Skinning Cats
It took a man of singular determination and self-confidence to propose a cyclotron capable of accelerating particles to 100 MeV. The substantial cost—projected at perhaps a million dollars—was not at first the major impediment. Nature, not money, seemed to set a limit to the size of cyclotrons. The difficulty, that the increase of mass with speed claimed by the theory of relativity would destroy the synchronism expressed in the cyclotron equation (2.1), had been noticed in 1931, by Livingston and by Feenberg; but the limit, whatever it might be, evidently did not affect the performance of the first cyclotrons, and the menace faded from view. When presenting Lawrence with the Comstock prize in November 1937, W.D. Coolidge saw no obstacles: "The limit to the particle energies which can be generated in this way is not yet in sight."[4] Precisely at that moment, however, Bethe and his student Morris E. Rose declared that in their calculations relativity limited the maximum energies obtainable in a cyclotron to about those achievable with the 37-inch machine. They observed that to
compensate for the continually rising mass, the magnetic field must increase toward the periphery of the orbit to keep the circulating particles in phase with the oscillator. But to focus the particles in the median plane, the field must decrease from the center outward. The cyclotroneer wants both resonance and focusing; nature requires a choice.
According to Bethe and Rose, the best that can be done is to sacrifice exact resonance; but even so, and with the best field design they could contrive, maximum energy would be 5.5 MeV for protons, 8 MeV for deuterons, and 16 MeV for alpha particles. This estimate supposed 50 kV on the dees; with 100 kV something more could be done, since the accelerated particles would acquire energy more quickly and so have more of it when they finally fell out of phase with the accelerating voltage. Still the outlook was grim. With a magnetic field of 18 kilogauss and a final orbit 37 cm in radius, deuterons of 11 MeV would emerge, "the highest obtainable with as much as 100 kV dee voltage." For such a cyclotron, pole faces 34 inches in diameter would suffice. "Therefore it seems useless to build cyclotrons of larger proportions than the existing ones."[5]
When the Laboratory received this news, it was engaged in what, according to the Cassandras of Cornell, was a wasteful and useless task. But its experience with the 37-inch gave it confidence that the 60-inch could go beyond 10 MeV despite the most refined calculations to the contrary. Lawrence wrote Bethe that relativity had not yet begun to inconvenience cyclotroneers; the existing inhomogeneities in the magnetic field of the 37-inch defocused more menacingly than the mass increase and indicated considerable room for maneuver in the 60-inch. And if shimming were to fail, other possibilities existed, for example, placing wire mesh across the mouths of the dees so as to obtain by electrical force the focusing that would be lost by adjustment of the magnetic field to secure resonance. "We have learned from repeated experience that there are many ways of skinning a cat."[6]
This response was not bluff. For a year or so Robert Wilson had been poking around inside the cyclotron tank, determining empirically the strength of the vertical component of the electric field near the dee mouths and of the radial component of the fringing magnetic field, which drives ions toward the meridian plane. The investigation of the circulating current, which eventuated in the internal target for isotope production, was part of his study. Wilson painstakingly worked out the trajectories of ions beginning their courses at any distance from the median plane and reaching the center of the gap in phase with the maximum field there. His numerical integrations showed that from about 10 cm out, where the focusing effect of the electric field becomes negligible (it decreases with the particles' energy), the magnetic field swiftly reduced the vertical amplitude of the beam from a spread of some 5 cm near the ion source to about 1 cm at the exit slit. Probe measurements confirmed Wilson's semi-empirical deduction of beam width as a function of orbital radius. He therefore felt confident in recommending that the aperture of the dees also be made to decrease with radius, thereby reducing their capacitance and easing the performance requirements for power oscillators to accelerate protons.[7]
Wilson presented his results in a seminar about the time that Bethe and Rose's letter of November 24 was circulating in Berkeley. The circumstances inspired McMillan to estimate the defocusing effect of relativity. It was he who found that for the 37-inch defocusing arising from inhomogeneities in the magnetic field exceeded that from relativity by a factor of four. Also, McMillan calculated from experience at Berkeley that the beam could fall out of phase with the electric field by more than 60° and still get through the cyclotron; and on this basis he calculated that the maximum energy of deuterons achievable without altering basic cyclotron design was perhaps 
