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The Measurement of Chromaticity by Head-Tail Phase Shift Analysis. CARE Workshop on Schottky Measurement and Tune, Coupling & Chromaticity Measurement & Feedback Chamonix 11 th - 13 th December 2007 Rhodri Jones CERN Beam Instrumentation Group. Outline. - PowerPoint PPT Presentation
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The Measurement of ChromaticityThe Measurement of Chromaticityby Head-Tail Phase Shift Analysisby Head-Tail Phase Shift Analysis
CARE Workshop on
Schottky Measurement and Tune, Coupling & Chromaticity Measurement & Feedback
Chamonix11th - 13th December 2007
Rhodri Jones CERN Beam Instrumentation Group
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
OutlineOutline
● The Head-Tail measurement principle
● The Head-Tail monitor of the SPS (2000)
● Improvements & simulations since
● Results from the Tevatron
● Head-Tail: impedance & higher order terms
● Conclusions
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail Principle IThe Head-Tail Principle I
● The Principle:● Apply single transverse kick & observe resulting motion
● Chromaticity will determine the pattern of this motion● By following the time evolution of any two positions within the bunch a phase-
difference is obtained from which the chromaticity can be calculated
● Assumptions used in the Theory:● The displacement due to the kick is much larger than the betatron
oscillations performed by the particles in the unperturbed bunch● when the kick is applied all particles assumed to have the same initial
betatron phase
● The synchrotron frequency is the same for all particles in the bunch● assumption holds for measurements are performed close to centre of bunch
● The presence of higher order fields such as octupolar fields not taken into consideration
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail Principle IIThe Head-Tail Principle II
Negative Chromaticity (Above Transition)
Q < Q0
Q > Q0
p/p
Head Tail
-s
Longitudinal Phase-Space
Negative Chromaticity (Above Transition)
Q < Q0
Q > Q0
p/p
Head Tail
-sQ < Q0
Q > Q0
p/p
Head Tail
-s
Longitudinal Phase-Space
Positive Chromaticity (Above Transition)
Q > Q0
Q < Q0
p/p
Head Tail
-s
Longitudinal Phase-Space
Positive Chromaticity (Above Transition)
Q > Q0
Q < Q0
p/p
Head Tail
-s
Longitudinal Phase-Space
Negative Chromaticity (above transition) Positive Chromaticity (above transition)
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail PrincipleThe Head-Tail Principle
Signal - Longitudinal Bunch Profile
Time
Head Tail
Signal - Longitudinal Bunch Profile
Time
Head Tail
Time
Head Tail
Signal - Transverse Bunch Position
Time
Head Tail
a
h a
h
Signal - Transverse Bunch Position
Time
Head Tail
Time
Head Tail
a
h a
h
Response forZero Chromaticity
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail PrincipleThe Head-Tail Principle
Response forNon-Zero Chromaticity
Signal - Longitudinal Bunch Profile
Time
Head Tail
Signal - Longitudinal Bunch Profile
Time
Head Tail
Time
Head Tail
Signal - Transverse Bunch Position
Time
Head
Tail
Signal - Transverse Bunch Position
Time
Head
Tail
Time
Head
Tail
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail Principle VThe Head-Tail Principle V
Time
Head Tail
1 Synchrotron Period
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail Principle VThe Head-Tail Principle V
Time
Head Tail
1 Synchrotron Period
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
The Head-Tail EquationThe Head-Tail Equation
● For >> t chromaticity depends only on● Maximum phase shift measured between head and tail● The separation between the head & tail measurements
The phase difference as a function of the number of turns from an initial kick is given by
1nQ2cos)n( s
where is the chromatic frequency and is defined as
00Q
The maximum phase shift is obtained after half a synchrotron period, when nQs = ½
MAX 2
The relative chromaticity can therefore be written as
= relative chromaticity = head-tail phase difference = 1/()2 - = time between the sampling of head and tailQs = synchrotron tune Q0 = betatron tune0 = angular revolution frequency n = number of turns since the initial kick
00
MAX
s00 Q21Qn2cosQ)n(
