Einstein and the Quantum (16 page)

But this was not the end of the story. A much younger Cambridge physicist, James Hopwood Jeans, had just completed his studies (graduating as Second Wrangler) and took up a research position, devoting himself to the theoretical study of gases. Jeans was a more flamboyant personality than Rayleigh, later specializing in astrophysics and cosmology and introducing the steady-state model of the universe, which was invalidated by the big bang theory.
⁷
It was precisely to blackbody radiation that Jeans turned in 1904 when he published his treatise
The Dynamical Theory of Gases
. In this work he expressed a more definitive view of the situation than Rayleigh, leading to a shocking conclusion: the equipartition theorem is valid for all frequencies, and the ultraviolet catastrophe is happening. “
If an interaction between aether and matter
exists, no matter how small this interaction may be … we are
led to the conclusion that no steady state is possible until all of the energy of the gas has been dissipated by radiation into the aether.”

So why wasn't the entire earth cooled down to absolute zero as the infinity of radiation modes sucked all energy from matter? Easy. It's all happening very, very slowly.

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 law
⁸
in 1905:
ρ
(
υ
) = (8
πυ
²
/
c
³
)
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, (8
πυ
²
/
c
³
), 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,
E
_gas
=
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,
E
_mol
=
kT
/2 +
kT
/=
kT
. However, for the realistic case of a gas particle moving in all three directions in space,
E
_gas
=
kT
/2 +
kT
/2 +
kT
/2 = 3
kT
/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 finding 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,
E
_mol
=
kT
; for atoms in a solid there are three independent directions of vibration, so according to the equipartition relation one gets 3
kT
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 3
k
, 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 3
k
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!