Einstein and the Quantum

by Stone, A. Douglas

  However, to actually get a precise formula for the quantum specific heat of a solid that he could compare to data, Einstein decided to make a simplified model of a vibrating solid. Any system in mechanical equilibrium will oscillate back and forth when it is given a little energy; think of a pendulum pushed a bit to the side from the vertical. But the frequency of the oscillations depends on the details of the system, and for a solid made up of an enormous number of atoms there are many different types of oscillatory motions with many different frequencies, depending, for example, on the chemical bonding arrangements of the constituent atoms. This set of different frequencies was too complicated to work out at the time (modern quantum physicists can do it with incredible precision), so to compare his theory’s prediction to the measurements for diamond, Einstein assumes that it has only a single, primary frequency of vibration. He is quick to point out that, given this simplification, “of course an exact agreement with the facts is out of the question.”

  Nonetheless, this assumption gives him an approximate law for the temperature variation of specific heat based directly on Planck’s expression for the energy of a single oscillator of frequency υ; and this expression shows remarkably good agreement with Weber’s data. He remarks, “both above-mentioned difficulties3 are resolved by the new interpretation and I believe it likely that the latter will prove its validity in principle.” In fact, according to Einstein’s new theory, the specific heat of all solids decreases with decreasing temperature until, at the absolute zero of temperature, it completely disappears—a stunning prediction.

  But there was one further radical step to take. Throughout this paper Einstein assumes that the same molecular vibrations that store heat also exchange energy with radiation through emission and absorption, thus closely tying the specific heat formula to the blackbody law. But after submitting his paper he recalled that there are molecular vibrations that do not interact with radiation at all, and that such vibrations can still store heat and contribute to the specific heat.4 Einstein realized that this was an important observation and actually published a note of correction, stating, “most certainly there could exist uncharged heat carriers [vibrations], i.e. such ones that are not observable optically.” But if neutral vibrations, those that do not interact with radiation, were also subject to the law of quantization of energy, then whatever the quantum theory was, its domain was not merely the interaction of radiation with matter, as Planck had hoped. The disease of discontinuity was present in matter without radiation; Newtonian atoms had frozen to death.

  FIGURE 13.1. Graph from Albert Einstein’s 1907 paper which predicts that the specific heat of all solids should go to zero as the temperature is lowered, due to quantization of vibrational energy. Here the theory (dashed line) is being compared to Weber’s data for the temperature variation of the specific heat of diamond. Courtesy the Albert Einstein Archive.

  1 Thermal energy and temperature are distinct concepts in thermodynamics. Suppose I heat a glass of water and a bathtub full of water with a blowtorch for ten seconds. Each receives the same amount of thermal energy but the change in their temperature is very different.

  2 This factor of 3 is the same one that gives us 3kT/2 for atoms in a gas, coming from the fact that the atom can move in all three spatial dimensions.

  3 The disappearance of specific heat at low temperature, and the absence of “extra” specific heat due to optical-frequency electronic vibrations.

  4 These are vibrations that do not generate a net dipole moment.

  CHAPTER 14

  PLANCK’S NOBEL NIGHTMARE

  The two constants [h, k] … which occur in the equation for radiative entropy offer the possibility of establishing a system of units for length, mass, time and temperature which are independent of specific bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and non-human.

  —MAX PLANCK

  It was the fall of 1908, and Svante Augustus Arrhenius was determined to see that Max Planck received the Nobel Prize for Physics that year. Arrhenius, a scientist of impressively broad and bold speculations, had recently returned from a tour of Europe, where he was received warmly as befitted the first Swedish winner of the newly minted Nobel prizes. Arrhenius had won the Chemistry Prize in 1903 (two years after the establishment of the awards) for his groundbreaking work on electrolytic chemistry. He was widely recognized as a founder of the discipline of physical chemistry, which works at the boundary of the fields of physics and chemistry. In 1905 he had been offered a professorship in Berlin but had turned it down to remain in Sweden and head the new Nobel Institute for Physical Research; after receiving the prize he would be a member of the Nobel Award Committee in Physics and a de facto member of the Chemistry Committee for the remainder of his life. As such he had enormous influence over who received these awards, and he did not hesitate to use that influence.

  Arrhenius, like all his contemporaries, was blissfully unaware of the looming crisis in atomic physics, uncovered by the work of the young Einstein, who was now becoming known—not for challenging the Newtonian paradigm of continuous motion but instead for dismissing another Newtonian axiom, the concept of absolute time. While Einstein had quickly moved to the terra incognita of the nascent quantum theory, assuming that atoms existed and trying to figure out their laws of motion and their interactions with radiation, Arrhenius was still fighting the last war, the war to prove that atoms were real. The ensuing episode illustrated just how oblivious the scientific community was to the gathering storm.

