Linear Non-scaling FFAGs for Rapid Linear Non-scaling FFAGs for Rapid Acceleration using High-frequency Acceleration using High-frequency

( (≥≥100 MHz) RF100 MHz) RF

Cast of Characters in the U.S./Canada:Cast of Characters in the U.S./Canada:

C. Johnstone, S. Berg, M. CraddockC. Johnstone, S. Berg, M. Craddock

S. Koscielniak, B. Palmer, D. TrbojevicS. Koscielniak, B. Palmer, D. Trbojevic

Oct 12 – Oct 16, 2004Oct 12 – Oct 16, 2004

FFAG04FFAG04

KEK, KEK,

Tsukuba, JapanTsukuba, Japan

Rapid AccelerationRapid Acceleration

In an ultra-fast regime—applicable to In an ultra-fast regime—applicable to unstable particles—acceleration is unstable particles—acceleration is

completed in a few to a few tens of turnscompleted in a few to a few tens of turns Magnetic field cannot be rampedMagnetic field cannot be ramped RF parameters are fixed—no phase/voltage RF parameters are fixed—no phase/voltage compensation is feasiblecompensation is feasible operate at or near the rf crestoperate at or near the rf crest

Fixed-field lattices have been developed which can Fixed-field lattices have been developed which can contain up to a factor of 4 change in energy; typical contain up to a factor of 4 change in energy; typical is a factor of 2-3is a factor of 2-3

There are three main types of fixed field lattices There are three main types of fixed field lattices under development:under development:

Conventional Recirculating Linear Accelerators (RLAs)Conventional Recirculating Linear Accelerators (RLAs)Scaling FFAG (Fixed Field Alternating Gradient)Scaling FFAG (Fixed Field Alternating Gradient)Linear, nonscaling FFAGLinear, nonscaling FFAG

Current Baseline: Recirculating LinacsCurrent Baseline: Recirculating LinacsA Recirculating Linac Accelerator (RLA) consists of two A Recirculating Linac Accelerator (RLA) consists of two

opposing linacs connected by separate, fixed-field arcs for opposing linacs connected by separate, fixed-field arcs for each acceleration turneach acceleration turn

In Muon Acceleration for a Neutrino Factory:In Muon Acceleration for a Neutrino Factory: The RLAs only support The RLAs only support ONLYONLY 4 acceleration turns4 acceleration turns

due to the passive switchyard which must switch beam due to the passive switchyard which must switch beam into the appropriate arc on each acceleration turn and into the appropriate arc on each acceleration turn and the large momentum spreads and beam sizes involved.the large momentum spreads and beam sizes involved.

2-3 GeV2-3 GeV of rf is required per turn of rf is required per turn (NOT DISTRIBUTED)(NOT DISTRIBUTED) Again to enable beam separation and switching to Again to enable beam separation and switching to

separate arcsseparate arcs

Advantage of the RLAAdvantage of the RLA

Beam arrival time or M56 matching to the rf is independently Beam arrival time or M56 matching to the rf is independently controlled in each return arc, no rf gymnastics are controlled in each return arc, no rf gymnastics are involved; I.e. single-frequency, high-Q rf system is used.involved; I.e. single-frequency, high-Q rf system is used.

RLAs comprise about 1/3 the cost of the U.S. Neutrino FactoryRLAs comprise about 1/3 the cost of the U.S. Neutrino Factory

Mulit-GeV FFAGs: MotivationMulit-GeV FFAGs: Motivation Ionization cooling is based on acceleration Ionization cooling is based on acceleration

- (deacceleration of all momenum components then longitudinal - (deacceleration of all momenum components then longitudinal reacceleration)reacceleration)

THERE is a STRONG argument to let the accelerator do the bulk of THERE is a STRONG argument to let the accelerator do the bulk of the LONGITUDINAL AND TRANSVERSE COOLING (adiabatic the LONGITUDINAL AND TRANSVERSE COOLING (adiabatic cooling). cooling).

The storage ring can accept The storage ring can accept ~ ~ 4% 4% p/p @20 GeV p/p @20 GeV If acceleration is completely linear, so that absolute If acceleration is completely linear, so that absolute

momentum spread is preserved, momentum spread is preserved, @ ~400 MeV @ ~400 MeV

p/p =p/p = 200% 200%

implying no longitudinal coolingimplying no longitudinal cooling.. (Upstream Linear channels for TRANSVERSE Cooling currently accept a (Upstream Linear channels for TRANSVERSE Cooling currently accept a

maximum of maximum of 22% for the solenoidal sFOFO and -22% to +50% for 22% for the solenoidal sFOFO and -22% to +50% for quadrupoles)quadrupoles)

..

