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3 Machine specifications (1) Misalignment –Introduce additional bend for both H and V and create orbit distortion. – What is the alignment tolerance? Gradient error –Introduce additional focusing for both H and V and create optical mismatch. –Introduce additional bend for H, not for V, and create orbit distortion. – What is the gradient tolerance? – What is the specification of trim coil? Lumped (not every cell) rf cavities –Introduce orbit distortion.
Citation preview
1
EMMA Tracking Studies
Shinji MachidaASTeC/CCLRC/RAL
4 January, 2007http://hadron.kek.jp/~machida/doc/nufact/
ffag/machida_20070104.ppt & pdf
2
Contents • Tracking study as an independent check if beam opti
cs is what we expect.
• Tracking study to determine machine specifications.• Aperture • Magnet error tolerance• Trim coil of Quadrupole
– Orbit distortion due to misalignment– Orbit distortion due to gradient error (H only)– Orbit distortion due to lumped rf cavities (H only)– Optical mismatch due to gradient error
• Tracking study to predict beam behavior and prepare diagnostics– With gradient error– With misalignment– Strategy of “resonance” crossing study
3
Machine specifications (1)
• Misalignment– Introduce additional bend for both H and V and create orbit d
istortion.– What is the alignment tolerance?
• Gradient error– Introduce additional focusing for both H and V and create op
tical mismatch.– Introduce additional bend for H, not for V, and create orbit di
stortion.– What is the gradient tolerance?– What is the specification of trim coil?
• Lumped (not every cell) rf cavities– Introduce orbit distortion.
4
Machine specifications (2)
• Previous study results (http://hadron.kek.jp/FFAG/FFAG04_HP/)– Alignment of 30 m (r.m.s Gaussian) by Keil and Sessler.– Alignment of 50 m (100% uniform) and
gradient of 0.5% (100% uniform) by Machida.
• Keil and Sessler showed the longitudinal acceptance limit imposed by misalignment.
• I showed amplitude growth of single particle.
5
Orbit distortion due to misalignment (1)
• Distortion pattern changes during acceleration because phase advance is not constant.
• Magnitude of distortion is a function of acceleration rate.
• Orbit moves in horizontal plane without misalignment.
• Simple formula does not work to estimate orbit distortion.
€
y = 1sinπQ
β ...
6
8x10-3
6
4
2
04x10-33210
rms of horizontal cod [m]
Orbit distortion due to misalignment (2)• Misalignment: =0.050 mm, max=0.100 mm (2 Gaussian
– Reduction of aperture of 3 mm most likely and 5 mm in the worst case in horizontal.
– Reduction of aperture of 3 mm most likely and 6 mm in the worst case in vertical.
horizontal50 seeds 8x10-3
6
4
2
04x10-33210
rms of vertical cod [m]
vertical50 seeds
7
Orbit distortion due to misalignment (3)• Misalignment: =0.050 mm, max=0.100 mm (2, Gaussian
– Use 50 different seeds to see statistics.– 6 examples in vertical are shown.
-4x10-3-2024
20x103151050s [cm]
-4x10-3-2024
20x103151050s [cm]
-4x10-3-2024
20x103151050s [cm]
-4x10-3-2024
20x103151050s [cm]
-4x10-3-2024
20x103151050s [cm]
-4x10-3-2024
20x103151050s [cm]
8
Orbit distortion due to misalignment (4)
• The slower acceleration gives larger orbit distortion as expected.
10x10-3
8
6
4
2
05x10-343210
rms of horizontal od [m]
12 turn
24 turn
60 turn
9
Orbit distortion due to gradient error (1)• Gradient error: =0.1%, max=0.2% (2, Gaussian
– Reduction of aperture of 2 mm most likely and 3.5 mm in the worst case.
horizontal50 seeds
4x10-3
3
2
1
02.0x10-31.51.00.50.0
rms of horizontal od [m]
10
Orbit distortion due to lumped rf cavities (1)• Lumped rf cavities means not every cell has a cavity.• More than 14 rf cavities, the maximum distortion is le
ss than 1 mm.• Resonance structure with 7 rf cavities.
