Brennan, J. M.

N., Cook

Peggs, S.

UNT Digital Library

     BNL-101095-2013-CP    Harmonic ratcheting for fast acceleration   N. Cook Stony Brook University, Stony Brook, NY 11790 USA  M. Brennan, S. Peggs  BNL, Upton, NY 11973 USA            Presented at the North American Particle Accelerator Conference (NA-PAC 13) Pasadena, CA September 29 – October 4, 2013     Collider-Accelerator Department Brookhaven National Laboratory  U.S. Department of Energy DOE Office of Science  Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.  This preprint is intended for publication in a journal or proceedings.  Since changes may be made before publication, it may not be cited or reproduced without the author’s permission.       DISCLAIMER  This report was prepared as an account of work sponsored by an agency of the United States Government.  Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors.  The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.                                  HARMONIC RATCHETING FOR FAST ACCELERATION∗N. Cook† , Stony Brook University, Stony Brook, NY 11790 USAM. Brennan, S. Peggs, BNL, Upton, NY 11973, USAAbstractFrom medical synchrotrons to accelerator driven sys-tems, there is a need for fast acceleration of protons andlight ions. This can be a costly undertaking, requiring spe-cially designed ferrite loaded cavities to be tuned across awide range of frequencies. Ferromagnetic materials allowfor the precise adjustment of cavity resonant frequency, butrapid tuning changes and operation outside material spe-cific frequencies result in significant Q-loss. This leads toa considerable increase in power requirements. We intro-duce an acceleration scheme known as harmonic ratchetingwhich reduces the cavity frequency range when accelerat-ing an ion beam in a synchrotron. This scheme addressesthe needs of high repetition rate machines for applicationsin which low beam intensity is needed. We demonstratewith simulations the type of ramps achievable with ratchet-ing and consider its advantages.INTRODUCTIONThe potential for fast acceleration of low energy ionbeams remains rich, with a host of applications proposedto take advantage of rapid cycling synchrotrons or FFAGs.Nonetheless, conventional acceleration of low β chargedparticles remains expensive and inefficient. The primarychallenge of accelerating such particles lies in achievingrobust and flexible tuning. Using speciality ferrites, tuningcan be obtained at the cost of efficiency. Ferrite materi-als suffer dramatic loss effects when driven at high biasfields and high magnetic flux [1] [2]. These difficultieshave stemmed the advancement of fast and efficient accel-erators for low energy ion beams.The integer harmonic number h relates the RF and revo-lution frequencies through the relationship fRF = h fref .The relative RF frequency range can be significantly re-duced below the revolution frequency range by decreasingh in steps as the ions accelerate and frev increases. This isthe motivation and basic method of harmonic ratcheting.HARMONIC RATCHETINGSuppose that two (or more) RF cavities take turns accel-erating the beam – one turns on when the other turns off,at different RF frequencies – so that the RF frequency isalways constrained to remain in the rangefmin ≤ fRF ≤ fmax (1)where fmin and fmax are externally determined design pa-rameters. It is possible to make the transition back or forth∗Work supported by Brookhaven Science Associates, LLC under Con-tract No. DE-AC02-98CH10886 with the U.S. Department of Energy.† ncook@bnl.govbetween harmonic numbersh = n (2)and h = n+∆where n and ∆ are positive integers, iffmaxn+∆>fminn(3)Equivalently, we have∆ < nr (4)wherer ≡ fmaxfmin− 1 > 0 (5)is the “ratcheting parameter”. Equation 4 shows that ∆ hasa maximum permissible value, which must be greater than1 if a harmonic transition is to be possible. A ratchetingtransition is possible if the following conditions are met.n > 1 , r > 1 (6)n >1r, r < 1Several harmonic ratchets can take place during one ac-celeration ramp. Figure 1 show an example with r = 0.50,where the revolution frequency increases from 0.61 MHzto 3.35 MHz, while the RF frequency is constrained to liebetween 5.5 MHz and 8.25 MHz.Figure 1: An example of