Book: The Theory of Heat Radiation

Overview
Max Planck's The Theory of Heat Radiation (1906) synthesizes experimental facts and theoretical reasoning about thermal radiation and culminates in the expression now known as Planck's law. The text organizes a program that reconciles observed black-body spectra with thermodynamics and statistical ideas, treating electromagnetic radiation inside cavities and its exchange with matter. It is both a technical derivation and a conceptual reinterpretation of energy exchanges at microscopic scales.

Experimental background
Planck begins with the empirical behavior of black-body spectra, emphasizing two established regularities: Wien's displacement law and the observed deviations at long wavelengths where classical predictions failed. Careful comparison with available spectral measurements motivates the search for a universal formula describing energy density as a function of frequency and temperature. This empirical grounding frames the problem that classical electrodynamics and equipartition seemed unable to resolve.

Theoretical method and derivation
Planck models matter as an ensemble of idealized harmonic oscillators (resonators) that interact with the electromagnetic field, and he combines thermodynamic relations with statistical counting. Using Boltzmann's relation between entropy and probability, he introduces a discretization of energy into finite "elements" ε proportional to frequency, ε = hν, to make the combinatorial count of microstates tractable. From the resulting expression for the average energy per oscillator and the density of modes of the electromagnetic field in a cavity, he derives what is now written as u(ν,T) = (8πhν^3/c^3)/(e^{hν/kT} − 1), which reproduces the full shape of the observed spectrum.

Conceptual innovations
The most profound conceptual step is the introduction of the constant h and the idea that energy exchange between matter and radiation occurs in discrete quanta proportional to frequency. Planck originally presented quantization as a formal device to obtain correct numerical results, not immediately as a radical ontological claim about the continuity of energy. Still, the quantization assumption departs from classical continuous energy and provides the mathematical mechanism that avoids the ultraviolet divergence implied by classical equipartition. The derivation also clarifies limits: at low frequencies (hν << kT) the formula reduces to the Rayleigh–Jeans law, while at high frequencies it approaches Wien's law.

Style and structure
The presentation is methodical, combining thermodynamic reasoning with statistical arguments and careful attention to the emission and absorption mechanisms of resonators. Planck interleaves derivations with comparisons to experiment, discusses limiting cases and constants introduced in the theory, and articulates the mathematical steps leading from entropy considerations to spectral energy density. The tone is cautious and technical, reflecting Planck's original ambivalence about the physical status of discrete energy elements.

Reception and legacy
The formula for black-body radiation had immediate and far-reaching consequences: it resolved an outstanding empirical puzzle and opened the conceptual path to quantum theory. The introduction of the constant h established a new fundamental scale in physics, and the quantization idea was later reinterpreted and extended by Einstein, Bohr, and others into a broader quantum framework. Planck's analysis thus stands as a pivotal bridge between classical thermodynamics and the quantum mechanics that would follow, marking a turning point in the understanding of microscopic energy processes.