Focusing and Maximum Energy of Ions in the Cyclotron

The effect on the motion of the ions of the inhomogeneities of the electric and magnetic fields in the cyclotron has been investigated. It is shown that both fields exert a focusing action on the ions, the electric focusing predominating over the first part of the path and the magnetic focusing over the remaining part. For the magnetic fields hitherto used, magnetic focusing, in its domain, is more efficient and the boundary between electric and magnetic focusing lies at a radius of about half the exit slit radius. The most important factor in the electric focusing is the relative change of the radiofrequency field during the acceleration of the ion. If the electric field decreases during the time of acceleration the ion will oscillate slowly about the median plane, the amplitude of these oscillations increasing as the square root of the distance from the center of the cyclotron. For an increase of the electric field during the acceleration the ions are defocused except in the case of acceleration very near the peak dee voltage. Thus somewhat more than half of the radiofrequency cycle will be effective in producing obtainable ions. The usual magnetic fields give a strong focusing of the ions resulting in a decreasing amplitude—convergence of the beam. The only requirement for magnetic focusing is that the magnetic field near the median plane decrease radially outwards from the center of the cyclotron chamber. The effect of the deviation of the ions from resonance with the electric field oscillations is discussed. The magnetic field must be of such a form that the ions (or at least some of them) do not encounter a retarding electric field. This "resonance condition" on the magnetic field is in some conflict with the requirement of good focusing. The necessity of finding a compromise between these two effects explains the very great sensitivity of the beam intensity on "shimming." Magnetic fields are described which give the greatest energy with measurable beam intensities. The main feature of these fields is that they are greater than the field required for exact resonance near the center, and become smaller than the resonance field near the exit slit. Satisfactory results are obtained with a field which is, at the center, 0.9 percent larger than the resonance field and behaves as $1-4\times {10}^{-3\mathrm{}}{\left(\frac{r}{R}\right)}^2$ with increasing distance $r$ ($R$ the radius of the exit slit). This field will give protons of 13.9 Mev, deuterons of 19.7 Mev and $\alpha$-particles of 39.4 Mev energy if the peak voltage on the dees is 50 kv. With increasing dee voltage, $2V_0$, the maximum energies increase as ${V_0}^{\frac{1}{2}}$. A somewhat more complicated field is derived which gives protons of 14.9 Mev, deuterons of 21.1 Mev and $\alpha$-particles of 42.2 Mev with intensities as high as one-quarter of the intensities obtained at low energies. The absolute maximum energies obtainable are about 5 percent higher than these values. It is pointed out that these high energies can only be obtained with extremely careful "shimming."