Linear Non-scaling FFAGs for Linear Non-scaling FFAGs for Rapid Acceleration using High- Rapid Acceleration using High- frequency frequency ( ( ≥ ≥ 100 MHz) RF 100 MHz) RF Cast of Characters in the Cast of Characters in the U.S./Canada: U.S./Canada: C. Johnstone, S. Berg, M. C. Johnstone, S. Berg, M. Craddock Craddock S. Koscielniak, B. Palmer, D. S. Koscielniak, B. Palmer, D. Trbojevic Trbojevic July 26-July 31, 2004 July 26-July 31, 2004 NuFact04 NuFact04
Transcript
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
July 26-July 31, 2004July 26-July 31, 2004
NuFact04NuFact04
Osaka University, Osaka University,
Osaka, JapanOsaka, 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)Dogbone RLAsDogbone RLAsScaling 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
Dogbone RLAs*Dogbone RLAs* *First proposed; D. Summers, Publication: Pac01, S. Berg and C. *First proposed; D. Summers, Publication: Pac01, S. Berg and C.
JohnstoneJohnstone Optics condition to close off-momentum orbits: Optics condition to close off-momentum orbits: match dispersion to all match dispersion to all
significant orderssignificant orders Dispersion relations for muon lattices:Dispersion relations for muon lattices:
2) completely periodic FODO optics (no change in dipole 2) completely periodic FODO optics (no change in dipole strength/period: linear nonscaling FFAG) : strength/period: linear nonscaling FFAG) :
-- 00 and and ’’ (d (d/ds) can be matched using linear optics (dipoles/phase /ds) can be matched using linear optics (dipoles/phase
advance=dipoles/quad strength)advance=dipoles/quad strength) - - 11 can be matched by not violating periodicity or canceled using can be matched by not violating periodicity or canceled using
sextupoles sextupoles
(( scaling machine with individual correctors rather than field scaling; scaling machine with individual correctors rather than field scaling; sextupole is the first and largest nonlinearity in a scaling FFAG)sextupole is the first and largest nonlinearity in a scaling FFAG)
Dogbone RLAs, continuedDogbone RLAs, continued
Chromatic aberrations (nonlinear sextupole distortions of Chromatic aberrations (nonlinear sextupole distortions of phase space) are canceled only at cell phase advances ofphase space) are canceled only at cell phase advances of
6060 9090 180180 unstable unstable
If you’re clever you can ~ cancel these distortions to 2If you’re clever you can ~ cancel these distortions to 2ndnd order in two of the arcs (the momentum spread is so large, order in two of the arcs (the momentum spread is so large, the phase advance changes rapidly and the chromatic the phase advance changes rapidly and the chromatic cancellation deteriorates ; cancellation deteriorates ;
the third arc has no chromatic cancellation and there is a the third arc has no chromatic cancellation and there is a sextupole-distorted phase spacesextupole-distorted phase space
Dogbone RLA: concernsDogbone RLA: concerns 10% dp/p can be difficult as high-order terms become important; 10% dp/p can be difficult as high-order terms become important;
DA declinesDA declines I’ve not seen I’ve not seen 20% dp/p acceptance with any reasonable DA. 20% dp/p acceptance with any reasonable DA.
(Trbojevic has some generated sextupole-dominated lattices)(Trbojevic has some generated sextupole-dominated lattices)
Nonlinear phase space may be mis-matched to the elliptical, linear Nonlinear phase space may be mis-matched to the elliptical, linear phase space of downstream accelerators; emittance may blow up phase space of downstream accelerators; emittance may blow up in these machines or the storage ring.in these machines or the storage ring.
Dogbone RLA ~ low energy RLADogbone RLA ~ low energy RLA, which we know is difficult; further , which we know is difficult; further the dogbone Switchyard contains reverse bends relative to the the dogbone Switchyard contains reverse bends relative to the arcs, the RLA does not; nonlinear matching arcs, the RLA does not; nonlinear matching dipole bend dipole bend strength.strength.
Strong sextupoles will decrease DA and longitudinal acceptance; Strong sextupoles will decrease DA and longitudinal acceptance; ring cooling will be needed and will eliminate any cost savings.ring cooling will be needed and will eliminate any cost savings.
A dogbone upstream will sacrifice much of the advantages of the A dogbone upstream will sacrifice much of the advantages of the FFAGs which do not require longitudinal cooling.FFAGs which do not require longitudinal cooling.
We are still looking at the 2.5-5 GeV FFAG; corrections to cost We are still looking at the 2.5-5 GeV FFAG; corrections to cost profiling and normal conducting, pulsed-magnet optionsprofiling and normal conducting, pulsed-magnet options
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
Japanese N.F. : 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 linear 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 submillisecond or * In muon machines acceleration is completed in submillisecond or millesecond timescalesmillesecond timescales
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
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 linear nonscaling FFAGsfor linear 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 dispersion minimizing dispersion minimizing emittance in electron machines The term minimum emittance does not apply to muon applications, but the lattice The term minimum emittance does not apply to muon applications, but the lattice
approach is similar, hence the references in the literatureapproach is similar, hence the references in the literature
Linear nonscaling FFAG lattices are completely periodic*Linear nonscaling FFAG lattices are completely periodic*
C is N Lcell (cell), where N is the number of cells. .
