Rutherford, on the other hand, calculated that the only configuration capable of knocking an alpha particle backward was one in which the entire mass and positive charge were concentrated in a very small volume in the center of a relatively huge (atom-size) sphere. The nucleus! The electrons would be spaced throughout the sphere. In time and with better data, Rutherford's theory was refined. The central positive charge (nucleus) occupies a volume no more than one trillionth of the volume of the atom. According to the Rutherford model, matter is predominantly empty space. When we pound on a table, it feels solid, but it is the interplay of electrical forces (and quantum rules) among atoms and molecules that creates the illusion of solidity. The atom is mostly void. Aristotle would be appalled.
Rutherford's surprise at the alpha particles bouncing backward may be appreciated if we abandon his artillery shell and think instead of a bowling ball thundering down the alley toward an array of tenpins. Picture the bowler's shock if the ball were stopped by the pins and then rebounded, careening back to the bowler, who would then run for her life. Could this happen? Well, suppose somewhere in the middle of the triangular array of pins there was a special "fat pin" made out of solid iridium, the densest metal known. This pin is heavy! It weighs fifty times more than the ball. A sequence of time-lapse photos would show the ball impinging on the fat pin, deforming it but coming to rest. Then, as the pin re-forms to its original shape and, indeed, recoils just a little bit, it imparts a resounding force to the ball, which reverses its velocity. This is what happens in any elastic collision, say of a billiard ball and cushion. Rutherford's more picturesque military shell metaphor was derived from his preconception, and that of most physicists of his day, that the atom was a sphere of pudding tenuously spread over a large volume. For a gold atom, this was an "enormous" sphere of radius 10−9 meters.
To get a sense of the Rutherford atom, if we picture the nucleus as the size of a green pea (about a quarter inch in diameter) the atom is a sphere of radius 300 feet, something that can surround six football fields, packed into a rough square. Rutherford's luck held here too. His radioactive source just happened to produce alphas with an energy of about 5 million electron volts (we write it 5 MeV), which was ideal for discovering the nucleus. The energy was low enough that the alpha particle never got too close to the nucleus but was turned back by its strong positive electric charge. The surrounding cloud of electrons had much too little mass to have any appreciable effect on the alpha. If the alpha had had much more energy, it would have penetrated the nucleus, sampling the strong nuclear force (we'll learn about this later) and greatly complicating the pattern of scattered alpha particles. (The vast majority of alphas pass through the atom so far from the nucleus that their deflections are small.) As it was, the pattern of scattered alpha particles, as subsequently measured by Geiger and Marsden and eventually by a host of continental competitors, was mathematically equivalent to what would be expected if the nucleus were a point. We know now that nuclei are not points, but if the alpha parades don't get too close, the arithmetic is the same.
Boscovich would have been pleased; the Manchester experiments backed up his vision. The outcome of a collision is determined by the force fields around the "point" things. Rutherford's experiment had implications beyond the discovery of the nucleus. It established that very large deflections imply small "pointlike" concentrations, a crucial idea experimenters eventually employed when going after quarks, the real points. In the slowly emerging view of the structure of the atom, Rutherford's model was a clear milestone. It was very much a miniature solar system: a dense, positively charged central nucleus with a number of electrons in various orbits such that the total negative charge exactly canceled the positive nuclear charge. Maxwell and Newton were duly invoked. The orbiting electron, like the planets, obeyed Newton's commandment, F = ma. F was now the electrical force (Coulomb's law) between charged particles. Since this is also an inverse-square force like gravity, one would assume at first glance that stable, planetary orbits would follow. And there you have it, the nice neat solar system model of the chemical atom. Everything was fine.
Well, it was fine until the arrival in Manchester of a young Danish physicist of theoretical persuasion. "Name's Bohr, Niels Henrik David Bohr Professor Rutherford. I'm a young theoretical physicist and I'm here to help you." We can only imagine the reaction of the gruff, earthy New Zealander.
THE STRUGGLE
The evolving revolution known as quantum theory didn't spring fully grown from the foreheads of theorists. It was slowly induced from data that emerged from the chemical atom. One can look at the struggle to understand this atom as practice for the real contest, understanding the sub-atom, the subnuclear jungle.