The Linac/RLA has been the showstopper in this argumentThe Linac/RLA has been the showstopper in this argument

Mulit-GeV FFAGs for a Neutrino Factory or Muon Mulit-GeV FFAGs for a Neutrino Factory or Muon ColliderCollider

Lattices have been developed which, practically, support up Lattices have been developed which, practically, support up to a factor of to a factor of 4 change in energy4 change in energy, or, or

almost unlimited momentum-spread acceptancealmost unlimited momentum-spread acceptance, which , which has immediate consequences on the degree of ionisation has immediate consequences on the degree of ionisation cooling requiredcooling required

Practical, technical considerations (magnet apertures, Practical, technical considerations (magnet apertures, mainly, and rf voltage) have resulted in a chain of FFAGs mainly, and rf voltage) have resulted in a chain of FFAGs with a factor of 2 change in energywith a factor of 2 change in energy

2.5 -5 GeV 5-10 GeV 10-20 GeV

Currently proposal,

U.S. scenario

Scaling FFAGs (radial sector): The B field and orbit are constructed such that the B field scales with radius/momentum such that the optics remain constant as a function of momentum.

Scaling machines display almost unlimited momentum acceptance, but a more restricted transverse acceptance than linear nonscaling FFAGs and more complex magnets.

KEK, Nufact02, London

Perk of Rapid Acceleration*Perk of Rapid Acceleration*

Freedom to cross betatron resonances:Freedom to cross betatron resonances: optics can change slowly with energyoptics can change slowly with energy allows lattice to be constructed from linear allows lattice to be constructed from linear

magnetic elements (dipoles and quadrupoles only)magnetic elements (dipoles and quadrupoles only)

This is the basic concept for a linear non-scaling This is the basic concept for a linear non-scaling FFAGFFAG

* In muon machines acceleration is completed in milliseconds or * In muon machines acceleration is completed in milliseconds or tens of millisecondstens of milliseconds

Linear non-scaling FFAGs:Linear non-scaling FFAGs:

Transverse acceptance:Transverse acceptance: ““unlimited” due to linear magnetic elementsunlimited” due to linear magnetic elements Large horizontal magnet aperture Large horizontal magnet aperture

General characteristic of fixed-field accelerationGeneral characteristic of fixed-field acceleration Orbit changes as a function of momentum: beam travels from the inside of the Orbit changes as a function of momentum: beam travels from the inside of the

ring to the outsidering to the outside

Momentum Acceptance:Momentum Acceptance: FODO optics:FODO optics:

Large range in momentum acceptance:Large range in momentum acceptance:

defined by lower and upper limits of stabilitydefined by lower and upper limits of stability Limits depend on FODO cell parametersLimits depend on FODO cell parameters

Triplet, doublet (dual-plane focusing) optics:Triplet, doublet (dual-plane focusing) optics: Too achromatic; small momentum acceptance to achieve horizontal+vertical Too achromatic; small momentum acceptance to achieve horizontal+vertical

foci.foci.

Phase advance in a linear non-scaling Phase advance in a linear non-scaling FFAGFFAG

Stable range as a function of momentumStable range as a function of momentum Lower limit:Lower limit:

Given simply and approximately by thin-lens Given simply and approximately by thin-lens equations for FODO opticsequations for FODO optics

Upper limit:Upper limit: No upper limit in thin-lens approximationNo upper limit in thin-lens approximation Have to use thick lens modelHave to use thick lens model

)1(lens)(thin /where;/12

sin Lf In the thin-lens approximation, the phase advance, In the thin-lens approximation, the phase advance, , is given by, is given by

with f being the focal length of ½ quadrupole and L the length of a half with f being the focal length of ½ quadrupole and L the length of a half cell from quadrupole center to centercell from quadrupole center to center

In equation (3), In equation (3), B’B’ is the quadrupole gradient in is the quadrupole gradient in T/mT/m and and pp is the is the momentum in momentum in GeV/cGeV/c. Selecting . Selecting = 90 = 90 at p at p00, the reference , the reference momentum implies the following:momentum implies the following:

)2(3.0

since;3.0

2sin

p

lBkL

p

lB

)3(.2

1stability,oflimitlower ,2

1

2sin 0

0

ppp

p

)5(2/1

2

)4(22

cos2

1

220

2

0

20

ppp

p

dp

d

or

dpp

pd

Differentiating the above equation gives the dependence of phase Differentiating the above equation gives the dependence of phase advance on momentumadvance on momentum

There is a low-momentum cut-off, but at large p, the phase advance There is a low-momentum cut-off, but at large p, the phase advance varies more and more slowly, as 1/pvaries more and more slowly, as 1/p22, and there is no effective , and there is no effective high-momentum cut-off in the thin-lens approximation. high-momentum cut-off in the thin-lens approximation.

