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Considerations for Nonlinear Beam Dynamics in NSLS-II lattice design

Weiming Guo

05/26/08

Acknowledgement:

J. Bengtsson S. Kramer S. Krinsky B. Nash

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Outline

Method:

Simulation using Elegant

1. Introduction to the linear lattice

2. Nonlinear requirements: injection and Toschek scattering

3. Minimum needed sextupole families

4. Required physical aperture

5. Effects of the multipole field errors

Subjects will not be covered:

On-going optimization of the nonlinear lattice;

Misalignment errors; choice of the working point;

and effects of the damping wigglers and IDs

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Main Design Parameters

Beam Energy 3 GeV

Circumference 792 m

Circulating current 500 mA

Total number of buckets 1320

Lifetime 3 hours

Electron beam stability 10% beam size

Top-off injection current stability <1%

ID straights for undulators >21

Straight length 9.3/6.6 m

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Magnets and lattice

Quad. Family: 8, 10 magnets per cell

Sext. Family: 9, 10 magnets per cell

Chromaticity per cell: -3.4/-1.4

0

0.5

1

1.5

2

2.5

200 400 600 800 1000 1200 1400

Total Radiated Energy [KeV]

Em

itta

nc

e [

nm

]

Natural emittance with 0, 1, 2, 3, 5 and 8 DWs added

Courtesy of S. Kramer

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Linear Lattice design approaches

1. Large bending radius enhances the effect of the damping wigglers

2. Strict double-bend-acromat

sin'

)cos1(,25,4.0

x

x

D

DmTB

Two reasons NSLS-II is sensitive to the residual dispersion:

• 1nm emittance. For βx =2 m, ηx=4.5 cm, βxεx~(ηxσδ)2

• Damping wiggler has quantumn excitation effect

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Nonlinear Design Goal

1. Sufficient dynamic aperture (>11mm) for injection

On momentum particle

Sep

tum

Bum

ped

stor

ed b

eam

Inje

cted

bea

m

(σx=

0.39

mm

, σ´ x

=77

μra

d)

3+1

mm

3.5

mm

1+1.

2 m

m

(σx=

0.15

mm

)

2. Sufficient dynamic aperture to keep Touschek scattered particles with δ= ±2.5%

Off momentum particle

Minimum required dynamic aperture

δ

Courtesy of I. Pinayev

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Source of the nonlinearity

yx

yx

J ,

,

d

dh

d

dh

Dh

h

h

y

x

y

x

00201

20001

)2(10002

)1(00111

)1(11001

Frist order chromatic terms (5)

yx

yx

x

x

x

h

h

h

h

h

2

2

3

10200

10020

10110

30000

21000

Frist order geometric terms (5)

Amplitude tune dependence

Second order chromaticity

}][{8

1

2

1

}][{8

1

2

1

)1(21

)2(2

)1()2(

)1(21

)2(2

)1()2(

d

dDKKDKds

d

dDKKDKds

yyyy

xxxx

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Number of chromatic sextupole families

)2||2cos()()2sin(2

)2||2cos()()2sin(2

)|cos(|)sin(2

)|cos(|)2

()sin(2

)1(21

)1(21

)1()1(1

)1(

)1()1(21

)1()2(

yyy

xxx

x

DKKdsd

d

DKKdsd

d

cDdsKD

DDK

KdsDD

yxyx DdsK ,)1(

2,)1(

Conclusion:

Because the betatron phase advance in the dispersive region is small (x: ~ 10o, y~ 20o),

the five chromatic terms have only two degree of freedom. 7~8 sextupole families

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Stay-clear aperture

38x12.5 mm

)()()()(

00.... s

ssxsx ococ

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Dynamic Aperture with Multipole Error

Magnet Type

Magnet Aperture(mm)

Multipole Order

Relative Strength (x10-4)

Quad 66 6 1

Quad 66 10 3

Sext 68 9 1

Sext 68 15 2

DA collapse at negative momentum

The Feb.-08 multipole components at 25 mm

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Effects of Multipoles

Conclusion:

introducing systematic

multipole errors doesn’t add

resonance lines.

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Amplitude Tune Dependence

Conclusion: Multipole terms changes the tunes at large

amplitude dramatically, and makes the stable area smaller.

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Why Negative Momentum?

nnxx

n

noc

n

n

n

noc

noc

oc

yM

JxC

xxCxxx

xxx

yxxx

B

BH

cos)2(

)(

...)45(!10

11

2/10

0

10..10

10

0

10..10

10..

10

..

28109

9

,cos)2(!10

11 15

1

2210..

2109

9

1010

kx

k

kkkoc

ky

x

JxkCx

B

B

HJ

,cos)2(

!10

11

15

1

2210..

210

9

9

1010

kx

k

kkkoc

k

y

x

JxkC

x

B

B

HJ

xxx

yxxx

B

BH

oc

y

..

28109

9

10 ...),45(!10

11

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Minimizing 2nd Order Dispersion

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Tune Scan

Conclusion: moving the tune doesn’t solve

this problem.

Delta = -3.0%

Delta = 3.0%

Delta = 0.0% Delta = -3.0%

Area of DA

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Squeeze the Tune Excursion

Tunes moved, sextupoles reoptimized

CDR lattice with multipole errorsRed: Jan4th lattice

Black: re-tuned lattice

The re-tuned lattice

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New Specifications

Magnet Type Magnet Aperture (mm)

Multipole Order Relative Strength (x10-4)

Quad 90 6 1

Quad 90 10 0.5

Quad 90 14 0.1

Sext 76 9 0.5

Sext 76 15 0.5

Sext 76 21 0.5

Multipole components of high precision magnets (25 mm)

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Summary

•Nonlinear design goal to provide sufficient dynamic aperture for injection and Touschek scattered particles up to δ=2.5%

•There are 8 sextupole families in the lattice, one more might be added.

•The physical aperture is conservatively retained.

•The reduction of the dynamic aperture due to multipole errors is show to be the momentum related tune shift with amplitude.

•Identified the multipole errors in the high dispersion region, and required increasing the pole radius.

•Further optimization of the dynamic aperture is proceeding.