Download Scaling FFAGs

Linear Non-scaling FFAGs for Rapid
Acceleration using High-frequency
(≥100 MHz) RF
Cast of Characters in the U.S./Canada:
C. Johnstone, S. Berg, M. Craddock
S. Koscielniak, B. Palmer, D. Trbojevic
July 26-July 31, 2004
NuFact04
Osaka University,
Osaka, Japan
Rapid Acceleration
In an ultra-fast regime—applicable to
unstable particles—acceleration is completed
in a few to a few tens of turns
Magnetic field cannot be ramped
RF parameters are fixed—no phase/voltage
compensation is feasible
operate at or near the rf crest
Fixed-field lattices have been developed which can
contain up to a factor of 4 change in energy; typical
is a factor of 2-3
There are three main types of fixed field lattices
under development:
Conventional Recirculating Linear Accelerators (RLAs)
Dogbone RLAs
Scaling FFAG (Fixed Field Alternating Gradient)
Linear, nonscaling FFAG
Current Baseline: Recirculating Linacs
A Recirculating Linac Accelerator (RLA) consists of two opposing linacs
connected by separate, fixed-field arcs for each acceleration turn
In Muon Acceleration for a Neutrino Factory:

The RLAs only support ONLY 4 acceleration turns
 due to the passive switchyard which must switch beam into the
appropriate arc on each acceleration turn and the large
momentum spreads and beam sizes involved.

2-3 GeV of rf is required per turn (NOT DISTRIBUTED)
 Again to enable beam separation and switching to separate
arcs
Advantage of the RLA
Beam arrival time or M56 matching to the rf is independently
controlled in each return arc, no rf gymnastics are involved; I.e.
single-frequency, high-Q rf system is used.
RLAs comprise about 1/3 the cost of the U.S. Neutrino Factory
Dogbone RLAs*



*First proposed; D. Summers, Publication: Pac01, S. Berg and C. Johnstone
Optics condition to close off-momentum orbits: match dispersion to all
significant orders
Dispersion relations for muon lattices:
1) completely periodic scaling FFAG (radial sector)
(p) = 0
2) completely periodic FODO optics (no change in dipole
strength/period: linear nonscaling FFAG) :
(p) = 0 + 1 
3) non-periodic optics; nonlinear optics
(p) = 0 + 1  + 2 2 + 3 3 + . . .
- 0 and ’ (d/ds) can be matched using linear optics (dipoles/phase
advance=dipoles/quad strength)
- 1 can be matched by not violating periodicity or canceled using sextupoles
( scaling machine with individual correctors rather than field scaling;
sextupole is the first and largest nonlinearity in a scaling FFAG)
Dogbone RLAs, continued

Chromatic aberrations (nonlinear sextupole distortions of phase
space) are canceled only at cell phase advances of





60
90
180 unstable
If you’re clever you can ~ cancel these distortions to 2nd order in two
of the arcs (the momentum spread is so large, the phase advance
changes rapidly and the chromatic cancellation deteriorates ;
the third arc has no chromatic cancellation and there is a sextupoledistorted phase space
Dogbone RLA: concerns

10% dp/p can be difficult as high-order terms become important; DA
declines



I’ve not seen 20% dp/p acceptance with any reasonable DA. (Trbojevic
has some generated sextupole-dominated lattices)
Nonlinear phase space may be mis-matched to the elliptical, linear
phase space of downstream accelerators; emittance may blow up in
these machines or the storage ring.
Dogbone RLA ~ low energy RLA, which we know is difficult; further the
dogbone Switchyard contains reverse bends relative to the arcs, the
RLA does not; nonlinear matching  dipole bend strength.

Strong sextupoles will decrease DA and longitudinal acceptance; ring
cooling will be needed and will eliminate any cost savings.

A dogbone upstream will sacrifice much of the advantages of the FFAGs
which do not require longitudinal cooling.