A high-momentum stability limit is observed in the thick lens A high-momentum stability limit is observed in the thick lens representationrepresentation

Beta functions in a linear non-scaling Beta functions in a linear non-scaling FFAGFFAG

Momentum dependence described by thin-lens Momentum dependence described by thin-lens equations equations

Magnitude and variation:Magnitude and variation: Lower limit on momentum (injection) is raised Lower limit on momentum (injection) is raised

away from lower limit of stabilityaway from lower limit of stability Minimized using ultra-short cellsMinimized using ultra-short cells

(7)minimumafor0)1()1(

)1(

)6()1(

)1(

2/12/3

2max

2/12max

dp

dL

dp

d

L

Using thin-lens solutions, the peak beta function for a FODO cell is Using thin-lens solutions, the peak beta function for a FODO cell is given by: given by:

In the above equation In the above equation (7), (2 - - 1) can only be set to can only be set to 0 locally (at locally (at ~76~76), but this does not guarantee stability in the beta function ), but this does not guarantee stability in the beta function over a large range in momentum. The only approach that over a large range in momentum. The only approach that minimizes minimizes dmax/dp over a broad spectrum is to let over a broad spectrum is to let L approach 0..

Phase advance and beta function dependence (thick lens) for a short FODO cell (half-cell length: 0.9 m). The momentum p0 represents 90 of phase advance.

Acceptance is 40% p/p about 1.5 p0 (~65) for practical magnet apertures (~0.1x0.25m, VxH) and large muon emittances (5-10 cm, full, normalized) at 1-2 GeV. This corresponds to an acceleration factor of 2.3.

Travails of Rapid Fixed Field AccelerationTravails of Rapid Fixed Field Acceleration A pathology of fixed-field acceleration in recirculating-beam A pathology of fixed-field acceleration in recirculating-beam

accelerators (for single, not multiple arcs) is that the particle accelerators (for single, not multiple arcs) is that the particle beam transits the radial aperturebeam transits the radial aperture

The orbit change is significant and leads to non-isochronism, or The orbit change is significant and leads to non-isochronism, or a lack of synchronism with the accelerating rf a lack of synchronism with the accelerating rf

The result is an unavoidable phase slippage of the beam The result is an unavoidable phase slippage of the beam particles relative to the rf waveform and eventual loss of net particles relative to the rf waveform and eventual loss of net acceleration withacceleration with

The lattice completely determining the change in circulation time The lattice completely determining the change in circulation time (for ultra relativistic particles)(for ultra relativistic particles)

The rf frequency determining the phase slippage which The rf frequency determining the phase slippage which accumulates on a per turn basis:accumulates on a per turn basis:

turnperrf t

Moderating Phase Slip in a non-scaling FFAGModerating Phase Slip in a non-scaling FFAG

Lattice: source

Minimize pathlength change with momentum

minimum momentum compaction lattices

RF: choices

Low-frequency (<25 MHz): construction problems There is an optimal choice of for high rf frequency (~200 MHz) Adjust initial cavity phase to minimize excursion of reference

particle from crest Inter-cavity phasing to minimize excursions of a distribution

Minimum Momentum-compaction latticesMinimum Momentum-compaction latticesfor nonscaling FFAGsfor nonscaling FFAGs

Phase slippage of reference orbits can be described as a change in Phase slippage of reference orbits can be described as a change in circumference for relativistic particles:circumference for relativistic particles:

Minimizing the dispersion function in regions of dipole bend fields Minimizing the dispersion function in regions of dipole bend fields controls phase slip for a given net bend/cell. controls phase slip for a given net bend/cell.

Historical Note: For a fixed bend radius:Historical Note: For a fixed bend radius: minimizing minimizing minimizing dispersion minimizing dispersion minimizing dispersion minimizing dispersion minimizing emittance in electron machines minimizing emittance in electron machines

FFAG lattices are completely periodicFFAG lattices are completely periodic C is N Lcell (cell), where N is the number of cells. .

Since Since N Lcell = C, ring = cell.

dsC

p

p

C

C

ring

ringring

1

Linear Dispersion in thin-lens FODO opticsLinear Dispersion in thin-lens FODO optics

Dispersion can be expressed in standard thin-lens matrix formalism.Dispersion can be expressed in standard thin-lens matrix formalism.

