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U N C L A S S I F I E D Slide 1
Coherent Space Charge Tune Shift Measurements in the Los Alamos
Proton Storage Ring (PSR)
Jeff Kolski (LANL)
Mini-workshop on Methods of Data Analysis in Beam Measurements, Including ICA, MOGA, and other Modeling
Methods
3/13/2013
LA-UR-13-21726
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U N C L A S S I F I E D Slide 2
Outline
Coherent Tune Shift Theory
Motivation
Cornell Electron Cloud (EC) Tune Shift Study
Goals of PSR Experiment
Measurement
Analysis
Further Work
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U N C L A S S I F I E D
Coherent Tune Shift Theory
Single particle motion (Hill’s Eq)
• Simple Harmonic motion with spring constant that varies with longitudinal distance• If K(s) = k/m is constant, the oscillation frequency is
Slide 3
x s K s x s 0
k
mx
s, t
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U N C L A S S I F I E D
Coherent Tune Shift Theory (2)
Two particle motion• Particle 1 experiences a sum of forces
— restoring focusing force of the magnet lattice— resistive electro-magnet force from particle 2, approximate as linear
• If K(s) = k/m is constant, the oscillation frequency is• The frequency is less than the single particle case.
The frequency shift due to the space charge self-force always lowers the frequency of betatron oscillation (betatron tune)
Slide 4
x
s, t
1
2
x s K s Ksc x s 0
k Ksc
m
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U N C L A S S I F I E D
Coherent Tune Shift Theory (3)
Incoherent tune shift – tune shift felt by individual particles as described in the previous slide
Coherent tune shift – average tune shift of the beam (bunch, slice)• N - # of protons
• rp – classical proton radius
• β and γ are the relativistic factors
• βy – average beta function around ring or in dipoles
• Bf – bunching factor (ave current / peak current)
• h – radius of the beam pipe
• C2 – fraction of circumference occupied by dipoles
• g – the half gap of the dipoles
First term gives dependence on the instantaneous current
Second term gives contribution from the DC or average current.
For the PSR, the second term is ~15% of the tune shift Slide 5
2
222
222 24
)(
2
)()(
g
bendC
hB
ringrNcoh y
f
ypy
-
Log of the analysis of beam response to 1-turn kick Oct 9, 2006 - R. Macek, 1/14/08, 1/15/08 draft
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U N C L A S S I F I E D
Motivation
Slide 6
We measure an asymmetric tune distribution along the bunch.
The beam profile is symmetric.
We expect a symmetric tune distribution about the center of the bunch.
Why don’t we measure this?
Data taken by R. Macek in Oct. 2006;1225 μs accumulation, 200 μs store.
WM41 vd
WC41
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U N C L A S S I F I E D
Motivation (2)
Slide 7
We can simulate a single-turn kick and compare with measurement.
Likewise, we can simulate the tune distribution along the bunch and compare with measurement.
Simulation matches our intuition, so why does not the measurement?
Simulation preformed by R. Macek based on the Oct. 2006 data; 1225 μs accumulation, 200 μs store.
WM41 int vd
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U N C L A S S I F I E D
Motivation (3)
The measured tune distribution differs from simulation on the trailing edge.
What could be different on the trailing edge?• trailing edge multipactor?• neutralization caused by the EC?
Is it possible to relate the variation of measured and expected tune shifts with neutralization from EC?
Slide 8
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U N C L A S S I F I E D
Cornell EC Tune Shift Study
Ten 0.75 mA / bunch 5.3 GeV positron bunches with 14 ns spacing followed by “witness” bunch.
Use a one-turn kick to induce betatron oscillation.• Kick whole train• Kick individual bunch
Measure the tune of each bunch in the train via gated BPMs
Slide 9G. Dugan, M. Billing, et. al., Proceedings of IPAC2012, New Orleans, Louisiana, USA, WEYA02.
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U N C L A S S I F I E D
Cornell EC Tune Shift Study (2)
EC buildup codes POSINST and ECLOUD show good agreement with measured tune shifts.
The EC density can be intuited from the measurement and compared with output from POINST.
Slide 10
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U N C L A S S I F I E D
Goals of PSR Experiment
Inspired by the Cornell tune measurement for different bunches along a train and witness bunches
The PSR beam:• 290 ns long (358 ns revolution period)
Can be thought of as a train of 100s – 1000s of bunches to investigate EC build up
PSR bunch is long enough to divide into slices to investigate the longitudinal EC density distribution within the bunch.
Slide 11
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U N C L A S S I F I E D
Goals of PSR Experiment (2)
Measure the global average EC density
EC density growth rates along the bunch
Gain some understanding of the bunch profile on the EC pinch dynamics
Why is the coherent space charge tune shift along the bunch asymmetric?
Slide 12
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U N C L A S S I F I E D
What is needed in the PSR measurement
Calculate the tune using pinged beam (one-turn kick from PSR pinger)• Digitize SRWM41 vs and vd signals at 2 GS/s offline analysis• Use a nonlinear cosine fitting routine
— Fitting the tune is more accurate than FFT— View fitted tunes as a function of turn and slice (location in the bunch)
Independent measurement of EC density• Digitize electron detector (ED) signals
Perform measurements with • buncher on/off (different longitudinal beam profile)• vary intensity (pattern width and count down)
Data collection is the same as ICA experiments• Previous measurement experience• Alternate analysis using ICA
Slide 13
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U N C L A S S I F I E D
Measurement
The beam must undergo coherent betatron oscillation, induced by a vertical single-turn kick during a store time after accumulation.