10x10-3
8
6
4
2
0403020100
number of rf cavities
constant E gain out of backet
11
Orbit distortion due to lumped rf cavities (2)
• Distortion with 7 rf cavities occurs later in a cycle.
-4x10-3-2024
100x103806040200element number
21rf
-4x10-3-2024
100x103806040200element number
14rf
-4x10-3-2024
100x103806040200element number
7rf
-4x10-3-2024
100x103806040200element number
6rf
-4x10-3-2024
100x103806040200element number
3rf
-4x10-3-2024
100x103806040200element number
1rf
12
Orbit distortion due to lumped rf cavities (3)
• Distortion with 7 rf cavities (every 6 cells) is likely attributed to synchro-beta coupling:
(6Qx)-Qs=1, where Qx and Qs are cell tunes and Qs=0.• Qx becomes 1/6 later in a cycle.• Dispersion at rf also becomes larger later in a cycle.
0.5
0.4
0.3
0.2
0.1
0.00.50.40.30.20.10.0
Qx
10.5 MeV/c
20.5 MeV/c
15.5 MeV/c
12.5
14.516.5
18.5-4x10-3
-2024
100x103806040200element number
7rf
13
Orbit distortion due to lumped rf cavities (4)
• Failure of some rf cavities out of 14 rf excites the coupling.
-4x10-3-2024
100x103806040200element number
14rf (1 off)
-4x10-3-2024
100x103806040200element number
14rf (1 and 2 off)
-4x10-3-2024
100x103806040200element number
14rf (1 and 3 off)
-4x10-3-2024
100x103806040200element number
14rf (1 and 4 off)
-4x10-3-2024
100x103806040200element number
14rf (1 and 5 off)
#1 fail
#1 and 3 fail
#1 and 5 fail#1 and 4 fail
#1 and 2 fail
14
Orbit distortion due to lumped rf cavities (5)
• If there are rf cavities every 3 cells and a few cavities are failed,
8x10-3
6
4
2
076543210
number of failed cavities
max rms
This is the worst case when cavities fail alternatively.
15
Orbit distortion (summary1)
• For example,
source misalignment gradient error rf failure
magnitude 0.050 mm () 0.10% () 1 rf cavity
horizontalmax
(average)5 mm(3 mm)
3.5 mm(2.5 mm)
1 mm(0.5 mm)
verticalmax
(average)6 mm(3 mm)
0 mm 0 mm
16
Orbit distortion (summary2) • Vertical design aperture is 11mm , where y=0.8 m, and yun=0.15 mm.• Orbit distortion of 3 mm reduces the acceptance by (8/11)2 ~0.5
• Either correct the orbit distortion or enlarge aperture.– How we can correct the orbit? Phase advance is not constant !– Beam based alignment?
• Consider again if 3 mm acceptance is necessary.– What is the rationality behind?
• Otherwise, simply add margin: 10 mm in H and 6 mm in V.
€
= yεy
17
Beam loss due to gradient error (1)
• Gradient error: =0.1%, max=0.2% (2, Gaussian is necessary to suppress beam loss for nominal acceleration.
• For slow acceleration, tolerance should be tighter.
100806040200
5 6 7 8 90.1
2 3 4 5
gradient error [%]
12 turns100806040200
5 6 7 8 90.1
2 3 4 5
gradient error [%]
24 turns
100806040200
5 6 7 8 90.1
2 3 4 5
gradient error [%]
60 turns
100806040200
5 6 7 8 90.1
2 3 4 5
gradient error [%]
120 turns
18
Beam loss due to gradient error (2)
• Gradient errors introduce optical mismatch and a beam starts tumbling.
• When the beam size effectively large, even if emittance does not change, some particles are lost.
• When single particle emittance (either H or V),
becomes more than 1.5 times, the particle is lost.