At the symmetry points of the FODO cell the slope of optical At the symmetry points of the FODO cell the slope of optical parameters is zero, and correspond to points of maximum and parameters is zero, and correspond to points of maximum and minimum dispersion. For horizontal dispersion, the center of the minimum dispersion. For horizontal dispersion, the center of the vertically-focusing element is a minimum and horizontally-focusing vertically-focusing element is a minimum and horizontally-focusing

element is a maximum.element is a maximum.

constant since

1

'

1

';'' 0

0

Mx

x

1

0

1

0

max

2/1

min FODOM

Thin lens matrix solutions for different dipole Thin lens matrix solutions for different dipole options in a FODOoptions in a FODO

The transfer matrix for a dipole field centered in the drift The transfer matrix for a dipole field centered in the drift between focusing elements: 1/2F-drift-1/2D is:between focusing elements: 1/2F-drift-1/2D is:

For a dipole field centered For a dipole field centered in the vertically-in the vertically- focusing element:focusing element:

100

)2

11(1

2

11

100

011001

100

010

02/1

100

10

001

100

010

02/1

100

011001

2

2/1

fL

fL

fL

LLfL

f

LL

f

B

B

BFODO

M

100

1

01

22/1

BFODO

CF fL

fL

LfL

M

Dispersion and dipole locationDispersion and dipole location

Dispersion solution for conventional FODODispersion solution for conventional FODO

Dispersion solution for the dipole field located in the vertically-Dispersion solution for the dipole field located in the vertically-focusing element—clearly reducedfocusing element—clearly reduced

f

L

L

fB 2

11

2max

f

L

L

fB 2

11

2min

BCF L

f 2

max

f

L

L

fBCF 1

2min

““Optimum” Minimum Momentum-compaction Optimum” Minimum Momentum-compaction lattices for nonscaling FFAGslattices for nonscaling FFAGs

The optimum lattices are strictly FODO-based, with two candidates:The optimum lattices are strictly FODO-based, with two candidates: Combined Function (CF) FODOCombined Function (CF) FODO

• Horizontally-focusing quadrupole, and combined function horizontally-Horizontally-focusing quadrupole, and combined function horizontally-defocussing magnetdefocussing magnet

• The rf drift is provided between the quadrupole and CF elementThe rf drift is provided between the quadrupole and CF element

FODO – like tripletFODO – like triplet• The horizontally-focusing quadrupole is split and the rf drift is inserted between The horizontally-focusing quadrupole is split and the rf drift is inserted between

the two halves.the two halves.• The magnet spacing between the quadrupole and the CF magnet is much The magnet spacing between the quadrupole and the CF magnet is much

reduced.reduced.

All optical units have reflective symmetry, implyingAll optical units have reflective symmetry, implying

ring = cell = 1/2 cell

Special insertions for rf, extraction, injection, etc. have not been successfulSpecial insertions for rf, extraction, injection, etc. have not been successful

Triplet configuration or “modified” FODOTriplet configuration or “modified” FODO

An structure defined as FDF: An structure defined as FDF:

[1/2rfdrift-QF—short drift—CF-short drift-QF-1/2rf drift][1/2rfdrift-QF—short drift—CF-short drift-QF-1/2rf drift] produces significantly reduced momentum compaction and therefore produces significantly reduced momentum compaction and therefore

phase slip relative to the separated and CF FODO cells. phase slip relative to the separated and CF FODO cells.

where where equivalent equivalent is defined in terms ofis defined in terms of rf drift length,rf drift length, (2 m) (2 m)

identical bend angle per cell, identical bend angle per cell, intermagnet spacingintermagnet spacing (0.5 m) (0.5 m)

phase advance at injectionphase advance at injection (0.72 (0.72 , both planes), both planes)maximum poletip field allowedmaximum poletip field allowed. (. ( 7T ) 7T )

DFD arrangement does not perform as the FDFDFD arrangement does not perform as the FDF

Transfer matrices for triplet (FDF) FODO Transfer matrices for triplet (FDF) FODO cellscells

For an rf drift inserted at the center of the horizontally-focusing For an rf drift inserted at the center of the horizontally-focusing quadrupole:quadrupole:

- Note that the half cell contains only half the rf drift, hence the added drift - Note that the half cell contains only half the rf drift, hence the added drift matrix is Lmatrix is Lrfrf/2, rather than the half-cell length as in the FODO cell case. /2, rather than the half-cell length as in the FODO cell case.

