L.
J. Curtis
Models
for atomic structure are often associated with certain individuals, such as the
“Atoms” of Thomson, Rutherford, Bohr, Schrödinger, and Dirac. Although it is seldom specifically
mentioned, many aspects in the development of these of models were suggested or
heavily influenced by Albert Einstein in papers that are primarily remembered
for other applications.
Many
of the epoch-breaking papers of Einstein are today remembered as anecdotal
treatises dealing with isolated phenomena, rather than for the new unified
world view that they collectively comprised.
These papers are often classified by superficial tag lines such as
“Brownian motion,” “Photoelectric effect,” “Special Relativity,” “Specific
Heats of Solids,” “the Planck Radiation law,” “General Relativity,” etc. However,
rather than a series of separate solutions to diverse problems, these
papers represented the development of a new perspective that changed the way we
think about our physical universe.
The
year 1905 was Einstein's “annus mirabilis” -- a date to set beside 1543 (when
Nicolas Copernicus published De
Revolutionibus Orbium Coelestium) and 1686 (when Isaac Newton published Philosophiae
Naturalis Principia Mathematica). Between March and September of that year
Einstein produced four papers on three different subjects, any one of which
could have assured his scientific immortality.
A word of praise must be given to Paul Drude, then editor of Annalen der
Physik, who received and published with dispatch a battery of manuscripts from
an obscure Swiss bureaucrat whose application to become a Privatdozent had been
rejected.
It
is clear why Drude accepted these manuscripts without hesitation. There is a striking clarity of exposition in
these four papers that makes it apparent that these were not to Einstein a
series of isolated studies of separate problems. The papers were instead a
manifestation of a new enlightenment that had come into existence in the mind
of Einstein that made these seemingly different problems become recognizable
pieces of a puzzle that fell uniquely into place.
As
an example of these interrelationships, this article will trace various ways in
which ten papers selected from Einstein's 1905-1925 work have influenced the
development of the atomic hypothesis, or as Richard Feynman has called it “the
atomic fact” which he characterized as “the most important discovery ever
made.” The popular literature has an
unfortunate tendency to portray Einstein's discoveries as counter-intuitive,
implying that nothing is really quite as it seems. Another goal of this article is to show that the opposite is
true. Einstein has provided us with new perspectives from which to view the
world. When we perceive nature with
this enhanced vision it becomes clear that everything is exactly as it seems,
and could not be otherwise. As Einstein
said, “the most incomprehensible thing about the world is that it is
comprehensible.” This article will
attempt to illustrate ways in which Einstein's discoveries regarding atomic structure can made
intuitive, and simplify our understanding of the world around us.
Already
in the fourth century BC, Democritus proposed that all matter consists of then
inconceivably small particles, which he described as ``atoms,'' which meant
“indivisible.” The atomic picture was given added credence in 1662 when Robert
Boyle discovered that air is compressible, and that this compression affects
the pressure in an inverse relationship. From this Boyle concluded that air
must consist of discrete particles separated by a void. In 1808 John Dalton
extended these studies to chemical measures of combining proportions which
indicated that atoms differ from each other in mass. Thus Dalton's work elevated atomism from a philosophical to a
chemical theory. The discovery by
Joseph-Louis Gay-Lussac that all gases expand to the same extent with a rise in
temperature led Amedeo Avogadro to hypothesize in 1811 that equal volumes of
all gases contain the same number of particles. Avagadro also indicated that
these particles may be combinations of individual atoms, which he called
“molecules” (from the Latin ``moliculus,'' meaning ``small mass''). Unfortunately,
despite the fame of Avogadro's name today, his suggestion received little
attention at the time and was rejected by Dalton and ignored by Jöns Jakob
Berzelius (the preeminent chemist of the day, whose symbolic system became the
language of chemistry.) Despite early philosophical formulations and the
subsequent experimental development of atomism, a question remained whether
atoms were real, or only a mnemonic device for coding chemical reactions. The
acceptance by the scientific community of the existence of atoms did not occur
until Einstein's May 1905 paper, the significance of which is usually concealed
under the tag line ``Brownian motion.''
