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Fig. 2: Synchronism in a cyclotron between the particle rotation and the RF accelerating wave (courtesy of Frederic Chautard.) Here mp is the particle rest-mass and c is the speed of light. — Figure 2 Synchronism in a cyclotron between the particle rotation and the RF accelerating wave (courtesy of Frederic Chautard.) Here mp is the particle rest-mass and c is the speed of light.

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Fig. 1: At first order, a particle in a cyclotron rotates at constant frequency, independent of radius or energy This is illustrated in Fig. 1. Thus, the angular velocity is constant: it is independent of radius, elocity, energy (in the non-relativistic limit), or time.

Fig. 3: Owing to the relativistic mass increase and the small negative magnetic field gradient, the particle in a classical cyclotron runs steadily out of phase with respect to the RF. Nevertheless, acceleration can be obtained during a limited number of turns (courtesy of Frederic Chautard.)

Fig. 4: A simple Excel model can show how much energy can be reached in a weak-focusing rotational symmetric cyclotron, depending on the dee voltage, the field index, and the magnetic field value.

Fig. 5: Some properties of the IBA superconducting synchro-cyclotron (S2C2). Left: modulation of the RF frequency as a function of time within one pulse. Right: average magnetic field and vertical tuning as a function of radius.

Fig. 6: Longitudinal beam dynamics in a synchro-cyclotron. Left: flow lines and separatrix in longitudinal phase space. Right: equations of motion that govern this phase space.

Fig. 7: Magnet of a compact azimuthally varying field cyclotron. Left: the hill sectors and poles, the valleys, and the different parts of the yoke. Right: histogram of the magnetic field.

Fig. 8: Vertical focusing in an azimuthally varying field cyclotron. Left: scalloping of the orbit with respect to the geometrical circle. Right: the radial velocity component and the azimuthal field component 10 mm above the median plane. The product of these gives the stabilizing vertical Lorentz forces directed towards the median plane.

Fig. 9: Vertical focusing in a cyclotron relates closely to vertical focusing at the entrance and exit of a dipole magnet.

Fig. 10: Vertical focusing in an azimuthally varying field cyclotron can be strongly increased by spiralling the hill sectors.

Fig. 11: Simple model to estimate flutter in a hard-edge approximation where A,, and B,, are the normalized Fourier harmonics of the field. With this representation of the field, the flutter can be written as

Fig. 13: The most important cyclotron vendors or manufacturers. RP, radiopharmaceuticals (mostly isotope pro- duction); PT, proton therapy.

Fig. 14: Deep-valley cyclotron; the concept is used in many compact isotope production and proton therapy Papers

Fig. 16: Mapping system carrying a Hall probe that moves on a polar grid in the cyclotron median plane

Fig. 17: Upper left: isochronization of a cyclotron magnetic field is achieved by machining the profile of a remov- able pole edge. Lower left: a rough estimate of the shimming effect (change of the average magnetic field) can be obtained with a hard-edge model. Right: a better prediction of the shimming can be obtained by calculating (with a 3D finite-element code) the change of the radial profile of the average magnetic field due to a well-defined pole cut and repeating this for several cuts at varying radii. Fig. 17: Upper left: isochronization of a cyclotron magnetic field is achieved by machining the profile of a remov-

Fig. 18: Left: IBA C235 cyclotron for proton therapy. Right: in this cyclotron there are three removable pole edges for isochronization of the magnetic field.

Fig. 19: Left: upper poles of the IBA C18/9 cyclotron. The movable iron wedges (flaps) are used to change the average magnetic field profile from protons (flaps close to the median plane) to deuterons (flaps farther away from the median plane). Right: relativistic correction of the magnetic field, as needed for protons (about 2.1%) and for deuterons (about 0.53%).

Fig. 20: Finite-element Opera-3d simulation of the effect of the flaps in the IBA C18/9 cyclotron. Left: the neutron field with the flaps farther away from the median plane. Right: the proton field with the flaps close to the median plane.