We are still looking at the 2.5-5 GeV FFAG; corrections to cost profiling
and normal conducting, pulsed-magnet options
Mulit-GeV FFAGs: Motivation

Ionization cooling is based on acceleration
- (deacceleration of all momenum components then longitudinal
reacceleration)
THERE is a STRONG argument to let the accelerator do the bulk of the
LONGITUDINAL AND TRANSVERSE COOLING (adiabatic cooling).
The storage ring can accept ~  4% p/p @20 GeV
If acceleration is completely linear, so that absolute momentum
spread is preserved, @ ~400 MeV
p/p =  200%
implying no longitudinal cooling.
(Upstream Linear channels for TRANSVERSE Cooling currently accept a maximum
of 22% for the solenoidal sFOFO and -22% to +50% for quadrupoles)
.
The Linac/RLA has been the showstopper in this argument
Mulit-GeV FFAGs for a Neutrino Factory or Muon Collider

Lattices have been developed which, practically, support up to a
factor of 4 change in energy, or


almost unlimited momentum-spread acceptance, which has
immediate consequences on the degree of ionisation cooling
required
Practical, technical considerations (magnet apertures, mainly, and rf
voltage) have resulted in a chain of FFAGs with a factor of 2 change
in energy
Currently proposal,
U.S. scenario
2.5 -5 GeV
5-10 GeV
10-20 GeV
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*
Freedom to cross betatron resonances:


optics can change slowly with energy
allows lattice to be constructed from linear magnetic
elements (dipoles and quadrupoles only)
This is the basic concept for a linear non-scaling
FFAG
* In muon machines acceleration is completed in submillisecond or
millesecond timescales
Linear non-scaling FFAGs:
Transverse acceptance:


“unlimited” due to linear magnetic elements
Large horizontal magnet aperture


General characteristic of fixed-field acceleration
Orbit changes as a function of momentum: beam travels
from the inside of the ring to the outside
Momentum Acceptance:

FODO optics:



Large range in momentum acceptance:
defined by lower and upper limits of stability
Limits depend on FODO cell parameters
Triplet, doublet (dual-plane focusing) optics:

Too achromatic; small momentum acceptance to achieve
horizontal+vertical foci.
Phase advance in a linear non-scaling FFAG
Stable range as a function of momentum
 Lower limit:


Given simply and approximately by thin-lens
equations for FODO optics
Upper limit:


No upper limit in thin-lens approximation
Have to use thick lens model
In the thin-lens approximation, the phase advance, , is given by
sin

2
 1/  ;
where   f / L (thin lens)
(1)
with f being the focal length of ½ quadrupole and L the length of a half cell
from quadrupole center to center
0.3Bl
sin 
L;
2
p

0.3Bl
since k 
p
(2)
In equation (3), B’ is the quadrupole gradient in T/m and p is the momentum
in GeV/c. Selecting  = 90 at p0, the reference momentum implies the
following:
1
p0

sin  2 ,
2
p
lower limit of stability,
1
p
p0 .
2
(3)
Differentiating the above equation gives the dependence of phase advance
on momentum
p0
1

cos d  
dp
2
2
2
2p
or
2 p0
d

dp
p 2 1  p02 / 2 p 2
(4)
(5)
There is a low-momentum cut-off, but at large p, the phase advance varies
more and more slowly, as 1/p2, and there is no effective high-momentum
cut-off in the thin-lens approximation.
A high-momentum stability limit is observed in the thick lens representation
Beta functions in a linear non-scaling FFAG
Momentum dependence described by thin-lens
equations
 Magnitude and variation:


Lower limit on momentum (injection) is raised away
from lower limit of stability
Minimized using ultra-short cells
Using thin-lens solutions, the peak beta function for a FODO cell is given by:
 max
 (  1)
L 2
(  1)1/ 2
d max
( 2    1)
d
L
0
3/ 2
1/ 2
dp
(  1) (  1) dp
(6)
for a minimum
(7)
In the above equation (7), (2 -  - 1) can only be set to 0 locally (at ~76),
but this does not guarantee stability in the beta function over a large
range in momentum. The only approach that minimizes dmax/dp 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 Acceleration



A pathology of fixed-field acceleration in recirculating-beam
accelerators (for single, not multiple arcs) is that the particle
beam transits the radial aperture
The orbit change is significant and leads to non-isochronism, or
a lack of synchronism with the accelerating rf
The result is an unavoidable phase slippage of the beam
particles relative to the rf waveform and eventual loss of net
acceleration with