100

11

21

0121

100

010

021

100

11

01

2**

11

2*

12/1

Brf

rf

rf

BFDF

CF

fD

f

L

f

DfDL

fD

L

fD

f

DfD

MWhere D, the distance between quadrupole centers, Lrf/2 replaces the half-cell length

Dispersion function for modified FODO;Dispersion function for modified FODO;triplet quadrupole configurationtriplet quadrupole configuration

The combined focal length, f*, is the general result for a doublet The combined focal length, f*, is the general result for a doublet quadrupole lens system. quadrupole lens system.

With the rf drift placed at the center of the horizontally focusing With the rf drift placed at the center of the horizontally focusing element, element, the differences between them and from the FODO cell are the differences between them and from the FODO cell are not immediately obvious we unless we explore the possible values not immediately obvious we unless we explore the possible values for for ff11 and and ff22. .

2121*

1

*max

1

min

*max

111

11

ff

D

fffwhere

fDff

D

f

BCFFDF

BFDF

Limit of stabilityLimit of stability One can solve for focal lengths in the limits of stability and use their One can solve for focal lengths in the limits of stability and use their

relative scaling over the entire acceleration range as a basis for relative scaling over the entire acceleration range as a basis for comparison between FODO cell configurations.comparison between FODO cell configurations.

In the presence of no bend, 90 degrees of phase advance across a In the presence of no bend, 90 degrees of phase advance across a half cell represents the limit of stability for FODO-like optics (single half cell represents the limit of stability for FODO-like optics (single minimum). This implies for a initial position on the x axis (x,x’=0), minimum). This implies for a initial position on the x axis (x,x’=0), that its position will be 0 (x=0,x’) after a half-cell transformation, that its position will be 0 (x=0,x’) after a half-cell transformation, conversely for the y planeconversely for the y plane

01

12

1

121

'

0

2**

112/1 x

fD

f

L

f

DfDL

fD

x rf

rf

FDFCFM

'

0

11

21

121

02

****

112/1

yf

Df

L

f

DfDL

fD

y

rf

rf

FDFCFM

0;2;5.1

)(lim;2

)2(

21;111

2

22121

**

rf

rf

rf

rf

rf

LD

LDD

LDLDf

LD

L

Dfff

D

fff

Df

ff

D

fff

1

2121*

111

Closed orbit in the limit of stabilityClosed orbit in the limit of stability These are the only closed orbits at the limits of stability (180These are the only closed orbits at the limits of stability (180):):

There is no “area” transmitted, beta functions go to infinity, There is no “area” transmitted, beta functions go to infinity, 0, phase space is a line.0, phase space is a line.

[x,0]

[0,y’]

y’

y

y’

y

[0,x’]

[y,0]

[-x,0]

[0,-y’]

Solutions for the limit of stabilitySolutions for the limit of stability

For CF or separated FODO cells:For CF or separated FODO cells: In the modified FDF FODO:In the modified FDF FODO:

Dfso

ff

D

fffand

Df

*

2121*

1

,111

rfLff 21

01

12

1

121

'

0

2**

112/1 x

fD

f

L

f

DfDL

fD

x rf

rf

FDFCFM

'

0

11

21

121

02

****

112/1

yf

Df

L

f

DfDL

fD

y

rf

rf

FDFCFM

DflatticesourDffor

ff

D

fffand

LD

L

Dfrf

rf

3.2)(4.1

;111

)2(

21

**2

2121**

2

Final Comparison, CF vs. modified FODOFinal Comparison, CF vs. modified FODO One can now compare the decrease in dispersion in the limit One can now compare the decrease in dispersion in the limit

of stability (using L ~1.5 D for the rf drifts, magnet spacing and of stability (using L ~1.5 D for the rf drifts, magnet spacing and lengths we use in actual designs).lengths we use in actual designs).

FODOFODO FDFFDF

At this point, one invokes scaling in focal length and bend At this point, one invokes scaling in focal length and bend angle to generalize conclusions over the entire momentum angle to generalize conclusions over the entire momentum range in the thin-lens approximation.range in the thin-lens approximation.

