Genius (30 page)

Mathematics, in the form of probability theory, had barely begun to provide tools for handling such complex patterns; he discussed the problem with the Polish mathematician Stanislaw Ulam, and Ulam’s approach to it helped midwife a new field of probability called branching-processes theory. Feynman himself worked out a theory of fluctuations building upward from the easier-to-calculate probabilities of short chain reactions: a neutron splits one atom; a newly liberated neutron finds another target; but then the chain breaks. Some measurable fluctuations—audible bursts of noise on a Geiger counter—could be traced back to an origin in a single fission event. Others were combinations of chains. As with so many other problems, Feynman took a geometrical approach, considering the probability that a burst in a certain unit volume would lead to a burst in another unit volume at a given time later. He arrived at a practical method that reliably computed the chances of any premature reaction taking hold. It was suitable even for the odd-shaped segments of uranium that would be blasted into one another in the Hiroshima bomb.

Bethe found in Feynman the perfect foil and goad. This young man was quick, fearless, and ambitious. He was not satisfied to take away one problem and work on it; he wanted to work on everything at once. Bethe decided to make him a group leader, a position otherwise reserved for prominent physicists like Teller, Weisskopf, Serber, and the head of the British contingent at Los Alamos, Rudolf Peierls. For his part Feynman, who had lived through twenty-five years and a full formal education without ever falling under the spell of a mentor, began to love Hans Bethe.

Diffusion

Feynman did some recruiting for the project. He had invited one of his MIT fraternity friends to join the secret work. He even tried to recruit his father. Melville’s health had turned poor—his chronic high blood pressure affected him more and more—and Lucille wished he could afford to travel less. Richard wrote his mother that there might be a job available as a purchasing clerk. He wished, too, that Melville could see at close range the heady intellectual world toward which he had so long aimed his son. “He would be partly out of the rush, etc. of the business work, & would be among academic men to a great extent, which I’m sure he would enjoy … Purchasing these days is quite difficult, & everyone here is in a hell of a hurry for their stuff … it will be a damn important position in our project and scientific venture.”

Nothing came of that suggestion. In the spring of 1944 Feynman came across a familiar name on a list of available physicists: T. A. Welton. He filled out a requisition. His college friend, working as an instructor at the University of Illinois, had been trying to remain a civilian by teaching military-related courses and had watched unhappily as the more distinguished members of his department disappeared to mysterious locales. Feynman’s requisition rescued him. Welton, like so many physicists by then, had pieced together more than the army security officials liked to think possible. When he was invited to meet a stranger in a hotel room in Chicago, and then invited by the stranger to drop everything and move to New Mexico, he understood that this was, as he said later, the classic impossible-to-refuse offer. The day he arrived, Feynman took him on a long hike down into a gorge that had lately been named Omega Canyon. He was able to startle Feynman with an affirmative answer to his first question, “Do you know what we’re doing here?”

“Yes,” Welton said. “You’re making an atomic bomb.” Feynman recovered quickly. “Well,” he asked, “did you know we’re going to make it with a new element?” His friend admitted that the news of plutonium had not drifted as far as Illinois. While they walked—Welton’s lungs desperately drawing in the underpressurized air of 7,000 feet above sea level—Feynman intoxicated him with a briefing. They talked about the bomb. There were now two designs. A uranium bomb would take the form of a gun, creating a critical mass by firing a uranium bullet at a uranium target. A plutonium bomb would use another audacious method. A hollow sphere would be blown inward on itself by the shock from explosives packed all around it. The hot plutonium atoms would be compressed not through one dimension, as in the gun, but through three dimensions. The implosion method, as it was accurately named, was starting to look better and better—in part because so many problems had plagued the alternatives. (Feynman did not mention his own initial reaction when implosion’s inventor, Seth Neddermeyer, first reported experiments on explosives wrapped around steel pipes. He had raised his hand in the back row and announced, “It stinks.”)

