Einstein's Masterwork

by John Gribbin

  According to [the kinetic theory], a dissolved molecule differs from a suspended body only in size, and it is difficult to see why suspended bodies should not produce the same osmotic pressure as an equal number of dissolved molecules. We have to assume that the suspended bodies perform an irregular, albeit very slow, motion in the liquid due to the liquid’s molecular motion.

  And he went on to calculate both that osmotic pressure and the nature of that irregular motion.

  The osmotic pressure that comes into both this work and the doctoral dissertation is a curious phenomenon worth describing in a little more detail. If you have a container of water (or some other liquid), like a fish tank, it can be divided into two by putting in a barrier that has tiny holes in it, just big enough for water molecules to pass through. If you do this, the water can get from one side (either side) of the barrier to the other. This is known as osmosis. But if you now dissolve something (such as sugar) in the water on one side of the tank, the dissolved molecules are too big to get through the holes. The barrier in such a setup is then called a ‘semi-permeable membrane’, because it lets some molecules through but not others.e This is where things get interesting.

  You now have a solution of sugar in water on one side of the membrane, and pure water on the other side. The result is a pressure that moves water molecules from one side of the membrane to the other. You might guess (most people do, the first time they come across this) that the presence of the sugar molecules pushes water out of that half of the tank, making the solution stronger and raising the level of the water on the other side of the barrier. In fact, just the opposite happens. Water from the pure side of the tank passes through the membrane, making the sugar solution more dilute, and increasing the height of the liquid on the side of the barrier where the sugar is. The process only stops when the pressure of the extra height of liquid (the osmotic pressure) is enough to stop the flow of water molecules through the membrane.

  This counter-intuitive behaviour is an example of the famous second law of thermodynamics at work. I don’t have space to go into all the details here, but the relevant point, at the heart of that law, is that natural processes tend to even out irregularities in the Universe. On a grand scale, the Sun and stars are pouring out heat into the cold Universe; on a more homely scale, an ice cube in a glass of water melts, evening things out to produce an amorphous liquid. In the example of osmotic pressure, the water molecules that move into the sugar solution make the solution more dilute, more like the pure water, so that there is less contrast between the fluids on opposite sides of the semi-permeable membrane.

  In the Brownian motion paper, Einstein first covered some similar ground to parts of his doctoral dissertation, but using a different (and rather more elegant) mathematical approach. His calculations involved the relationship between osmotic pressure, viscosity and the way individual particles suspended in the liquid diffuse through the sea of molecules surrounding them. But this time, he was describing the behaviour of particles big enough to see under the microscope.

  The way Einstein set about his work, though, was just as important as the results he obtained. He realised that the kick produced by a single molecule hitting a particle as large as a pollen grain could not produce a measurable shift in the position of the large particle. But the large particle is constantly being bombarded by molecules, from all sides. On average, the kicks from one side are balanced by the kicks from the opposite side, so you might not expect the large particle to move at all. Einstein realised, however, that the important words are ‘on average’. If you take a very small time interval, then just by chance at that instant the particle will be receiving more kicks on one side and fewer on another. The combined effect will be to shift the particle by a minute amount in the direction of least resistance. Then, in the next instant the pattern will change, and the particle will shift in another direction, and so on. Einstein’s special insight into the nature of this kind of statistical fluctuation was that what happens during each of these small time intervals is entirely independent of what happens in any other time interval, even the one just before the one being considered.

  Because of this independence, and the statistical nature of the fluctuations, the particle doesn’t simply move to and fro around the same spot that it started from. Nor does it keep moving in one direction. Einstein discovered that it gradually moves further and further away from its starting point, but following a zigzag path that has become known as a ‘random walk’. He showed that wherever the particle starts from, the distance it moves away from its starting point depends on the square root of the time that has passed. So if it moves a certain distance in one second, it will move twice as far in four seconds (because 2 is the square root of 4), four times as far in sixteen seconds, and so on. But it doesn’t keep going in the same direction – after four seconds it will be twice as far away from the start as it was after one second, but in a random and unpredictable direction.

  This is called a ‘root mean square’ displacement, and the equation Einstein worked out for the displacement involves the temperature of the liquid, its viscosity, the radius of the particle and Avogadro’s number. He used this equation and a value for Avogadro’s number inferred from other experiments to predict that a particle with a diameter of 0.001 mm in water at a temperature of 17°C would shift a distance of 6 millionths of a metre from its starting point in one minute. But he also realised that if the predicted displacement could be measured accurately enough, the same equation could be used the other way around, to give a value for Avogadro’s number.

  The prediction provided a classic example of the scientific method at work, since measuring the way a particle moved away from its starting point would answer the question of whether the theory it was based on was right or wrong. As Einstein put it in his paper:

  If the prediction of this motion were to be proved wrong, this fact would provide a far-reaching argument against the molecular-kinetic conception of heat.