harmonic ratcheting with r =0.50. The plot shows one possible solution – with the har-monic number sequence h = 9, 7, 5, 4, 3, 2 – that maxi-mizes RF frequency while minimizing ratcheting number.Designing a Ratcheting RampThe selection of the initial harmonic number in the ratch-eting ramp is determined by considering the minimum re-quired bucket length, tinj to accept particles into a singlebucket at injection. This length sets a maximum possibleinitial harmonic. Moreover, the energy acceptance at injec-tion also scales inversely with hmax2. These requirementscan be summarized byhmax ≤1tinjfrev(0)(7)and hmax2 ≤ 2QVgβ2Esπ|ηs|∆E02Here, Q is particle charge, Es is synchronous energy,and ηs is the phase slip factor. As the initial harmonic in-creases, a corresponding increase in voltage is necessary tokeep the acceptance the same. Ferrite tuning time poses anadditional constraint on the ratcheting permutation. Tuningthe inactive cavity from a high frequency down to a lowfrequency requires circumnavigating the hysteresis curve.This tuning time, τt, is on the order of a few ms, whichsets a lower limit on the active time for any cavity betweenratcheting transitions τactive > τt.Emittance growth at ratcheting transitionDuring the harmonic transition, the voltage on the cavityoperating near fmax at harmonic h = n + ∆ is reducedwhile a second cavity operating at harmonic h = n has itsvoltage raised from a feedback level to provide the desiredgap voltage at the new harmonic. Assume that this changehappens quickly and that both the synchronous phase andtotal accelerating voltage are smoothly varying when theratcheting takes place.The result of the harmonic ratcheting is a relaxation ofthe bucket potential. As a result, large amplitude particlesstray further in phase from the design particle in order tofeel the same restoring force, thereby increasing longitudi-nal emittance S. Similarly, the bucket area A will grow ash is reduced. Both quantities grow asS2S1∝ A2A1∝(n+∆n) 32(8)The emittance growth due to the harmonic number tran-sition matches with the bucket area growth during theratcheting process. Thus, for small or slow changes in har-monic number, the ratcheting process is inherently stable,as illustrated in figure 4. This transition is calculated non-adiabatically. The transfer of voltage takes place over sometime period τr. A particle will make nt = βcτr/C turnsin that time. If τr = 100µs, a β = 0.1 particle makes 50turns of a 60 m synchrotron.FERRITE-LOADED CAVITIESThe resonant frequency of a ferrite-loaded cavity is f =12π1√LCwhere L, the inductance of the cavity, is ad-justable between minimum and maximum values Lmin andLmax. Consider a cavity with a total ferrite length ofl, made of many rings with inner and outer radii ra andrb. The inductance is L = l2π µ′ ln rbra where the in-cremental complex permeability of the ferrite is writtenµ = µ′ + jµ′′. The component µ′ and the inductanceof the cavity are tuned by biasing the ferrite with a pseudo-constant azimuthal magnetic field driven by a tuning cur-rent. Ignoring the stray inductance, the required dynamicrange of L and of µ′ isLminLmax=µ′minµ′max=1(1 + r)2 (9)The voltage across the gap of a cavity isVgap = 2πf Bmax l ra lnrbra∼ f l (10)where Bmax ≈ 0.01 T is the maximum RF magnetic fieldthat is allowed at the inner radius of the ferrite rings. Thisshows that for a fixed maximum field Bmax, the ferritelength l can be reduced if the RF frequency in the cavityis increased. Thus, harmonic ratcheting allows a given ac-celeration waveform to be achieved with shorter cavities.The total length of ferrite required with a ratcheting pa-rameter r, relative to a non-ratcheting scheme in which ionsare accelerated over the full dynamic range is given approx-imately bylratchetlnon−ratchet=fmin,non−ratchetfmin,ratchet≈ 2 1 + rD(11)where D = βmaxβmin , and the factor of 2 recognizes that onlyhalf the cavities are active at any one time. For example,Figure 1 illustrates acceleration over a dynamic range ofD = 5.5, using a ratcheting parameter of r = 0.50. Inthis case ratcheting makes it possible to decrease the to-tal length of ferrite by a factor of 0.55. Conversely, thegap voltage can be nearly doubled for a fixed cavity length,almost doubling