€
x = 1β x
x 2 + β x x'+α x x( )2
[ ]
19
Beam behavior and diagnostics (1)
• Gradient errors induce optical mismatch and a beam starts tumbling.
• Within 10 turns, it does not smear out. • Although emittance is constant, beam size oscillates.
-2x10-3
0
2
-2x10-3-1 0 1 2x
0 turn -2x10-3
0
2
-2x10-3 0 2x
3 turn
-2x10-3
0
2
-2x10-3 0 2x
6 turn -2x10-3
0
2
-2x10-3 0 2x
9 turn
-2x10-3
0
2
-2x10-3 0 2y
0 turn -2x10-3
0
2
-2x10-3 0 2y
3 turn
-2x10-3
0
2
-2x10-3 0 2y
6 turn -2x10-3
0
2
-2x10-3 0 2y
9 turn
0 turn
6 turn
3 turn
9 turn
20
Beam behavior and diagnostics (2)
-1.0x10 -3-0.50.00.51.0
-1.0x10 -30.0 1.0x
-1.0x10 -3-0.50.00.51.0
-1.0x10 -30.0 1.0x
• Another result of a muon 10 to 20 GeV ring with gradient errors.
• When emittance is small, there is only tumbling. When emittance is large, nonlinear distortion appears.
-1.0x10 -3-0.50.00.51.0
-1.0x10 -30.0 1.0x
-1.0x10 -3-0.50.00.51.0
-1.0x10 -30.0 1.0x
-1.0x10 -3-0.50.00.51.0
-1.0x10 -30.0 1.0x
-10x10 -3-505
10
-10x10 -3 0 10x
-10x10 -3-505
10
-10x10 -3 0 10x
-10x10 -3-505
10
-10x10 -3 0 10x
-10x10 -3-505
10
-10x10 -3 0 10x
-10x10 -3-505
10
-10x10 -3 0 10x
-100x10 -3-50
050
100
-100x10 -3 0 100x
-100x10 -3-50
050
100
-100x10 -3 0 100x
-100x10 -3-50
050
100
-100x10 -3 0 100x
-100x10 -3-50
050
100
-100x10 -3 0 100x
-100x10 -3-50
050
100
-100x10 -3 0 100x
0 turn 4 turn 8 turn 12 turn 16 turn
0.003 mm
0.3 mm
30 mm
21
Beam behavior and diagnostics (3)• Another result of a muon 10 to 20 GeV ring with gradi
ent errors.• Beam beta defines as does not have any
clear correlation with total tune.
12
8
4
0
35 34 33 32 31 30
tune
0.003 rad (1
12
8
4
0
30 29 28 27 26 25
tun
0.003 rad (2
12
8
4
0
25 24 23 22 21 20
tun
0.003 rad (3
12
8
4
0
20 19 18 17 16 15
tun
0.003 rad (4
€
=xi
2
ε x, rms
22
Beam behavior and diagnostics (4)
• Alignment errors introduce orbit distortion.• On the frame of distorted orbit, beam shape does not
change.
-2x10-3
0
2
-2x10-3 0 2x
0 turn -2x10-3
0
2
-2x10-3 0 2x
3 turn
-2x10-3
0
2
-2x10-3 0 2x
6 turn -2x10-3
0
2
-2x10-3 0 2x
9 turn
-2x10-3
0
2
-2x10-3 0 2y
0 turn -2x10-3
0
2
-2x10-3 0 2y
3 turn
-2x10-3
0
2
-2x10-3 0 2y
6 turn -2x10-3
0
2
-2x10-3 0 2y
9 turn
0 turn
6 turn
3 turn
9 turn
23
Beam behavior and diagnostics (5)
• When acceleration is fast, there is no resonance behavior in a conventional sense.
• What we would see is – orbit distortion– beam tumbling– a bit deformation due to nonlinearity
• Measuring emittance is not a right way to study “resonance” crossing.
