by Colin Pask
With equation (9.6) Einstein shows that electrons can escape the metal with positive energy E only when the frequency is large enough to make hν – P greater than zero. Beyond that cut-off point, equation (9.7) tells us that the energy of the electron E increases linearly with the frequency ν.
If, for a fixed frequency ν, the intensity of the light is increased, it will consist of photons each with energy hν but there will be more of them. This means that there will be more photons to cause more electrons to be emitted from the metal in accordance with equation (9.7). Thus increasing the intensity of light should not change the energy of the individual emitted electrons, but it should increase their number.
By 1916, Robert Millikan had experimentally demonstrated the accuracy of Einstein's theory (his results for electron energy versus radiation frequency are reproduced by Rigden), but he was not convinced by the underlying assumptions and wrote that
despite then the apparently complete success of the Einstein equation, the physical theory on which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it, and we are in the position of having built a very perfect structure and then knocked out entirely the underpinning without causing the building to fall.17
Millikan was wrong about Einstein, and the validity of the light quanta concept was becoming widely accepted. With his little calculation, Einstein has completely explained the three key observations on the photoelectric effect. As a measure of the importance of this work, we can note that when Einstein was awarded the 1922 Nobel Prize it was given to Albert Einstein “for his services to theoretical physics and especially for his discovery of the law of the photoelectric effect.”18 (If you are surprised that the prize was not given for Einstein's work in relativity, there is an interesting background to the subject—see Pais chapter 30.)
9.5.2 Compton Changes Photons with Electrons
If the electron is not tightly bound in some way, when a photon hits it, there will be a scattering process, and the output will be another photon and an electron moving in some particular way. This process is known as Compton scattering, and the dynamics involved are shown in figure 9.12. In this particular process, the radiation used in experiments is not light but x-rays, and, for x-rays, the electrons are loosely bound in ordinary atoms. The dynamics of the process are worked out using the conservation laws for energy and momentum. The explanation of the photoelectric effect is basically an argument about energies, but now the photon momentum as in equation (9.6) must also be used. (For more details and history see the books by Gamow and Williams.)
Figure 9.12. Compton scattering: the collision of a photon with a stationary electron. Figure created by Annabelle Boag.
If the incoming photon has frequency ν0, and the outgoing, or scattered, photon has frequency ν, according to equation (9.6) the electron gains an amount of energy hν0 – hν. We must also make sure that the momentum is conserved in the two directions involved, that is the direction of the incoming photon and the direction perpendicular to that, or the horizontal and vertical directions in figure 9.12. Letting the outgoing electron have mass m and speed V, and using equation (9.6) for the photon momentum, leads us to
Using these conservation-laws equations allows us to calculate the result of the scattering process. The startling conclusion is that the outgoing photon has a different frequency for different scattering angles θ (as in figure 9.12). The result is usually expressed in terms of wavelengths λ rather than frequency ν with λ = c/ν. The outgoing photon has wavelength λ larger than the incoming photon wavelength λ0 by an amount
Equation (9.8) tells us that if we look at the scattered photons at various angles, we will see a wavelength change varying from zero at θ = 0, to (h/mc) at θ = 90º, to (2h/mc) at θ = 180º.
The calculation of the effects of radiation-electron scattering using the photon-electron picture comes up with a clear prediction involving the quantum constant h. The detailed experiments carried out by Arthur Holly Compton in 1922 confirmed the results of the photon theory, and, in 1927, he was awarded the Nobel Prize for his work. In his 1923 Physical Review paper, Compton wrote that his results convinced him “that the radiation quantum carries with it directed momentum as well as energy.”19 So calculation 35, photons exist is based on work that merits not one but two Nobel Prizes! But does it really answer the question about whether photons give us the true nature of light? I return to that in section 9.7.
9.6 THE BENDING OF LIGHT
If light consists of particles, as Newton supposed, it is natural to ask whether these particles are affected by gravitational forces. In fact, Newton raised the matter in his Opticks:
Query 1. Do not Bodies act upon Light at a distance, and by their action bend its rays; and is not this action (caeteris paribus) [other things being equal] strongest at the least distance?20
Finding the answer to this question has proved to be of great importance in the development of science.
9.6.1 The First Deliberations
In section 7.5.2, we saw that the astronomer Rev. John Michell followed Newton's suggestion and showed that the assumption of a gravitational pull on light led to the idea of a “dark star.” In 1783, Michell wrote to Henry Cavendish (1731–1810) (famous for his measurement of the gravitational constant), suggesting gravity might reduce the speed of light, so that a method for measuring the mass of a star might be devised by exploiting that effect. That stimulated Cavendish to calculate by how much the path of light might be bent as it closely passed by a massive body. Cavendish never published his calculation. (For further details, see the book and 1988 paper by Will.) Pierre-Simon Laplace also wrote about the possibility of a “dark star,” and this probably stimulated the Bavarian astronomer Johann Georg von Soldner (1776–1833) to independently calculate the bending of a light ray as it passed by the sun. Soldner did publish his findings in 1804.
