A Brief History of the T⁴ Radiation Law

Since the 1700s, natural philosophers understood that heat exchange between two bodies was not precisely linearly dependent on the temperature difference, and that at high temperatures the discrepancy became greater. Over the years, many models were developed with varying degrees of success. The lack of success was due to the difficulty obtaining accurate experimental data, and a lack of knowledge of the fundamental mechanisms underlying radiation heat exchange. Josef Stefan, of the University of Vienna, compiled data taken by a number of researchers who used various methods to obtain their data, and in 1879 proposed a unique relation to model the dependence of radiative heat exchange on the temperature: the T⁴ law. Stefan’s model was met with some skepticism and was not widely accepted by his colleagues. His former student, Ludwig Boltzmann, who by then had taken a position at the University of Graz in Austria, felt that there was some truth to the empirical model proposed by his mentor. Boltzmann proceeded to show in 1884, treating electromagnetic radiation as the working fluid in a Carnot cycle, that in fact the T⁴ law was correct. By the time that Boltzmann published his thermodynamic derivation of the radiation law, physicists became interested in the fundamental nature of electromagnetic radiation and its relation to energy, specifically determining the frequency distribution of blackbody radiation. Among this group of investigators was Wilhelm Wien, working at Physikalisch-Technische Reichsanstalt in Charlottenburg, Berlin. He proposed a relation stating that the wavelength at which the maximum amount of radiation was emitted occurred when the product of the wavelength and the temperature was equal to a constant. This became known as Wien’s Displacement Law, which he deduced this by imagining an expanding and contracting cavity, filled with radiation. Later, he combined his Displacement Law with the T⁴ law to give a blackbody spectrum which was accurate over some ranges, but diverged in the far infrared. Max Planck, at the University of Berlin, built on Wien’s model but, as Planck himself stated, “the energy of radiation is distributed in a completely irregular manner among the individual partial vibrations...” This “irregular” or discrete treatment of the radiation became the basis for quantum mechanics and a revolution in physics. This paper will present brief biographies of the four pillars of the T⁴ radiation law, Stefan, Boltzmann, Wien and Planck, and outline the methodologies used to obtain their results.