• Because of tumbling and unknown beta function, beam size measurement does not give emittance.
24
Beam behavior and diagnostics (6)
• Possible alternative is– To survey initial phase space with a pencil beam.– When acceptance is 3 mm, Sqrt[3/0.01]xSqrt[3/0.01]=17x17 gri
d points in phase space can be surveyed.– Measure beam loss at accurate timing.– Make sure a pencil beam remains as a pencil.
• Diagnostics – 1 pass beam position monitor at every cell for both H and V.– Beam current or beam loss monitor (at every cell).– Beam profile monitor or slits at extraction line to make sure small
pencil beam size.– Collimator (at several place) to define or reduce machine apertur
e.
25
Summary
• In addition to misalignment, lumped rf cavities introduce orbit distortion in horizontal plane.
• We need a margin of a few mm in aperture to provide a room for orbit distortion and beam tumbling.
• Magnitude of margin depends on the alignment and gradient tolerance and rf cavity configuration. For example, 6 mm in vertical if is 0.05 mm.
• Trim coil to make 1% focusing error is enough to excite “controlled” error.
• Identification of beam loss at accurate timing should be more emphasized than emittance or beam size measurement to study “resonance” crossing.
26
Backup slides
•
27
Beam dynamics parameters (1)lattice function
• function at different momentum.
1.00.80.60.40.20.0
403020100s [cm]
horizontal vertical
1.00.80.60.40.20.0
403020100s [cm]
horizontal vertical
1.00.80.60.40.20.0
403020100s [cm]
horizontal vertical
10.5 MeV/c
15.5 MeV/c
20.5 MeV/c
28
Beam dynamics parameters (2)tune and time of flight
• Nominal operation
55.50x10-3
55.45
55.40
55.35
55.3020x10-31816141210
momentum [GeV/c]
0.5
0.4
0.3
0.2
0.1
0.00.50.40.30.20.10.0
Qx
10.5 MeV/c
20.5 MeV/c
15.5 MeV/c
12.5
14.516.5
18.5
29
Beam dynamics parameters (3)difference in tune
• Relatively large discrepancy around injection momentum.
0.5
0.4
0.3
0.2
0.1
0.00.50.40.30.20.10.0
Qx
10.5 MeV/c
20.5 MeV/c
Enge, g=0.02 Berg, multipole
30
Beam dynamics parameters (4)emittance evolution
• Initial emittance 3 mm, waterbag• Acceleration with constant energy gain
12 turns (nominal) 120 turns
1.20
1.10
1.00
0.9020x10-31816141210
kinetic energy [GeV]
horizontal vertical
3.0
2.0
1.0
0.020x10-31816141210
kinetic energy [GeV]
horizontal vertical
31
Beam dynamics parameters (5)end field modeling
• Tracking code adopts “Multipole symmetry” with Enge fall off.
• Keep the leading order and truncate the rest. becomes
that is same as Berg’s assumption except the form of G2,0(z).
€
P2 r,θ,z( ) = r2 sin2θ2
G2,0 z( ) + G2,2 z( )r2 + ⋅⋅⋅[ ]
€
Bθ = rcos2θ ⋅G2,0(z)
€
Bθ
32
Beam dynamics parameters (6)magnet (thin lens) and trajectory of
8th December lattice
2 cell LS enlarged
33
Beam dynamics parameters (7)x and x’ at the center of LS, QF, and QD
• Based on the 13th December lattice with hard edge.P [MeV/c] x [mm] x’ [mrad] tof [ns] Qx Qy
10.5 LS -3.098 -31.190 55.4744 0.35606 0.34111
QF -15.437 +23.330
QD -31.006 +16.090
15.5 LS +0.180 +0.230 55.3353 0.21601 0.18997
QF -8.723 +0.404
QD -31.253 -1.617
20.5 LS +10.322 +27.614 55.4610 0.16458 0.12364
QF +4.410 -18.715
QD -24.155 -17.785