The phase difference as a function of the number of turns from an initial kick is given by
1nQ2cos)n( s 1nQ2cos)n( s
where is the chromatic frequency and is defined as
00Q
00Q
The maximum phase shift is obtained after half a synchrotron period, when nQs = ½
MAX 2 2
The relative chromaticity can therefore be written as
= relative chromaticity = head-tail phase difference = 1/()2 - = time between the sampling of head and tailQs = synchrotron tune Q0 = betatron tune0 = angular revolution frequency n = number of turns since the initial kick
00
MAX
s00 Q21Qn2cosQ)n(
00
MAX
s00 Q21Qn2cosQ)n(
The phase difference as a function of the number of turns from an initial kick is given by
1nQ2cos)n( s
where is the chromatic frequency and is defined as
00Q
The maximum phase shift is obtained after half a synchrotron period, when nQs = ½
MAX 2
The relative chromaticity can therefore be written as
= relative chromaticity = head-tail phase difference = 1/()2 - = time between the sampling of head and tailQs = synchrotron tune Q0 = betatron tune0 = angular revolution frequency n = number of turns since the initial kick
00
MAX
s00 Q21Qn2cosQ)n(
The phase difference as a function of the number of turns from an initial kick is given by
1nQ2cos)n( s 1nQ2cos)n( s
where is the chromatic frequency and is defined as
00Q
00Q
The maximum phase shift is obtained after half a synchrotron period, when nQs = ½
MAX 2 2
The relative chromaticity can therefore be written as
= relative chromaticity = head-tail phase difference = 1/()2 - = time between the sampling of head and tailQs = synchrotron tune Q0 = betatron tune0 = angular revolution frequency n = number of turns since the initial kick
00
MAX
s00 Q21Qn2cosQ)n(
00
MAX
s00 Q21Qn2cosQ)n(
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
CERN-SPS System Set-upCERN-SPS System Set-up
StraightStriplineCoupler
Beam PipeBeam
Hybrid
VMEAcquisitionvia GPIB
Sum
Difference
Bunch SynchronousTrigger
GPIB linkUNIX
User Interface
Fast (2GS/s per channel)Digital Oscilloscope
SPS Tunnel
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Coupler long enoughfor bunch length
Signal
Reflected Signal
Signal and Reflectionfully resolved
Time
Pick-up● Straight stripline coupler - 37cm long● Completely resolves a bunch < 2.5ns in length
● NOT the case in the CERN-SPS where bunch length is ~4ns
Useable signal ~2.5ns measure head & centre
NOT head & tail
Signal and ReflectionNOT fully resolved
Coupler too shortfor bunch length
Real response
The Initial CERN-SPS Head-Tail MonitorThe Initial CERN-SPS Head-Tail Monitor
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Measuring Q’Measuring Q’
Qs-1 = 97 turns
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Measuring Q’Measuring Q’
Qs-1 = 230 turns
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Measuring Q’Measuring Q’
Comparison of Head-Tail Chromaticity Measurementswith Radial Steering Measurements at 115GeV in the SPS
head-tail = 0.45
radial-steering + orbit dependent offset
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Radial Steering Chromaticity ()
Hea
d-T
ail C
hrom
atic
ity ( )
-2.1mm Orbit+0.1mm Orbit+2.6mm Orbit
Scaling factor of 2.2 (=0.45-1) at 115GeV
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Measuring Q’’ and Q’’’Measuring Q’’ and Q’’’
Radial Position versus Chromaticity (115GeV)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-6 -4 -2 0 2 4 6
Radial Position (mm)
Chr
omat
icity
( )
Radial SteeringScaled Head-Tail
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Improvements and DevelopmentsImprovements and Developments
● Added 60cm long coupler● can fully resolve bunches up to 4ns in length
● Added low loss cables & reduced cable length● increase in the overall system bandwidth
● Performed more complete simulations (Stephane Fartoukh)
● originally intended to find source of missing factor● Turned out to be due to hardware bandwidth limitations● Added better cables and introduced deconvolution algorithm to
eliminate this factor
● developed into a robustness study for the technique● Effect of accelerating buckets● Effect of Q’’ and Q’’’
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Measuring Q’ (long coupler)Measuring Q’ (long coupler)