the potential repetition rate of a rapid cy-cling synchrotron.EXAMPLE: A RAPID CYCLINGMEDICAL SYNCHROTRONConsider a rapid cycling synchrotron for acceleration ofC6+ ions for radiation therapy applications. Assume a 15Hz repetition rate and a circumference C = ∼ 65 m. Ionsmust accelerated through a range 8 to 400 MeV/µ.Once the initial harmonic is determined, the initial RFfrequency is set at fRF (0) = h0 βc/C. Appropriate se-lection of a frequency ceiling limits the ferrite to operatewithin a more efficient permeability range. In this instance,we choose a ratcheting parameter of r = 0.5 via a final har-monic transition of h = 3 to h = 2. Figure 1 shows the RFfrequency and figure 2 plots the active harmonic.When the frequency ceiling is reached at the current har-monic, RF power is transferred to the second cavity sys-tem at a lower harmonic number. Each time the harmonicFigure 2: Active cavity harmonic for the ∼65 m Carbonsynchrotron harmonic ratcheting ramp.number is reduced, the RF frequency is reduced by a fac-tor hnhn−1 where hn is the new harmonic, and hn−1 is theprevious value. The maximum frequency swing during theramp is determined by the maximum value of this ratio, orequivalently the ratcheting parameter r, according to∆ff0= r (12)For example, if the largest transition in a particular ratch-eting scheme is from h = 3 to h = 2, then the ratchetingparameter is r = 0.5, and that particular ramp is limited toa 50% frequency swing.Figure 3: Output voltage for each cavity system. The volt-age is calculated to meet the constraint ϕs ≤ 33◦.Figure 3 outlines the output voltage needed for each cav-ity system in our example ramp. The harmonic transitionand transfer of power to the inactive cavity cannot be per-formed instantaneously; the voltage may be raised over∼ 100µs. Detailed feedforward and diagnostic systemsare required in order to maintain the proper voltage. Theemittance evolution for our example case is shown in 4.Ratcheting may also reduce losses due to anomalous ef-fects in ferrite. At sustained high fields, ferrites demon-strate a Q-loss effect (QLE) which can result in a reductionof 20% of maximum field. High bias fields can increaseFigure 4: Estimated emittance and bucket area is shown fora simulated ramp with fixed synchronous phase.this effect by reducing the threshold of Q-loss onset. How-ever, at high bias rates the threshold for this effect is sig-nificantly raised [1]. A ratcheting scheme should mitigateQLE by reducing total biasing. Rapid bias field changesmay also induce a dynamic loss effect in the ferrite. Theselosses can lower a cavity’s Q value by up to 40%. Unlikethe QLE, this effect appears to be more dependent on thetemperature and duration of the biasing. By decreasing thetime spent biasing at high fields, this effect can be reducedconsiderably [2].CONCLUSIONA new scheme for accelerating low energy ions quicklyand over a large frequency range is presented. By adjust-ing the harmonic number during acceleration in a manorspecifically tailored to the desired acceleration cycle, a sig-nificant reduction in the frequency range is obtained, easingthe ferrite’s tuning requirement. Moreover, the frequencyincrease allows potentially several times larger gap volt-ages to be obtained for the same amount of ferrite, mak-ing higher machine repetition rates possible. Alternatively,cavity length could be reduced by an equivalent factor. Thisapproach is particularly suited to low intensity ion beamswhich do not require filling a ring but necessitate fast ac-celeration across a range of energies.ACKNOWLEDGMENTWe thank D. Trbojevic for many encouraging discus-sions on this topic.REFERENCES[1] K. Kaspar, H. G. König, and T. Winnefeld., ”Studies on themaximum RF voltages in ferrite-tuned accelerating cavities,”TUPKF013, Proceedings of EPAC 2004, 985-7.[2] J.E. Griffin and G. Nicholls. IEEE Trans, Nucl. Sci. 26 (1979)3965-7.[3] Y. Zhao, ”A Study of Ferrite Cavity,” BNL Informal ReportCAP-341-RF-R, April 2002[4] N. Cook, Dissertation, Stony Brook University. To Be Pub-lished.

Harmonic ratcheting for fast acceleration

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Harmonic ratcheting for fast acceleration

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