  Had Arrhenius known the story of the checkered career of the German/Swiss Jew, who was still not recognized formally by the conservative professoriate of Switzerland in 1908, he likely would have recognized a kindred spirit. Arrhenius grew up near Uppsala, Sweden, where his father was a surveyor for the University of Uppsala, the oldest and among the most prestigious of the Nordic universities. A science and math prodigy, he had matriculated at the university at age seventeen, and received his degree in two years, before moving on to graduate studies in physics. However, in a striking parallel to Einstein, he alienated the senior members of the faculty, Tobias Thalen (physics) and Per Theodor Cleve (chemistry), and left after three years to complete his doctorate at the new Physical Institute of the Academy of Sciences in Stockholm. Unfortunately for Arrhenius the new institute was not yet allowed to grant PhDs on its own. Thus when, in 1884, he produced a monumental 150-page work on the conductivity of electrolytic solutions, explaining, for example, the high conductivity of salt in water by its dissociation into ions, it was received with great skepticism by a committee consisting mainly of faculty whom he had spurned at Uppsala. In the end the thesis was approved with the lowest possible passing grade, non sine laude approbatur, (“accepted, not without praise”). Forty years later Arrhenius would recount bitterly that Cleve and Thalen even refused to offer him the customary congratulations after the doctoral ceremony, saying that they had decided to “sacrifice him.” Although this work and its extensions would eventually earn him the Nobel Prize, the grade it had received was so poor that he was at least nominally disqualified from pursuing an academic career in Sweden at the time.

  Here, however, his story diverges from that of Einstein, for he boldly sent the devalued thesis to the leading lights of European chemistry and physics, Clausius (inventor of the concept of entropy), van ‘t Hoff in Amsterdam (who would be the first Nobel Laureate in Chemistry), and Ostwald in Riga (the ninth Nobel chemistry laureate). One of these men, Ostwald, immediately recognized its innovativeness, to the extent that he even traveled personally to Uppsala to offer Arrhenius a job at his own institution.1 Arrhenius did not cut a particularly impressive figure, according to Ostwald: “[Arrhenius] is somewhat corpulent with a red face and a short mustache, short hair; he reminds me more of a student of agriculture than a theoretical chemist with unusual ideas.” But a brilliant chemist he was, and eventually Arrhenius did move to
Europe and trained with Ostwald, van ‘t Hoff, and even with Boltzmann before returning to Sweden to become the unquestioned leader of Swedish physical chemistry, and the person who defined the international scope of the Nobel prizes at their inception.

  A decade later, at the turn of the century, there was still a major movement in chemistry and physics that regarded atoms as somewhat suspect heuristic entities, a movement led by Arrhenius’s former mentor, Wilhelm Ostwald. This school of thought was known as “energetics” and also had adherents in the Swedish physics community, which maintained an attitude of distrust toward theory in general and of “pronounced hostility toward atomism and toward atomic theory” in particular. Arrhenius had decided to put this movement to final rest and make 1908 the Nobel Year of the Atom. Max Planck would receive the physics prize for the manner in which his radiation law had led to an accurate determination of Avogadro’s number and the elementary unit of atomic charge, e. The chemistry prize would be awarded to the British physicist Ernest Rutherford, who had shown that atoms disintegrated (i.e., emitted doubly ionized helium atoms, known as alpha particles) during radioactive decay. In a very recent experiment with Geiger, Rutherford had deduced a value of the elementary charge from alpha particles in excellent agreement with that calculated by Planck using his radiation law, tying the two prizes neatly together.

  The fact that Rutherford considered himself a physicist and would be very surprised to know that he had been reclassified a chemist2 did not deter Arrhenius from his plan. Arrhenius had nominated Rutherford for both the physics and chemistry prizes that year, but it is likely that he had planned all along to support Planck in the Physics Committee, of which he was a member. By the time of the crucial meeting on September 18, 1908, he knew that the Chemistry Committee (based on an internal report he had apparently ghostwritten) was committed to awarding the prize to Rutherford. Planck and Wien had been jointly nominated in physics for the theory of heat radiation by Ivar Fredholm, a Swedish mathematician and mathematical physicist, and Arrhenius swung his support to this nomination, but with the intention of splitting the ticket and engineering a prize for Planck alone.

  Why did Arrhenius think that Planck alone should be recognized? Because at that time Arrhenius was not interested in the physical principles behind the law of thermal radiation3 so much as in its connection to the fundamental constants in molecular chemistry. This is an aspect of Planck’s work of 1900 that is barely mentioned in modern times, but at that time it overshadowed his radical quantum hypothesis. Planck’s radiation law depended on the two newly discovered physical constants that he introduced, h, the “quantum of action” (Planck’s constant), and k, Boltzmann’s constant (the constant associated with entropy through the equation S = k log W and thermal energy through the equipartition relation Emol = kT.) From a careful fit of blackbody radiation data one can extract quite precise values for both h and k, and Planck did so immediately after deriving his radiation law in 1900. The constant h appeared to him completely enigmatic and was not put to any immediate use, but the constant k, which only later became known as Boltzmann’s constant,4 was instantly recognized as providing a theoretical microscope for studying the atom.

