We can now trace the course of events when one or more masses of gas are left to themselves in the undisturbed aether [i.e., in contact with radiation] … a transfer of energy is taking place between the principal degrees of freedom of the molecules and the vibrations of low frequency in the aether. This … endows the aether with a small amount of energy…. After this a third transfer of energy begins to show itself, but the time required for this must be measured in millions and billions of years unless the gas is very hot.
In the technical language of thermodynamics, Jeans was dropping the assumption that matter is “in equilibrium” with radiation—the idea that radiation and matter have interacted long enough to find the most probable distribution of energy and that if one waited a long time, and measured blackbody radiation over and over again, the energy distribution would not change.
In 1905, at just about the same time that Einstein was writing his paper on quanta, Jeans and Rayleigh argued this question in a series of letters to the journal Nature. At this point Rayleigh, dropping his unjustified fudge factor, published what became known as the Rayleigh-Jeans law8 in 1905: ρ(υ) = (8πυ2/c3)kT. As before, ρ(υ) is the mathematical function expressing the radiation law—Rayleigh’s classical version of the Planck law. What does it say? Well first it says that energy density of radiation at frequency υ is proportional to kT. Since the equipartition theorem says that each radiation mode must have kT of energy, the factor in front, (8πυ2/c3), must represent the number of such modes per unit volume (hence their density). But note that this factor has a crazy feature, the one that Einstein noticed: it is proportional to the square of the frequency, implying that more and more energy is held by radiation of higher and higher frequencies. In fact the proposed law is identical to the one that Einstein wrote down and immediately rejected in his 1905 paper on light quanta, because if matter were in equilibrium with radiation the law leads to the ultraviolet catastrophe. This catastrophe is only being postponed temporarily, in the view of Jeans; the entire material universe is fighting a losing battle against radiation. The only reason we haven’t all frozen to death is that we are losing the battle at an imperceptible pace.
You don’t get away with this kind of maneuver scot-free. Things are interconnected in physics; in thermodynamics we almost always assume that nature is in thermal equilibrium in order to explain how things work. If you assume blackbody radiation is not in equilibrium, how, for example, do you explain the well-verified Stefan-Boltzmann law for the total energy radiated by a blackbody, which requires thermal equilibrium? A lucky accident?
Moreover the detailed agreement of the Planck formula with measurement would also have to be a lucky accident. And if you think hard, the coincidences required by the Jeans hypothesis multiply rapidly.9 Planck, steeped in detailed experimental data, hardly took the Jeans idea seriously when he mentioned it in his textbook on thermodynamics in 1906. Privately he was even more scathing, commenting about Jeans in a letter to Wien, “he is the model of the theorist as he should not be, just as Hegel was in philosophy; so much the worse for the facts if they don’t fit.” Nonetheless, the Jeans “slow catastrophe” model remained under serious discussion for the remainder of the decade.
A puzzle always looks simpler after you know the answer. It hard for us now to believe that many outstanding scientists could accept such a flimsy explanation. But, from the perspective of physicists at the time, was a radical failure of Newtonian mechanics at the atomic level really a more attractive option than dropping the assumption of thermal equilibrium? Somehow Einstein intuited that it was. And he set out to substantiate his view shortly after his acceptance of the Planck law in 1906. He did this by looking at the very same physical property that had troubled Maxwell and Rayleigh, and had led Jeans to embrace his radical alternative to the Planck law. He reexamined the specific heat of matter, but not in the gaseous state; in the solid state instead. Ultimately this work would sweep away any hope that atoms might obey classical mechanics.
1 Routh was known to be so sparing in his praise that when another First Wrangler, Lord Fletcher Moulton, produced an almost unheard-of perfect exercise, the only comment on the paper from Routh was “Fold neatly.”
2 He was recognized for his discovery of the element argon in the atmosphere.
3 See appendix 2 for a graph showing the three radiation laws as a function of frequency.
4 For a gas particle moving in a single dimension, Egas = kT/2, half that of the oscillator. This is because the gas particle has only kinetic energy and no potential energy, which for the oscillator gives an equal contribution, Emol = kT/2 + kT/= kT. However, for the realistic case of a gas particle moving in all three directions in space, Egas = kT/2 + kT/2 + kT/2 = 3kT/2.