0min

max

FDF

BFDF D

DL

L

fBBBCF 5.1

2max

0

12

min

f

L

L

fBCF

10-20 GeV “Nonscaling” FFAGs: Examples10-20 GeV “Nonscaling” FFAGs: Examples FDF-triplet FODOFDF-triplet FODOCircumference 607m 616mCircumference 607m 616m#cells 110 108#cells 110 108Rf drift 2m 2mRf drift 2m 2mcell length 5.521m 5.704mcell length 5.521m 5.704mD-bend length 1.89m 1.314mD-bend length 1.89m 1.314mF-bend length 0.315m (2!) 0.390F-bend length 0.315m (2!) 0.390F-D spacing 0.5 mF-D spacing 0.5 m 2 m 2 mCentral energy** 20 GeV 18.65 GeVCentral energy** 20 GeV 18.65 GeVF gradiF gradient 60 T/m 60 T/ment 60 T/m 60 T/mD gradient 20 T/m 18 T/mD gradient 20 T/m 18 T/mF strength 0.99 mF strength 0.99 m -2-2 0.9486 m 0.9486 m-2-2

D strength 0.300 mD strength 0.300 m -2 -2 0.300 m 0.300 m-2-2 Bend-field (central energy) 2.0 T 2.7 TBend-field (central energy) 2.0 T 2.7 TOrbit swingOrbit swing Low -7.7 -9.5Low -7.7 -9.5 High 0 3.3High 0 3.3C (pathlength) 16.6 27.3C (pathlength) 16.6 27.3xmaxxmax//ymaxymax (10 GeV) 6.5/13.8 14.4/11.44 (10 GeV) 6.5/13.8 14.4/11.44(injection straight) 6.5 5.8(injection straight) 6.5 5.8Tune, (x,y)Tune, (x,y) Inject / Extract 0.36 / 0.36 (130Inject / Extract 0.36 / 0.36 (130 ) 0.36 / 0.36 (130) 0.36 / 0.36 (130) ) Extract 0.18 / 0.13 (~56Extract 0.18 / 0.13 (~56 ) 0.14 / 0.16 (~54) 0.14 / 0.16 (~54) )

** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in ** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect.effect.

Note pathlength difference

High-frequency (~200 MHz) RF accelerationHigh-frequency (~200 MHz) RF acceleration

In a nonscaling linear FFAG, the orbital pathlength, or In a nonscaling linear FFAG, the orbital pathlength, or T, is T, is parabolic with energy. At high-frequency, parabolic with energy. At high-frequency, 100 MHz, the 100 MHz, the accumulated phase slip is significant after a few turns,accumulated phase slip is significant after a few turns,

The phase-slip can reverse twice with an implied potential for the The phase-slip can reverse twice with an implied potential for the beam’s arrival time to cross the crest three times, given the beam’s arrival time to cross the crest three times, given the appropriate choice of starting phase and frequencyappropriate choice of starting phase and frequency

6-20 GeV Nonscaling FFAG

-20

-10

0

10

20

30

40

50

0 5 10 15 20 25

Momentum (GeV)

Cir

cum

fere

nce

Chang

e (

cm

)

harmonic of rf = point of phase reversal

We know from the parabolic dependency of the circumference the We know from the parabolic dependency of the circumference the explicit dependence of explicit dependence of (and therefore (and therefore ) on ) on must be must be

This impliesThis implies

Where the coefficients correspond to momentum compaction at Where the coefficients correspond to momentum compaction at the lower and upper momentum, respectively and the lower and upper momentum, respectively and is taken as the is taken as the momentum offset to the central energy, or bottom of the parabola. momentum offset to the central energy, or bottom of the parabola. coefficiencts are kept postive so that a negative momentum coefficiencts are kept postive so that a negative momentum compaction is clearercompaction is clearer

constant for 10

10

0;0

)()( 10max10max

lu

uuuullll CCC

oror

At transition (the bottom of the parabola)At transition (the bottom of the parabola)

For our rings with For our rings with ppuu/p/pll = 2, = 2, ll =0.5, =0.5, uu =-0.25,=-0.25,

2

1100 ;

l

uul

l

uul

u

ucucl

lclc

uc

uulc

ll

lll

p

pp

p

ppfor

also

CC

;

2;2

0)2(

1010

10max

uull and 1010 5.0

This impliesThis implies

One can study the first coefficient, One can study the first coefficient, 00, to determine , to determine

the behavior of the behavior of C as a function of free parametersC as a function of free parameters

One can study the behavior at one momentum; the One can study the behavior at one momentum; the lower limit of stability, for example lower limit of stability, for example

which makes the problem substantially easier to which makes the problem substantially easier to parameterizeparameterize

Circumference change in the thin-lens modelCircumference change in the thin-lens model

In the thin-lens model, the total change in circumference (and In the thin-lens model, the total change in circumference (and therefore the total phase slip, or therefore the total phase slip, or T) can be estimated from the T) can be estimated from the solutions found for dispersion about the central orbit (solutions found for dispersion about the central orbit (=0). (The =0). (The contributing term in contributing term in is managed by symmetrizing the parabola.) is managed by symmetrizing the parabola.)

