Magnetic and Electrostatic Focusing in the Cyclotron

Abstract

A quantitative study of the magnetic and electrostatic focusing of ions being accelerated in the cyclotron is made on the basis of electric and magnetic field measurements. The relative importance of the two effects is found to depend on the energy or path radius of the circulating particles. For small energies, only the electrostatic action is effective; and it is found that actually it constitutes a defocusing action which causes the ions to move away from the central plane between the magnet pole faces and so to become lost. At larger energies the electrostatic field becomes a focusing action but also the effect becomes rapidly smaller and, if acting alone, would cause the particles to oscillate back and forth across the central plane with increasing amplitude so that a very diffuse beam would result. On the other hand, at the larger energies, the focusing due to the outward curvature of the magnetic lines of force soon predominates and becomes rapidly larger and causes the ions to oscillate about the central plane, but now with decreasing amplitude so that it is by the magnetic effect that the diffuse starting ions are focused to the concentrated beam which is actually obtained. It is shown that the amplitudes of vertical oscillation due to the electrostatic field increases with $r^{\frac{3}{4}}$, and that the amplitudes due to the magnetic field vary as ${(-\frac{r\partial H}{\partial r})}^{-\frac{1}{4}}$ for ions in phase. Calculations of the vertical motion of the ions as they circulate out from the center by means of a step by step process of integration indicate that the electrostatic defocusing action decreases with $\frac{d}{h}$, the ratio of electrode separation to electrode height. Thus for maximum output currents the electrode height should be as large and the separation as small as possible. The width of the beam and distribution of ions with height was measured at various distances from the center for the case of the cyclotron in use at Berkeley up to August 1937. This was done by measuring the radioactivity induced in a probe which was introduced into the beam. The widths so obtained agree within one or two mm with the widths calculated from the theory presented. Various changes in accelerator electrode design are suggested which would decrease their capacity and so enable larger high frequency input voltages to be obtained.