Fig. 21: Alternative method for isochronization of a dual-particle cyclotron. In this case, the flaps are in a fixed position and connected magnetically to the base of the return yoke by a solid iron cylinder. The magnetic flux into the flaps is controlled by the solenoidal coil around the cylinder. This method has not yet been realized but has been patented by IBA.

Fig. 22: Yet another method for isochronization of a dual-particle cyclotron is to split the pole into three layers and to wind coils around the middle layers. As explained in the right figure, these coils push magnetic field from the outer pole region towards the centre or vice versa, depending on the polarity of the coil current. Note that in the upper left figure, the cover has been removed.

Fig. 23: A method similar to the one outlined in Fig. 22 is used in the AGOR superconducting cyclotron [20, 21.

Fig. 24: A more general method is used to isochronize multiparticle variable-energy cyclotrons. Here, an array of concentric circular coils is placed on the pole and each coil uses its own independent power supply. In this way, there is a lot of flexibility in creating the required average magnetic field profile. Left: correction coils for the Berkeley 88-inch cyclotron [24]. Right: correction coils of the Philips 30 MeV azimuthally varying field cyclotron [23].

Fig. 25: Left: the IBA C70 as an example of an industrial cyclotron for medical isotope production, showing the small spiral in the pole to adjust vertical focusing and the removable pole edge for isochronization. Right: average magnetic field and flutter of this cyclotron. ‘ig. 25: Left: the IBA C70 as an example of an industrial cyclotron for medical isotope production, showing the

Fig. 26: Left: RF phase slip per turn and integrated RF phase slip in the IBA C70 cyclotron. Right: calculated phase slip per turn can be related to the required shimming of the removable pole edges.

Fig. 27: Left: radial (v,.) and vertical (v,) tune function of the IBA C70 cyclotron as a function of radius. Right: related tuning diagram, showing the working curve and some important resonance lines.

Fig. 29: Stability diagram of a cyclotron with three-fold rotational symmetry, assuming that the flutter is relatively small, as it is produced by the iron poles. The solid black line shows the maximum kinetic energy (relativistic +) that can be obtained, taking into account the requirements of vertical focusing, as well as the 2v,. = 3 stop band.

Fig. 30: Left: in a cyclotron, the radial betatron oscillation can be represented as a (slow) motion of the orbit centre. Right: the betatron oscillations take place with respect to the magnetic centre of the cyclotron. Owing to a first-harmonic field error, the magnetic centre shifts away from the geometric centre of the cyclotron. Fig. 30: Left: in a cyclotron, the radial betatron oscillation can be represented as a (slow) motion of the orbit

Fig. 31: Amplitudes of the (most important) harmonic field errors in the measured field map of the IBA C70 cyclotron.

Fig. 32: Left: Monarch S250 synchro-cyclotron from Mevion (Couresy Mevion medical systems Ref. [26]). Right: superconducting synchro-cyclotron (S2C2) from IBA.

Fig. 33: Mismatch between the cyclotron eigenellipse and the beam ellipse injected into the machine leads, after many ture ton an increace af the circulating emittance

Fig. 34: Left: cold-cathode Penning ion gauge source. Right: two chimneys, two cathodes, and a puller. The chimney on the right shows an eroded slit.

Fig. 35: Central region of the IBA C18/9 cyclotron showing the dees and dummy dees. One of the two ion sources has been removed to show the puller. The figure also shows the four hill sectors. The removable circular disc underneath the central region is the central plug; it is used to fine tune the magnetic field bump in the centre.

Fig. 36: Example of a 3D finite-element model of the IBA C18/9 central region. Fine meshing is used in regions with small geometrical details, such as the source—puller gap. The graded mesh size allows modelling of the full dee-structure. Complete parameterization of the model is used for fast modification and optimization.

Fig. 37: Calculated orbits in the IBA C18/9 central region. The D™ source is shifted farther outward because its orbit is larger. Note that parts of the D™ chimney have been cut away, to give sufficient clearance for the H~ beam. Assessment of longitudinal matching is illustrated: the red dots and the green dots give the particle position when the dee voltage is zero and a maximum, respectively. Ideally, the red dots should be on the dee centre line. Electric fields are obtained from Opera. Magnetic fields from Opera or from a measured map.