Despite
the importance of the atomic nature of matter to the fields of chemistry and
physics, Einstein's proof of their existence occurred as an outgrowth of
studies by the Scottish botanist Robert Brown. In 1827 Brown was viewing under a microscope a suspension of
pollen grains in water, and noticed that the individual grains were moving
about irregularly. He initially thought
that this might be a result of animate life hidden within the grains, but found
that the same erratic motion occurred when the pollen grains were replaced with
ground glass or dye particles. Although
he had no explanation for the phenomenon, he reported the result, which has
since been called ``Brownian motion.''
Brownian
motion found an explanation in the kinetic theory of gases developed in around
1860 by the Scottish mathematician James Clerk Maxwell. Amazingly the kinetic theory itself owes its
initial formulation not to physics and chemistry, but to the social sciences.
After the French revolution the great mathematician Pierre-Simon Laplace was
required to adapt his work to serve the revolutionary goals, and to educate the
populace through a series of public lectures.
To this purpose Laplace adapted his studies of probability theory
(initiated to rescue “Laplacian determinism” from the measurement imprecision
that he attributed to “human weakness” rather than to innate indeterminacy) to
demography and actuarial determination.
Laplace's lectures were attended by the Belgian astronomer Adolphe
Quetelet, who was inspired by them to formulate the study of
``Staatswissenschaft,'' the forerunner of the modern statistical social sciences.
Quetelet's work was heralded as a cure for societal ills, and was championed by
the social reformer Florence Nightingale.
This subsequently inspired James Clerk Maxwell, through his reading of
an essay on Quetelet's work written by John Herschel, to adopt a strategy using
Laplace's probabilistic methods as a basis for his kinetic theory of gases.
Maxwell's formulation of statistical mechanics marked a turning point in
physics, since (in contrast to Laplacian determinism) it presupposed the operation
of chance in nature. Thus in this case the “exact sciences” borrowed from the
“social sciences.”
It
should be noted that the contributions of Florence Nightingale were of great
significance. Although usually
remembered as a pioneer in nursing, she was also one of the leading
mathematicians of her time. She
developed new techniques of analysis and innovations in the collection,
tabulation, interpretation, and graphical display of statistical data. During the Crimean War she invented the now
familiar ``polar-area diagram'' (pie-chart) to dramatize the needless deaths
caused by unsanitary conditions in military hospitals. Although she lived to see the Einstein era
(she died in 1910) her mathematical interest can be traced to the
post-revolution lectures of Laplace (she was 6 years old when Laplace died in
1827).
Although the kinetic theory of matter
provided a qualitative explanation of Brownian motion, a quantitative
formulation was still lacking. This was
provided by Einstein in his 1905 Doctoral Dissertation and in the May paper of
his annus mirabilis. Einstein attacked this problem using the same
probabilistic methods developed by Laplace, here the “Random Walk,” which is a
sequence of discrete steps, each in a random direction. For a large object, the
number of molecules striking on all sides and from all angles is approximately
equal, so there is no overall effect.
For a smaller object, the number of molecules striking it during a short
interval can be reduced by statistical fluctuations, and small differences in
bombardment can buffet the object about to an observable degree. The larger the
size and mass of the molecules, the larger the size of the object for which
this difference in bombardment can produce detectable results. Using the methods of probability, Einstein
was able to compute the distribution of distances by which the pollen grains
would be expected to migrate as a function of the size of the molecules. Theodor Svedberg at the University of
Uppsala had also suggested a molecular explanation for Brownian motion, but it
was Einstein who produced the mathematical formulation that demonstrated its
correctness.
Publication
of the Brownian motion paper quickly led to the determination of Avogadro's
number. Before Einstein, the number of molecules in a gram molecular weight was
assumed to be constant, but had no definite quantity. By 1908, Jean Perrin used
Einstein's paper to make the first estimate of the number of molecules in a
mole of any substance. Within a decade, 6.02 x 1023 atoms per
gram-mole was on its way to becoming one of the most widely known of the
fundamental constants.
In
the period prior to 1905, a leading proponent of the atomic hypothesis was
Ludwig Boltzmann. Boltzman extended
Maxwell's work in statistical mechanics to obtain the Maxwell-Boltzmann
distribution, which connected atoms to macroscopic phenomena. Boltzmann's atomistic ideas were bitterly
attacked by scientists such as Wilhelm Ostwald, and this has been cited as a
possible contributing cause of the depression and mental breakdown that led to
Boltzmann's suicide in 1906. If so, it
is ironic that the means which would the vindicate his work were already
available at the time of his death.