  In his December 1900 magnum opus Planck states, “To conclude, I may point to an important consequence of this theory which at the same time makes possible a further test of its reliability.” He goes on to show by straightforward steps that the Boltzmann constant satisfies the simple relationship k = R/Na, where R is the constant in the ideal gas law PV = RT for a mole of gas, and Na is Avogadro’s number (which has struck fear into so many beginning chemistry students), the number of atoms contained in a mole of any gas. This number was imperfectly known in 1900, whereas R was very well known. Hence by extracting k very precisely from the radiation law, Avogadro’s number could be determined to unprecedented precision. Planck found the value Na = 6.175 × 1023, which is within 2.5 percent of the currently accepted value 6.022 × 1023. Using the same information, he could determine the mass of a hydrogen atom, again with high accuracy. Finally, in a coup that must have impressed the physical chemist, he used considerations from electrolytic chemistry, Arrhenius’s own field, to find the elementary charge on a proton, obtaining a value within 2.5 percent of the modern value. In contrast the best-known value of e, the charge on an electron, measured by J. J. Thomson from electron studies, was off by 35 percent! Planck concluded his 1900 analysis with the confident declaration, “If the theory is at all correct, all of these relations should not be approximately, but absolutely valid. The accuracy of the calculations … is thus much better than all determinations up to now.”

  Planck had always been fascinated by fundamental constants as expressions of the absolute and eternal in physics. Even before his work of 1900 he had realized that the radiation law involved two distinct and new fundamental constants. Fundamental constants allow one to define what are called absolute units, units of measurement relating to the basic laws of physics. For example, the speed of light, c, provides a natural unit of velocity, because no signal can travel faster than c and all relativistic phenomena become more and more important as this speed is approached. In the famous twin paradox of special relativity, your identical twin ages more and more slowly compared with you as her relative velocity approaches c. Planck pointed out that his two newly discovered constants, when combined with the speed of light and the gravitational constant, would allow fundamental units to be defined for all physical quantities (length, time, temperature, etc.). Transported by this revelation, the staid professor allowed his inner geek to emerge in print, rhapsodizing that these units would be valid for “all times and civilizations … even extraterrestrial ones.” Later, when Planck became embroiled in a philosophical debate with the Viennese philosopher-scientist Ernst Mach, Mach would lampoon his exuberance over fundamental units: “concern for a physics valid for all times and all peoples, including Martians, seems to me very premature and even almost comic.”

  Nonetheless in 1900 it was these fundamental constants, which had emerged from his radiation law, that most excited Planck, and not his unexamined introduction of discontinuity into the laws of physics. To his disappointment, the rest of the physics community did not immediately appreciate even this aspect of his breakthrough. He later recounted:

  I could derive some satisfaction from these results. But matters were viewed quite differently by other physicists. Such a calculation of an elementary electrical [charge] from measurements of thermal radiation was not even given serious consideration in some quarters. But I did not allow myself to become disturbed by such a lack of confidence in my constant k. Nevertheless, I only became completely certain on learning that Ernest Rutherford had obtained a [very similar] value by counting alpha particles.

  This spectacular agreement between completely disparate physical phenomena, all pointing to a single consistent atomic picture of the world, had convinced Arrhenius that Planck alone should be recognized with the physics Nobel Prize in 1908. It was the connection to fundamental constants that distinguished Planck’s work from Wien’s in Arrhenius’s mind, and in his report to the Nobel Physics Committee he barely mentioned Planck’s derivation of the radiation law and completely omitted any mention of “quanta of energy.” Planck’s use of the constant k, he said, had “made it extremely plausible that the view that matter consists of molecules and atoms is correct…. No doubt this is the most important offspring of Planck’s magnificent work.”

  Arrhenius’s enthusiasm did not sweep through the conservative Physics Committee unchallenged. Among its members was the distinguished experimentalist Knut Angström, who had actually done experiments on heat radiation and was aware of the experimental prehistory leading up to Planck’s “act of desperation.” With much justice he wrote, “it is very far from being that the theoretical works have guided the experimental ones, but rather that one could justly make a completely contrary statement.” However, there was a small problem with his argument that an experiment
er should receive or share the prize: none had been nominated that year. Angström and the other skeptics on the Physics Committee were reluctantly convinced by Arrhenius to join the Planck bandwagon.

  And so the modest, upright Planck (who had himself nominated Rutherford for the physics prize that year) might have received this honor, not because of a deep appreciation of the true significance of his work, elucidated by Einstein from 1905 to 1907, but rather because of a general ignorance of its full implications. After the Physics Section of the Swedish Academy had approved Planck as the awardee, rumors of the result quickly traveled around the continent, apparently reaching Planck himself, who stated to the press, “[if true] I presume that I owe this honor principally to my works in the area of heat radiation.” But the full Swedish Academy would still have to approve the recommendation of the Physics Committee, and in the interim between these votes something had changed the mood in Stockholm. The most famous theoretical physicist of his generation, the man Einstein admired the most, had finally spoken publicly on the Planck law, and his opinion would derail Arrhenius’s well-laid plans.