Einstein and the Quantum (17 page)

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, h
υ
, 2h
υ
, 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 noted, for a mass oscillating periodically back and forth in each of three directions the equipartition principle predicts 3
kT
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
, 2
hυ
, 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.

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 difficulties
³
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 3
kT
/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.