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Understanding the Scaling FactorUnderstanding the Scaling Factor
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5 0.6 0.7 0.8 0.9 1Sextupole Trim Value
Mea
sure
d C
hrom
atic
ity
Radial steering chromaticity
Raw head-tail chromaticity
Head-Tail results x 1.4
Factor down from 2.2 to 1.4From 2000 to 2001
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Signal Output – Bandwidth LimitationsSignal Output – Bandwidth Limitations
-100
-80
-60
-40
-20
0
20
40
60
4 6 8 10 12 14 16Time (ns)
Am
plitu
de (
arb)
Original Delta signalOutput Delta signal (2GS/s)Output Delta signal (8GS/s)
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Signal Output - DeconvolutionSignal Output - Deconvolution
-100
-80
-60
-40
-20
0
20
40
60
80
100
4 6 8 10 12 14 16Time (ns)
Am
plitu
de (
arb)
Original Delta signalOutput Delta signal after deconvolution (2GS/s)Output Delta signal after deconvolution (8GS/s)
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Effect of AccelerationEffect of Acceleration
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Use of Kicked Head-Tail in the TevatronUse of Kicked Head-Tail in the Tevatron● Intensity of 2.5-3x1011 particles per bunch with 3ns bunch rms length
(coalesced bunches)● Main instrument for chromaticity measurement on the Tevatron ramp
● RF modulation not feasible● change in RF required to resolve the chromaticity causing losses during the ramp● tune moved a lot during snapback & RF frequency modulation too slow to resolve
chromaticity
Courtesy of Vahid Ranjbar
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Use of Kicked Head-Tail in the TevatronUse of Kicked Head-Tail in the Tevatron
● Calibration with RF modulation at injection energy● Head-Tail measurements at the start of the ramp
Courtesy of Vahid Ranjbar
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Effects of Impedance & Higher Order TermsEffects of Impedance & Higher Order Terms● Addition of higher order terms
● Octupoles induce tune spread● Coherence maintained for much shorter time● Phase difference over 1 synchrotron period more difficult to extract
Courtesy of Vahid Ranjbar
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Effects of Impedance & Higher Order TermsEffects of Impedance & Higher Order Terms● Effect of impedance observed early on in the SPS
● Phase shift at low energy depended on single bunch intensityChange of Head-Tail Phase Difference with Intensity
(MESPS-short at 26GeV on the P2 cycle)
-1.5
-1
-0.5
0
0.5
1
0 50 100 150 200 250 300 350
Turn
Ph
ase
Dif
fere
nce
(ra
d)
8.9e10 ppb
3.4e10 ppb
2.0e10 ppb
1.1e10 ppb
0.5ns
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Effects of Impedance & Higher Order TermsEffects of Impedance & Higher Order Terms● Effect of impedance confirmed by data & simulations at FNAL
● Results of multi-particle simulation● N=1000 particles with resistive wall wake field 4.4x105cm-1 (Zeff = 7 Mm-1)
● The behaviour is almost identical to measured response
Courtesy of Vahid Ranjbar
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
Effects of Impedance & Higher Order TermsEffects of Impedance & Higher Order Terms● Adding second order chromaticity results in even better fits
Measured dataFit with impedanceFit with impedance & Q’’
Courtesy of Vahid Ranjbar
Rhodri Jones – CERN Beam Instrumentation GroupRhodri Jones – CERN Beam Instrumentation Group CARE Meeting – Chamonix 2007 CARE Meeting – Chamonix 2007
ConclusionsConclusionsExperimental● Operational Head-Tail Q’-Measurement system demonstrated
● Used for studies in SPS● limited to use at high energy due to impedance
● Used routinely in the Tevatron● Deconvolution required to remove perturbations due to hardware
bandwidth limitations (mainly cables)● Useful instrument for other applications
● Routinely used in the SPS for transverse instability studies
Theoretical● Method applicable for both stationary and accelerating buckets
● Experimentally verified with the constraint that the measurement be performed symmetrically about the bunch centre
● Data can be used to fit for other machine parameters● Good agreement between simulations and measurements in the Tevatron● Allows estimate of
● Resistive wall impedance● Second order chromaticity