1. Acceleration of Protons in FFAG with Non-Scaling Lattice and Linear Field Profile Alessandro G. Ruggiero Semi-Annual International 2006 FFAG Workshop November 6 - 10, 2006 KURRI - Osaka - Japan

2. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory2/20 Acceleration of Protons in FFAG Accelerators with Non-Scaling Lattice and Linear Field Profile Few examples: (1) FFAG injector for AGS Upgrade AGS-FFAG (2) Low Energy Proton Driver (Neutron Source, Energy Production, Tritium Production, Nuclear Waste Transmutation, ….) MA-LE-PD (3) Neutrino Factory (and similar ….) AGS-FFAGMA-LE-PD Injection Energy400 MeV200 MeV  0.7130.566 Extraction Energy1.5 GeV1.0 GeV  0.9230 0.875 Circumference809 m202 m Repetition Rate2.5-5 Hz1 kHz

3. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory3/20 Acceleration by RF-Modulation A.G. Ruggiero “RF Acceleration with Harmonic-Number Jump”, BNL Internal Report, C-A/AP 237, May 2006 In the case of acceleration of Protons the particle velocity varies considerably. Conventional Method of acceleration considered so far is Frequency- Modulated RF system. FFAG accelerators have the good (excellent) property of constant bending and focusing field that does not impose any limit (in principle) to the acceleration rate. Unfortunately this is set by the rate of the RF Modulation. For lighter particles (electrons, muons, … ) this is not a problem because  = 1 at all times and the RF accelerating system can be taken with fixed frequency (typically superconducting)

4. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory4/20 Required RF Modulation AGS-FFAGMA-LE-PD Repetition Rate5 Hz1 kHz Acceleration Period7 ms0.5 ms RF frequency6 – 9 MHz (AGS)30 – 50 MHz Modulation Rate0.5 MHz/ms40 MHz/ms Fermilab Booster:30 – 50 MHz in 30 ms ->1 MHz/ms RF system is operating in heavy beam loading condition with large average/peak power

5. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory5/20 AGS-FFAG A.G. Ruggiero, “Feasibility Study of a 1.5-GeV Proton FFAG in the AGS Tunnel”, BNL Internal Report C-A/AP/157, June 2004 Acceleration in the AGS-FFAG Circumference807.091 m Harmonic Number, h24 Energy Gain 0.5 MeV / turn Transition Energy,  T 105.5 i Peak RF Voltage1.2 MVolt Number of full Buckets22 out of 24 Total Number of Protons1.0 x 10 14 Bunch Area, full0.4 eV-sec Protons / Bunch4.6 x 10 12 Injection Period1.0 ms Acceleration Period7.0 ms Total Cycle Period8.0 ms RF Cavity System No. of RF Cavities30 No. of Gaps per Cavity1 Cavity Length1.0 m Internal Diameter10 cm Peak Voltage / Cavity40 kVolt Power Amplifier / Cavity250 kW Energy Range, MeV4001,500  0.7130.923 Rev. Frequency, MHz0.2650.343 Revolution Period, µ s3.782.92 RF Frequency, MHz6.3578.228 Peak Current, Amp4.245.49 Peak Beam Power, MW2.122.75

6. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory6/20 MA-LE-PD A.G. Ruggiero, “FFAG Accelerators for High-Intensity Proton Beams”, Proceedings ICFA-HB2004 -- Bensheim, Germany -- October 18-22, 2004 Acceleration in the MA-LE-PD Circumference201.773 m Harmonic Number, h36 Energy Gain 1.2 MeV / turn Transition Energy,  T 53.755 i Peak RF Voltage1.8 MVolt Number of full Buckets26 out of 36 Total Number of Protons0.94 x 10 14 Bunch Area, full0.4 eV-sec Protons / Bunch3.6 x 10 12 Injection Period0.5 ms Acceleration Period0.5 ms Total Cycle Period1.0 ms RF Cavity System for the MA-LE-PD No. of RF Cavities40 No. of Gaps per Cavity1 Cavity Length1.0 m Internal Diameter10 cm Peak Voltage / Cavity45 kVolt Power Amplifier / Cavity1.0 MW Energy Range, MeV2001,000  0.5660.875 Rev. Frequency, MHz0.8411.300 Revolution Period, µ s1.1890.769 RF Frequency, MHz30.2846.80 Peak Current, Amp12.6519.55 Peak Beam Power, MW15.223.5

7. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory7/20 Is there a way of RF acceleration that does not require Frequency Modulation of the RF cavity system? Turn after turn the path length is adjusted so that the beam crosses the accelerating RF system always on phase (cyclotron condition). This requires that the harmonic number changes (increases) also. The accelerating RF cavity system is at constant frequency (and voltage). RF One example is acceleration in a Cyclotron (a Microtron)

8. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory8/20 What can we do But the isochronous condition cannot be satisfied in a FFAG with Non- Scaling Lattice and Linear Field Profile for a low-energy proton beam. So what can we do? Leave the path length turn-after-turn as it is (there is a change, but it is small) and vary properly the transit time between cavity crossing. This is made easier when one operates at energies below the ring transition energy (with a surprise … ).  T/T=  C/C –  /  =(1/  T 2 – 1/  2 )  p/p  T >>  Vary properly the transit time between cavity crossing so that the RF cavity is always crossed in phase. This leads to the concept of HNJ (or HNH)

9. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory9/20 Harmonic Number Jump or Hopping A.G. Ruggiero, “RF Acceleration with Harmonic Number Jump”, BNL Internal Report C-A/AP/237, May 2006 A.G. Ruggiero, “rf acceleration with harmonic number jump”, Phys. Review Special Topics – Accelerators and Beams 9, 100101 (2006) h h –  h

10. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory10/20 Acceleration of Synchronous Particles A ssume the beam as a sequence of point-like bunches (synchronous, reference). T he energy gain is adjusted for a change in the travel period T n in the following arc so that the reference particle is pushed forward or back exactly by  h harmonics. T n = h n T RF T n-1 = h n-1 T RF h n – h n-1 = –  h  E n =  n 2  n 3 E 0  h / h n (1 –  pn  n 2 ) T he ring is made of N RF cavities equally spaced. E n = total energy T n = h n T RF  E n = (Q eV n / A) sin (  RF t n ) = (Q eV n / A) sin (  n )

11. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory11/20 Acceleration of Non-Synchronous Particles A ny other particlet n = t n +  n  E n = (Q eV n / A) sin (  RF t n )  n = E n – E n  n = (Q eV n / A) [ sin (  n +  RF  n ) – sin (  n ) ] ~ (Q eV n / A) (cos  n )  RF  n   n =  n –  n – 1 = – (1 –  pn  n 2 ) T n  n /  n 2  n 3 E 0 S mall-Amplitude Oscillations  2  n /  n 2 +  n 2  n = 0with  n 2 = 2 π   h / tg  n

12. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory12/20 T he Hamiltonian H =(Q eV n / A  RF ) [ cos (  n +  RF  n ) +  RF  n sin (  n ) ] + – (1 –  pn  n 2 ) T n  n 2 / (2  n 2  n 3 E 0 ) cos (  n +  1,2n ) + cos (  n ) + (  1,2n – π + 2  n ) sin (  n ) = 0 RF Buckets with Harmonic-Number Jump nnnn  RF  n 

13. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory13/20 RF Buckets with Harmonic-Number Jump B ucket Area B n = (8 /  w RF )[ 2 Qe V n b n 2 g n 3 E 0 / A  h n (1 – a pn g n 2 ) ] 1/2 I(  1n,  2n ) I(  1n,  2n ) = ∫[cos (  n +  ) +  sin (  n ) + G(  n )] 1/2 / 4  2 d  G(  n )= cos(  n ) – (π – 2  n ) sin (  n ) B ucket Height  2 = 2 Qe V n  n 2  n 3 E 0 F(  n ) / A  (1 –  pn  n 2 ) h n F(  n ) = cos(  n ) – (π/2 –  n ) sin (  n )

14. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory14/20 Consequences of Harmonic-Number Jump T o avoid beam losses, the number of bunches ought to be less than the harmonic number at all time. On the other end, because of the change of the revolution period, the number of RF buckets will vary. There is a difference between the case of acceleration below and above transition energy. Below transition energy the beam extension at injection ought to be shorter than the revolution period. That is, the number of injected bunches cannot be larger than the RF harmonic number at extraction. The situation is different when the beam is injected above the transition energy. In this case the revolution period decreases and the harmonic number increases during acceleration. Below TransitionAbove Transition

15. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory15/20 Energy Gain Programming E nergy gain at the n-th cavity  E n = eV n sin (  n ) = A  n 2  n 3 E 0  h / Q h n (1 –  pn  n 2 ) V n = n c g  n TTF(  0 /  n c ) TTF(x  1)= sin(πx/2) /(πx/2) g =  0 / 2  n = average axial field  E n (MeV/m) vs. no. of Cavity Crossings Example: First of three FFAG Rings of NuFact identical to AGS-FFAG  h = 1 Constant RF frequency of 805.2 MHz with 2 diametrically opposite groups each of 8 equally-spaced, independently-phased, superconducting, single-gap cavities made of one single elliptically-shaped cell with gap g operating in half- wavelength mode. T a = 174 revolutions = 0.5 ms A. G. Ruggiero, “ FFAG Accelerator Proton Driver for Neutrino Factory ”, Nucl. Phys. B Proc. Suppl. 155, 315 (2006); BNL Report C-A/AP/219, 2005

16. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory16/20 Energy Gain Programming F our Programming Methods: 1. Constant RF Phase  n It requires the design of a RF Cavity with proper radial field profile 2. Constant average axial Field  n It requires a RF phase modulation 3. Modulation of the Harmonic Number Jump  h 4. Matching of the Acceleration period with the cavity Filling Time

17. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory17/20 Constant RF Phase Average Axial Field  n (MV/m) versus Radial Beam Position x (cm) with  n = 60 o The realization of such field profile across the radial aperture is problematic but not impossible. It could be made with ordinary pill-box cavities resonating in TM010 mode but displaced horizontally. A cavity that provides a longitudinal kick proportional to the radial displacement of the beam is the one operating in TM110 mode. Such cavity introduces also transverse deflecting modes that should be evaluated first and their impact to the beam compensated or at least reduced.

18. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory18/20 Constant Average Axial Field RF Phase  n (degrees) versus number n of Cavity Crossings T a = 174 revolutions  n = 15.74 MVolt/m Phase changes less than 1 degree per crossing

19. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory19/20 Modulation of Harmonic Number Jump Average Axial Field  n (MV/m) vs. radial Beam Position x (cm) with  n = 60 o by varying the amount  h of the harmonic number jump during acceleration Below transition energy, the energy gain increases with the cube of the particle total energy, for constant harmonic number jump. If on the other side the latter is also programmed as  h ~  n -2  n -3, that is a large value at injection that then decreases to unit at top energy, it is possible to reduce the variation of the axial field along the radius of the orbit. When this method is applied to our example, we found that the required axial field is about unchanged, but the range of the change is considerably reduced to only a factor of two, instead of ten. This method is more useful above transition energy. The energy gain varies linearly with the total particle energy. It could be flattened by allowing the amount  h of the harmonic number jump also to decrease, but this time only linearly. T a = 63 revolutions

20. November 6-10, 2006A.G. Ruggiero -- Brookhaven National Laboratory20/20 Matching of Acceleration Period with the Cavity Filling Time It takes a finite amount of time to fill up with power the RF cavities T F =1.4 (Q 0 /  ) / (2 + P b / P w ) Typically T F is a fraction of a millisecond and it can be made, with a proper choice of parameters, to match in magnitude the time T a required for acceleration over the desired energy range. In our example the required energy gain is about an order of magnitude in the field variation over about half a millisecond. The beam could be injected just a little after RF power is poured in the cavities. As the beam is accelerated the cavities are filled with more power until they are topped at the end of the acceleration cycle. During the filling the axial field will increase correspondingly as required. This method sounds more feasible than it may be suggested here. Although the cavity time constant is both a tool and an impediment to tailoring the voltage profile versus time, all cavity-modulator systems are equipped with amplitude and phase loops, or full vector feedback, and so following a voltage program with high accuracy is eminently possible – even if it calls for changes which are faster than the cavity time constant, provided that they are modest and the required over-power is available.