Harmonic ratcheting for fast acceleration

A major challenge in the design of rf cavities for the acceleration of medium-energy charged ions is the need to rapidly sweep the radio frequency over a large range. From low-power medical synchrotrons to high-power accelerator driven subcritical reactor systems, and from fixed focus alternating gradient accelerators to rapid cycling synchrotrons, there is a strong need for more efficient, and faster, acceleration of protons and light ions in the semirelativistic range of hundreds of MeV/u. A conventional way to achieve a large, rapid frequency sweep (perhaps over a range of a factor of 6) is to use custom-designed ferriteloaded cavities. Ferrite rings enable the precise tuning of the resonant frequency of a cavity, through the control of the incremental permeability that is possible by introducing a pseudoconstant azimuthal magnetic field. However, rapid changes over large permeability ranges incur anomalous behavior such as the “Q-loss” and “f-dot” loss phenomena that limit performance while requiring high bias currents. Notwithstanding the incomplete understanding of these phenomena, they can be ameliorated by introducing a “harmonic ratcheting” acceleration scheme in which two or more rf cavities take turns accelerating the beam—one turns on when the other turns off, at different harmonics—so that the radio frequency can be constrained to remain in a smaller range. Harmonic ratcheting also has straightforward performance advantages, depending on the particular parameter set at hand. In some typical cases it is possible to halve the length of the cavities, or to double the effective gap voltage, or to double the repetition rate. This paper discusses and quantifies the advantages of harmonic ratcheting in general. Simulation results for the particular case of a rapid cycling medical synchrotron ratcheting from harmonic number 9 to 2 show that stability and performance criteria are met even when realistic engineering details are taken into consideration.


INTRODUCTION
The potential for fast acceleration of low energy ion beams remains rich, with a host of applications proposed to take advantage of rapid cycling synchrotrons or FFAGs. Nonetheless, conventional acceleration of low β charged particles remains expensive and inefficient. The primary challenge of accelerating such particles lies in achieving robust and flexible tuning. Using speciality ferrites, tuning can be obtained at the cost of efficiency. Ferrite materials suffer dramatic loss effects when driven at high bias fields and high magnetic flux [1] [2]. These difficulties have stemmed the advancement of fast and efficient accelerators for low energy ion beams.
The integer harmonic number h relates the RF and revolution frequencies through the relationship f RF = h f ref .
The relative RF frequency range can be significantly reduced below the revolution frequency range by decreasing h in steps as the ions accelerate and f rev increases. This is the motivation and basic method of harmonic ratcheting.

HARMONIC RATCHETING
Suppose that two (or more) RF cavities take turns accelerating the beam -one turns on when the other turns off, at different RF frequencies -so that the RF frequency is always constrained to remain in the range where f min and f max are externally determined design parameters. It is possible to make the transition back or forth * Work supported by Brookhaven Science Associates, LLC under Equivalently, we have is the "ratcheting parameter". Equation 4 shows that Δ has a maximum permissible value, which must be greater than 1 if a harmonic transition is to be possible. A ratcheting transition is possible if the following conditions are met.
Several harmonic ratchets can take place during one acceleration ramp. Figure 1 show an example with r = 0.50, where the revolution frequency increases from 0.61 MHz to 3.35 MHz, while the RF frequency is constrained to lie between 5.5 MHz and 8.25 MHz.

Designing a Ratcheting Ramp
The selection of the initial harmonic number in the ratcheting ramp is determined by considering the minimum required bucket length, t inj to accept particles into a single bucket at injection. This length sets a maximum possible initial harmonic. Moreover, the energy acceptance at injection also scales inversely with h max 2 . These requirements can be summarized by and Here, Q is particle charge, E s is synchronous energy, and η s is the phase slip factor. As the initial harmonic increases, a corresponding increase in voltage is necessary to keep the acceptance the same. Ferrite tuning time poses an additional constraint on the ratcheting permutation. Tuning the inactive cavity from a high frequency down to a low frequency requires circumnavigating the hysteresis curve. This tuning time, τ t , is on the order of a few ms, which sets a lower limit on the active time for any cavity between ratcheting transitions τ active > τ t .

Emittance Growth at Ratcheting Transition
During the harmonic transition, the voltage on the cavity operating near f max at harmonic h = n + Δ is reduced while a second cavity operating at harmonic h = n has its voltage raised from a feedback level to provide the desired gap voltage at the new harmonic. Assume that this change happens quickly and that both the synchronous phase and total accelerating voltage are smoothly varying when the ratcheting takes place.
The result of the harmonic ratcheting is a relaxation of the bucket potential. As a result, large amplitude particles stray further in phase from the design particle in order to feel the same restoring force, thereby increasing longitudinal emittance S. Similarly, the bucket area A will grow as h is reduced. Both quantities grow as The emittance growth due to the harmonic number transition matches with the bucket area growth during the ratcheting process. Thus, for small or slow changes in harmonic number, the ratcheting process is inherently stable, as illustrated in figure 4. This transition is calculated nonadiabatically. The transfer of voltage takes place over some time period τ r . A particle will make n t = βcτ r /C turns in that time. If τ r = 100μs, a β = 0.1 particle makes 50 turns of a 60 m synchrotron.

