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)
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
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
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