Annus Mirabilis · Interactive critical edition in preparation
The radiation spectrum and regime comparison
Compare a hot body's glow with Wien's formula and the classical rule.
Radiation spectrum and regime comparison
The radiation spectrum and regime comparison
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Model note
- Primary outputs frequencyEnergyDensity, peakFrequency, peakWavelength: Host calculation (radiation.spectra). Owner radiation.spectra.
- Primary output bandEnergy: Host calculation (radiation.bandIntegration). Owner radiation.bandIntegration.
- Accepted input revision 1.
- Snapshot version 1.
- Not modeled: emissivity and cavity imperfections; detector and spectrometer response; polarization; non-equilibrium radiation; any photon or quantum-statistical model beyond the displayed formulas.
What does a measured radiation spectrum look like at a given temperature, in which regime is Wien's law or the classical law an accurate description, and what does a density plot actually measure?
- Planck 1900
- Wien 1896
- Classical §1
Spectral densities at the probe frequency
| Law | uν (J·m⁻³·Hz⁻¹) | uλ (J·m⁻⁴) |
|---|---|---|
| Planck 1900 | 4.223940 × 10−16 J/(m³ Hz) | 5.072237 × 105 J/(m³ m) |
| Wien 1896 | 4.210617 × 10−16 J/(m³ Hz) | 5.056239 × 105 J/(m³ m) |
| Classical §1 | 2.318107 × 10−14 J/(m³ Hz) | 2.783655 × 107 J/(m³ m) |
Band energy: identical across representations
From the frequency integral: 0.1286853 J/m³. From the wavelength integral over the same physical band: 0.1286853 J/m³. These agree because the Jacobian is applied; a relabeled axis without it would not agree (see “Show the code”).
Peak locations: representation-dependent, and why
| Frequency-density peak νpeak | 2.939463 × 1014 Hz |
| Wavelength-density peak λpeak | 5.795544 × 10−7 m |
| c / λpeak (NOT the frequency-density peak) | 5.172810 × 1014 Hz |
| Per-natural-log-interval peak (x = 3.9206904) | 4.084697 × 1014 Hz |
The frequency-density peak and c divided by the wavelength-density peak are different numbers on purpose: uλ and uν are different functions related by a Jacobian, so their maxima do not correspond under λ = c/ν.
Wien and classical regime verdict
At this temperature and probe frequency, x = hν/(kBT) = 5.759092. Wien's law's relative error here is 0.0031540; the classical law's relative error is 53.880, so it gives 54.880 times Planck's value. In the Wien regime at this temperature and frequency.
This pointwise error is not a certificate for the light paper's integrated entropy argument (§4), which holds only within its own stated limits.
Not modeled: emissivity and cavity imperfections; detector and spectrometer response; polarization; non-equilibrium radiation; any photon or quantum-statistical model beyond the displayed formulas.
The same action without dragging, color, or a canvas
Every action here is typed text entry and a text result: type the temperature, the band edges and the probe frequency, choose the representation from the select lists above, and read the band-energy and peak tables and the regime verdict sentence. The spectrum drawing shows the same densities; its description names the peak, and the three laws differ by line pattern, not colour. No control depends on dragging a handle, distinguishing color alone, or reading a canvas.
A hot body glows at every frequency, most brightly in a middle range that moves higher as the body gets hotter. Wien's formula matches the glow closely at high frequencies and the older classical rule matches it at low ones, and each fails badly where the other works.
Section 1 derives what Maxwell's theory and the electron theory predict for radiation in equilibrium with resonators and a gas: each resonator carries a mean energy (R/N)T, and Planck's relation between resonator and radiation then gives ρν = (R/N)(8πν2/L3)T. Einstein says this law disagrees with experience and makes the total energy infinite. Section 2 takes Planck's formula, ρν = αν3/(eβν/T − 1), which fits all experience so far; for large T/ν it becomes the law of Section 1, and matching the two gives N = 6.17 × 1023. Section 4 works in the opposite limit with Wien's law, ρ = αν3e−βν/T, confirmed for large ν/T, keeping in mind that its results hold only within limits. The instrument draws all three. At 5000 K and a probe at 600 THz, where x = hν/kBT = 5.76, Wien's law is 0.32 percent below Planck's and the classical law gives 54.9 times Planck's value; within a 1 percent tolerance Wien's law holds for x above 4.61 and the classical law for x below 0.0199. The peak depends on how the density is measured: per unit frequency at 294 THz, per unit wavelength at 580 nm, per logarithmic interval at 408 THz. The energy in a band, 0.129 J/m3 between 400 and 600 THz, is the same whichever axis is used.
Write x = hν/kBT, the ratio of one quantum's energy to the thermal energy. At 5000 K, kBT/h = 1.042 × 1014 Hz, so 600 THz is x = 6.00/1.042 = 5.76. Planck's law is u = (8πhν3/c3)/(ex − 1). Wien's law has ex in place of ex − 1, so Wien divided by Planck is (ex − 1)/ex = 1 − e−x: Wien's law is low by the fraction e−x, and e−5.76 = 0.0032, 0.32 percent. The classical law replaces ex − 1 by its first term, x, which gives u = 8πν2kBT/c3; classical divided by Planck is (ex − 1)/x. At x = 5.76 that is (316.9 − 1)/5.76 = 54.9, so the classical law gives 54.9 times Planck's value; at x = 0.01 it is (1.01005 − 1)/0.01 = 1.005, only 0.5 percent high. For a 1 percent tolerance, Wien's law needs e−x ≤ 0.01, so x ≥ ln 100 = 4.61, which at 5000 K means above 480 THz; the classical law needs (ex − 1)/x ≤ 1.01, so x ≤ 0.0199, below 2.07 THz. Between the two neither simple law will do. The classical law also has no finite total: ν2 grows without limit, so ∫ν2dν over all frequencies is infinite, which is Section 1's objection. The peak of Planck's curve per unit frequency is where x3/(ex − 1) is largest, at x = 2.821, that is 2.821 × 104.2 THz = 294 THz. Per unit wavelength the density picks up a factor c/λ2 and peaks at x = 4.965, a wavelength of 580 nm, and c divided by 580 nm is 517 THz, not 294 THz. Per logarithmic interval the peak is at x = 3.921, 408 THz. The energy in a band does not depend on the axis, because uλdλ and uνdν describe the same energy: from 400 to 600 THz both give 0.129 J/m3.
Einstein wrote L for the speed of light and ρν for the density per unit frequency, with Planck's constants α and β, β = 4.866 × 10−11 (the exponent of α on the plate is read as −56 on a single witness, where the N he states needs −57). He gave the divergence no name, the textbook name for it being Ehrenfest's of 1911, and he names no one for the classical law: Rayleigh had argued for a density growing as ν2T at long wavelengths in June 1900, and Jeans's correction of the coefficient came in 1905. Wien's law dates from 1896 and Planck's formula from October 1900. The value N = 6.17 × 1023 is Planck's own; Einstein's point in Section 2 is that it follows from the long-wavelength limit alone, without Planck's theory of resonators.
The explanation
Full explanation
A hot body glows at every frequency, most brightly in a middle range that moves higher as it gets hotter. Wien's formula matches the high frequencies and the classical rule the low ones, and each fails badly where the other works.
Show every step of the investigation
Set the temperature and read where each law agrees with the full spectrum. Then check what a density plot measures: per unit frequency and per unit wavelength put the peak in different places.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.