M.M. Gordon

Performance estimates for injector cyclotrons

Abstract Available evidence on performance to be expected from injector cyclotrons is collected and reviewed. On the basis of this information, estimates are made of current, spot size, angular divergence, energy spread and duty cycle for the extracted beam of possible 40 MeV and 200 MeV injector cyclotrons.

The electric gap-crossing resonance in a three-sector cyclotron†

The theory of accelerated orbits in the median plane of a sector-focused cyclotron is developed using a Lagrangian formalism. The equation for the radial oscillations about the instantaneous equilibrium orbit is reduced, by suitable transformation, to that for a simple oscillator of frequency vr driven by perturbations arising from the electric gap-crossing mechanism. For a three-sector magnet and a “two-sector” dee-geometry, these perturbations have Fourier components in resonance with the radial oscillation frequency vr, which is always close to unity in a medium-energy machine. The effects of these perturbations are calculated and discussed. The results of computer studies of accelerated orbits in three-sector fields are presented which demonstrate the effects of the gap-crossing resonance under various conditions. These results are discussed with the aid of the theory. The data indicate that a first-harmonic correction field of 3 to 10 gauss can be used to counteract the gap-crossing resonance effects.

Effects of spiral electric gaps in superconducting cyclotrons

Abstract The K = 500 MeV superconducting cyclotron being built here has an RF system containing three dees with spiral electric gaps. These gaps produce a significant force component transverse to the ion orbits. An analysis is presented which shows nevertheless that the effect of such gaps does not alter the relevant orbit period, in accordance with an unpublished theorem due to McMillan. The structure of the “Spiral Gap” orbit program is then described. This program calculates ion orbits in a given magnetic field and provides a systematic treatment of the transverse as well as the longitudinal electric force resulting from spiral electric gaps. Sample results are presented which show that there is no significant difference in the phase-energy history of ion orbits accelerated with spiral or non-spiral electric gaps. A discussion of central ray orbits and accelerated equilibrium orbits is also presented together with analytical and computer results.

DESIGN CONSIDERATIONS FOR A SEPARATED TURN ISOCHRONOUS CYCLOTRON.

Abstract The objective of this machine is the production of large proton currents at energies from 200 to 1000 MeV with clear turn separations so that 100% beam extraction is achieved. Each stage of the cyclotron consists of a “ring” of identical magnet sectors interpersed with an appropriate set of rf acceleration cavities. The beam is injected from a linac thereby eliminating the central region problems associated with present isochronous cyclotrons. As a result of the narrow magnet gaps, the machine has relatively strong axial focusing throughout. The orbital frequency is high enough so that the beam can be accomodated within the space charge limits. Additional small third harmonic cavities provide for a “flat-top” on the resultant voltage wave form thereby allowing a relatively large phase width in the accelerated beam. Conditions for achieving clear turn separation are analysed and found feasible. For a reasonable total number of turns, the isochronous field trimming requirements are quite tractable. Tentative design figures are presented and compared with the present Separated Orbit Cyclotron (SOC) designs.

Design Study for a Compact 200 MeV Cyclotron

A preliminary study has been carried out for a variable energy, 200 MeV cyclotron which is designed for both nuclear physics and medical research. The operational and structural features of this machine are modeled closely on our present 50 MeV cyclotron. The magnet would have the same gap, but twice the diameter (3m) and twice the number of trim coils (17), and it would have four (instead of three) sectors with sufficient additional spiral. The rf system would operate in the 10–20 MHz range, with two 90° dees carrying up to 70 kV. This cyclotron would produce protons at energies up to 200 MeV, and other ions to corresponding energies. By extending our present phase selection and single turn extraction techniques to this cyclotron, the energy resolution of the extracted beam would be: E/(ΔE)=7000. With proton currents of about 2 μA, the beam emittances would be 0.3 mm‐mrad radially and 2.5 mm‐mrad axially. The estimated cost of the cyclotron itself would be around two million dollars.