As Welton listened, trying to keep up along the narrow canyon walls, he understood that Feynman was also saying that he had worked hard to establish himself as a smart kid to be reckoned with—that a young researcher had to impress the senior people with his usefulness, that he, Feynman, had been through that process, and that he had succeeded. They talked only briefly about Arline. She was not well, spending most of her days in a wooden bed in the Presbyterian Sanatorium, a small, poorly staffed facility by the side of a highway in Albuquerque. Feynman, visiting her almost every weekend, hitchhiked or borrowed a car to head down the unpaved road toward Santa Fe on Friday afternoon or Saturday. Away from the laboratory he would turn his thoughts back to the pure theory of quantum mechanics. He used the long trip, and the hours when Arline slept, to push his thesis work further. Welton remembered how obstinately his friend had resisted the Lagrangian simplification of dynamical problems when they were a pair of precocious sophomores in MIT’s theory course. He was amused and impressed to hear how far Feynman had taken the Lagrangian method in reformulating the most basic quantum mechanics. Feynman sketched out his idea of expressing quantum behavior as a sum of all the possible space-time trajectories a particle could take, and he told Welton frankly that he did not know how to apply it. He had a wonderful recipe that had not gelled.

Welton became the fourth physicist in the group Feynman headed, now formally known as T-4, Diffusion Problems. As a group leader Feynman was ebullient and original. He drove his team hard in pursuit of his latest unorthodox idea for solving whatever problem was at hand. Sometimes one of the scientists would object that a Feynman proposal was too complex or too bizarre. Feynman would insist that they try it out, computing in groups with their mechanical calculators, and he had enough unexpected successes this way to win their loyalty to the cause of wide-ranging experimentation. They all tried to innovate in his fashion—no idea too wild to be considered. He could be ruthless with work that did not meet his high standards. Even Welton experienced the humiliation of a Feynman rebuke—“definitely ungentle humor” to which “only a fool would have subjected himself twice.” Still Feynman managed to build esprit. He had taught himself to flip a pencil in one motion from a table into his hand, and he taught the same trick to his group. One day, amid a typical swirl of rumors that military uniforms were going to be issued to scientists working in the technical area, Bethe walked in to talk about a calculation. Feynman said he thought they should integrate it by hand, and Bethe agreed. Feynman swiveled around and barked, “All right, pencils, calculate!”

A roomful of pencils flipped into the air in unison. “Present pencils!” Feynman shouted. “Integrate!” And Bethe laughed.

Diffusion, that faintly obscure and faintly pedestrian holdover from freshman physics, lay near the heart of the problems facing all the groups. Open a perfume bottle in a still room. How long before the scent reaches a set of nostrils six feet away, eight feet away, ten feet away? Does the temperature of the air matter? The density? The mass of the scent-bearing molecules? The shape of the room? The ordinary theory of molecular diffusion gave a means of answering most of these questions in the form of a standard differential equation (but not the last question—the geometry of the containing walls caused mathematical complications). The progress of a molecule dependedon a herky-jerky sequence of accidents, collisions with other molecules. It was progress by wandering, each molecule’s path the sum of many paths, of all possible directions and lengths. The same problem arose in different form as the flow of heat througha metal. And the central issues of Los Alamos, too, were problems of diffusion in a newguise.

The calculation of critical mass quickly became nothing more or less than a calculation of diffusion—the diffusion of neutrons through a strange, radioactive minefield, where now a collision might mean more than a glancing, billiard-ball change of direction. A neutron might be captured, absorbed. And it might trigger a fission event that would give birth to new neutrons. By definition, at critical mass the creation of neutrons would exactly balance the loss of neutrons through absorption or through leakage beyond the container boundaries. This was not a problem of arithmetic. It was a problem of understanding the macroscopic spreading of neutrons as built up from the microscopic individual wanderings.

For a spherical bomb the mathematics resembled another strange and beautiful diffusion problem, the problem of the sun’s limb darkening. Why does the sun have a crisp edge? Not because it has a solid or liquid surface. On the contrary, the gaseous ball of the sun thins gradually; no boundary marks a division between sun and empty space. Yet we see a boundary. Energy diffuses outward from the roiling solar core toward the surface, particles scattering one another in tangled paths, until finally, as the hot gas thins, the likelihood of one more collision disappears. That creates the apparent edge, its sharpness more an artifact of the light than a physical reality. In the language of statistical mechanics, the mean free path—the average distance a particle travels between one collision and the next—becomes roughly as large as the radius of the sun. At that point photons have freed themselves from the pinball game of diffusion and can fly in a straight line until they scatter again, in the earth’s atmosphere or in the sensitive retina of one’s eye. The difference in brightness between the sun’s center and its edge gave an indirect means of calculating the nature of the internal diffusion. Or should have—but the mechanics proved difficult until a brilliant young mathematician at MIT, Norbert Wiener, devised a useful method.