Since Since N Lcell = C, ring = cell
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 Modified FODO – quadrupole tripletModified FODO – quadrupole 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 failed* Special insertions for rf, extraction, injection, etc. have failed
Triplet configuration or “modified” FODOTriplet configuration or “modified” FODO
An structure defined as FDF: An structure defined as FDF:
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
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.
1for
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
Transfer matrices for modified (FDF) Transfer matrices for modified (FDF) FODO cellsFODO cells
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;4.1
)(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:These are the only closed orbits at the limits of stability:
There is no “amplitude” transmitted, beta functions go to infinity, There is no “amplitude” 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]
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
0;2;4.1
)(lim;2
)2(
21
;111
2
2
2121**
rf
rf
rf
rf
rf
LD
LDD
LDLDf
LD
L
Df
ff
D
fff
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.
** 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
Summary: minimizing momentum Summary: minimizing momentum compaction in a FODO cellcompaction in a FODO cell
For a fixed bend/cell, minimizing momentum compaction For a fixed bend/cell, minimizing momentum compaction requires:requires:
Center the dipole field at Center the dipole field at minmin = min = minxx,, • MinMinxx (center of vertically-focusing quad length, (center of vertically-focusing quad length, lldd) is always the ) is always the
position of position of minmin in a periodic structure and this positioning, in a periodic structure and this positioning, minimizes momentum compactionminimizes momentum compaction
• As was derived, this location of the dipole field also minimizes As was derived, this location of the dipole field also minimizes dispersion.dispersion.
= {= {min min l l dd}/L}/Lcellcell = = minmin B B (thin lens)(thin lens)
{< ½{< ½maxmax + + min min > > BB }/ L }/ Lcellcell (current (current
In addition, In addition, BB==/N, so /N, so is dependent on the focal length and the is dependent on the focal length and the number of cells; giving a circumference change/phase slip ofnumber of cells; giving a circumference change/phase slip of
The focal length scales with half cell length for a given phase advance, (sin /2 = L / f) so the dependence is linear.
The focal length dependence is critical in discriminating between The focal length dependence is critical in discriminating between optical structures and optimizing the lattice.optical structures and optimizing the lattice.
* # cells in FDF scaled to give * # cells in FDF scaled to give C of FODO using C of FODO using C C f/N; f=1/(kl), using k and l values in table. Gradients f/N; f=1/(kl), using k and l values in table. Gradients were similar so only F lengths were used for scaling. Other parameters remain identical to FDF.were similar so only F lengths were used for scaling. Other parameters remain identical to FDF.
** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect.** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect.
Scaling laws work
Scaling with energy/momentumScaling with energy/momentumlower energy rings*lower energy rings*
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
If you want is If you want is C/N to remain constant (phase-slip per cell)C/N to remain constant (phase-slip per cell) The scaling law is then approximately:The scaling law is then approximately:
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
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
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
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
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
** 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
Conclusions from 6-20 GeV FFAG Conclusions from 6-20 GeV FFAG (Snowmass/KEK studies, 2001):(Snowmass/KEK studies, 2001):
Using single-frequency, but different initial phases for the cavities,Using single-frequency, but different initial phases for the cavities,
andand imposing a conserved output phase spaceimposing a conserved output phase space
one can expect to transmit 1-2 eV-s for 20-40% overvoltages, with the one can expect to transmit 1-2 eV-s for 20-40% overvoltages, with the approximate turn dependence given below:approximate turn dependence given below:
RF freqRF freq # turns# turns
25 MHz25 MHz 40? (extrapolation is approximate) 40? (extrapolation is approximate)
50 MHz50 MHz 20 20
100 MHz100 MHz 10 10
200 MHz200 MHz 5 5
Further studies also indicated that only 100 cells were required to Further studies also indicated that only 100 cells were required to achieve these transmissions; ie more cells do not improve machine achieve these transmissions; ie more cells do not improve machine dynamics. (multiple-frequency beating was investigated, but dynamics. (multiple-frequency beating was investigated, but dismissed because of the bunch train.dismissed because of the bunch train.
Summary: FFAGs and high-frequency rfSummary: FFAGs and high-frequency rfNuFact04: Osaka, JapanNuFact04: Osaka, Japan
Limiting number of turns: Limiting number of turns: CF FODO ~8 @200 MHz due to phase slippageCF FODO ~8 @200 MHz due to phase slippage FDF FODO ~10-15 @200 MHz due to phase slippageFDF FODO ~10-15 @200 MHz due to phase slippage
Rf voltage requirements at 200 MHz: Rf voltage requirements at 200 MHz: ≥≥2 GV/turn, 8 turns, CF FODO or triplet2 GV/turn, 8 turns, CF FODO or triplet ~1-1.5 GV/turn, 10-15 turns, FDF FODO~1-1.5 GV/turn, 10-15 turns, FDF FODO
Improved phase space transmissionImproved phase space transmission Optimal variation of initial cavity phasingOptimal variation of initial cavity phasing
Addition of higher harmonicsAddition of higher harmonics 22ndnd and 3 and 3rdrd improve area and linearity of transmitted phase space improve area and linearity of transmitted phase space
Lattice and rf work is concludingLattice and rf work is concluding Detailed simulation and magnet designDetailed simulation and magnet design Electron prototype of a nonscaling FFAG is now appropriate Electron prototype of a nonscaling FFAG is now appropriate
