Fig. 1 "Oh thee spiraling hills and valleys !"

The Rutgers 12-Inch Cyclotron is uniquely suited to study the beam dynamics under different pole tip configurations. So far, a set of pole tips with a slight constant radial decrease of the vertical field focusing, known as weak focusing, have been used to bring the accelerated beam to the periphery. Pole tips which cause the vertical magnetic field to vary with azimuth at a given radius also provide axial focusing which can significantly exceed the weak focusing field. Here we present a synopsis of the simulations and measurements of the magnetic fields of both the trial radial sector Azimuthally Varying Field (AVF) pole tips and an optimized set of spiral sector AVF pole tips expected to successfully bring a proton beam to the periphery. This page is just a quick overview of the AVF project, please refer to the paper for the full gory details.

The axial restoring force in the constant gradient - weak - focusing poles results from the interaction of the ions' angular velocity component with the radial magnetic field component above and below the median plane. The angular velocity is proportional to the particles' kinetic energy and the radial component of the magnetic field is proportional to the magnetic field gradient, on which stability requirement places limits. In the radial sector AVF, the vertical restoring force arises from the ion's radial velocity interacting with the magnetic field's azimuthal component. There are then two parameters contributing to the AVF axial restoring force: the ion's radial velocity and the azimuthal component of the magnetic field. First, the ion's radial velocity component is generated by the change in vertical field strength between hill and valley regions - this will be seen as scalloping of circular orbits. The rms variation of the vertical field is called flutter and is denoted as F in the literature. The azimuthal magnetic field component is also proportional to the flutter. The axial focusing can be further enhanced by instantaneously tilting the sector edge. Continuously applying an edge angle will result in a spiral sector edge from the center to the periphery. AVF focusing can be used to supplement weak focusing and to gain a deeper illumination on the focusing properties of azimuthally varying fields, the Rutgers 12-Inch cyclotron has, to-date, constructed two sets of AVF poletips for experimentation.

The first set was a simple, pure-radial sector design of periodicity four and are shown installed with the cyclotron chamber removed in figure ?. Their geometry was the least expensive to machine and provided the maximum field variation achievable within practical constraints. With a periodicity of four, the hills and valley are each 45 degrees wide. They maintain a constant thickness out to the pole edge, except for a quarter-inch chamfer to break the sharp corners. Not expected to host beam, their purpose was to benchmark simulations, measurement techniques, and test analysis code.

Fig. 2 First set of AVF pole tips.

The data from the first scan is plotted in the figure below. The four hills are prominent, however a small central bump is observed from the slug that ties the four vanes together. This bump is required to promote a centrally localized weak focusing field since the flutter will be too small to be effective at the central convergence.

Fig. 3 2-D measurement of the first set of AVF pole tips.

The figure of merit used to quantify stability of each simulated magnetic field and compare between geometries is trace space, a time-independent record of the particle behavior. Trace space correlates a particles position and angle with respect to the ideal orbit tangent (a measurement of transverse momentum) each time it crosses a reference plane. In order to determine the orbit characteristics of a magnetic field, ions with varying initial conditions are launched in the simulated field. The ions' subsequent motion is recorded to establish the limits of the stability in both the radial and axial directions. The code chosen for the task is SIMION, a full 4-D E&M code. SIMION has the capacity to fully model our 12-Inch cyclotron, including the RF aspects and physical apertures. In out stability investigation, the accelerating RF was turned off, but the aperture of the DEE structure was retained. Such simulations are referred to as 'static'. After launching the ions, SIMION tracks the particle through the simulated magnetic field. A typical display is shown in Figure ? where four 50 keV ions were launched in the median plane, each at different radii.

Fig. 4 Four 50 keV ion trajectories.

Trace space is related to the more standard phase space, where position and transverse momentum are correlated. Since the Rutgers 12-Inch Cyclotron is far from the relativistic regime, trace space and phase space are equivalent sans scaling. SIMION provides a suite of predefined data recording options and criteria and was configured to record each particle.s angle and velocity. This data is recorded at the ions start, and every instance of crossing a defined reference plane, such as the z=0 plane. Radial trace space was explored in order to identify the radial equilibrium orbits.

Fig. 5 Trace space determined from the four 50 keV ion trajectories.

