1 EMMA Tracking Studies Shinji Machida ASTeC/CCLRC/RAL 4 January, 2007 ffag/machida_20070104.ppt...

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