For a periodic ringFor a periodic ring

In the thin lens,In the thin lens, is linear across the half cell: is linear across the half cell:

dsL

with

NLCC

cellcell

cellcellring

1

sD

)( minmax0

D is the half cell length and s D is the distance from the center of the CF quadrupole

Integrating over the length of the CF quadrupole, Integrating over the length of the CF quadrupole, llBB, (for , (for

a half cell) and noting that in the thin-lens limit of stability a half cell) and noting that in the thin-lens limit of stability minmin=0=0

2max

00

2max

2/1

0

max

2/10

0

2/10

2212

2

1

11

Blc

lllc

B

cell

l

cell

l

cell

l

DNNC

D

l

L

dsD

s

Lds

L

BB

Inserting Inserting maxmax==BBf*f* and and BB==l l BB//

stability oflimit lower for thesince

22

2

*

*

2max

2

l

BlB

lc

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Further near the acceleration range we have chosen, 2:1, the Further near the acceleration range we have chosen, 2:1, the dependence goes as the square of the range. The original dependence goes as the square of the range. The original designs were designs were 3, with either the 3, with either the C or the circumference a C or the circumference a factor of 9 larger. This represents the biggest factor in the factor of 9 larger. This represents the biggest factor in the phase-slip profile/design.phase-slip profile/design.

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Naively one would hope that circumference would scale with momentum. Naively one would hope that circumference would scale with momentum. However, we know that However, we know that T or T or C must be held at a certain value for C must be held at a certain value for successful acceleration. If successful acceleration. If C is set or scaled relative to the C is set or scaled relative to the High High Energy Ring (HER)Energy Ring (HER), then a , then a Low Energy Ring (LER)Low Energy Ring (LER) would follow: would follow:

*see FFAG workshop, TRIUMF, April, 2004, C. Johnstone, “Performance Criteria and *see FFAG workshop, TRIUMF, April, 2004, C. Johnstone, “Performance Criteria and Optimization of FFAG lattices for derivationsOptimization of FFAG lattices for derivations

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This is somewhat optimistic because you are simply keeping the This is somewhat optimistic because you are simply keeping the number of turns, and number of turns, and T ~ constant.T ~ constant.

For our rings this implies the 2.5-5 GeV ring is only ~60% the For our rings this implies the 2.5-5 GeV ring is only ~60% the size of the 10-20 GeV ring. S. Berg’s optimizer finds 80% so this size of the 10-20 GeV ring. S. Berg’s optimizer finds 80% so this is fairly close for an approximate descriptionis fairly close for an approximate description

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Lattice conclusions: TRIUMF FFAG workshopLattice conclusions: TRIUMF FFAG workshop Need revisions in cost profileNeed revisions in cost profile

Magnet cost scales linearly with magnet aperture, magnet Magnet cost scales linearly with magnet aperture, magnet cost cost 0 as aperture 0 as aperture 0. 0.

No differentiation between 7T multi-turn and 4T single-turn No differentiation between 7T multi-turn and 4T single-turn SC magnetsSC magnets

Better cost profiling to be provided for KEK FFAG workshop, Better cost profiling to be provided for KEK FFAG workshop, Oct, 2004.Oct, 2004.

Large-aperture 7T magnets are prohibitively expensiveLarge-aperture 7T magnets are prohibitively expensive

Optimum for the two higher energy rings may be 4TOptimum for the two higher energy rings may be 4T

The lower energy ring The lower energy ring higher-energy rings in cost higher-energy rings in cost Large cost for small energy gain (2.5 GeV). Large cost for small energy gain (2.5 GeV). The next jump in magnet cost would be large-aperture normal The next jump in magnet cost would be large-aperture normal

conducting and pulsed, 1.5T. (Refer to the large-aperture Fermi conducting and pulsed, 1.5T. (Refer to the large-aperture Fermi proton driver design for costingproton driver design for costing

Asynchronous AccelerationAsynchronous Acceleration

The number of phase reversals (points of sychronicity The number of phase reversals (points of sychronicity with the rf) = number of fixed points in the Hamiltonianwith the rf) = number of fixed points in the Hamiltonian

Scaling FFAGs with a linear dependence of pathlength Scaling FFAGs with a linear dependence of pathlength on momentum have 1 fixed pointon momentum have 1 fixed point

Linear nonscaling FFAGs with a quadratic pathlength Linear nonscaling FFAGs with a quadratic pathlength dependence have 2dependence have 2

