Cavity radiation and the pre-Planck laws

Put two bodies of any materials inside a sealed enclosure at temperature T and wait. Each must absorb exactly what it emits, or its temperature would drift and heat would move between objects already at the same temperature. Slide a narrow-band filter between them and the balance must hold frequency by frequency, which forces e_ν = a_ν I_ν(T) with the same I_ν for every body: a good absorber is a good emitter, and the ratio is universal. A body with a_ν = 1 at every frequency is a blackbody, and no real surface is one — lamp soot reaches about 0.95, polished tungsten far less. A small hole in a large cavity is one: light entering is scattered many times before it can find the hole again. For walls of emissivity 0.5 and a hole one thousandth of the internal area, the standard isothermal-cavity estimate ε/[ε + (1 − ε)Aₕ/A] gives an effective emissivity of 0.999. That is why the radiation standard is a hole and not a paint.

To do thermodynamics on the field you need one mechanical fact. Radiation carries momentum p = E/c, and a beam striking a wall at angle θ delivers normal momentum with one factor of cos θ from the projection and another from the flux; averaging cos²θ over an isotropic hemisphere gives 1/3, so P = u/3. Compare a monatomic ideal gas, where E = p²/2m rather than pc and the same average returns P = 2u/3. The factor is small; its consequences are not. At 300 K, u = aT⁴ = 6.13 × 10⁻⁶ J m⁻³ and P = 2.0 × 10⁻⁶ Pa, some 5 × 10¹⁰ times below atmospheric — which is why nobody stumbled on radiation pressure by accident. At 1.5 × 10⁷ K, the temperature of the Sun's core, the same formula gives P = 1.3 × 10¹³ Pa. The two pressures differ by a factor of 6 × 10¹⁸, almost nineteen orders of magnitude, because T⁴ is a violent function.

Write U = u(T)V and use dU = T dS − P dV. The Maxwell relation that follows is (∂U/∂V)_T = T(∂P/∂T)_V − P. The left side is simply u, since doubling the volume at fixed T doubles the energy. The right side, with P = u/3, is (T/3)(du/dT) − u/3. Rearranged, 4u/3 = (T/3)(du/dT), so du/u = 4 dT/T and u = aT⁴; the same integration gives S = (4/3)aT³V. Only the outward hemisphere escapes a small hole and the cos θ projection costs another half, so the flux is M = (c/4)u = σT⁴ with σ = ac/4. Notice what was never used: no model of the walls, no counting of modes, no h. Notice too where the 4 came from. For P = γu the exponent is (1 + γ)/γ, so radiation's 1/3 gives 4 where a monatomic gas's 2/3 would have given 5/2.

Now compress the cavity quasi-statically between perfect mirrors. Entropy is constant and S = (4/3)aT³V, so T³V is constant and T ∝ V^(−1/3) ∝ 1/L. Meanwhile each standing mode satisfies ν = nc/2L with n fixed — a slow squeeze creates and destroys no modes — so ν ∝ 1/L as well; equivalently, each bounce off the retreating mirror Doppler-shifts the light by exactly that amount. So ν/T survives untouched, and so does the classical adiabatic invariant E/ν of each mode. Combine those with a mode count going as ν² and the spectrum is forced into u_ν = ν³f(ν/T), or u_λ = λ⁻⁵g(λT) after the Jacobian c/λ². Two consequences come free. The peak of λ⁻⁵g(λT) sits wherever λT takes one fixed value, whatever g is, so λₘₐₓ T = b exists as a theorem even though its value 2.898 mm K does not. And ∫ν³f(ν/T) dν = T⁴∫x³f(x) dx returns Stefan–Boltzmann, a check on the whole chain.

Thermodynamics cannot supply f, and the reason is dimensional: pinning the shape of a spectrum takes a constant carrying the dimensions of action, and neither Maxwell's equations nor the two laws contain one. Wien filled the gap in 1896 by analogy. He supposed each frequency was emitted by molecules of one speed and weighted them by the Maxwell distribution's e^{(−mv²/2kT)}; to make that exponent depend on ν/T rather than on ν²/T he had to take the emitted frequency proportional to the molecule's kinetic energy, ν ∝ v², which turns the factor into e^(−βν/T). The result u_ν = αν³e^(−βν/T) already has the right scaling form, so it passes automatically every test thermodynamics can set — and it fitted Paschen's 1897 photometry from roughly 1 to 8 µm at 700–1500 K to within the errors of the day. Its displacement constant comes out at hc/5k = 2.878 mm K against the true 2.898, low by 0.70%; its Stefan constant is low by the factor 6/(π⁴/15) = 0.924, or 7.6%. Neither number is a fair test, because neither reaches the region where the guess is wrong: the peak sits at hν/kT ≈ 5, where Wien's curve runs only 0.7% below the truth, and the Stefan integral draws a mere 18% of its weight from hν/kT < 2.

Wien's law falls below the truth by exactly the factor 1 − e⁻ˣ, with x = hν/kT: 0.7% at x = 5, 5% at x = 3, 37% at x = 1. The defect only becomes obvious below x ≈ 1, which means long wavelengths or high temperatures, and Rubens and Kurlbaum reached them — reststrahlen mirrors isolated a band near 51.2 µm while the source was swung from 85 K to about 1770 K. At 51.2 µm and 1500 K, x = 0.187 and Wien's formula returns 17% of the energy actually measured. The shape of the failure matters more than its size. Hold ν fixed and raise T: e^(−hν/kT) → 1, so Wien's law saturates at a ceiling αν³. The data instead ran dead straight, u_ν ∝ T, with no ceiling in sight. That linear law is what classical equipartition predicts, and it is the next topic — Wien's guess and Rayleigh–Jeans fail at opposite ends of the same axis, which is the clue Planck read.