Boiling Electrons

David Kaiser

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A decade ago, digging through a physicist’s archive, I stumbled on a document that has haunted me ever since: a hand-typed table of integrals seemingly little different from the ones I’d kept by me as a student. The familiarity of the contents jarred with the table’s front page. Only 31 copies of the table had been printed, the recipients listed on the cover. The table, dated 24 June 1947, had been prepared to accompany a classified report. The distribution lists for the two documents were a close match; nearly everyone who was issued with the table had security clearance to handle secret defence-related materials.

How to reconcile the tables’s banal contents with its restricted circulation? What disaster would have befallen the US government if enemies of the state had learned that the integral of x/(1 + x)2 between x = 0 and x = 1 equalled 0.1931? How could the authorities have hoped to limit access to basic mathematical results? Wouldn’t anyone schooled in the routines of calculus arrive at the same answers, whether or not they appeared on the table’s special distribution list?

The classified report was written by the physicist and Nobel laureate Hans Bethe. (I found both the report and the table among Bethe’s papers at Cornell University in upstate New York.) Bethe had become one of the world’s experts on nuclear physics in the 1930s; by 1938 he had pieced together the complicated nuclear reactions that make stars shine. He served as the director of the Theoretical Division at wartime Los Alamos, reporting directly to Robert Oppenheimer. After the war, he returned to Cornell, but he remained active as a consultant to the nuclear weapons programme, as well as to the budding nuclear power industry.

In 1947, Bethe was asked to tackle the problem of shielding for nuclear reactors. In trying to work out how best to block or absorb the radiation released when heavy nuclei like uranium or plutonium are blasted apart by neutrons, Bethe kept finding that he needed to evaluate integrals of a particular form. A colleague – another Manhattan Project veteran and by then a senior researcher at a nuclear-reactor facility – prepared the table of integrals so that selected co-workers would be able to perform calculations like Bethe’s.

Similar mathematical handbooks and tables had been produced for centuries. Around the time of the French Revolution, as Lorraine Daston has written, leading civil servants produced mammoth tables of logarithms and trigonometric functions calculated to 14 or more decimal places – far greater precision than any practical application would then have required. Gaspard Riche de Prony’s tables were a deliberate demonstration of Enlightenment mastery, one more testament to the triumph of Reason, to be admired more than used.

Though the 1947 table of integrals was not prepared in anticipation of public fanfare (just the opposite, as its distribution list made clear), the table stood closer to Prony’s time than to ours. Indeed, the introduction explained that most of the integrals had been evaluated by making clever changes of variables so that the functions of interest matched forms that had been reported in the venerable Nouvelles tables d’intégrales définies, published in Leyden in 1867 by the wealthy Dutch mathematician David Bierens de Haan.

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