The lattice completely determining the change in circulation time
(for ultra relativistic particles)
The rf frequency determining the phase slippage which
accumulates on a per turn basis:
  rf t per turn
Moderating 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 lattices
for linear nonscaling FFAGs

Phase slippage of reference orbits can be described as a change in
circumference for relativistic particles:
C
p
  ring
  ring
C
p
1 
 ring   ds
C 

Minimizing the dispersion function in regions of dipole bend fields
controls phase slip for a given net bend/cell.
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
approach is similar, hence the references in the literature
Minimum Momentum-compaction lattices
for nonscaling FFAGs

Linear nonscaling FFAG lattices are completely periodic*
C is N Lcell (cell ), where N is the number of cells.
Since

N  Lcell = C,
ring = cell
The optimum lattices are strictly FODO-based, with two candidates:

Combined Function (CF) FODO
• Horizontally-focusing quadrupole, and combined function horizontallydefocussing magnet
• The rf drift is provided between the quadrupole and CF element

Modified FODO – quadrupole triplet
• The horizontally-focusing quadrupole is split and the rf drift is inserted between
the two halves.
• The magnet spacing between the quadrupole and the CF magnet is much
reduced.

All optical units have reflective symmetry, implying
ring = cell = 1/2 cell
* Special insertions for rf, extraction, injection, etc. have failed
Triplet configuration or “modified” FODO

An structure defined as FDF:
[1/2rfdrift-QF—short drift—CF-short drift-QF-1/2rf drift]
produces significantly reduced momentum compaction and therefore
phase slip relative to the separated and CF FODO cells.
where equivalent is defined in terms of
rf drift length,
identical bend angle per cell,
intermagnet spacing
phase advance at injection
maximum poletip field allowed.

(2 m)
(0.5 m)
(0.72 , both planes)
( 7T )
DFD arrangement does not perform as the FDF
Linear Dispersion in thin-lens FODO optics

Dispersion can be expressed in standard thin-lens matrix formalism.
 x      
 x'   '     '  M
     
      1 

0 
 ' for   1
 0
 1 
At the symmetry points of the FODO cell the slope of optical
parameters is zero, and correspond to points of maximum and
minimum dispersion. For horizontal dispersion, the center of the
vertically-focusing element is a minimum and horizontally-focusing
element is a maximum.
 min 


1 / 2 FODO
0

M


 1 


 max 


0


 1 


Thin lens matrix solutions for different dipole
options in a FODO

The transfer matrix for a dipole field centered in the drift
between focusing elements: 1/2F-drift-1/2D is:
 1

M1/ 2 FODO   1
f
 0

0 0 1 L / 2 0 1 0 0  1 L / 2 0  1


1 0 0
1
0 0 1 B  0
1
0   1
f


0
1 0 0 1  0
0
1  0
0 1 0
1
 L

1

L
L

B


f
2

1L 
L
L
 
1
B (1 
)
2
f
f
f
2


0
1
 0




For a dipole field centered
in the verticallyM1CF/ 2 FODO 
focusing element:
1  L
L
f

 L 2 1  L
f
f

 0
0

0 0

1 0
0 1
0

B 

1

Dispersion and dipole location



Dispersion solution for conventional FODO
max
f2  1L 

 B 1 
L
 2 f 
 min
f2 
1 L


B 1 
L
2 f 

Dispersion solution for the dipole field located in the verticallyfocusing element—clearly reduced

2
max
CF
f
 B
L
min
CF
f2 
L

B 1  
L
f

Transfer matrices for modified (FDF)
FODO cells

For an rf drift inserted at the center of the horizontally-focusing
quadrupole:
M
1 / 2 FDF
CF
1  D
D
0  1 Lrf
f

1

2
1
D



 
1
B 0
1
f2
f*


0
 0
0
1  0



 D
1  f 1

1
  *

f
 0



0
0

1

Where D, the distance between
quadrupole centers, Lrf/2
replaces the half-cell length

1  D   D
0


2
f1 

Lrf  1   D 
   1

 
2  f *  
f 2  B 
0
1


Lrf
- Note that the half cell contains only half the rf drift, hence the added drift
matrix is Lrf/2, rather than the half-cell length as in the FODO cell case.
Dispersion function for modified FODO;
triplet quadrupole configuration