This slow unfolding of the real world is probably a blessing. What would Galileo or even Newton have done if the full data emerging from Fermilab had somehow been revealed to them? A colleague of mine at Columbia, a very young, very bright, articulate, enthusiastic professor, was given a unique teaching assignment. Take the forty or so freshmen who had declared physics as their major and give them two years of intensive instruction: one professor, forty aspiring physicists, two years. The experiment turned out to be a disaster. Most of the students switched to other fields. The reason came later from a graduating mathematics major: "Mel was terrific, best teacher I ever had. In those two years not only did we get through the usual—Newtonian mechanics, optics, electricity and so on—but he opened a window on the world of modern physics and gave us a glimpse of the problems he was facing in his own research. I felt there was no way I could ever handle such a difficult set of problems, so I switched to mathematics."
This raises a deeper question, whether the human brain will ever be prepared for the mysteries of quantum physics, which in the 1990s continues to disturb some of the very best physicists. Theoretician Heinz Pagels (who died tragically a few years ago in a mountain-climbing accident) suggested, in his fine book The Cosmic Code, that the human brain may not have evolved enough to understand quantum reality. Perhaps he's right, although a few of his colleagues seem convinced that they have evolved much more than the rest of us.
The overriding point is that quantum theory, as a highly refined and dominant theory of the 1990s, works. It works in atoms. It works in molecules. It works in complex solids, metals, insulators, semiconductors, superconductors, and anywhere it has been applied. The success of the quantum theory accounts for a significant fraction of the industrial world's total gross national product (GNP). But more important for us, it is the only tool we have as we proceed down into the nucleus, into its constituents and down, down into the vast minuteness of primordial matter—where we will confront the a-tom and the God Particle. And it is there that quantum theory's conceptual difficulties, dismissed by most working physicists as mere "philosophy," may play a significant role.
BOHR: ON THE WINGS OF A BUTTERFLY
Rutherford's discovery, coming after several experimental results that contradicted classical physics, was the last nail in the coffin. In the ongoing contest between experiment and theory, this would have been a good time to rub it in: "How clear do we experimenters have to make it before you theorists are convinced you need a new thing?" It appears that Rutherford didn't realize how much havoc his new atom was going to wreak on classical physics.
And then along came Niels Bohr, who would play Maxwell to Rutherford's Faraday, Kepler to his Brahe. Bohr's first position in England was at Cambridge, where he went to work for the great J. J., but the twenty-five-year-old kept irritating the master by finding mistakes in his book. While studying at the Cavendish Lab, on a Carlsberg Beer fellowship no less, Bohr heard Rutherford lecture about his new model of the atom in the fall of 1911. Bohr's thesis had been a study of "free" electrons in metals, and he was aware that all was not well with classical physics. He knew of course about Planck and about Einstein's more dramatic deviation from classical orthodoxy. And the spectral lines emitted by certain elements when they were heated was an
other clue to the quantum nature of the atom. Bohr was so impressed by Rutherford's lecture, and his atom, that he arranged to go to Manchester for a four-month visit in 1912.
Bohr saw the real significance in the new model. He realized that to satisfy Maxwell's equations, the electrons in circular orbits around a central nucleus would have to radiate energy, just like an electron accelerating up and down an antenna. To satisfy the laws of energy conservation, the orbits would shrink, and in no time flat the electron would spiral into the nucleus. If all these conditions were met, matter would be unstable. The model was a classical disaster! Yet there really was no alternative.
Bohr had no choice but to try something very new. The simplest atom of all is hydrogen. So Bohr studied the available data, such as how alpha particles slow down in hydrogen gas, and concluded that hydrogen has one electron in a Rutherford orbit around a positively charged nucleus. In facing up to a break with classical theory, Bohr was encouraged by other curiosities. He noted that nothing in classical physics determines the radius of the electron's orbit in the hydrogen atom. In fact, the solar system is a good example of a variety of planetary orbits. According to Newton's laws, any planetary orbit can be imagined; all it needs is to be started off properly. Once a radius is fixed, the planet's speed in orbit and its period (the year) are determined. But all hydrogen atoms, it would seem, are exactly alike. The atom shows none of the variety exhibited by the solar system. Bohr made the sensible but absolutely anticlassical assertion that only certain orbits are allowed in atoms.
