Einstein's Contributions to Atomic Physics

 

L. J. Curtis

 

 

Introduction

 

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.

 

 

 

II.  The Existence of Atoms

 

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.”