MeV with 100 kV on the dees. With Lawrence's grids, he thought, any amount of electrostatic focusing could be attained.[8] On this last point McMillan had a short and victorious duel by mail with Bethe, who had thrust forward the opinion that "no change of the shape of the dees, no
insertion of grids at the dee openings, etc., can have any appreciable effect on the electric focusing." McMillan parried that Bethe had mistakenly assimilated a dee with grid to an open dee of smaller aperture; Bethe concurred, and allowed the possibility of doubling the energy limit.[9]
Meanwhile Rose, who had also been working for a long time on cyclotron focusing, sweated to get his theory ready for the press. Whereas Wilson and McMillan relied on their experience with a single machine, Rose began with general equations of motion in changing electric and magnetic fields and deduced, by clever substitutions, a differential equation for the excursion of an ion from the median plane as a function of the phase of the radio frequency voltage it met as it crossed between the dees. His treatment of the general case—which "had been considered much too complicated for solution by many"—agreed with the conclusions about electric and magnetic focusing reached in Berkeley.[10] Rose could do more: from his differential equation he could deduce the maximum energy obtainable without defocusing the beam when the gradient of the magnetic field compensates for relativity. He ended more generous than he and Bethe had begun. They allowed, in a note added to their initial announcement on December 4, that a field giving an angular velocity too large for resonance at the start and too small at the finish could deliver deuterons of 17 MeV with V = 50 kV, a number Rose raised to 21.1 MeV. These numbers would be multiplied by 
if 100 kV were placed across the dees. Rose thought that no greater potential could be reached without severe difficulty and that grids would not have the power Lawrence supposed. "It seems very possible that the energies mentioned [21.1 MeV deuterons] represent the natural upper limit for the cyclotron with the given dee voltage of 50 kV, at least without very radical changes in design."[11]
As Bethe conceded to McMillan, after explaining that he and Rose had published hurriedly because "we considered the existence of a relativistic limit so important that we thought we should communicate it to cyclotronists as quickly as possible, without endeavoring to give accurate figures," "it makes all the difference in the world whether the limit is 8 MV or 20."[12] Or 100. The question came before that high tribunal of science, Time , whose investigative science editor, Walter Stackley, drew from Lawrence a firm rejection of the 20 MeV limit. There was new work under way at Berkeley, Lawrence said, "which may increase the energy maxima materially." "We believe that there are experimental possibilities of improving focusing conditions which remove the limitation on energy to some unknown point."[13] And, just at this point, L.H. Thomas, known to physicists as the discoverer of a relativistic effect important in atomic theory (the "Thomas precession"), described a novel way to achieve both focusing and resonance by a magnetic field that had notably different strengths in several pie-shaped sections into which he divided the median plane. Thomas's ingenious suggestion received some attention at Berkeley and more at Stanford, where Oppenheimer's former student Leonard Schiff continued the calculations. Although the scheme, which is difficult to put into practice, was not exploited at the time, it gave ample evidence that nature did allow for several methods of cat-skinning.[14]
The opinion of the experienced cyclotroneer about Bethe's limit is nicely reflected in notes by the Rockefeller Foundation's Tisdale. After recording that "Joliot's cyclotron, by a lucky chance, is designed just to the limit of the theoretical voltage," which would have been at once a triumph and an end to the Foundation's investments in cyclotrons, Tisdale reported that Paxton would have none of it. "P[axton] considers that the mass effect is not very important."[15] The cyclotroneer did not doubt that so refined a thing as a relativity effect could be beaten by brute force.
Thornton: "Difficulties in reaching high voltages seem to me quite real. . . . But of course there are a number of ways [by] which one may get around [Bethe's] objections." Wells: "It can probably be compensated by applied magnetic inhomogeneities of the field or by properly chosen electrostatic fields." Compton: "[It] can be passed (theoretically) by altering pole pieces and electrostatic focusing. Thus no limit is now assignable." Oliphant: "I am not deterred by papers which have been written on the maximum energy obtainable from a cyclotron."[16] Gentner looked to Thomas's method. Cockcroft preferred to follow Lawrence, who thought azimuthally changing magnetic fields impractical and no longer favored fitting the dees with wires. Instead, he bruited a solution in the style of the Old West: put a million or two million volts on the dees and drive the beam home before it knows that it has been defocused.[17]