by Colin Pask
There are two difficulties here. That “just a little imagination and thinking” is not so easy to supply, and we shall see that it is calculations coupled as usual with experiments that are needed. But the obvious first and more fundamental difficulty for most people relates to the very existence of atoms. As Lucretius says in De Rerum Natura, “the original particles, although themselves invisible,” may give a plausible explanation for properties of matter, but how do we know that the major assumption that atoms exist is a sound one? It is only in more recent times that tools like x-ray diffraction and electron microscopes have become available for exploring matter at fine scales.
The nineteenth century saw a blossoming of science at these fundamental levels. For example, chemists like John Dalton showed the value of the atomic hypothesis in explaining an array of reactions and chemical properties of substances. The spread of thin layers of oil over the surface of water was seen to have a limited extent, and, assuming a monomolecular layer, Lord Rayleigh and others estimated the size of fundamental molecules and atoms. Some of the most productive “imagination and thinking” saw significant advances in the kinetic theory of gases, that model of a gas as an enormous collection of rapidly moving atoms or molecules as first envisaged by Newton and Daniel Bernoulli. James Clerk Maxwell used statistical methods to show that the kinetic theory led directly to Boyle's Law PV = RT governing how gas pressure P and volume V are linked to the temperature T (R is the gas constant). Maxwell also found a “very startling”3 prediction that the coefficient of viscosity of a gas is independent of its density (in certain ranges). The experimental verification of that prediction was one of many triumphs for the kinetic theory—and for the atomic hypothesis, of course. (See Brush for the history of this work.)
As great as these advances were, and despite the overwhelming weight of evidence in favor of the atomic hypothesis, there were still some respectable scientists (notably Ernst Mach and Wilhelm Ostwald) who found it hard to accept that atoms existed. For them, the evidence was just too remote.
(The history of this part of science is told in many books. I recommend the “popular” book by George Gamow, the book by Holton and Brush, and the definitive articles by Pais and Rechenberg for the physics and extensive references to key papers.)
10.1 DIRECT EVIDENCE FOR THE EXISTENCE OF ATOMS
Atoms and molecules are not visible, and the tests for their existence were just too indirect for many people. What was needed was a critical test that was somehow visible yet closer to molecular effects or mechanisms. The clinching test was provided by Albert Einstein and Jean Perrin. In 1905, along with his wonderful papers on relativity and quantum theory, Einstein published a paper titled “On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat.” (This paper and others mentioned below are available in Einstein's collected papers—see bibliography.) The paper opens with the sentence:
It will be shown in this paper that, according to the molecular-kinetic theory of heat, bodies of microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitude that these motions can easily be detected by a microscope.4
Here was a physical effect that (it was suggested) could be observed, and Einstein was in no doubt about its vital importance as he explains just what was at stake:
If it is really possible to observe the motion discussed here, along with the laws it is expected to obey, then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces, and an exact determination of the real sizes of atoms becomes possible. Conversely, if the prediction of this motion were to be proved wrong, this fact would provide a weighty argument against the molecular-kinetic conception of heat.
Einstein could not be sure that the predicted phenomenon was already known as Brownian motion (see section 10.11):
It is possible that the motions to be discussed here are identical with the so-called “Brownian molecular motion”; however, the data available to me on the latter are so imprecise that I could not form a definite opinion on this matter.
In his doctoral thesis, “A New Determination of Molecular Dimension,” Einstein considered particles dissolved in a liquid and developed a theory for their diffusion using known laws for osmotic pressure and motion in a viscous medium. In his 1905 paper, he carried over these ideas to large, observable particles suspended in a liquid; in a tour de force, Einstein came to section 4, “On the Random Motion of Particles Suspended in a Liquid and Their Relation to Diffusion” and then the big triumph, section 5, “Formula for the Mean Displacement of Suspended Particles: A New Method of Determining the True Size of Atoms.” In 1906, Einstein published a paper “On the Theory of Brownian Motion” in which he derives the same results but with a theory more closely related to “the foundations of the molecular theory of heat.”5
(Details of Einstein's work may be found in many statistical physics textbooks. Pais, chapter 5, “The Reality of Molecules” is an excellent general coverage of the topic, including Einstein's mathematics, and Rigden gives a simpler account of Einstein's 1905 paper. The papers by Bernstein and by Newburgh, Peidle, and Rueckner are also recommended. The latter gives a brief review of the mathematics and alternative derivations of Einstein's results. Feynman's lecture 41 provides a lovely pedagogical approach to Brownian motion showing how the concept of resisted motion and simple averaging lead to Einstein's results.)
10.1.1 Einstein's Prediction for Brownian Motion
The sort of particle movements Einstein refers to are called Brownian motions after the distinguished botanist Robert Brown (1773–1858), who used a microscope to watch tiny pollen grains as they jerked around in water. At first, Brown thought that the seemingly random jumps must be from some life form (“animalcules”) propelling itself through the liquid. As a good experimenter, he tried inert substances, like particles of glass and rock, and still saw the same phenomenon. The vital addition made by Einstein was to establish the law Brownian motion is expected to obey. Here are real predictions put forward for testing.