  It wasn’t proved wrong. Although Einstein didn’t know it when he wrote the paper, in the early 1900s microscopists were already developing improved instruments, known as ultramicroscopes, that would be able to measure the kind of motion he was describing accurately enough to test the prediction.

  The Brownian motion paper was published in July 1905, and almost immediately Henry Siedentopf, a German working with the new ultramicroscope, wrote to Einstein to tell him that the kind of motion described in his paper almost certainly was Brownian motion. It still wasn’t possible at that time to test Einstein’s detailed predictions, but he was sufficiently encouraged to write another paper, this time plainly titled ‘On the Theory of the Brownian Motion’, which he sent off to the Annalen der Physik in December; it was published in 1906. In this paper he developed his ideas further and also predicted that particles suspended in a liquid would experience a rotary movement, dubbed ‘Brownian rotation’, although he did not expect this to be observable.

  It was extremely difficult to make the observations required with enough accuracy to test Einstein’s predictions, and several researchers tried and failed over the next couple of years; but in 1908 the French physicist Jean Baptiste Perrin finally succeeded. Instead of trying to measure the displacement of tiny individual particles from their starting point, Perrin used another result that had by then emerged from Einstein’s theoretical model, which predicted the way particles suspended in a solution would be arranged vertically.

  In such a suspension, the particles would be tugged downward by gravity, gradually sinking to the bottom of the liquid. But superimposed on this very slow downward drift would be Einstein’s random walk. The overall effect would be a vertical distribution of particles, with more at the bottom and fewer at the top, obeying a precise mathematical law (a specific exponential, decreasing with increasing height). Perrin’s results exactly matched the predictions from Einstein’s theory, and he even went one better by measuring the Brownian rotation that Einstein had predicted befor
e anyone had seen it. He also used the observations to make an accurate measurement of the value of Avogadro’s number.

  The whole package finally established the reality of atoms and molecules, and the validity of the kinetic theory, silencing the few (by then, very few) remaining doubters. This work was so important that Perrin received the Nobel Prize ‘in particular for his discovery of the equilibrium of sedimentation’, as the citation put it, in 1926.

  But Einstein’s work had even wider applications. The kind of statistical methods he used, coupled with the idea of random events occurring independently in individual tiny intervals of time, proved fruitful across a whole range of topics in physics. In his second paper on Brownian motion (still written in, though not published in, the annus mirabilis), Einstein had pointed out the possibility of applying the same approach to the study of fluctuations in electric circuits (the phenomenon now known as ‘noise’), and in years to come the technique would be widely applied in the new field of quantum physics. There, for example, exactly the same combination of statistical effects and random changes occurring in independent time intervals leads to an understanding of the nature of the half-life associated with radioactive processes.

  Particles of light

  This link with quantum physics is particularly appropriate, because the next paper I shall discuss from Einstein’s miraculous year saw him laying one of the foundation stones on which the whole edifice of quantum theory was built. Curiously, although this was the one paper that Einstein himself referred to as ‘very revolutionary’ (in that letter to Conrad Habicht), in many ways it builds from his other work, and to modern eyes looks less revolutionary than the work on the Special Theory of Relativity. But that is because we have become used to the idea that light exists in the form of tiny particles, called photons. In 1905, that really was a revolutionary concept – although it wasn’t exactly new.

  Isaac Newton thought of light as a stream of tiny particles, and used this model in his attempts to explain his observations of the way light is bent when it passes through a prism, how it is reflected by mirrors and why it can be broken up into all the colours of the rainbow, forming a spectrum. His 17th-century Dutch contemporary, Christiaan Huygens, had argued for a different interpretation of the same phenomena, based on the idea that light is a form of wave; but Newton’s model held sway (largely because of the god-like status that his successors gave to Newton) until the work of the Englishman Thomas Young and the Frenchman Augustin Fresnel early in the 19th century. Although their contributions were equally important, what became regarded as the definitive proof that light travels in the form of a wave, like ripples on a pond, comes from what has become known as Young’s double-slit experiment.

  The experiment is based on shining light of a single colour (this would later be interpreted as meaning light of a single wavelength) coming from a light source through two holes in a screen. These could be two parallel thin slits, made with a razor, or two pinholes. We have all seen the pattern of circular ripples that spreads out from a point when a pebble is dropped into still water, and the more complicated pattern of ripples that is produced if two pebbles are dropped into the water simultaneously. The complications in the second pattern are caused by two sets of ripples interacting with one another – as physicists put it, ‘interfering’ with one another. The experiments showed that light spreads out from the holes (or slits) in Young’s experiment in just the same way, and that two sets of waves, one from each hole, are interfering with one another. Young (and then many other people) proved this by placing a second screen on the other side of the two slits from the light source, and looking at the pattern of bright and dark stripes made on the second screen. Bright stripes are places where the two sources of light combine with one another to make an extra high wave, and dark stripes are places where the two sets of ripples cancel each other out, with one going up while the other goes down. This is quite different from the pattern that would be expected if light travelled in the form of tiny particles, like little bullets. The experiments are so precise that the spacing of the stripes on the second screen can be used to calculate the wavelength of the light involved – proof, indeed, that light travels as a wave.

