Study of Slow Extraction Relevant to the FAIR

ProjectMarkus Kirk

Fair-Synchrotrons GroupGSI mbH

Blockseminar, Riezlern, 8th March 2007

Overview

1) Theoretical introduction to resonant extraction

2) Technical aspects of the RF „Knock Out“ method

3) Power supply ripple and its effect on the spill

4) Summary and outlook

Effect of a sextupole perturbation on the linear motion

Equation of motion (horizontal “sheet” beam)

Theory – 3rd integer resonance

2222

2

2

)(2

)( xsKxB

xBxsK

ds

xds

y

22'

2

1yxgB y

where the B-field of the sextupole is

The focussing strength K(s) changes along the lattice. We wish remove this variation by a change of variables…

Taken from G. Rees, CERN school, Vol. II

…normalized coordinates. Viz.,

Theory – 3rd integer resonance

Considering just the nth harmonic of the driving term

Equation of motion can be reduced to SHM and sextupole driving term

xxu )(1

0

sQQ

dsx

s

x

Qp

ds

dxxQ

d

dux

x

x

)cos(222

2

nQuAuQd

udn

0

25 )cos()(n

nn nAsKQ

Change variables again to enable us to find the unstable fixed points,

i.e. (u,p)(r,)

Theory – 3rd integer resonance

)/(tan3/ 1

22

upn

pur

Thus eventually obtaining

3cos83

3sin82

rAQndd

rAddr

n

n

The unstable fixed points (A,B,C in figure) are given by

0 ddddr

Which yield the conditions on r and …

Theory – 3rd integer resonance

… ...,34,32,03

nAQnr 83

x

un xxA

d

dx

8

22

ddxturnsx 6)3(

For optimum condition, 2 , one obtains

which should be greater than the thickness of the septum wires 0.1mm thick. Furthermore, it may be shown that the area of the separatrix is given by

Thereby allowing the jump in x per turn to be calculated

Simulation of SIS-300

22

348resQQ

SA

B

LBS

''23

2

1where

Excitation of the resonance

Aim: with as little power as possible create a strong enough resonance whereby we do not shift away from the natural chromaticity. A suitable configuration is to alternate the sextupole strengths according to

1,1,

,,,22

ijij

ijijij D

DLKLK

which follows from

B

LBD

''

4

1

where subscript variables denotethe ith sextupole in the jth arc

the change in chromaticity due to a sextupole

SIS-300 lattice

~12mSX11SX12

SV

SH

SV

SH

Quadrupole layoutmodified to accomodatefor injection (below)

SIS-300 lattice – beta function in x

SIS-300 lattice – beta function in y

SIS-300 lattice – dispersion in x

Magnet parameters of SIS-300

Dipoles12 short dipoles, 24 longBending radius 67 mMax. rigidity 300 TmPhysical field length 7.76 m (3.88 m short)Pole gap 1 m

QuadrupolesFODO structure40 focusing and 38 defocusing (2 super periods)Max. B’=45 T/mPhysical/effect field length 1.0/0.9 mPole radius 50 mmAlso…4 long F-quads and 4 D-quads (3m each)

Fast quadrupolemax. 30T/m, 0.5 m effective length, pole radius 55 mm

Magnet parameters of SIS-300

Sextupoles

Resonance excitation2 per straight, 12 in totalMax. B’’=750 T/m2

Physical/effect field length 0.75/0.75 mPole radius 47.5 mm

Chomaticity control4 per arc (V-H-H-V), 24 in totalMax. B’’=750 T/m2

Physical/effect field length 0.75/0.75 mPole radius 47.5 mm

Ripple in power supplies

tfQtfp

pQtQ rippleripples 2sin2sin)( 0

Time variation in tune of a bunched beam subject to ripple from the power supplies to the quadrupoles

Therefore area of separatrix will also oscillate (effect of ripple in sextupoles is much smaller)

dt

dQQQ

Sdt

dAres 2

3482

Thus, to minimize sensitivity to ripple in the quadrupoles, extract with as high S as possible without distortion to the separatrix.

dsssK )()(4

11

where

RF exciter - parameters

Technical details

• Stripline electrodes in horizontal plane

• Pseudo random noise from a maximum length sequence register

• RF carrier digitally modulated with Binary Phase Shift Keying

• Carrier frequency at Qff0

• Full width of centre lobe in spectrum 2fbit

• Spectrum

2

)(

)()(

SC

SCS Tff

TffSinATfG

RF K.O./BTF system architecture

In operationin SIS-18

P. Moritz, GSI

Simulation – Particle tracking

Micromap library (Franchetti, GSI)

Thin lens sextupoles

Thick lens quadrupoles and dipoles

Closed orbit bump made with thin dipoles

Dispersion and chromaticity treated exactly

Kicks from RF exciter treated as thin elements

Sextupole strengths for resonance control calculated from the amplitude and phase

RF exciter amplitude modulation with commissioned ramps

Kick from RF exciter

Thin element approximation, viz.

UqEE '|||| pp

xmL

qUx

t

R

Txxxxx

2; 0

E

UxL

where

13.3268

13.3270

13.3272

13.3274

13.3276

13.3278

13.3280

13.3282

13.3284

13.3286

13.3288

13.3290

0 0.00002 0.00004 0.00006 0.00008 0.0001 0.00012

dK/K1

Qx

9.2888

9.2890

9.2892

9.2894

9.2896

9.2898

9.2900

9.2902

9.2904

Qy

Qx

Qy

Ripple in power supplies

RF-Gap (kV) 0

dp/p|HW 1.6E-5

Chromaticities (-1.49,~1.4)

Tunes (13.327,9.289)

0

10

20

30

40

50

60

70

80

90

0E+00 2E-05 4E-05 6E-05 8E-05 1E-04 1E-04

dK/K1

Rel

ativ

e ch

ang

e in

are

a o

f se

par

atri

x [%

]

SIS-300: Tune sensitivity to focussing strength K1

Ripple in power supplies

SIS-300:Spill under ideal conditions

dI/I in quads 0

Gap-Voltage (kV) 0

dp/p|HW 1.6E-5

x -1.49

Ripple in power supplies

SIS-300:Spill degradation due to ripple

dI/I in quads 5E-5

Gap-Voltage (kV) 0

dp/p|HW 1.6E-5

x -1.49

Tripple

Ripple in power supplies

RF-Gap (kV) 0

dp/p|HW 1.6E-5

Chromaticities (-1.49,~1.4)

Tunes (13.327,9.289)

0

50

100

150

200

250

0E+00 1E-05 2E-05 3E-05 4E-05 5E-05 6E-05 7E-05 8E-05 9E-05 1E-04

Ripple amplitude on Quadrupoles dI/I

RM

S/A

VG

[%

]

0

2

4

6

8

10

12

14

16

18

MA

X/A

VG,

AV

G

RMS/AVG

MAX/AVG

AVG

SIS-300:Systematic survey on ripple amplitude

140

142

144

146

148

150

152

154

156

158

0 2 4 6 8 10 12 14 16 18

Average counts per bin (AVG)

RM

S/A

VG

[%

]

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

MA

X/A

VG

RMS/AVG

MAX/AVG

Summary and outlook

SummarySlow extraction model with RF-K.O. developedSystematic survey of ripple in quadrupoles undertaken

OutlookHardt conditionStochastic extractionIntroduction of B-clock (all rings)

Acknowledgements:P. Spiller, N. Pyka, J. Stadlmann, G. Franchetti

from GSI Darmstadt