Figure 10.1. Brownian motion of three mastic particles as recorded by Jean Perrin. The positions are given every 30 seconds, and the joining straight lines suggest the zigzag motion involved. From Wikipedia, user MiraiWarren.
Particles suspended in a fluid jump around because there are fluctuations in the total force imposed by the molecules colliding with them. See figure 10.1. Thus the motion depends on the properties of the fluid, as Einstein expressed in his use of osmotic forces and diffusion. It is obvious from the results shown in figure 10.1 that no formula can be given for the details of the distances traveled by any particular particle. What we can do is to ask about how far a particle undergoing Brownian motion will have moved from its original position and calculate the mean square displacement
The y and z components are similar, and hence the total averaged distance is obtained. Einstein's first prediction is that, on average, the distance moved varies as the square root of the time. (Of course, for uniform, free motion, the distance varies directly as the time.)
Einstein also gave a formula for the proportionality constant a introduced above in equation (10.1):
In equation (10.2), T is the temperature, R is the gas constant, η is the liquid's coefficient of viscosity, N is Avogadro's number (the number of molecules in a mole of a substance), and r is the radius of the particle undergoing the motion. The gas constant is known and the values of T, η, and r are readily measured. Thus, remarkably, knowing the parameter a would mean knowing Avogadro's number N. (In his doctoral thesis, Einstein had used viscosity data for sugar solutions to estimate N as 2.1 × 1023; the current value replaces 2.1 with 6.022.)
As scientist and Einstein biographer Abraham Pais puts it:
One nev
er ceases to experience surprise at this result, which seems, as it were, to come out of nowhere: prepare a set of small spheres which are nevertheless huge compared with simple molecules, use a stopwatch and a microscope, and find Avogadro's number.6
10.1.2 Experimental Confirmation
Einstein's predictions were confirmed by Jean Perrin (1870–1942) using a series of experiments with a microscope arrangement to record the motion of various different particles. (See Perrin's book for a description and interpretation of his work.) In 1926, Perrin was awarded the Nobel Prize in physics in part for “his work on the discontinuous structure of matter.”7 Molecules had not been observed directly, but the immediate effects of their motions had been confirmed. The link between these molecular motions and the Brownian motion observations could only be made using a theory resulting in calculations like Einstein's. Perrin found the square root of time variation, and his results gave a parameter leading to Avogadro's number N = 7.15 × 1023. Using a theory to link the micro- and macroworlds has allowed us to use measurements of things such as Brownian motion to determine the nature of the atomic or molecular system. (The paper by Newburgh and colleagues shows how Perrin's experiments can be reproduced in a modern laboratory.)
These findings removed the last barriers for the acceptance of the atomic hypothesis. The importance of this work was summed up by Max Born when he wrote that Einstein's theory of Brownian motion did “more than any other work to convince physicists of the reality of atoms and molecules, of the kinetic theory of heat, and of the fundamental part of probability in the natural laws.”8 Surely nobody could fail to add calculation 37, atoms really do exist to any list of important calculations.
10.2 LIGHT AND ATOMS
John Dalton (1766–1844) published A New System of Chemical Philosophy in 1808, and throughout the nineteenth century the idea of atoms became more and more firmly entrenched in chemistry. The ideas of chemical reactions and valence theory were developed, and the nature of the chemical elements gradually became clear. Dmitri Ivanovich Mendeleev (1834–1907) published Foundations of Chemistry and presented his periodic table organizing elements into a system based on their atomic weights, which revealed their chemical properties and identified groups of elements behaving in the same way. However, there was no underlying theory for the nature of the atoms themselves. (Dalton suggested that atoms had a center surrounded by an atmosphere of what he called “caloric,” so that neighboring atoms in gases had touching atmospheres.) While the chemists were developing their theories, some physicists were discovering a different—and what proved in some ways to be a more valuable—method of labeling atoms with a characteristic signature. This work was a key step in the lead up to the theory of the atom.
10.2.1 Atoms and Light
It was long known that burning different substances produces light of different colors. It was discovered that in addition to the continuum of wavelengths in radiation (for example, that seen from very hot bodies and black bodies, the subject of Planck's work much later), a heated gas or vapor also emitted light at a set of discrete wavelengths. These are the line emission spectra, and here was a wonderful new tool in science. The collaboration of Gustav Kirchhoff and Robert Bunsen produced the seminal 1860 paper “Chemical Analysis by Observation of the Spectrum.” They wrote that
it is a well-known characteristic of some substances that when placed in a flame certain bright lines appear in their spectrum. These lines open a method of qualitative analysis, extending the field of chemical reactions, and also leading to the solution of problems which have previously been considered inaccessible.9
They go on to point out the value of this analysis and an advantage it has over chemical or analytical analysis:
In spectral analysis, on the other hand, the colored lines stay unchanged and are not influenced by the presence of impurities; the position of the lines in the spectrum is a chemical characteristic of such fundamental nature as the atomic weight of the substance and can be measured with astronomical exactitude.
