In his previous post and the one before that, John Howard explored the history of blackbody radiation. Here, he describes how Max Planck was persuaded to derive a formula for blackbody radiation. Planck presented his formula in December of 1900 to the German Physical Society in Berlin, ushering in the quantum era.

In the late summer of 1900, Otto Lummer and Ernst Pringsheim carefully measured the spectral distribution of the thermal radiation from a blackbody radiator, and H. Rubens and his colleague Kurlbaum made a similar set of measurements at various temperatures. They then plotted their results and compared the results with the two theoretical predictions—of Wilhelm Wien for high frequencies, and Lord Rayleigh for lower frequencies. They found good agreement with the Wien formula, except that the Rayleigh formula was definitely better at low frequencies.

I have read two different accounts of how H. Rubens related to the young thermodynamicist Max Planck that the Wien formula did not fit well at low frequencies. According to one story, Rubens attended a seminar at the University of Berlin shortly after plotting his data.  At the tea and social hour before the event, he saw Planck, joined him for tea, and reported his results. In the other version, Planck invited Rubens and his wife to a Sunday lunch at Planck’s home; after their meal, the two physicists discussed the partial failure of the Wien formula.

Planck was indeed very interested, as less than a year earlier he had carefully worked to put Wien’s derivation of his formula on to a more solid thermodynamic foundation. After the seminar (or the lunch) was over, Planck spent the rest of that day looking for an “interpolation formula” that would reduce to the Rayleigh prediction at low frequencies, and to the Wien formulation at higher frequencies.

After several hours, he succeeded in finding such a formula. It was generally similar to the Wien formulation, but with an additional exponential term in the denominator. He sent a note with his proposed formula to Rubens, who returned to Planck two days later and said that interpolation formula fits everything, so it must be right! Planck said later that finding that formula was “just a lucky guess.”

 At a Berlin meeting of the German Physical Society in mid-October of 1900, Kurlbaum gave a short paper on the Rubens-Kurlbaum measurements, following which Planck arose with some comments and sketched his modified formula on the blackboard. The attendees were pleased with this ad hoc formula; now Planck was faced with the more daunting challenge of producing a satisfactory scientific derivation of that “interpolation formula.” Planck labored over that derivation for about two months, calling it the “hardest labor of my life,” before presenting his detailed derivation to a meeting of the Physical Society in mid-December 1900.

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