The physical situation is shown in figure 9.13. The problem is to calculate the bending angle θ when the light ray passes a distance Δ from the center of the sun. Clearly, the maximum deviation occurs when the ray just grazes the sun and Δ is equal to the sun's radius.
Figure 9.13. The path of the light from a star S is bent through an angle θ as it passes the sun. An observer O would interpret the light as coming from the apparent star AS. The direct line from S to O may pass through the sun so S would not be observed if it was not for the illustrated bending phenomenon. Figure created by Annabelle Boag.
If we treat the particles of light like any other particle, the mathematical problem is to solve their equation of motion (from Newton's second law),
for the position vector r of the particle as it moves past the mass M situated at r = 0. Solving equation (9.9) is a standard problem, which gives the elliptical orbits for planets as discussed in chapter 6. For the situation in figure 9.13, the particle is moving so fast that it goes by the sun in the same way that we get fly-by data for a spacecraft moving past a planet. The orbit is a hyperbola (see the 1988 paper by Will for details) and gives the light bending as θ = 0.875". Soldner gives 0.84" in his 1804 paper. These are very small angles, and so experimental tests seemed most unlikely.
(A technical aside: Equation (9.9) involves the mass M of the sun, but not the mass of the particle passing by it. This is because the inertial mass and the gravitational mass are equal, and so the particle mass occurs on both sides of the equation of motion and thereby may be cancelled out. Thus no question about the mass of the particles of light is involved.)
9.6.2 Einstein Changes the Physics
Einstein's definitive paper The Foundations of the General Theory of Relativity appeared in 1916. In it he argued that Newtonian mechanics, based on the interaction of bodies by means of forces, should be replaced by a new theory in which gravitational effects manifest themselves by changing the metric properties of space around bodies. Newton's theory emerges as the first approximation to Einstein's theory, and in the very simplest
case, the equation to be solved reduces to Newton's equation (9.9). It is only in extreme cases (for example, very massive bodies and speeds approaching the speed of light) that the results from Einstein's theory differ substantially from those given by Newton. (The books by Lambourne and Weinberg are recommended for the technical details.) The phenomenon of light bending as shown in figure 9.13 offers one case where those extreme effects might come into play.
Some details of the metric theory for the region around a single massive body (like the sun) were given in section 7.5.2. The problem now is to calculate the geodesics, which will be the paths taken by light near such a body. The resulting differential equations turn out to be much like Newton's equations of motion with an additional term bringing in Einstein's gravitational effects. (See Lambourne and Weinberg.) The importance of this additional term was apparent in section 6.6.3 where we saw that it is responsible for the rotation of planetary orbits.
In his 1916, paper Einstein gives the light bending angle as θ = 1.75"/Δ where Δ is the closest approach to the sun's center measured in sun radii. (Actually, Einstein gives 1.7 rather than the more accurate 1.75.) We now have a very clear difference for the case of light just grazing the sun (Δ = 1):
classical Newtonian theory (Soldner) θ = 0.875",
relativity theory (Einstein) θ = 1.75".
Thus the calculations made a very clear prediction: the bending of light is twice as great if Einstein's theory should replace Newton's. These are crucially important calculations, and I choose them as calculation 36, bending light. Here was a very clear prediction, a reference to a phenomenon that had never been observed, and so there was no possibility of the theory being manipulated to fit the data. This is the ideal scientific situation. (For a useful discussion of this point see Brush's paper “Prediction and Theory Evaluation: The Case of Light Bending.”) But could light bending really be detected?
9.6.3 The Story of a Triumph
The situation shown in figure 9.13 suggests an experiment: check the positions of stars seen close to the sun against their positions when the light from them does not pass close to the sun. But obviously the faint light from stars is completely swamped by the light from the sun itself. However, if the observations could be made during a total eclipse of the sun, everything changes. The apparent position of stars could be measured during that six-minute window and compared with their known positions already recorded by astronomers. So it was that as World War I ended, scientists were able to leave aside national differences and make expeditions to places where a May 29, 1919 eclipse could be fully observed. An expedition to Principe (off the coast of Spanish Guinea) was led by Arthur Eddington and another to Sobral (in northern Brazil) was led by Andrew Crommelin. (This is one of science's great stories, and to read more, I recommend the book by Will, the biography by Pais, the paper by Kennefick, and Eddington's own account in his 1920 book.)
The remote locations, weather worries, and difficulties involved in making the observations mean that we should not expect great accuracy. Nevertheless, analysis of the experimental data shows that
θ = 1.98 ± 0.16" (Sobral data), θ = 1.61± 0.40" (Principe data).
While the accuracy is not exceptional, these results clearly favor Einstein's theory. Later measurements gave a similar spread in accuracy (see Weinberg's table 8.1) and even a “modern” 1973 expedition to Mauritania did little better. Later experiments with radio waves and other astronomical situations have confirmed that Einstein's theory is indeed the one to use.
