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UBC Theses and Dissertations

Properties of ion orbits in the central region of a cyclotron Louis, Robert John


The behaviour of ion orbits in the magnetic and electric fields at the centre of a cyclotron is studied in detail. The objective is to optimize the phase acceptance and beam quality for a 500 MeV H¯ isochronous cyclotron. Since accurate electric fields are necessary for orbit calculations, a numerical method for calculating these fields is examined in detail. The method is suitable for complicated electrode shapes and converges rapidly, yielding potentials in three dimensions with average errors of less than 0.01%. The magnetic fields used in the orbit calculations are measured on model magnets. The axial motions are examined using a thick lens approximation for the accelerating gaps. A method is demonstrated for calculating the axial acceptance of the cyclotron as a function of RF phase. This method is used to evaluate the merits of various central geometries and injection energies. This method is also used to examine the effects of flat-topping the RF voltage by adding some third harmonic to the fundamental waveform. It is found that addition of the optimum amount of third harmonic increases the phase acceptance by about 20 deg. Finally, the effects of field bumps on the axial motions are investigated. To allow accurate radial motion calculations to high energy, an approximate formula is developed which yields accurate (<1%) values for the changes in orbit properties of an ion crossing a dee gap. The geometry of the orbit on the first turn is discussed in detail. The radial centring is studied by tracking ions from injection to 20 MeV, and a method is described for choosing the starting conditions of the beam so as to minimize the radial betatron amplitude over a desired phase range. The problems associated with using a three-fold symmetric magnetic field with a two-fold symmetric electric field are also discussed. Besides the well-known gap-crossing resonance, a previously ignored phase-oscillation effect is found to be important for cyclotrons operating on a high harmonic of the ion rotation frequency.

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