Bohr also proposed that in these very special orbits the electron doesn't radiate. This, in historical context, was an incredibly audacious hypothesis. Maxwell rotated in his grave, but Bohr was simply trying to make sense of the facts. One important fact concerned the spectral lines that Kirchhoff had found shining out of atoms decades earlier. Glowing hydrogen, like other elements, emits a distinctive series of spectral lines. To get spectral lines, Bohr realized he must allow the electron to have the option of a number of different orbits corresponding to different energy contents. So he gave hydrogen's single electron a set of allowed radii representing a set of states of higher and higher energy. To explain spectral lines, he postulated (out of the blue) that radiation occurs when an electron "jumps" from one energy level to a lower one; the energy of the radiating photon is the difference of the two energy levels. He then proposed an absolutely outrageous rule for these special radii that determine the energy levels. Orbits are allowed, he said, in which the angular moment, a well-known quantity that measures the rotational momentum of the electron, takes on only integer values when measured in a new quantum unit. Bohr's quantum unit was nothing but Planck's constant, h. Bohr later said that "it was in the air to try to use the preexisting quantum ideas."
Now what is Bohr doing in his attic room late at night in Manchester with a sheaf of blank paper, a pencil, a sharp knife, slide rule, and some reference books? He is searching for nature's rules, rules that will correspond to the facts listed in his reference books. What right does he have to make up rules for the behavior of invisible electrons orbiting the nucleus (also invisible) of the hydrogen atom? His legitimacy is ultimately established by his success in explaining the data. He starts with the simplest atom, hydrogen. He understands that his rules ultimately have to emerge from some deep principle, but first the rules. This is how theorists work. Bohr in Manchester was, in the words of Einstein, trying to know the mind of God.
Bohr soon returned to Copenhagen to allow his seminal idea to germinate. Finally, in three papers published in April, June, and August of 1913 (the great trilogy), he presented his quantum theory of the hydrogen atom—a mixture of classical laws and totally arbitrary assertions (hypotheses) clearly designed to get the right answer. He manipulated his model of the atom so that it would explain the known spectral lines. Tables of these spectral lines, a series of numbers, had been painstakingly compiled by the followers of Kirchhoff and Bunsen, checked in Strasbourg and Göttingen, in London and Milan. What kind of numbers? Here are a few from hydrogen: λ1 = 4,100.4, λ2 = 4,339.0, λ3 = 4,858.5, λ4 = 6,560.6. (Sorry you asked? Don't worry. No need to memorize them.) How do these spectral vibrations come about? And why only these, no matter how the hydrogen is energized? Oddly, Bohr later minimized the importance of spectral lines: "One thought that spectra are marvelous. But it is not possible to make progress there. Just as if you had the wing of a butterfly, then certainly it is very regular with its colors and so on. But nobody thought that you could get the basis of biology from the coloring of the wing of a butterfly." And yet it turned out that the spectral lines of hydrogen, the wing of the butterfly, provided a crucial clue.
Bohr's theory was crafted to give the numbers for hydrogen that were on the books. Crucial to his analyses was the overriding concept of energy, a term that was precisely defined in Newton's time, then evolved and enlarged. An understanding of it is necessary for the educated person. So let's take two minutes for energy.
TWO MINUTES FOR ENERGY
In high school physics, an object with a certain mass and a certain velocity is said to have kinetic energy (energy by virtue of motion). Objects have energy also by virtue of where they are. A steel ball on top of the Sears Tower has potential energy because someone worked hard to get it up there. If you drop it off the tower, it will, in falling, trade in its potential energy for kinetic energy.