So Einstein's theory passed one of its great tests, and the publicity was enormous (see Pais). But what if it had failed? What would Einstein say then? Einstein believed that his theory of relativity had logical consistency and necessity. Establishing such things was a driving force in his life; he said, “What I am really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all.”21 When asked what he would have said if the observations had not confirmed his theory of light bending, Einstein replied, “Then I would have to pity the dear Lord. The theory is correct anyway.”22
The bending of light by distant objects is now an important factor in cosmology. It is known as gravitational lensing (see Lambourne section 7.3.2, for example). The deflecting body may be a galaxy or even a cluster of galaxies.
9.6.4 A Sour Note
Einstein was Jewish, so for some people in Germany, he was far from popular, and efforts were made to discredit his work. He was attacked by those supporting the nationalist movement for “Deutsche Physik” or “Aryan Physics.” A leading figure in that movement was Philipp Lenard, who won the 1905 Nobel Prize for experiments in photoelectric effects to which I referred in section 9.5.1. Lenard accused Einstein of plagiarism and claimed that he took his light bending results from Soldner, although there is no evidence that Einstein knew of the earlier work. Lenard published a critical paper in the 1921 Annalen der Physik. It was a sad time for Einstein and German science, and, of course, he moved to the United States for the rest of his life.
There is another little twist in this story. In 1911—and before he had fully developed his theory of general relativity—Einstein published a paper “On the Influence of Gravitation on the Propagation of Light” in the Annalen der Physik. In that paper, he left out some of the relativistic effects and came up with the result θ = 0.83", that is, with the classical or Newtonian result. There is no mention of Soldner in that paper (nor would there be for the devious Einstein portrayed by Lenard), but he does suggest the experiment in which stars are observed during a solar eclipse. Perhaps luckily for Einstein, a 1912 experiment in Brazil failed because of rain, and a 1914 expedition to observe an eclipse in Crimea was aborted because of war. Perhaps the “dear Lord” was sparing Einstein some embarrassment until his 1916 paper appeared!
Surely there are few other calculations in science which are surrounded by such remarkable scientific and social outcomes. Calculation 36, bending light is truly a landmark in science and in its overall impact.
9.7 AN END TO THE STORY?
We have come full circle, from Newton's corpuscles to Young's light waves, on to Maxwell's electromagnetic waves, and back to particles again with Einstein's photons. What does this say in answer to the question: What is light? To go into this more deeply could take another whole book. Today, we still use the wave theory of light (and other waves like radio waves), and the theories presented in sections 9.3 and 9.4 continue to be used and impress with the range of phenomena so described. There are things that are only explained using a quantum approach. It is a question of choosing the appropriate theory to apply. In a popular article published in 1924, Einstein wrote:
The positive result of the Compton experiment proves that radiation behaves as if it consisted of discrete energy projectiles, not only in regard to energy transfer but also in regard to momentum transfer.23
Notice Einstein's use of the words “as if” because that is the way many people like to state the position about light: sometimes it behaves as if it is a wave; at other times, it behaves like a stream of particles. (A wonderful attempt to communicate how quantum theory is used to explain everything about light may be found in Richard Feynman's book QED.)
Finally, what should we say about the photon? You might enjoy the various articles collected together by Roychoudhuri and Roy under the heading “The Nature of Light: What is a Photon?” But perhaps it is best to finish with the opinion of Einstein himself toward the end of his life in 1951:
All these fifty years of conscious brooding have brought me no nearer to the answer to the question “what are light quanta (photons)?” Nowadays every Tom, Dick, and Harry thinks he knows it, but he is mistaken.24
I have a feeling that Dr. Samuel Johnson would have really appreciated that statement!
in which we see some of the important calculations used when trying to determine the nature of matter and identify the fundamental building blocks.
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Questions about the properties of matter and what happens when we continue to divide things into smaller and smaller pieces go back at least to the ancient Greeks. Already there was a form of what today we call the atomic hypothesis, although not everyone believed there were final, discrete entities forming the world. The ancient position is beautifully summed up in Lucretius's De Rerum Natura (The Poem on Nature) written around 50 BCE:
The whole of nature consists of two elements:
There are material bodies, and there is empty space,
In which they are situated and through which they move.
…
The bodies themselves are of two kinds: the particles
And complex bodies constructed of many of these;
Which particles are of an invincible hardness
So that no force can alter or extinguish them.1
Lucretius tries to explain a variety of natural phenomena using that atomic hypothesis and even refers to animal reproduction and the need for “immutable matter” in order for the next generation to maintain the appearances of its parents. Today molecular biology and the study of DNA dominate much of our natural science. In fact it would be hard to overestimate the importance of atoms in the modern world. The Feynman Lectures in Physics, given in 1962 to students at the Californian Institute of Technology, are a wonderful source of knowledge, insight, and wisdom. In the very first lecture Richard Feynman talks about atoms:
If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.2