FERRITE-LOADED CAVITIES
The resonant frequency of a ferrite-loaded cavity is f = 1 2π 1 √ LC where L, the inductance of the cavity, is ad-justable between minimum and maximum values L min and L max . Consider a cavity with a total ferrite length of l, made of many rings with inner and outer radii r a and r b . The inductance is L = l 2π μ ln r b ra where the incremental complex permeability of the ferrite is written μ = μ + jμ . The component μ and the inductance of the cavity are tuned by biasing the ferrite with a pseudoconstant azimuthal magnetic field driven by a tuning current. Ignoring the stray inductance, the required dynamic range of L and of μ is The voltage across the gap of a cavity is where B max ≈ 0.01 T is the maximum RF magnetic field that is allowed at the inner radius of the ferrite rings. This shows that for a fixed maximum field B max , the ferrite length l can be reduced if the RF frequency in the cavity is increased. Thus, harmonic ratcheting allows a given acceleration waveform to be achieved with shorter cavities. The total length of ferrite required with a ratcheting parameter r, relative to a non-ratcheting scheme in which ions are accelerated over the full dynamic range is given approximately by where D = βmax βmin , and the factor of 2 recognizes that only half the cavities are active at any one time. For example, Figure 1 illustrates acceleration over a dynamic range of D = 5.5, using a ratcheting parameter of r = 0.50. In this case ratcheting makes it possible to decrease the total length of ferrite by a factor of 0.55. Conversely, the gap voltage can be nearly doubled for a fixed cavity length, almost doubling the potential repetition rate of a rapid cycling synchrotron.

EXAMPLE: A RAPID CYCLING MEDICAL SYNCHROTRON
Consider a rapid cycling synchrotron for acceleration of C 6+ ions for radiation therapy applications. Assume a 15 Hz repetition rate and a circumference C = ∼ 65 m. Ions must accelerated through a range 8 to 400 MeV/μ.
Once the initial harmonic is determined, the initial RF frequency is set at f RF (0) = h 0 βc/C. Appropriate selection of a frequency ceiling limits the ferrite to operate within a more efficient permeability range. In this instance, we choose a ratcheting parameter of r = 0.5 via a final harmonic transition of h = 3 to h = 2. Figure 1 shows the RF frequency and figure 2 plots the active harmonic.
When the frequency ceiling is reached at the current harmonic, RF power is transferred to the second cavity system at a lower harmonic number. Each time the harmonic  number is reduced, the RF frequency is reduced by a factor hn hn−1 where h n is the new harmonic, and h n−1 is the previous value. The maximum frequency swing during the ramp is determined by the maximum value of this ratio, or equivalently the ratcheting parameter r, according to For example, if the largest transition in a particular ratcheting scheme is from h = 3 to h = 2, then the ratcheting parameter is r = 0.5, and that particular ramp is limited to a 50% frequency swing.  Figure 3 outlines the output voltage needed for each cavity system in our example ramp. The harmonic transition and transfer of power to the inactive cavity cannot be performed instantaneously; the voltage may be raised over ∼ 100μs. Detailed feedforward and diagnostic systems are required in order to maintain the proper voltage. The emittance evolution for our example case is shown in 4.
Ratcheting may also reduce losses due to anomalous effects in ferrite. At sustained high fields, ferrites demonstrate a Q-loss effect (QLE) which can result in a reduction of 20% of maximum field. High bias fields can increase this effect by reducing the threshold of Q-loss onset. However, at high bias rates the threshold for this effect is significantly raised [1]. A ratcheting scheme should mitigate QLE by reducing total biasing. Rapid bias field changes may also induce a dynamic loss effect in the ferrite. These losses can lower a cavity's Q value by up to 40%. Unlike the QLE, this effect appears to be more dependent on the temperature and duration of the biasing. By decreasing the time spent biasing at high fields, this effect can be reduced considerably [2].

CONCLUSION
A new scheme for accelerating low energy ions quickly and over a large frequency range is presented. By adjusting the harmonic number during acceleration in a manor specifically tailored to the desired acceleration cycle, a significant reduction in the frequency range is obtained, easing the ferrite's tuning requirement. Moreover, the frequency increase allows potentially several times larger gap voltages to be obtained for the same amount of ferrite, making higher machine repetition rates possible. Alternatively, cavity length could be reduced by an equivalent factor. This approach is particularly suited to low intensity ion beams which do not require filling a ring but necessitate fast acceleration across a range of energies.