Acceleration of Protons in FFAG with Non-Scaling Lattice and Linear Field Profile Alessandro G....

Acceleration of Protons in FFAG with Non-Scaling Lattice and Linear Field Profile

Alessandro G. RuggieroSemi-Annual International

2006 FFAG Workshop November 6 - 10, 2006

KURRI - Osaka - Japan

November 6-10, 2006 A.G. Ruggiero -- Brookhaven National Laboratory

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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-FFAG MA-LE-PD

Injection Energy 400 MeV 200 MeV0.713 0.566

Extraction Energy 1.5 GeV 1.0 GeV0.9230 0.875

Circumference 809 m 202 mRepetition Rate 2.5-5 Hz 1 kHz


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Acceleration by RF-ModulationA.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)


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Required RF Modulation

AGS-FFAG MA-LE-PD

Repetition Rate 5 Hz 1 kHz

Acceleration Period 7 ms 0.5 ms

RF frequency 6 – 9 MHz (AGS) 30 – 50 MHz

Modulation Rate 0.5 MHz/ms 40 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


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AGS-FFAGA.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

Circumference 807.091 mHarmonic Number, h 24Energy Gain 0.5 MeV / turnTransition Energy, T 105.5 iPeak RF Voltage 1.2 MVoltNumber of full Buckets 22 out of 24Total Number of Protons 1.0 x 1014

Bunch Area, full 0.4 eV-secProtons / Bunch 4.6 x 1012

Injection Period 1.0 msAcceleration Period 7.0 msTotal Cycle Period 8.0 ms

RF Cavity System

No. of RF Cavities 30No. of Gaps per Cavity 1Cavity Length 1.0 mInternal Diameter 10 cmPeak Voltage / Cavity 40 kVoltPower Amplifier / Cavity 250 kWEnergy Range, MeV 400 1,500 0.713 0.923Rev. Frequency, MHz 0.265 0.343Revolution Period, µs 3.78 2.92 RF Frequency, MHz 6.357 8.228 Peak Current, Amp 4.24 5.49 Peak Beam Power, MW 2.12 2.75


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MA-LE-PDA.G. Ruggiero, “FFAG Accelerators for High-Intensity Proton Beams”,

Proceedings ICFA-HB2004 -- Bensheim, Germany -- October 18-22, 2004

Acceleration in the MA-LE-PD

Circumference 201.773 mHarmonic Number, h 36Energy Gain 1.2 MeV / turnTransition Energy, T 53.755 iPeak RF Voltage 1.8 MVoltNumber of full Buckets 26 out of 36Total Number of Protons 0.94 x 1014

Bunch Area, full 0.4 eV-secProtons / Bunch 3.6 x 1012

Injection Period 0.5 msAcceleration Period 0.5 msTotal Cycle Period 1.0 ms

RF Cavity System for the MA-LE-PD

No. of RF Cavities 40No. of Gaps per Cavity 1Cavity Length 1.0 mInternal Diameter 10 cmPeak Voltage / Cavity 45 kVoltPower Amplifier / Cavity 1.0 MWEnergy Range, MeV 200 1,000 0.566 0.875Rev. Frequency, MHz 0.841 1.300 Revolution Period, µs 1.189 0.769 RF Frequency, MHz 30.28 46.80 Peak Current, Amp 12.65 19.55 Peak Beam Power, MW 15.2 23.5


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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

RF

One example is acceleration in a Cyclotron (a Microtron)


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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)


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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


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Acceleration of Synchronous Particles

Assume the beam as a sequence of point-like bunches (synchronous, reference).

The energy gain is adjusted for a change in the travel period Tn in the following arc so that the reference particle is pushed forward or back exactly by h harmonics.

Tn = hn TRF Tn-1 = hn-1 TRF hn – hn-1 = – h

En= n2 n

3 E0 h / hn (1 – pn n2)

(n+1)-th Cavity

An+1 An n-th Cavity An–1 (n–1)-th Cavity

The ring is made of N RF cavities equally spaced.

En = total energy

Tn = hn TRF

En= (Q eVn / A) sin (RF tn)

= (Q eVn / A) sin (n)


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Acceleration of Non-Synchronous Particles

Any other particle tn = tn + n En = (Q eVn / A) sin (RF tn)

n = En – En

n = (Q eVn / A) [ sin (n + RF n) – sin (n) ]

~ (Q eVn / A) (cos n) RF n

n = n – n–1

= – (1 – pn n2) Tn n / n

2 n3 E0

Small-Amplitude Oscillations

2 n / n2 + n2 n = 0 with n

2 = 2 πh / tg n


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The Hamiltonian

H = (Q eVn / A RF) [ cos (n + RF n ) + RF n sin (n) ] +

– (1 – pn n2) Tn n

2 / (2 n2 n

3 E0)

cos (n + 1,2n) + cos (n) + (1,2n – π + 2 n) sin (n) = 0

RF Buckets with Harmonic-Number Jump

n n

RF n

n

n

n


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RF Buckets with Harmonic-Number Jump

Bucket Area

Bn = (8 /wRF)[ 2 Qe Vn bn2 gn

3 E0 / A hn (1 – apn gn2) ]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)

n

F( n)

Bucket Height

2 = 2 Qe Vn n2 n

3 E0 F(n) / A (1 – pn n2) hn

F(n) = cos(n) – (π/2 – n) sin (n)


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Consequences of Harmonic-Number Jump

To 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 Transition Above Transition


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Energy Gain Programming

Energy gain at the n-th cavity

En = eVn sin (n) = A n2 n

3 E0 h / Q hn (1 – pn n2)

Vn = nc g n TTF(0/nc) TTF(x1)= sin(πx/2) /(πx/2)

g = 0/ 2 n = average axial field

En (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.

Ta = 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


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Energy Gain Programming

Four 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


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Constant RF Phase

Average Axial Field n (MV/m) versus Radial Beam Position x (cm) with n = 60o

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.


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Constant Average Axial Field

RF Phase n (degrees) versus number n of Cavity Crossings

Ta = 174 revolutions

n = 15.74 MVolt/m

Phase changes less than 1 degree per crossing


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Modulation of Harmonic Number Jump

Average Axial Field n (MV/m) vs. radial Beam Position x (cm) with n = 60o 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.

Ta = 63 revolutions


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Matching of Acceleration Period with the Cavity Filling Time

It takes a finite amount of time to fill up with power the RF cavities

TF = 1.4 (Q0 / ) / (2 + Pb / Pw)

Typically TF is a fraction of a millisecond and it can be made, with a proper choice of parameters, to match in magnitude the time Ta 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.

Acceleration of Protons in FFAG with Non-Scaling Lattice and Linear Field Profile Alessandro G....

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