The combined focal length, f*, is the general result for a doublet
quadrupole lens system.
max
 FDF
 B f *
min
max
 FDF
 1  D CF
 B f * 1  D 
f1 
f1 


where
1
1 1
D



f*
f1 f 2 f 1 f 2
 With the rf drift placed at the center of the horizontally focusing
element, the differences between them and from the FODO cell are
not immediately obvious we unless we explore the possible values
for f1 and f2.
Limit of stability


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
comparison between FODO cell configurations.
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
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,
conversely for the y plane
0
1 / 2 FDF
 x'  M CF
 
Lrf  D 
 D

1

1


D



  x
f1
2
f1 


Lrf  1   D  0
  1
   1

2  f *  
f 2 
 f *

 y
1 / 2 FDF
 0   M CF
 
Lrf  D 
 D

1
D 


1  f1
2
f1 


Lrf  1   D 
 1
   1

2  f **  
f 2 
 f **

Lrf


2
f 2  D 1 
 ( D  Lrf
2

Lrf  D (lim L  )
1
1 1
D




;
f **
f1 f 2 f 1 f 2
1
1 1
D



f*
f1 f 2 f 1 f 2
f 2  2 D;
 1.4 D; D 
f1  D
 D;
Lrf
Lrf  0
2
0
 y '
 



)

Closed orbit in the limit of stability

These are the only closed orbits at the limits of stability:
[y,0]
[x,0]
[0,y’]

[0,x’]
There is no “amplitude” transmitted, beta functions go to infinity,
0, phase space is a line.
y’
y’
y
y
Solutions for the limit of stability


For CF or separated FODO cells:
In the modified FDF FODO:
0
1 / 2 FDF
 x'  M CF
 
Lrf  D 
 D

1

1


D



  x
f1
2
f1 

  



1
1
L
 
 rf  *   1  D  0
*
2f  
f 2 
 f

f1  D
and
1
1 1
D



,
*
f
f1 f 2 f 1 f 2
so f *  D
f1  f 2  Lrf
 y
1 / 2 FDF
 0   M CF
 
Lrf  D 
 D

1

1


D




f1
2
f1 




1
1
L



 rf  **   1  D 
**
2f  
f 2 
 f

0
 y '
 
1
1 1
D




;
**
f
f1 f 2 f 1 f 2
Lrf




2
f 2  D 1 

L
rf
 (D 
)
2 

f 2  2 D;
Lrf  D (lim L  )
 1.4 D; D 
 D;
Lrf
Lrf  0
2
Final Comparison, CF vs. modified FODO

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 lengths we use in actual designs).

max
CF
f2
 B  LB  B1.5D
L
2

f
L
min
CF 
B 1  
L
f

0
FODO

max
 FDF
 B D
min
 FDF
0
FDF
At this point, one invokes scaling in focal length and bend
angle to generalize conclusions over the entire momentum
range in the thin-lens approximation.
10-20 GeV “Nonscaling” FFAGs: Examples
FDF-triplet
Circumference
607m
#cells
110
Rf drift
2m
cell length
5.521m
D-bend length
1.89m
F-bend length
0.315m (2!)
F-D spacing
0.5 m
Central energy**
20 GeV
F gradient
60 T/m
D gradient
20 T/m
F strength
0.99 m-2
D strength
0.300 m-2
Bend-field (central energy)
2.0 T
Orbit swing
Low
-7.7
High
0
C (pathlength)
16.6
xmax/ymax (10 GeV)
6.5/13.8
(injection straight)
6.5
Tune, (x,y)
Inject / Extract
0.36 / 0.36 (130)
Extract
0.18 / 0.13 (~56)
FODO
616m
108
2m
5.704m
1.314m
0.390
2m
18.65 GeV
60 T/m
18 T/m
0.9486 m-2
0.300 m-2
2.7 T
-9.5
3.3
27.3
14.4/11.44
5.8
Note pathlength difference
0.36 / 0.36 (130)
0.14 / 0.16 (~54)
** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in
effect.
Summary: minimizing momentum
compaction in a FODO cell

For a fixed bend/cell, minimizing momentum compaction
requires:

Strong horizontal focusing, short focal lengths
• Horizontally focusing quadrupole fields focus horizontal dispersion

