Adventures of a Mathematician

Stanislaw Ulam

Adventures of a Mathematician

In memory of my parents

Preface to the 1991 edition

by William G. Mathews and Daniel Hirsch

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.

This remark of Stanislaw Ulam's is particularly appropriate to his own career. Our world is very different today because of Ulam's contributions in mathematics, physics, computer science, and the design of nuclear weapons.

While still a schoolboy in Lwów, then a city in Poland, he signed his notebook "S. Ulam, astronomer, physicist and mathematician." Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam's early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day. The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam's highly verbal and collaborative style early on. Ulam's early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam's lifelong, highly creative quest for new mathematical and scientific problems.

As conditions in prewar Poland deteriorated, Ulam welcomed opportunities to visit Princeton and Harvard, eventually accepting a faculty position at the University of Wisconsin. As United States involvement in World War II deepened, Ulam's students and professional colleagues began to disappear into secret government laboratories. Following a failed attempt to contribute to the Allied war effort by enlisting in the U. S. military, Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. It was at Los Alamos that Ulam's scientific interests underwent a metamorphosis and where he made some of his most far-reaching contributions.

On his very first day at Los Alamos he was asked to work with Edward Teller's group on the "Super" bomb project, an early attempt to design a thermonuclear or hydrogen bomb. Except for Teller's small group, the scientists at Los Alamos were working on the design and construction of an atomic bomb based on the energy released by the fission or breakup of uranium or plutonium nuclei. Although there was a general consensus at Los Alamos that the fission bomb would have to precede the Super for which it would serve as an ignition device, Teller was already preoccupied with the Super and re-fused to work on the fission bomb calculations. As a means of retaining Teller at Los Alamos, Robert Oppenheimer as lab director allowed Teller to work on the Super bomb with several scientists and assistants. Teller's assignment for Ulam on his arrival at Los Alamos was to study the exchange of energy between free electrons and radiation in the extremely hot gas anticipated in thermonuclear bombs. Ironically, this first-day problem for Ulam in 1943 would later become a critical part of Ulam's work with Cornelius Everett in 1950 in which he demonstrated that Teller's design for the Super bomb was impractical.

This first problem in theoretical physics was the beginning of a major scientific transition for Ulam from the esoteric, abstract world of pure mathematics to a quite different kind of applied mathematics necessary to visualize and solve problems in physics. The mathematics relevant to the physical problems at Los Alamos involved differential and integral equations that describe the motion of gas, radiation, and particles. The transition from pure mathematics to physics is seldom attempted and very rarely accomplished at Ulam's level. The creative process and the initial guesswork that lead to significant new ideas in physics involve an added dimension of taste and judgment extending beyond the rigorous logic of mathematics alone. Physical intuition which "very few mathematicians seem to possess to any great degree" is constrained by knowledge of natural phenomena determined from experiment. Ulam claims not to have experienced this "gap between the mode of thinking in pure mathematics and the thinking in physics." Indeed, in these memoirs Ulam discusses his transition from pure mathematics to mathematical physics and hopes that his analysis ''of thinking in science is one of the possible interests of this book."

Ulam could hardly have been in better company to learn physics. During the war years the scientists assembled at Los Alamos represented a Who's Who of modern physical science. The large number of eminent physicists — Hans Bethe, Niels Bohr, Enrico Fermi, Richard Feynman, Ernest Lawrence, J. Robert Oppenheimer, and so on — formed an intellectual powerhouse of physics that has not been surpassed before or since.

During the war years Ulam contributed to the development of the fission bomb with statistical studies on the branching and multiplication of neutrons responsible for initiating and sustaining the chain reaction and energy release in uranium or plutonium. A critical problem on which Ulam worked with von Neumann was the detailed calculation of the implosion or compression of a sphere of uranium effected by an external chemical detonation. When uranium is compressed the small number of naturally occuring neutrons created by random fissions of uranium nuclei collide more easily with other uranium nuclei. Some of these collisions result in further fissions, multiplying further the number of neutrons until a rapid chain reaction ensues, ultimately releasing an extraordinary amount of energy in a powerful explosion. In order to predict the amount of energy released, Los Alamos scientists needed to estimate the detailed behavior of the uranium as it was being compressed. Although this problem was conceptually straightforward, accurate solutions were not possible using standard mathematical analyses. This problem was quite literally at the secret core of atomic bomb research at Los Alamos — even the word "implosion" was classified during the war.

But Ulam's most remarkable achievement at Los Alamos was his contribution to the postwar develoment of the thermonuclear or hydrogen bomb in which nuclear energy is released when two hydrogen or deuterium nuclei fuse together. Ulam was a participant at a Los Alamos meeting in April, 1946, at which the wartime efforts on the Super bomb were discussed and evaluated. The conceptual idea of the "Classical Super" was to heat and ignite some part of a quantity of liquid deuterium by using an atomic bomb. The thermal energy deposited in this part would initiate deuterium reactions which would in turn heat adjacent regions, inducing further thermonuclear reactions, until the detonation would propagate through the entire amount of deuterium fuel. Deuterium, a heavier osotope of hydrogen having an extra neutron in its nucleus, was preferred since it reacts at significantly lower temperatures than ordinary hydrogen. Tritum, a third and even heavier form of hydrogen with two neutrons, reacts at even lower temperatures but, unlike deuterium, is virtually nonexistent in nature and was extremely expensive to make in nuclear reactors.

The evaluation of Teller's Super project at the 1946 meeting was guardedly optimistic, but the participants were aware of major technical uncertainties and potential difficulties with the Super design. In discussing the conclusions reached at the meeting J. Carson Mark has written, "The estimates available of the behavior of the various steps and links in the sort of device considered were rather qualitative and open to question in detail. The main question of whether there was a specific design of that type which would work well was not answered." Studies prior to 1946 had established that the net balance of energy gains over losses in the Super bomb was marginal; there was no large margin of design flexibility for which a successful detonation could be guaranteed. According to Mark,

As it was, the studies of this question had merely sufficed to show that the problem was very difficult indeed; that the mechanisms by which energy would be created in the system and uselessly lost from it were compar ...

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