Eine Beziehung zwischen dem elastischen Verhalten under der spezifischen Wärme bei festen Körpern mit einatomigen Molekül. Offprint from Annalen der Physik, 4. Folge, 34. Bd., 1911.

Leipzig: Barth, 1911.

First edition, rare author’s presentation offprint with “Überricht von dem Verfasser” printed on the front wrapper, of Einstein’s second important paper on specific heats (the first was published in 1907). The problem of specific heat – the measure of heat or thermal energy required to increase the temperature of a unit quantity of a substance by one unit – was investigated by several physicists in the nineteenth century, beginning with Pierre Louis Dulong and Alexis Petit, who formulated the Dulong-Petit law stating that the specific heat of all monatomic solids was close to constant. The classical theoretical interpretation of this law was given by Boltzmann in the 1870s, but he was unable to explain some puzzling exceptions to the law, in particular the specific heats of diamond (carbon), boron and silicon. “By the end of the first decade of the twentieth century, three major quantum theoretical discoveries had been made. They concern the blackbody radiation law, the light-quantum postulate, and the quantum theory of the specific heat of solids. All three arose from statistical considerations. There are, however, striking differences in the time intervals between these theoretical advances and their respective experimental justification. Planck formulated his radiation law in an uncommonly short time after learning about experiments in the far infrared that complemented earlier results at higher frequencies. It was quite a different story with the light-quantum. Einstein’s hypothesis was many years ahead of its decisive experimental test … the story is quite different again in the case of specific heats. Einstein’s first paper on the subject, submitted in November 1906, contains the qualitatively correct explanation of an anomaly that had been observed as early as 1840: the low value of the specific heat of diamond at room temperature. Einstein showed that this can be understood as a quantum effect” (Pais, p. 389). The fundamental quantity characterizing the specific heat of a particular solid in Einstein’s theory was the frequency v of atomic vibrations. In the present paper Einstein was able to derive an equation expressing v in terms of the compressibility of the solid along with its density and molecular weight, and hence to determine v from the elastic properties of the solid. “Einstein’s work on specific heats is above all important because it made clear for the first time that quantum concepts have a far more general applicability” (Pais, p. 394).

“Remarkably little experimental evidence in support of the consequences of the light quantum hypothesis had accumulated by 1910. Experiments on the photoelectric effect had not yet produced unequivocal proof for the validity of Einstein’s equation. Although Einstein had suggested in 1905 that photochemical processes should provide a testing ground for the light quantum hypothesis, in 1909 he thought that such processes were not well suited to test his ideas because they seemed to show the presence of a threshold for excitation. But significant support for quanta suddenly arrived from an unexpected quarter. In March 1910 Einstein received a visitor from Berlin – Walther Nernst – an important visitor with important news. Nernst was Professor of Physical Chemistry, a distinguished researcher who had recently formulated a new heat theorem that would acquire the status of a third law of thermodynamics. He was also a force to be reckoned with in German science. To provide evidence for his thermodynamic theory and to explore its implications, Nernst had mounted a major research program in his laboratory to measure thermodynamic properties of many substances as functions of temperature. His theorem required in particular that the molar specific heats of all elements should approach a definite limit as the temperature approaches absolute zero. The measurements of the specific heat of solids at low temperatures required the development of new methods by Nernst and his coworkers, and it was only in the early months of 1910 that results were obtained. These results were indeed consistent with Nernst’s theorem, but they went beyond it. All the specific heat curves obtained down to liquid air temperatures had shown a marked decrease with decreasing temperature, and as Nernst remarked, ‘one gets the impression that the specific heats are converging to zero as required by Einstein’s theory.’ The theory he referred to had been published by Einstein at the beginning of 1907, but this was the first indication that an experimenter had tested it seriously.

“Nernst, a man for direct action, decided to have a look at the Einstein who had so impressively predicted the result of his measurements on the basis of what Nernst later called ‘a rule for calculation, and indeed one may say a very odd, even a grotesque one.’ Since he was going to Lausanne for a brief holiday, Nernst made a stop at Zürich on the way, called on Einstein, and told him about the new results that he had just presented to the Prussian Academy of Sciences. Einstein was delighted with the news. A week later he wrote to Laub, mentioned Nernst’s visit, and added: ‘The quantum theory is established, as far as I am concerned. My predictions concerning specific heats seem to be brilliantly confirmed.’ For his part Nernst was deeply impressed by Einstein; just how deeply is apparent in this quotation from a letter he wrote to a friend:

‘I believe that so far as the development of physics is concerned, we can be very happy to have found such an original young thinker, a ‘Boltzmann redivivus’; the same intensity and rapidity of comprehension – a great theoretical boldness, which, however, cannot do any harm because the most intimate contact with experiment is maintained.

‘Einstein's ‘quantum hypothesis’ is probably one of the most remarkable ever devised; … if it is false, well, then it will remain for all time ‘a beautiful memory’.’

“Einstein had developed his theory of the specific heats of solids in 1907 on the basis of a simple model: the N atoms of a solid vibrate independently about their positions of equilibrium, and all 3N of these vibrations have the same frequency v. The average energy of each such oscillation was then assumed to be given by the same formula Planck had used in his theory of black-body radiation. The frequency v was the only quantity characterizing a given solid, and the key question, therefore, was how to determine this frequency from some measurable property of this solid, a property other than the specific heat. For those substances that absorb infrared radiation, Einstein suggested that the frequency at which this absorption is a maximum – the frequency of the residual rays (Reststrahlen) – should be identified with the frequency of atomic vibrations. Since not all substances exhibit this phenomenon, there was no independent general method available for finding the atomic vibration frequency of Einstein’s theory.

“When Nernst’s experiments confirmed beyond any doubt that Einstein’s theory captured the essential features of the behavior of specific heats with temperature, Einstein returned to the problem of determining v, the frequency of atomic vibrations. Since the same forces give rise to both the atomic vibrations and the elastic properties of the solid, it is not surprising that Einstein was able to derive an equation expressing v in terms of the compressibility of the solid along with its density and molecular weight. Einstein was not alone in making this connection between an elastic constant and an optical property (the absorption frequency) of a solid. The Australian physicist William Sutherland had been working for some time on a theory of solids composed of particles interacting through electrical forces, and Einstein built on Sutherland’s work. The recent work of Sutherland on the connection between optical and elastic properties of solids was made possible by Heinrich Rubens’s measurements of the frequencies of the residual rays, measurements that involved techniques for working further into the infrared than had previously been possible. What was new and unique in Einstein’s approach was the linking of specific heats to the optical and elastic properties, a linking made possible by his introduction of quantum ideas. In fact, despite some rather rough approximations in his calculations, Einstein found a ‘truly surprising’ agreement between his frequency calculated from the compressibility and one determined from Nernst’s specific heat data for silver, the one substance so far for which both compressibility and the temperature dependence of the specific heat had been measured” (Papers, pp. xxi-xxiv).

Weil *39. Shields, Bibliography of the writings of Albert Einstein to October 1949, 38. The Collected Papers of Albert Einstein, Vol. 3: The Swiss Years: Writings 1909-1911. Pais, Subtle is the Lord, 1982.

8vo (224 x 145 mm), pp. 170-174. Original printed wrappers (lightly creased horizontally).

Item #5066

Price: $15,000.00