5 When Rayleigh published his law with a more detailed and careful discussion in 1905, he clearly pointed out that if equipartition held for all frequencies, one would get an infinite blackbody radiation energy, which is absurd and implies some type of failure of the equipartition principle.
6 Editing his paper for a collection in 1902, Rayleigh added a footnote claiming that in 1900 he really meant his law to apply only to the low-frequency behavior anyway, and thus took the experiments of Rubens and Kurlbaum as vindicating his guess. The fudge factor was not mentioned. He was by then aware of Planck’s correct guess and stated it.
7 Ironically, the big bang theory was itself definitively validated by the observation of the blackbody radiation it produced.
8 Rayleigh published this formula with a trivial error: he neglected to include both polarizations of light, which gave too small a result by a factor of eight. Jeans corrected the error, and the law is now universally named after the pair. Einstein derived the same law independently in 1905, publishing it about a month earlier than Rayleigh, but his name is never associated with the law. Until Rayleigh’s initial error was found, his formula did not agree with the low-frequency limit of Planck’s law. Rayleigh noted that a comparison of the approaches would be helpful, but “not having succeeded in following Planck’s reasoning, he declared himself “unable to undertake it.” Once the error was corrected, the two laws agreed perfectly at low frequencies.
9 Rayleigh, to his credit, never fully endorsed Jeans’s slow catastrophe theory. Saying merely that for short wavelengths “there must be some limitation on the principle of equipartition.” This limitation was provided by Planck and Einstein, but Rayleigh was not convinced, writing in 1911: “Since the date of these [1905] letters further valuable work has been done by Planck, Jeans, Lorentz, … Einstein and others. But I suppose the question can hardly be considered settled.”
CHAPTER 13
FROZEN VIBRATIONS
The whole thing started with a kind of interpolation formula by Planck. Nobody wanted to accept it because it did not appear logical … half the argument was continuous and the other half was based … on quanta of energy. The only man who appeared sensible was Einstein. He had the feeling that if there was anything to Planck’s idea it must appear in other parts of physics.
—NOBEL LAUREATE PETER DEBYE, 1964
Einstein’s nemesis in his student days, Professor Heinrich Weber, may have come close to depriving posterity of Einstein’s historic genius, but now Herr Weber would indirectly play a crucial role in the flowering of that genius. In 1875 the young Weber, then an assistant to Helmholtz in Berlin, had just completed the best experiments extant on the specific heat of solids. The effect he was studying was an apparent violation of the empirical law noted by Pierre Dulong and Alex Petit fifty-six years earlier in 1819. These French researchers had discovered that pretty much every solid they measured had the same specific heat, once one took into account the difference of the atomic weight of the constituents. For example, a copper atom’s weight is about 60 percent that of a silver atom, so 0.6 grams of copper and one gram of silver have the same number of atoms, and would then also be found to have the same specific heat. Even at that early date Dulong and Petit interpreted their find
ing in terms of the properties of underlying atoms, stating boldly that “one is allowed to infer … the following law: the atoms of all simple [elements] have exactly the same heat capacity.” Later they would restrict this optimistic assessment to atoms in solids; as already noted, gases were behaving in a strange manner, which would puzzle Maxwell, Rayleigh, and others for another eighty-seven years.
What exactly is specific heat, and why did it suggest something about atoms? Specific heat is a number that characterizes a chunk of stuff (solid, liquid, gas); it is the amount the thermal energy in a gram of that stuff changes when you change its temperature by one degree centigrade.1 Thus it measures how thermal energy varies with temperature. We have already learned that if one trusts Newtonian mechanics on the atomic scale, and the laws of statistics, then the energy of each vibrating structure bears the simplest possible relation to temperature, Emol = kT; for atoms in a solid there are three independent directions of vibration, so according to the equipartition relation one gets 3kT of energy per atom.2 If this relation holds, then if you change the temperature by one degree, the energy per atom changes by exactly 3k, independent of the type of atom involved. This is exactly the law found by Dulong and Petit. But by 1906 Einstein had seen the red flag waving here; this argument relies completely on the equipartition principle, precisely the notion that he realized had failed for blackbody radiation. Thus the specific heat of solids would provide his next opportunity to extend quantum concepts, bolstered by the experimental work of his erstwhile opponent.