Center the dipole field at min = minx,
• Minx (center of vertically-focusing quad length, ld) is always the
position of min in a periodic structure and this positioning,
minimizes momentum compaction
• As was derived, this location of the dipole field also minimizes
dispersion.
 = {min l d}/Lcell = min B
(thin lens)
  {< ½max + min > B }/ Lcell
(current lattices; long
magnets)
Scaling laws:

phase slip/circumference
change
In addition, B=/N, so  is dependent on the focal length and the
number of cells; giving a circumference change/phase slip of
f2 2
f
C  2 N  L1/ 2 cell1/ 2 cell  N
B 
L
N
since 1/ 2 cell 
1
 
L1/ 2 cell

l1/ 2 dipole 
 
B
L1/ 2 cell
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
optical structures and optimizing the lattice.
10-20 GeV “Nonscaling” FFAGs: Examples
FDF
Circumference
607m
#cells
110
Rf drift
2m
cell length
5.521m
D-CF length (l)
1.89m
F-Quad length (l)
0.315m (2!)
F-D spacing
0.5 m
Central energy**
20 GeV
F gradient
60 T/m
D gradient
20 T/m
F strength (k)
0.99 m-2
D strength (k)
0.300 m-2
Bend-field (central energy)
2T
Orbit swing
Inject / Extract
-7.7 / 0 cm
C (pathlength)
16.6 cm
xmax/ymax (10 GeV)
6.5 / 13.8 m
x(injection straight)
6.5
Tune, (x,y)
Inject
0.36 / 0.36 (130)
Extract
0.18 / 0.13 (~56)
FODO
616m
108
2m
5.704m
1.314m
0.390
0.5m
18.65 GeV
60 T/m
18 T/m
0.949 m-2
0.300 m-2
2.7 T
FDF scaled to FODO*
375m
68
-9.8 / 3.8 cm
27.3 cm
14.4 / 11.44 m
5.8
-12.4 / 0 cm
24.9 cm
Scaling laws work
0.36 / 0.36 (130)
0.14 / 0.16 (~54)
* # cells in FDF scaled to give C of FODO using C  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.
** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect.
Scaling with energy/momentum
lower energy rings*

Naively one would hope that circumference would scale with momentum.
However, we know that T or C must be held at a certain value for
successful acceleration. If C is set or scaled relative to the High
Energy Ring (HER), then a Low Energy Ring (LER) would follow:
C  p
u
1
( HER)
2
N H BB
p
u
1
( LER )
2
N L BB
NH
NH
or N L 
rather than N L 
R
R
p u ( HER)
with R  u
, the scaling ratio
p ( LER )
*see FFAG workshop, TRIUMF, April, 2004, C. Johnstone, “Performance Criteria and
Optimization of FFAG lattices for derivations
Scaling Law: Phase-slip/cell


If you want is C/N to remain constant (phase-slip per cell)
The scaling law is then approximately:
NH
NL  3
R

This is somewhat optimistic because you are simply keeping the
number of turns, and T ~ constant.

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
is fairly close for an approximate description
Lattice conclusions: TRIUMF FFAG workshop

Need revised cost profile

Magnet cost scales linearly with magnet aperture, magnet
cost  0 as aperture  0.

No differentiation between 7T multi-turn and 4T single-turn
SC magnets

Better cost profiling to be provided for KEK FFAG workshop,
Oct, 2004.

Large-aperture 7T magnets are prohibitively expensive

Optimum for the two higher energy rings may be 4T

The lower energy ring  higher-energy rings in cost


Large cost for small energy gain (2.5 GeV).
The next jump in magnet cost would be large-aperture normal
conducting and pulsed, 1.5T. (Refer to the large-aperture Fermi
proton driver design for costing
High-frequency (~200 MHz) RF acceleration

In a nonscaling linear FFAG, the orbital pathlength, or T, is
parabolic with energy. At high-frequency, 100 MHz, the
accumulated phase slip is significant after a few turns,
The phase-slip can reverse twice with an implied potential for the
beam’s arrival time to cross the crest three times, given the
appropriate choice of starting phase and frequency
6-20 GeV Nonscaling FFAG
50
Circumference Change (cm)