Effects of field imperfections on radial stability in a three-sector cyclotron†

In a medium-energy, three-sector cyclotron the radial oscillation frequency is close to unity; as a result, the stability of the radial motion is sensitive to relatively small field imperfections of particular types. In the absence of field imperfections, the radial stability limits are defined by the location of the three unstable fixed-point orbits associated with the 3/3 resonance. A theoretical analysis is given which specifies the movement of these fixed-point orbits and the consequent evolution of the radial phase space diagram as a function of the imperfection field strength. Formulae are given for the critical strengths of these perturbations at which complete loss of stability sets in. These results can be used to set tolerances on certain undesirable field harmonics for any given three-sector magnet geometry. Radial phase plots and fixed point orbits obtained from extensive computer calculations using various kinds of field imperfections are presented and analysed. In most cases the agreement between the theory and the computer data is relatively good. The results also demonstrate that the sensitivity of the radial stability to field imperfections increases rapidly with the amount of spiral in the main three-sector field.

ANALYSIS OF ISOCHRONOUS CYCLOTRONS WITH SEPARATED HARD-EDGE MAGNETS.

Abstract The orbit properties of isochronous cyclotrons with separated magnets are analyzed on the basis of the hard-edge approximation. This analysis extends previous work by treating magnets having a quite general shape. This treatment results in a generalized expression for the field index. A computer program (“Spyring Code”) is described which uses this analysis to calculate orbit properties for a given magnet geometry. Results are presented for a 600 MeV cyclotron having a symmetric six sector geometry with a radially decreasing magnet fraction. These results demonstrate that this type of magnet geometry produces greater vertical focusing than a comparable design with a constant magnet fraction.

Resonant extraction from three-sector low-spiral cyclotrons†

Results of MSU studies of resonant extraction systems for a 40 MeV variable energy, multi-particle cyclotron are reviewed. Initial sections give a qualitative account of extraction principles. Following this, results of numerical studies of both radial and axial motion under many varying resonant conditions are considered and, for comparison, also a corresponding non-resonant condition. Final sections briefly review (a) initial design studies relative to deflector hardware, (b) performance estimates for the system, and (c) preliminary studies of techniques for achieving improved duty cycle.

Nonlinear Effects of Focusing Bars Used in the Extraction Systems of Superconducting Cyclotrons

Sets of rectangular iron bars are used in combination with electostatic deflectors to focus and bend the beam along its path out of the cyclotron field. Since the design of this system was based on conventional orbit computations that include only linear vertical motion, additional studies have been undertaken to evaluate the nonlinear effects on the phase space of the extracted beam. These studies are carried out with a special orbit code that uses exact equations of motion, and treats the main cyclotron field correctly to fourth order in z. In addition, the rectangular focusing bars are assumed to be uniformly magnetized so that the resultant field can be calculated exactly on and off the median plane using simple formulas. Some preliminary results from this orbit code are presented here that show how the final radial and vertical phase space properties depend on the initial emittance values, and how they are changed by possible small displacements of the focusing bars.

THEORETICAL AND EXPERIMENTAL BEAM STUDIES FOR THE MICHIGAN STATE UNIVERSITY CYCLOTRON.

Abstract Numerical orbit studies using Michigan State University cyclotron fields have been performed to determine the validity of use of the separated longitudinal equations of motion under conditions obtained in actual cyclotron operation. These results have been applied in a systematic study of phase measurement procedures using frequency and trim coil detuning. Direct experimental measurements were performed to determine the phase and phase width of the beam, as well as the betatron oscillation frequencies v r and v z , as a function of radius. From the phase history data the radial dependence of the rf dee voltage has been obtained, and the nominal initial phase ϕ 0 has been determined to be +20° ± 1° for the MSU central region geometry. Use of necessary computer calculations with this result gives an rf starting time τ 0 of −28° ± 1° for maximum beam intensity. Results of experimental measurements are compared with pre-computed orbit behavior where possible.