It is unlikely that the young Heinrich Weber would even have known the statistical theory underpinning the Dulong and Petit law when, between 1872 and 1875, he decided to test it carefully. However, earlier measurements on solids had hinted that the relation was not quite as trustworthy as its discoverers had originally thought. One elemental solid was a particularly “bad actor,” one that had required “difficulty and expense” to study: diamond. Diamond, the hardest of the elemental solids, refused to give up its full quota of energy when its temperature was lowered one degree, registering a specific heat less than 30 percent of the expected Dulong-Petit value. Not only was diamond miserly with its heat energy; the measured values of its specific heat reported by various experimenters did not even agree. That is where Weber came in.
Weber, the eventual staid professor, in his youth was not averse to bold hypotheses, and so he made one: the specific heat of solids is not constant at all but can vary widely as the overall temperature is varied. This conjecture was in complete disagreement with the equipartition principle, of course, but given his distrust of theory, this would not likely have swayed Weber even had he known of it. With this hypothesis the different values of diamond’s specific heat could be reconciled, as they corresponded to measurements made at rather different starting temperatures. Weber suspected that somehow the Dulong-Petit value of 3k per atom was only reached at high enough temperatures, and for some reason, in the case of diamond, room temperature wasn’t high enough. He thought that if he could cool diamond samples well below room temperature, he would find even larger deviations from that value. His work predated all the breakthroughs in cryogenics that now make it possible routinely to lower the temperature of a solid to hundredths of a degree above absolute zero (−273°C). Poor Weber had to rely on natural ice to do his measurements at low temperature, and needed to suspend them in March of 1872 due to the lack of available snow!
By 1875 Weber had pushed his experimental technology to a higher level and was able to present beautiful measurements of the specific heat of diamond, varying the temperature from −100°C to +1,000°C. Sure enough, at the highest temperatures the specific heat of diamond increased until it attained the Dulong-Petit (DP) value and then stopped increasing, whereas as the temperature was lowered below normal room temperature it continued to decrease down to one-fifteenth of the DP value. Moreover other elemental solids showed a similar but less dramatic variation with temperature. Weber’s basic hypothesis was right. For some reason, for most materials, room temperature is hot enough that the DP law initially appeared universal; but for diamond and a few others it is not. And most puzzling of all, at very low temperatures diamond and other materials appeared to lose completely the ability to emit or absorb heat energy when the temperature was changed; their specific heat seemed to disappear. Walther Nernst, who studied with Weber prior to becoming the preeminent physical chemist of his generation, described the situation thus: “through the diamond experiment one has therefore found that the atomic vibrations can be brought to a standstill. As soon as this happens, the concept of heat does not any longer exist for the ‘dead body.’ ” Weber had made a great experimental discovery, the greatest of his career; eventually it landed him a full professorship at the Poly, leading to his fateful encounters with Einstein.
Recall that Einstein lauded Weber’s course on heat during their brief honeymoon period. Einstein’s course notes from that time have actually survived, but they contain no evidence that Weber discussed his own discovery, the strong temperature variation of specific heat. Nonetheless we have already noted that by 1901 Einstein, in a letter to Mileva, announced that he had been considering “the latent [specific] heat of solids” in connection with Planck’s radiation formula, and that his views on latent heat had changed because his views on radiation theory had “sunk back into the sea of haziness.” Thus it is safe to assume he was by then aware of Weber’s systematic demonstration of anomalous behavior. Now, in early 1906, Einstein’s views on radiation were no longer hazy: Planck’s formula was right, equipartition was wrong, and Newtonian mechanics was in jeopardy. It was time to see if the heretical ideas relating to quanta could clean up the specific heat anomalies just as they had explained the odd behavior of the photoelectric effect. By November of 1906, eight months after his paper announcing that the Planck formula required light quanta, Einstein submitted his second great work on quantum theory to Annalen Der Physik, titled “Planck’s Theory of Radiation and the Theory of Specific Heat.”