While the constructed radial sector pole tips were not expected to support acceleration, we could not resist trying non-static SIMION simulations in which the accelerating RF was turned on. The incurred phase slippage at normal operating conditions - namely a DEE voltage of 8 kV-peak - was indeed too severe to successfully bring ions to the full radius. However, a DEE voltage of 20 kV peak accepted ions over 20 degrees of the RF cycle and accelerated them to the periphery. This suggested that our first attempt is not too far from a practical design. Figure ? shows a single proton accelerated by a DEE voltage of 20 kVpeak.

Fig. 6 Single proton trajectory with RF on in radial sector pole tips.

The second set of AVF poletips are the product of an ambitious design program assigned to the Spring 2011 Cyclotron Students. Using the design tools developed with the pure radial sector poles, the second set, employing an Archimedes spiral, is expected to successfully transport beam from the ion source to the periphery. The students' result was a four sector Archimedean spiral sweeping 270 degrees, and are referred to as AKG270 (for students Aaron, Kiersten, and George). Less the weak focusing central region, these pole tips aimed to satisfy the isochronous condition, in order to minimize the phase slippage, and reduce the minimum DEE voltage while preserving axial stability throughout the accelerating region.

Fig. 7 AKG270 Spiral sector pole tips.

Fig. 8 AKG270 Spiral sector pole tips.

The AKG270 radial trace space was explored first to identify the equilibrium orbits in 50 keV increments. Even, at 50 keV, non-linear behavior is noted in the larger stable orbits, exhibiting four-sided contours, behavior which is consistent with pole tips that have a sector periodicity of four. The corners of the four-sided nonlinear orbits become more pronounced and bulbous with increasing energy. The following is a compilation of the radial trace space for a range of energies. Each plot displays six contours. The inner most contour identifies the equilibrium orbit, and the outer most contour is within 5% of the greatest stable orbit.

Fig. 9 AKG270 Radial trace space for a range of ion energies.

By 250 keV four closed contours formed in the protracted corners, displayed in Figure ?. Thus, in addition to the primary Equilibrium Orbit, there are four off-center stable orbits. These satellite orbits were confirmed with SIMION, and are shown in Figure ?.

Fig. 10 Radial Phase Space of 250 keV shows five stable orbits !

Fig. 11 Five stable orbits verified by SIMION.

The off-center equilibrium orbits were experimentally verified using the wire-loop orbit technique. A 30 AWG wire loop, with a circumference of 71 mm, was energized with a current of 2.5 amps and placed in the magnet gap. Myriad other stable orbits made it difficult to perform this experiment in the median plane; instead a clear acrylic sheet was placed on the bottom pole tip to provide a flat surface on which the wire loop could lay. The energized wire loop simply needed to be tossed towards the gap and it would reproducibly snap to the nearest stable orbit, one such off-center orbit is show in figure ?. An overlay of five loop images demonstrating five stable orbits is shown in Figure ?. This technique found another four orbits (for a total of nine) located even further away from the center. Since the wire-loop technique does not discriminate based circumference (only the loop's tension will vary), the further outlaying orbits are most likely lower energy equilibrium orbits.

Fig. 12 A stable orbit found with the "wire loop" method .

Fig. 13 Five stable orbits shown with "wire loop"

After locating the equilibrium orbits, a complete comparison of the focusing between the AKG270 poles and the weak focusing pole tips was performed using the simulated fields. The DEEs were removed from both cases to observe, if any, non-linear effects at large excursions. The radial and axial results are respectively shown in the following figure. In most of the radial cases the AKG270 radial trace space is bounded by the weak focusing pole tips. At the lower energies of 50 and 100 keV, the vertical trace space (not shown here - see paper) of the AKG270 poletips is larger than that of the weak focusing poles, indicating a greater angular acceptance from the ion source. This is due to the enhanced central focusing from the weak focusing bump.

Fig. 14 Radial Weak Focusing vs AVF focusing

The final phase of this effort was to measure the vertical field in the median plane and compare with the 3D simulation. Good agreement was found.

Fig. 15 2-D field map of AKG270 pole tips.

Protons were flown with RF in SIMION with the AKG270 magnetic field. The trajectory of a single proton is shown in Figure ?. The driving frequency and amplitude were iteratively tuned to locate the minimum peak DEE voltage necessary to successfully accelerate the proton to the target. This lowest practical voltage found in the simulation was 6 kVpeak at a frequency of 15.534 MHz.

Fig. 16 SIMION simulation of a single proton flown in the AKG270 field.

We will put more data up as soon as we try to accelerate beam with these pole tips. Again, please refer to the more complete write up on this work.

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