John D. Cockcroft and Ernest Walton at the Cavendish Laboratory in Cambridge, England, sought a way into the nucleus through a prediction of quantum mechanics. George Gamow had suggested that a particle with too little energy to overcome the electrical repulsion of the nucleus through the barrier. (The trick was that the energy of the particle was not actually well-defined, according to Heisenberg's Uncertainty Principle). In 1930 Cockcroft and Walton used a 200-kilovolt transformer to accelerate protons down a straight discharge tube, but they concluded that Gamow's tunnelling did not work and decided to seek higher energies.
To penetrate the nucleus, Cockcroft and Walton built a voltage multiplier that used an intricate stack of capacitors connected by rectifying diodes as switches. By opening and closing switches in proper sequence they could build up a potential of 800 kilovolts from a transformer of 200 kilovolts. They used the potential to accelerate protons down an evacuated tube eight feet long. In 1932 they put a lithium target at the end of the tube and found that protons disintegrated a lithium nucleus into two alpha particles. A Soviet team in Kharkov found the same result several months later.
Robert Van de Graaff worked as an engineer for the Alabama Power Company before obtaining his Ph.D. in physics at Oxford. While a postdoctoral fellow at Princeton he conceived a device to build up a high voltage using simple principles of electrostatics. A belt of insulating material carries electricity from a point source to a large insulated spherical conductor. Another belt likewise delivers electricity of the opposite charge to another sphere. The spheres build up a potential until the electric field breaks down the air and a huge spark "arcs" across. By 1931 Van de Graaff could charge a sphere to 750 kilovolts, giving 1.5 megavolts differences between two oppositely charged spheres.
By increasing the radius of the spheres, Van de Graaff could reach higher voltages without arcing. The maximum voltage in theory, in megavolts, roughly equalled the radius of the sphere in feet. He was soon planning a pair of spheres 15 feet across.
The difficulties of maintaining high voltages led several physicists to propose accelerating particles by using a lower voltage more than once. Lawrence learned of one such scheme in the spring of 1929, while browsing through an issue of Archiv für Elektrotechnik, a German journal for electrical engineers. Lawrence read German only with great difficulty, but he was rewarded for his diligence: he found an article by a Norwegian engineer, Rolf Wideröe, the title of which he could translate as "On a new principle for the production of higher voltages." The diagrams explained the principle and Lawrence skipped the text.
Particles with a positive electric charge are drawn into the first cylindrical electrode by a negative potential; by the time they emerge from the tube the potential has switched to positive, which propels them away from the electrode with a second boost. Adding gaps and electrodes can extend the scheme to higher energies.
Lawrence soon brought David Sloan to Berkeley. He was a young expert in electronics from the General Electric Laboratories in Schenectady with experience in handling high voltages. While Lawrence was building the cyclotron, Sloan pursued Wideröe's linear accelerator. Sloan's device eventually had a series of thirty electrodes. By May 1931 it accelerated mercury ions to energies of a million volts. This work gave Lawrence and his students experience with oscillators and beam focusing, knowledge they would later apply to cyclotrons. Sloan, however, put the linear accelerator aside to develop a resonant transformer, which turned out to provide a powerful source of X-rays which was of great interest to hospitals.
The linear accelerator proved useful for heavy ions like mercury, but lighter projectiles (such as alpha particles) required a vacuum tube many meters long. Lawrence judged that impractical. Instead he thought of bending the particles into a circular path, using a magnetic field, in order to send them through the same electrode repeatedly.
A few quick calculations showed that such a device might capitalize on the laws of electrodynamics. The centripetal acceleration of a charged particle in a perpendicular magnetic field B is evB/c, where e is the charge, v the particle's velocity, and c the velocity of light. The mechanical centrifugal force on the particle is mv2/r, where m is the mass and r the radius of its orbit. Balancing the two forces for a stable orbit yields what is now known as the cyclotron equation: v/r = eB/mc.
Lawrence was surprised to find
that the frequency of rotation of a particle is independent of the radius of
the orbit: f = v/2 
r
= eB/2mc, with
r disappearing from the equation. The circular method would thus allow an electric
field alternating at a constant frequency to kick particles to ever higher energies.
As their velocities increased so did the radius of their orbit. Each rotation
would take the same amount of time, keeping the particles in step with the alternating
field as they spiralled outward.
An electric field with a frequency of about four million cycles per second lay in the realm of short radio waves. Lawrence's experience with these waves would come in handy, and recent advances in high-power vacuum-tube oscillators would be indispensable. Combined with a reasonable magnetic field, a potential on the electrodes of only ten thousand volts could accelerate an alpha particle to one million electron volts. Bigger magnets promised higher energies. In theory, the scheme offered the long-sought route to study the nucleus. Lawrence pressed students and professors to confirm his calculations and sketched out a device.
Putting the plan into practice, however, meant facing daunting obstacles. It required vacuum seals that could withstand the stresses of the alternating electric field and the magnet. Too poor a vacuum and the circulating particles might be bumped from their paths by air molecules. The particles might also go astray crossing the gap or, what would be the hardest problem, deviate from the horizontal plane of their orbits and crash into the floor or ceiling of the electrodes.
Lawrence bypassed these obstacles with the help of two cut-and-try discoveries. The first was to remove a metal grid from the entrance to the dees, which he had thought necessary for electrical shielding. Sloan's work had demonstrated that instead it interfered with focusing by the electric field that kept particles in the horizontal plane. The second trick was to insert small iron shims into the magnetic field to coax particles back into the orbital plane. A bit of tinkering gave Lawrence his million-volt projectiles.