Einstein’s papers in general have a more philosophical tone than typical physics papers, even those of the time. And so after an introductory review of his 1905 and 1906 papers on light quanta, he presents the following ontological dictum to the (in all likelihood dumbfounded) reader:
For although one has thought before that the motion of molecules obeys the same laws that hold for the motion of bodies in our world of sense perception … we must now assume … that the diversity of states that they can assume is less than for bodies within our experience. For we make the additional assumption that the mechanism of energy transfer is such that the energy of elementary structures can only assume the values 0, hυ, 2hυ, etc.
This is the statement of quantization of energy at the atomic scale, as clear and unequivocal as one would find in a modern physics textbook. Einstein, not Planck, said it first. Discontinuity is not a mathematical trick; it is the way of the atomic world. Get used to it.
Einstein continues:
I believe we must not content ourselves with this result. For the question arises: If the elementary structures … cannot be perceived in terms of the current molecular-kinetic theory [of heat], are we then not obliged also to modify the theory for other periodically oscillating structures considered in the molecular theory of heat? In my opinion the answer is not in doubt. If Planck’s radiation theory goes to the root of the matter, then contradictions between the current molecular-kinetic theory and experience must be expected in other areas of the theory of heat as well, which can be resolved along the lines indicated. In my opinion this is actually the case, as I now shall attempt to show.
The argument from here is remarkably straightforward. Atoms form a solid when they arrange themselves in a regular pattern in space, held together by electrostatic interactions. Einstein states that the simplest picture one may have of heat energy stored in a solid is that all the atoms “perform [periodic] oscillations around their equilibrium positions.” As already not
ed, for a mass oscillating periodically back and forth in each of three directions the equipartition principle predicts 3kT of energy per atom, yielding the DP value for the specific heat. But, Einstein notes, several elements (diamond, boron, silicon) have smaller specific heat than expected from this law, and compounds containing oxygen and hydrogen also show similar violations. Finally, he notes that Drude identified other kinds of oscillations in solids, involving the electrons, which appear to be important in how solids absorb light but don’t seem to contribute to the specific heat. But the equipartition principle requires that all oscillations get their share of energy, so these “extra” oscillations should cause the specific heat of solids to actually exceed the DP value, which was not observed. So something is out of kilter.
The atoms in a solid were really no different from the “elementary resonators” in Planck’s blackbody radiation theory (which were held in place by electric forces but could vibrate in all three directions around their equilibrium values), and Einstein had already announced in his 1906 paper that such vibrating structures can only have energies equal to an integer times their frequencies, E = 0, hυ, 2hυ, etcetera. Thus each atomic vibration has a ladder of allowed energies separated by hυ. But the typical amount of energy available to each atom from its thermal environment is just the equipartition value, kT (per direction of vibration). So what happens if the quantized energy of the atomic vibration, hυ, is much larger than kT? The atom then is like a man trying to climb a ladder whose rungs are much farther apart than his reach. It can never get off the lowest “rung”; its vibrational energy remains stuck at zero.
Thus some modes of vibration are “frozen out”; their first nonzero quantized energy level is too high to absorb the amount of energy dictated by the Dulong-Petit (equipartition) law. Moreover, it makes sense that these “missing vibrations” would disappear first in materials that are very hard, like diamond. Roughly speaking, a material is harder if its atomic constituents are more tightly bound in place, so that they vibrate very rapidly when disturbed from equilibrium. But if they vibrate very rapidly, then their frequency is unusually high, so that the energy-level spacing of that material, hυ, is unusually large. Thus, when compared over the same range of (decreasing) temperature, their vibrations freeze into the lowest level before those of a softer solid. This paucity of high-frequency atomic vibrations is of course conceptually linked to the “missing” high-frequency modes of thermal radiation that characterizes the Planck law and that so puzzled Rayleigh. Einstein had now realized that quantum freezing of vibrations is also the ultimate explanation for the strange behavior of the specific heat of solids.
Einstein and the Quantum Page 12