serving opposite purposes: then a means of exhibiting an unknown universal function, now a means of realizing a defined unit. The section should be explicit that the present arrangement is not evidence for Planck's law and say what is: the mutual consistency of primary thermometers resting on different physics, and the cosmic microwave background spectrum of Experiment: The Cosmic Microwave Background, which is the most exactly Planckian spectrum known. Also the closing of the historical circle — the constant Planck read off the Reichsanstalt data in 1900 [Planck:1901] is now a defining constant of the SI [BIPM:2019], so that his 1901 values may be quoted as a percent-level verification of a definition.]

Primary references

[BIPM:2019] [Mohr:2025]. These are the normative and the recommended-values documents; the primary literature of absolute cryogenic radiometry and of the filter-radiometer realizations is not cited here because references.bib carries no entry for it, and no key has been invented to stand in its place.

What the measurements settle

Experiment	Quantity fixed	What it establishes
Stefan 1879	neither	total emission rises as a high power of $T$, consistent with the fourth but not resolving it
Lummer–Kurlbaum 1898	—	an artefact whose emissivity is calculable and near unity; the universal function becomes observable
Lummer–Pringsheim 1899	$T$ fixed, $\lambda$ scanned	the accepted distribution law is wrong at long wavelength, by more than any error of measurement
Rubens–Kurlbaum 1900	$\lambda$ fixed, $T$ swept	the radiance grows without bound and nearly linearly in $T$; Wien's ceiling does not exist
Radiometric kelvin 2019	all three constants	Planck's law realizes the unit of temperature rather than being tested by it

The measurements reported in this chapter and what each establishes. The third column states the claim the experiment is entitled to, which in the first case is markedly weaker than the claim it is usually credited with.

The sequence in Table 73.2 has a shape that is unusual in this treatise and worth naming. Elsewhere a theory predicts and an experiment checks; here the experiments arrive first and the theory is built to fit them, and the constant $h$ enters physics not as a postulate but as the number required to reconcile two limits that were each measured before either was understood. Wien's exponential is the short-wavelength limit and was measured by 1896; the proportionality to $T$ is the long-wavelength limit and was measured in October 1900; the formula joining them was written within days of the second, and the hypothesis that explains the formula came two months later [Planck:1900a] [Planck:1900b].

What the chapter does not settle should be as clear. None of these measurements shows that light is granular. They show that the energy exchanged between a cavity wall and the field comes in units proportional to the frequency, which is what Planck claimed and no more; the light quantum is a further and independent step, and the experiments that force it are those of The Photon: Photoelectric and Compton Effects. Nor does the chapter settle the exponent Stefan is remembered for — Phenomenon 73.1 gives it as an inference from a ratio whose uncertainty allows several tenths either way, and it is Boltzmann's derivation [Boltzmann:1884] together with the cavity work of Section 73.2 that makes it secure. The one thing established here beyond argument is negative and decisive: no distribution law derivable from classical electrodynamics and equipartition can fit the curve that Rubens and Kurlbaum measured at $51.2\,\mu\mathrm{m}$, because equipartition gives the linear rise at long wavelength only at the cost of an infinite total energy [Rayleigh:1900] [Jeans:1905]. The measurement that forced quantisation is a measurement of the far infrared, made by a standards laboratory, in the service of the lighting industry.

$\lambda T$	excess of Planck over Wien
\(3000\,\mu\mathrm{m}\,\mathrm{K}\)	\(0.8\,\mathrm{\%}\)
\(6000\,\mu\mathrm{m}\,\mathrm{K}\)	\(10\,\mathrm{\%}\)
\(9000\,\mu\mathrm{m}\,\mathrm{K}\)	\(25\,\mathrm{\%}\)
\(12000\,\mu\mathrm{m}\,\mathrm{K}\)	\(43\,\mathrm{\%}\)