the measurement of chromaticity by head-tail phase shift analysis

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


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


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


Positive Chromaticity (Above Transition)

Q > Q0

Q < Q0

p/p

Head Tail

-s


Positive Chromaticity (Above Transition)

Q > Q0

Q < Q0

p/p

Head Tail

-s


Negative Chromaticity (above transition) Positive Chromaticity (above transition)


The Head-Tail PrincipleThe Head-Tail Principle

Signal - Longitudinal Bunch Profile

Time

Head Tail


Time

Head Tail

Time

Head Tail

Signal - Transverse Bunch Position

Time

Head Tail

a

h a

h


Time

Head Tail

Time

Head Tail

a

h a

h

Response forZero Chromaticity


The Head-Tail PrincipleThe Head-Tail Principle

Response forNon-Zero Chromaticity


Time

Head Tail


Time

Head Tail

Time

Head Tail


Time

Head

Tail


Time

Head

Tail

Time

Head

Tail


The Head-Tail Principle VThe Head-Tail Principle V

Time

Head Tail

1 Synchrotron Period


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(


1nQ2cos)n( s 1nQ2cos)n( s


00Q

00Q


MAX 2 2



00

MAX

s00 Q21Qn2cosQ)n(

00

MAX

s00 Q21Qn2cosQ)n(


1nQ2cos)n( s


00Q


MAX 2



00

MAX

s00 Q21Qn2cosQ)n(


1nQ2cos)n( s 1nQ2cos)n( s


00Q

00Q


MAX 2 2



00

MAX

s00 Q21Qn2cosQ)n(

00

MAX

s00 Q21Qn2cosQ)n(


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


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


Measuring Q’Measuring Q’

Qs-1 = 97 turns



Qs-1 = 230 turns



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


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


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’’’


Measuring Q’ (long coupler)Measuring Q’ (long coupler)


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


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)


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)


Effect of AccelerationEffect of Acceleration


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


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



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



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


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



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’’



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

the measurement of chromaticity by head-tail phase shift analysis

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