Eugene
Wigner has recounted that a book from which he studied that was written before
1905 stated that “atoms and molecules may exist, but this is irrelevant from
the point of view of physics.” After
Einstein's analysis, it was not only known that atoms exist,
but
anyone with a ruler and a stop watch could measure their size. While this paper never captured the popular
imagination, in many ways it is the one that had the most profound effect on
contemporary thought.
III.
The Discrete Nature of the Photon
Of
the annus mirabilis papers, only the one of March 1905 was considered by
Einstein himself to be “revolutionary.” This paper is given the tag line
“Photoelectric Effect,” although that subject plays only a minor role. The primary impact of the paper was to
unequivocally establish the integrity of the photon as a localizable particle
possessing a discrete amount of energy. Einstein began with the observation
that the entropy of a system (the ratio of energy to temperature) varied with
the volume of a closed cavity for light in the same way as it did for an ideal
gas. Since the gas entropy relationship
was deduced from the assumption that the gas existed as discrete molecules,
Einstein reasoned that light was also emitted in discrete entities. The paper then went on to apply
this
revolutionary conclusion to the Stokes rule of photoluminescence, to the
photoelectric effect, and to the ionization of gases by ultraviolet light. The
historical emphasis given to the photoelectric effect application rather than
to the new physics that Einstein proposed is therefore misleading. It was for
this discovery that Einstein was awarded the Nobel Prize in Physics in 1921.
In
placing this discovery by Einstein in the context of our understanding of
atomic structure, it is important to clarify historical shifts in the meaning
of certain words. The standard textbook
presentation of this discovery by Einstein is clouded by multiple meanings
associated with the word “quantum” as used in different contexts. It is often
incorrectly asserted that light quanta had been proposed in 1900 by Max Planck
in his formulation of blackbody emission. Planck's formulation did not
contain any suggestion concerning the particle nature of light, since he
instead associated the denumerable discreteness to fictitious oscillators that
he assumed produced the light. Had Planck suspected that he was counting
discrete light particles rather than discrete resonators, it would have been
more reasonable to denote them as “corpuscles”' instead of “quanta.” In 1899 J. J. Thomson had used the word
“corpuscle” (the diminutive of the latin word “corpus,” or body) to describe
his discovery that the electron is a discrete particle. Newton had used “corpuscule” for light
particles prior to the development by Thomas Young of the wave model. In contrast, Planck assumed that discrete
resonators produced quanta of energy, but the electromagnetic waves so-produced
were considered to be continuously distributed over space.
Following
Planck's nomenclature, Einstein used the words “Raumpunkten lokalisierten
Energiequanten” to describe what we would now call photons. The name “photon” was first suggested in
1926 by the American chemist Gilbert N. Lewis, as a way to differentiate
between discrete light particles and the quantum numbers that prescribe the
discrete energy levels of a bound system which were introduced in 1913 by Niels
Bohr. It is interesting to note that
the Planck-Einstein use of the word “quantum” corresponds to the modern concept
of “second quantization” (the
quantization of the field in quantum electrodynamics), whereas the
Bohr-Sommerfeld quantization is now called “first quantization” (the discrete
units of action characteristic of a bound state of an atom). Thus, these two
important concepts are introduced using a historical framework in which “second
quantization” was postulated before “first quantization.”
The
continuing discussions in modern physics textbooks of the ficticious Planckian
oscillators and the use of the word quanta to denote photons are unfortunate. Clearly blackbody radiation involves
continuum collisions of free electrons, and should not be confused with
transitions between the bound states of a quantum mechanical harmonic
oscillator. Similarly, the use of the word quantum to denote a photon confuses
its usage as the unit of action that quantizes a stationary state.
One
cannot leave the subject of the photoelectric effect without mentioning the
relationship between Einstein and Phillipp Lenard, upon whose data the
application to the photoelectric effect at the end of the paper was based.
Lenard and Johannes Stark were probably the two most vehement Nazi supporters
among German scientists, and their savage anti-Semitic attacks on Einstein's
theories were a factor causing him to leave Germany. In an interview, Robert Shankland discussed Lenard with
Einstein. Shankland cited as a true measure of Einstein's objectivity the fact
that he referred to Lenard's work “with complete fairness and not the slightest
trace of malice or bitterness.”