The only thing that makes energy interesting is that it is conserved. Picture a complex system of billions of atoms in a gas, all in rapid motion, colliding with the walls of the vessel and with one another. Some atoms may gain energy; others lose it. But the total energy never changes. It wasn't until the eighteenth century that scientists discovered that heat is a form of energy. Chemicals can release energy via reactions such as the burning of coal. Energy can and does continually change from one form to another. Today we recognize mechanical, thermal, chemical, electrical, and nuclear energy. We know that mass can be converted to energy via E = mc2. In spite of these complexities, we are still a hundred percent convinced that in complex reactions the total energy (including mass) always remains constant. Example: slide a block along a smooth plane. It stops. Its kinetic energy was changed into heat in the ever so slightly warmer plane. Example: you fill your car with gasoline, knowing that you have bought 12 gallons of chemical energy (measured in joules), which you can use to give your Toyota a certain kinetic energy. The gasoline goes away, but its energy can be accounted for—320 miles, from Newark to North Hero. The energy is conserved. Example: a waterfall crashes onto the rotor of an electric generator converting natural potential energy to electrical energy to warm and illuminate a distant town. In nature's ledger it all has to add up. You end up with what you brought.
SO?
Okay, how does this relate to the atom? In Bohr's picture, the electron must confine itself to specific orbits, with each orbit defined by its radius. Each of the allowed radii corresponds to a well-defined energy state (or energy level) of the atom. The smallest radius corresponds to the lowest energy, which is called the ground state. If we pour energy into a sample of hydrogen gas, some of it is used in shaking up the atoms so that they move faster. Some of the energy, however; is absorbed by the electron in a very specific bundle (remember the photoelectric effect), which allows the electron to reach another of its energy levels, or radii. The levels are numbered 1, 2, 3, 4, ..., and each one has its energy, E1, E2, E3, E4, and so on. Bohr constructed his theory to include the Einstein idea that the energy of a photon determines its wavelength.
If photons of all wavelengths rain down on a hydrogen atom, the electron will eventually swallow the appropriate photon (light bundle of some particular energy) and jump up from E1 to E2 or E3, say. In this way electrons populate the higher energy levels of the atom. This is what happens, for example, in a discharge tube. When electrical energy goes in, the tube glows with the characteristic colors of hydrogen. The energy induces some electrons in the trillions of atoms to jump
to higher energy states. If the input electrical energy is large enough, many of the atoms will have electrons occupying essentially all possible higher energy states.
In Bohr's picture the electrons in higher energy states spontaneously jump down to lower levels. Now remember our little lecture on the conservation of energy. If electrons jump down, they lose energy, and that lost energy has to be accounted for. Bohr said, "No problem." A dropping electron emits a photon of energy equal to the difference in energy of the orbits. If the electron jumps from level 4 to level 2, for example, the photon's energy is equal to E4 minus E2. There are lots of jump possibilities, such as E2→E1, E3→E1, or E4→E1. Multilevel jumps are also allowed, such as E4→E2, then E2→E1. Every change of energy results in the emission of a corresponding wavelength, and a series of spectral lines is observed.
Bohr's ad hoc, quasi-classical explanation of the atom was a virtuoso, if unorthodox, performance. He used Newton and Maxwell when they were convenient. He discounted them when they weren't. He used Planck and Einstein when they worked. It was outrageous. But Bohr was smart, and he got the right answer.
Let's review. Thanks to the work of Fraunhofer and Kirchhoff back in the nineteenth century, we knew about spectral lines. We knew that atoms (and molecules) emit and absorb radiation at specific wavelengths and that each atom has its own characteristic pattern of wavelengths. Thanks to Planck, we knew that light is emitted in quanta. Thanks to Hertz and Einstein, we knew that it is also absorbed in quanta. Thanks to Thomson, we knew there are electrons. Thanks to Rutherford, we knew that the atom has a dense little nucleus, lots of void, and electrons scattered throughout. Thanks to my mother and father, I got to learn this stuff. Bohr put this data—and much more—together. The electrons are allowed only certain orbits, said Bohr. They absorb energy in quanta, which forces them to jump to higher orbits. When they drop back down to lower orbits, they emit photons, quanta of light. Scientists observe these quanta as specific wavelengths—the spectral lines peculiar to each element.
The God Particle Page 21