Fig. 38: Energy gain per turn depends on several parameters, such as the dee voltage Vace, the number of dees, NV, the harmonic mode, h, the dee angle, a, and the RF phase ®pp. For a dee angle of 45°, the harmonic mode h = 4 is most efficient. Fig. 38: Energy gain per turn depends on several parameters, such as the dee voltage Vace, the number of dees, N,

Fig. 39: Vertical electrical focusing in the cyclotron accelerating gap. This focusing is important only in th cyclotron centre.

Fig. 40: Normalized vertical electrical forces obtained from orbit tracking of a particle during five turns in a central region with two dees (four accelerating gaps). Dee crossings are indicated at the top of the curve (in blue) and gap crossings (two per dee) at the bottom of the curve (in green). Each gap is focusing at the entrance and defocusing at the exit.

Fig. 41: Left: calculated orbits in the central region of the IBA S2C2. Right: 3D view of the Penning ion gauge source and puller used in this central region.

Fig. 42: The ion source and central region can be extracted from the cyclotron as one assembly, for easy mainten- ance and precise repositioning.

Fig. 43: Simulation of the beam capture process in the S2C2 central region (CR).

Fig. 44: Small bits of thin paper are placed in the pole gap and burned by the beam, to find the beam position and size during the first few turns.

Fig. 45: Horizontal (trochoidal) injection. The beam travels along the hill—valley pole edge. In the centre, an electrostatic device places the beam on the equilibrium orbit. Figure taken from Ref. [28]

Fig. 46: Left: mirror inflector. Right: hyperboloid inflector. Figure taken from Ref. [28]

Fig. 47: 1:1 model of the spiral inflector and central region of the IBA Cyclone 30 cyclotron

Fig. 48: Left: axially injected 3D orbits calculated with the cyclotron tracking code AOC are imported in the

Fig. 49: C30HC injection line installed on top of the cyclotron. The ion source is on top of the vacuum box. This box is pumped by two turbo pumps (front and back).

Fig. 50: Left: layout of the C30HC injection line. Right: beam envelopes in this beam line, calculated with TRANSPORT.

Fig. 51: Left: IBA C18+ cyclotron with an internal rhodium target for the production of palladium. Right: detail of an internal target showing the profiling of the target surface, optimized to maximize the beam spot and minimize beam reflection on the target. Fig. 51: Left: IBA C18+ cyclotron with an internal rhodium target for the production of palladium. Right: detail

Fig. 52: Left: H™ extraction by stripping. Energy is selected by moving the stripper foil to the correct radius. Right: a simple carbon stripper foil is used to extract the beam (typically 50-200 ug/cm?).

Fig. 53: Left: H™ extraction by stripping. Several targets can be placed around the machine. Right: median plane of a typical IBA isotope production cyclotron, showing the most important subsystems.

Fig. 54: Mechanisms that may contribute to an increase of the turn separation in a cyclotron

Fig. 55: Harmonic coil used in the IBA self-extracting cyclotron (note that the ruler measures centimeters)

Fig. 56: Top: principle function of an electrostatic deflector. Bottom: electrostatic deflector installed in the IBA C235 cyclotron.

Fig. 57: Close-up of C235 ESD, showing the septum and the electrode

Fig. 58: Left: vertical cross-section of the passive gradient corrector channel used in the small ILEC cyclotron at Technical University Eindhoven. Right: calculated magnetic field and field gradient. The field is increasing with radius, increasing the radial focusing. Figure taken from Ref. [49]

Fig. 59: Extraction scheme used for the IBA C235 proton therapy cyclotron

Fig. 60: Left: definition of the elliptical pole gap used in the IBA C235 cyclotron. Right: the magnetic field shows a very sharp cut-off near the radius of the pole.

Fig. 61: Left: gradient corrector installed in the C235 cyclotron. Right: magnetic field at two different azimuths, showing the field profile experienced by the extracted beam.

Figure 2 – uploaded by Wiel Kleeven

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