  It’s worth putting this dramatic discovery in its historical perspective. Newton had spelled out his ideas about light and colour in a great book, Opticks, published in 1704. His image of light as a stream of tiny particles held sway for almost exactly 100 years, until the work of Thomas Young at the beginning of the 19th century. That’s almost the same as the time interval from Einstein’s work in 1905 to the present day – and the interval from Young to Einstein is the same as the span from Newton to Young. To suggest at the beginning of the 19th century that Newton had made a major blunder was very much as if evidence were uncovered today showing that Einstein had made a major blunder. The discovery was dramatic, and it took time for people to be convinced. But in the 100 years from Young to Einstein, a great deal more evidence did come in to show that light travels as a wave.

  We have already mentioned the most important piece of that evidence – James Clerk Maxwell’s discovery of the equations that describe how electromagnetic waves (or ‘vibrations of the ether’, as they would have put it then) move through space. Maxwell’s equations describe waves, and they predict the speed with which those waves move. This speed is exactly the same as the speed of light. What more proof could be needed that light travels as a wave? By 1900, the idea that light, and other forms of electromagnetic radiation, existed in the form of waves seemed as solid a foundation of science as the idea that apples fall downwards from trees. But then the first crack appeared in this foundation.

  Two of the big areas of scientific interest in the middle and late 19th century were thermodynamics (which dealt with energy) and light (which had been identified as a wave and was also a form of radiant energy) – ‘light’ in this context refers to all kinds of electromagnetic radiation, including invisible infrared heat, radio waves and ultraviolet light. It was clear that there is a relationship between heat (energy) and light. A piece of iron that is just warm to the touch doesn’t radiate any visible light at all, but as it is heated further it glows first red, then orange, then white as its temperature increases. Indeed, the relationship between colour and temperature was so well known in a qualitative way that in the days before accurate scientific measurements were possible, potters used to gauge the temperature of their kilns by looking at the colour of the pots they were firing. But what was the precise, quantitative relationship between light and energy? What were the equations that could describe, or predict, the colour of a hot object from its temperature alone?

  The first person to tackle this puzzle in a quantitative way was the German physicist Gustav Robert Kirchoff, at the beginning of the 1860s. Kirchoff was especially interested in spectroscopy, studying the distinctive patterns of lines in a spectrum (looking not unlike a modern barcode) corresponding to different elements. But he also developed a thermodynamic approach to understanding the relationship between light and energy through his idea of a ‘black body’. A black body would be an object which absorbed entirely all the radiation that fell on it – a perfect absorber. Of course, it was impossible for experimenters to make a perfect black body to study in the lab, but Kirchoff came up with a very close approximation. He devised an experiment involving a closed container painted black inside, with a tiny pinhole as the only opening to its interior. Any radiation that entered through the pinhole would be absorbed, and only a tiny amount of radiation would escape through the pinhole, making it very nearly a perfect black body.

  But this was only the first step. According to the thermodynamic rules, an object that absorbed all kinds of radiation should also radiate all kinds of radiation, with no complications involving things like spectral lines. If the container were heated, the glowing walls inside would produce light which would bounce around inside and get thoroughly mixed up before escaping from the pinhole, in the form that became know
n as ‘cavity radiation’, or (more commonly today) ‘black-body radiation’. According to the thermodynamic principles, this would be a pure form of light, with a colour that depended only on the temperature of the container – the black body. Crucially, the colour of the radiation did not depend on what the container was made of.

  But the black-body radiation is not light of a pure single colour. It is always a mixture of different colours – that is, different wavelengths – of light. For any particular temperature, however, there is always more energy radiated in one group of wavelengths, with less energy radiated at both longer and shorter wavelengths. As the cavity is heated, the peak intensity of the light shifts from the longer wavelength end of the spectrum (red) through the familiar colours of the rainbow (orange, yellow, green, blue and so on). So a red-hot piece of iron (or anything else) radiates mostly red light, but also some infrared radiation (at longer wavelengths) and some yellow and orange light (at shorter wavelengths).f

  A graph representing the spectrum of a black body radiating in this way looks like a little hill, with a peak at a particular wavelength corresponding to the temperature of the black body, and slopes rolling down on either side. Even without any understanding of why this should be so, the discovery had immediate practical uses. For example, the shape of the spectrum of the Sun very closely follows this ‘black-body curve’ for an object with a temperature of about 6,000°C; so astronomers could measure the temperature of the surface of the Sun (and, indeed, of other stars) without ever leaving the Earth. What was needed to complete Kirchoff ’s work, though, was to find an equation that described the shape of this hill, and a physical basis for that equation. That proved extremely difficult, and Kirchoff, who died in 1887, didn’t live to see it.