Kirchhoff and Bunsen went on to use this analysis to extend the alkali metal group beyond lithium, sodium, and potassium to include cesium and rubidium. Here, they said, was a new tool “to discover the smallest traces of an element on the earth.” An example of spectral lines is given in figure 10.2.
Figure 10.2. Visible spectral lines for a few elements. Each element has its own particular set of lines. The spectrum labeled “?” comes from a gas, and, comparing lines, it can be deduced that it contains helium and sodium. Reprinted with permission of HarperCollins Publishers Ltd., from Simon Singh, The Big Bang (New York: HarperCollins, 2004).
It was also in going beyond the confines of Earth that spectral analysis created great interest. In 1814, Joseph von Fraunhofer had detected dark lines in light from the sun, and, in 1859, Kirchhoff had linked these to the emission spectra discussed above. So it came to be known that gases could both emit and absorb light at very particular wavelengths, and this data would allow the responsible elements to be identified. At last, astronomers could find out about the composition of stars and other objects observed in the sky. Husband and wife team William and Margaret Huggins were pioneers in the field. As an example, working in the 1860s, they found dark lines in light from the star Betelgeuse indicating absorption by elements sodium, magnesium, calcium, iron, and bismuth.
Spectral analysis became an essential tool for various scientists, but there remained the question of whether there existed an underlying order and rationale behind the spectral lines.
10.2.2 Bringing Order to the Spectrum
The great step was taken not by someone with a household name like Albert Einstein, Max Planck, or Niels Bohr, but by an obscure Swiss schoolmaster Johann Jakob Balmer (1825–1898). In 1885, Balmer published a paper entitled “Notice Concerning the Spectral Lines of Hydrogen.” (It is easy and most worthwhile to read this paper—see bibliography.) He began with the four well-known visible lines in the hydrogen spectrum and came up with a formula for the wavelength λ of the hydrogen lines:
This remarkable formula gives wavelength values for a whole series of spectral lines. Balmer gives wavelength values for the first four lines, which he could compare against experimental values found by Anders Jonas Ångström:
In his paper, Balmer also gives comparisons for the next lines using data supplied by Huggins. The fit is almost as good, but later work has shown that Huggins's data contains small errors. Balmer pointed out that the higher spectral lines become more closely spaced and approach the limiting value of 3645.6.
Balmer also speculated about other series of lines obtained by changing n2/(n2- 22) to n2/(n2- p2) and taking other integer values for p. Later it became conventional to write the hydrogen spectra in terms of the inverse of the wavelength (which gives the frequency when multiplied by the speed of light c) with the result that
R is the Rydberg constant which has the value 1.09678 × 107 m–1. Choosing a value for p and then taking n to be p plus 1, 2, 3, 4,…generates the wavelengths for the whole set of series of lines which are labeled by their discoverer and year of discovery as follows:
p = 1 Lyman series (1906–1914)
p = 2 Balmer series (1885)
p = 3 Paschen series (1908)
p = 4 Brackett series (1922)
p = 5 Pfund series (1924)
Of course most of these wavelengths refer to radiation not in the visible range.
Equations (10.3) and (10.4) give beautifully simple formulas which describe the hydrogen spectrum with great accuracy. They tell us that there is a definite pattern and mechanism involved in light emission from atoms. Like all good work of this type, they point the way to the future by posing some clear questions: Why do the integers p and n occur in that particular manner? What is the origin of the constants like R at the front of these formulas? Answering these questions led to a revolution in physics as I will describe in the next section.
The work of Balmer and others led to many new results in sp
ectral analysis. For example, in 1896, E. C. Pickering discovered absorption spectral lines in the light from the star ζ Puppis corresponding to the formula
In 1912, Fowler saw the same lines emitted from a gas mixture of hydrogen and helium. The link to the existence of helium had to wait for Bohr's theory described in the next section.
It is not clear how Balmer did his calculations to obtain his hydrogen spectrum formula—in his paper he says, “I gradually arrived at a formula.”10 However, his work is enormously important in the development of physics and I add calculation 38, spectral line patterns to my list. Certainly Niels Bohr, the founder of the quantum theory of the atom, was in no doubt: “As soon as I saw Balmer's formula the whole thing was immediately clear to me,”11 he said.
10.3 ATOMS AND THE DAWN OF QUANTUM MECHANICS
Ernest Rutherford (1871–1937) reported on his wonderful experiments and deductions in his 1911 paper “The Scattering of α and β Particles by Matter and the Structure of the Atom.” (The book by Ter Haar is an excellent introduction to this part of physics and includes the seminal original papers by Planck, Einstein, Rutherford, Bohr, and others.) Rutherford came to the conclusion that atoms consist of a heavy, positively charged nucleus surrounded by N electrons when the nucleus carries a charge Ne, where –e is the charge of an electron. In the simplest case of hydrogen, N=1, and the atom consists of a single proton with one electron accompanying it. Clearly the place to start in understanding the structure of the atom is with hydrogen, and in fact, according to nuclear scientist Victor Weisskopf (1908–2002), “To understand hydrogen is to understand all of physics.”12 Furthermore, the results given by Balmer and his followers show precisely what that understanding must explain.