The number of fixed points = number of asynchronous The number of fixed points = number of asynchronous modes of accelerationmodes of acceleration

Asynchronous Modes of AccelerationAsynchronous Modes of Acceleration

Single fixed point acceleration: half synchrotron oscillation

Two fixed point acceleration: half synchrotron oscillation + path between fixed points

Scaling FFAGScaling FFAGLinear nonscaling FFAGLinear nonscaling FFAG

½ Synchrotron osc.½ Synchrotron osc. Libration pathLibration path

TimeTime TimeTime

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Optimal Longitudinal DynamicsOptimal Longitudinal Dynamics

Optimal choice of rf frequency:Optimal choice of rf frequency: TT11 = 3 = 3TT22

recent results: recent results: TT11 = 5 = 5TT22

for phase space linearityfor phase space linearity

Optimal choice of initial cavity phasingOptimal choice of initial cavity phasing Min Min for reference particle for reference particle (p) = phase slip/turn relative to rf crest(p) = phase slip/turn relative to rf crest

Optimal initial phasing of individual cavitiesOptimal initial phasing of individual cavities Minimizes Minimizes (())22 of a distribution of a distribution

Phase space transmission of a FODO nonscaling Phase space transmission of a FODO nonscaling FFAGFFAG

Optimal frequency, optimal Optimal frequency, optimal initial cavity phasing initial cavity phasing (tranmission of ~0.5 ev-sec)(tranmission of ~0.5 ev-sec)

Optimal frequency, optimized Optimal frequency, optimized initial phasing of individual initial phasing of individual cavities : cavities : improved linearityimproved linearity

Out put emittance and energy Out put emittance and energy versus rf voltage for versus rf voltage for acceleration completed in acceleration completed in 4(black), 5(red), 6(green), 4(black), 5(red), 6(green), 7(blue), 8(cyan), 9(magenta), 7(blue), 8(cyan), 9(magenta), 10(coral), 11(black), 12(red).10(coral), 11(black), 12(red).

Next: Electron Prototype of a nonscaling Next: Electron Prototype of a nonscaling FFAGFFAG

Test resonance crossingTest resonance crossing Test multiple fixed-point accelerationTest multiple fixed-point acceleration Output/input phase spaceOutput/input phase space Stability, operationStability, operation Error sensitivity, error propagationError sensitivity, error propagation Magnet design, correctors?Magnet design, correctors? DiagnosticsDiagnostics

Example 10-20 MeV electron prototype Example 10-20 MeV electron prototype nonscaling FFAG*nonscaling FFAG*

FDF-triplet FODOFDF-triplet FODOCircumference 13.7m 12.3mCircumference 13.7m 12.3m#cells 28 28#cells 28 28cell length 0.49m 0.44mcell length 0.49m 0.44mCF length 7.6cm 6.9cmCF length 7.6cm 6.9cmF-bend length 1.24 cm (2!) 2 cmF-bend length 1.24 cm (2!) 2 cmF-D spacing 0.05 m 0.15mF-D spacing 0.05 m 0.15mCentral energy** 20 MeV 18.5 MeVCentral energy** 20 MeV 18.5 MeVF gradiF gradient 12 T/m 12 T/ment 12 T/m 12 T/mD gradient 3.9 T/m 3.5 T/mD gradient 3.9 T/m 3.5 T/mF strength 175.6 mF strength 175.6 m -2-2 194.6 m 194.6 m-2-2

D strength 57.3 mD strength 57.3 m -2 -2 50.8 m 50.8 m-2-2 Bend-field (central energy) 0.2 T 0.2 TBend-field (central energy) 0.2 T 0.2 TOrbit swingOrbit swing Low -2.8 -2.5Low -2.8 -2.5 High 0 0.9High 0 0.9C (pathlength) 5.8 6.8C (pathlength) 5.8 6.8xmaxxmax//ymaxymax (10 GeV) 0.6/1 1/0.8 (10 GeV) 0.6/1 1/0.8(injection straight) 0.6 0.9(injection straight) 0.6 0.9Tune, (x,y)Tune, (x,y) Inject / Extract 0.34 / 0.33 (130Inject / Extract 0.34 / 0.33 (130 ) 0.36 / 0.36 (130) 0.36 / 0.36 (130) ) Extract ~ 0.18 / 0.13 (~56Extract ~ 0.18 / 0.13 (~56 ) ~ 0.14 / 0.16 (~54) ~ 0.14 / 0.16 (~54) )

** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in ** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect.effect.

Note pathlength difference