40
harmonic of rf =
point of phase reversal
30
20
10
0
-10
0
5
10
15
-20
Momentum (GeV)
20
25
Asynchronous Acceleration

The number of phase reversals (points of sychronicity
with the rf) = number of fixed points in the Hamiltonian

Scaling FFAGs with a linear dependence of pathlength
on momentum have 1 fixed point

Linear nonscaling FFAGs with a quadratic pathlength
dependence have 2

The number of fixed points = number of asynchronous
modes of acceleration
Asynchronous Modes of Acceleration
Libration path
 Energy
 Energy
½ Synchrotron osc.
 Time
Single fixed point acceleration:
half synchrotron oscillation
Scaling FFAG
 Time
Two fixed point acceleration:
half synchrotron oscillation +
path between fixed points
Linear nonscaling FFAG
Optimal Longitudinal Dynamics

Optimal choice of rf frequency:
T1 = 3T2

Optimal choice of initial cavity phasing
  Min    for reference particle
 (p) = phase slip/turn relative to rf crest

Optimal initial phasing of individual cavities
Minimizes ()2 of a distribution
Phase space transmission of a FODO nonscaling
FFAG
Optimal frequency, optimal
initial cavity phasing
(tranmission of ~0.5 ev-sec)
Optimal frequency, optimized
initial phasing of individual
cavities : improved linearity
Out put emittance and energy
versus rf voltage for
acceleration completed in
4(black), 5(red), 6(green),
7(blue), 8(cyan), 9(magenta),
10(coral), 11(black), 12(red).
Next: Electron Prototype of a nonscaling
FFAG







Test resonance crossing
Test multiple fixed-point acceleration
Output/input phase space
Stability, operation
Error sensitivity, error propagation
Magnet design, correctors?
Diagnostics
Example 10-20 MeV electron prototype
nonscaling FFAG*
FDF-triplet
FODO
Circumference
13.7m
12.3m
#cells
28
28
cell length
0.49m
0.44m
CF length
7.6cm
6.9cm
F-bend length
1.24 cm (2!)
2 cm
F-D spacing
0.05 m
0.15m
Central energy**
20 MeV
18.5 MeV
F gradient
12 T/m
12 T/m
D gradient
3.9 T/m
3.5 T/m
-2
F strength
175.6 m
194.6 m-2
D strength
57.3 m-2
50.8 m-2
Bend-field (central energy)
0.2 T
0.2 T
Orbit swing
Low
-2.8
-2.5
High
0
0.9
Note pathlength difference
C (pathlength)
5.8
6.8
xmax/ymax (10 GeV)
0.6/1
1/0.8
(injection straight)
0.6
0.9
Tune, (x,y)
Inject / Extract
0.34 / 0.33 (130) 0.36 / 0.36 (130)
Extract
~ 0.18 / 0.13 (~56) ~ 0.14 / 0.16 (~54)
** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in
effect.
Conclusions from 6-20 GeV FFAG
(Snowmass/KEK studies, 2001):


Using single-frequency, but different initial phases for the cavities,
and
imposing a conserved output phase space
one can expect to transmit 1-2 eV-s for 20-40% overvoltages, with
the approximate turn dependence given below:
RF freq
# turns
25 MHz
40? (extrapolation is approximate)
50 MHz
20
100 MHz
10
200 MHz
5

Further studies also indicated that only 100 cells were required to
achieve these transmissions; ie more cells do not improve
machine dynamics. (multiple-frequency beating was investigated,
but dismissed because of the bunch train.
Summary: FFAGs and high-frequency rf
NuFact04: Osaka, Japan
C. Johnstone, et al

Limiting number of turns:



Rf voltage requirements at 200 MHz:



Optimal variation of initial cavity phasing
Addition of higher harmonics


≥2 GV/turn, 8 turns, CF FODO or triplet
~1-1.5 GV/turn, 10-15 turns, FDF FODO
Improved phase space transmission


CF FODO ~8 @200 MHz due to phase slippage
FDF FODO ~10-15 @200 MHz due to phase slippage
2nd and 3rd improve area and linearity of transmitted phase space
Lattice and rf work is concluding


Detailed simulation and magnet design
Electron prototype of a nonscaling FFAG is now appropriate