If the sun were a coolly radioactive metal ball a few inches across, with neutrons rattling about inside, it would start to look like a miniaturized version of the same problem. For a while this approach proved useful. Past a certain point, however, it broke down. Too many idealizing assumptions had to be made. In a real bomb, cobbled together from mostly purified uranium, surrounded by a shell of neutron-reflecting metal, the messy realities would defy the most advanced mathematics available. Neutrons would strike other neutrons with a wide range of possible energies. They might not scatter in every direction with equal probability. The bomb might not be a perfect sphere. The difference between these realities and the traditional oversimplifications arose in the first major problem assigned to Feynman’s group. Bethe had told them to evaluate an idea of Teller’s, the possibility of replacing pure uranium metal with uranium hydride, a compound of uranium and hydrogen. The hydride seemed to have advantages. For one, the neutron-slowing hydrogen would be built into the bomb material; less uranium would be needed. On the other hand, the substance was pyrophoric—it tended to burst spontaneously into flame. When the Los Alamos metallurgists got down to the work of making hydride chunks for testing, they set off as many as half a dozen small uranium fires a week. The hydride problem had one virtue. It pushed the theorists past the limits of their methods of calculating critical masses. To make a sound judgment of Teller’s idea they would have to invent new techniques. Before they considered the hydride, they had got by with methods based on an approximation of Fermi’s. They been able to assume, among other things, that neutrons would travel at a single characteristic velocity. In pure metal, or in the slow reaction of the water boiler, that assumption seemed to work out well enough. But in the odd atomic landscape of the hydride, with its molecules of giant uranium atoms bonded to two or three tiny hydrogen atoms, neutrons would fly about at every conceivable velocity, from very fast to very slow. No one had yet invented a way of computing critical mass when the velocities spread over such a wide range. Feynman solved that problem with a pair of approximations that worked like pincers. The method produced outer bounds for the answer: one estimate known to be too large and another known to be too small. The experience of actual computation showed that this would suffice: the pair of approximations were so close together that they gave an answer as accurate as was needed. As he drove the men in his group toward a new understanding of criticality (poaching sneakily, it seemed to them, on the territory of Serber’s group, T-2), he delivered up a series of insights that struck even Welton, who understood him best, as mystical. One day he declared that the whole problem would be solved if they could produce a table of so-called eigenvalues, characteristic values of energies, for the simplified model that T-2 had been using. That seemed an impossible leap, and the group said so, but they soon found that he was right again. For Teller’s scheme, the new model was fatal. The hydride was a dead end. Pure uranium and plutonium proved far more efficient in propagating a chain reaction.

In this way, amid these clusters of scientists, the theory of diffusion underwent a kind of scrutiny with few precedents in the annals of science. Elegant textbook formulations were examined, improved, and then discarded altogether. In their place came pragmatic methodologies, gimmicks with patches. The textbook equations had exact solutions, at least for special cases. In the reality of Los Alamos, the special cases were useless. In Feynman’s Los Alamos work, especially, an accommodation with uncertainty became a running theme. Few other scientists filled the foreground of their papers with such blunt acknowledgments of what was not known: “unfortunately cannot be expected to be as accurate”; “Unfortunately the figures contained herein cannot be considered as ‘correct’”; “These methods are not exact.” Every practical scientist learned early to include error ranges in their calculations; they learned to internalize the knowledge that three miles times 1.852 kilometers per mile equals five and a half kilometers, not 5.556 kilometers. Precision only dissipates, like energy in an engine governed by the second law of thermodynamics. Feynman often found himself not just accepting the process of approximation but manipulating it as a tool, employing it in the creation of theorems. Always he stressed ease of use: “… an interesting theorem was found to be extremely useful in obtaining approximate expressions … it does permit, in many cases, a simpler derivation or understanding …”; “… in all cases of interest thus far investigated … accuracy has been found ample … extremely simple for computation and, once mastered, quite simple to use in thinking about a wide variety of neutron problems.” Theorems as theorems, or objects of mathematical beauty, had never been so unappealing as at Los Alamos. Theorems as tools had never been so valued. Again and again the theorists had to devise equations with no hope of exact solution, equations that sentenced them to countless hours of laborious computation with nothing at the end but an approximation. When they were done, the body of diffusion theory had become a hodgepodge. The state of knowledge was written in no one place, but it was more practical than ever before.