Beam is• accumulated• stored• kicked during storage• Stored after kick
Large beam loss resulting from ep instability• We should take into account the
varying current when we intuit the EC density via the measured tune shift.
Slide 14
CM42
Data taken by R. Macek in Oct. 2006; 1225 μs accumulation, 200 μs store.
The beam is stored for 420 turns (150 μs) after the single-turn kick.
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U N C L A S S I F I E D Slide 15
Analysis
Integrate WM41 vs signal [V, Current derivative] and the WM41 vd signal [V, Current derivative * position] -> WM41 int vs [Vs, ~Current] and WM41 int vd [Vs, ~Current * position]
Stack the digitized data from WM41 int vs, vd, and int vd signals turn-by-turn to obtain matrices [# slices, # turns]
Calculate the position int vd / int vs [~position]
For each time slice of the WM41 vd and ~position signals• Fit a cosine to determine the tune as a function of turn after kick• Computer the average current over the fit
Each tune fit has• Number of turns included in fit• Number of turns shifted between fits• Typically used 30 turns in a fit, shifting
10 turns for the next
Also need to calculate the average current over the region of fit.
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U N C L A S S I F I E D
Analysis (2)
Can we actually measure a change in the space charge tune shift and relate it to changing neutralization from EC?
It is important to study correlations produced by the cosine fit amongst the fitting parameters and initial parameters.
Most interested in how the tune correlates with• The average current over the fit• The fitted amplitude
Also interested in correlations between• Fitted amplitude and average current over the fit• Fitting error on the fitted tune and average current over the fit
Slide 16
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U N C L A S S I F I E D
Analysis (3)
Cosine fit using SRWM41 vd signal [V]• Quality tune fit for current range 7-26 ~Vs (minimum amplitude that yields low fitting
error)• Amplitude fit has large correlation with current (frequency blossoming of central
slices effecting sensitivity of SRWM41 which peaks around 400 MHz)• Tune fit is also correlated with amplitude fit
Cosine fit using ~position (int vd / int vs) [~m]• Quality tune fit for current range 5 – 27 ~Vs (include more slices in the tune fit)• No amplitude fit correlation with current except for extreme head and tail slices
where current signals are in the noise.• Tune fit is not correlated with amplitude fit (tune fitting error is uncorrelated with
fitted amplitude)
Slide 17
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U N C L A S S I F I E D Slide 18
Fitting results differs using vd signal and ~position (1)
Correlation between fitted tune and average current during fit.
Both WM41 vd and ~position signals yield fitted tunes that depend linearly on average current (coherent tune shift).
WM41 vd has more spread in the tune fit for the largest average currents and for smaller currents than ~position
WM41 vd: Quality fit range 7- 26 ~Vs
~Position: Quality fit range 5 - 27 ~Vs
WM41 vd ~position
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U N C L A S S I F I E D
Fitting results differs using vd signal and ~position (2)
Correlation between the fitting error on the tune and the average current of the fit.
WM41 vd has larger tune fitting error for largest and for smaller average currents.
Most fitting errors are less than 0.005.• We are looking for trends (changes in
the tune shift) at this precision.
Slide 19
WM41 vd ~position
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U N C L A S S I F I E D
Fitting results differs using vd signal and ~position (3)
Correlation between the fitted amplitude and the average current.
Amplitude of WM41 vd fits very correlated with the average current.
We believe this to be due to WM41’s 400 MHz peak frequency response.• We have observed frequency “blooming”
for the central slices resulting in high frequencies.
No dependence in ~position fit.
Slide 20
WM41 vd ~position
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U N C L A S S I F I E D
Fitting results differs using vd signal and ~position (4)
Correlation between fitted tune and fitted amplitude.
Obvious correlation in the WM41 vd fits
No correlation in the ~position fits.
Slide 21
SRWM41 ~position
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U N C L A S S I F I E D Slide 22
Tune as a Function of Turn
Can we actually measure a change in the space charge tune shift and relate it to changing neutralization from EC?
Measure a clear linear trend in the tune for slice 300 (~25 ns upstream of bunch peak)• ~0.01 change in the tune
WM41 vd ~position
Single-turn kick
Single-turn kick
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U N C L A S S I F I E D
Tune as a Function of Turn (2)
Tune fit for all slices
Slide 23
WM41 vd
~position
Single-turn kick
At some point signal becomes too small to reliably fit the tune.
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U N C L A S S I F I E D
Tune Shift as a Function of Turn
Tune shift = bare tune – tune fit
Measure a changing tune shift, which could be due to• Instantaneous current change from
current profile change or beam loss• Neutralization by EC
Slide 24
~position
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U N C L A S S I F I E D
Interpretation
Tune fit or tune shift for different time slices and turns can be misleading due to beam loss and other changes in instantaneous current.
For a better handle on any changing neutralization due to EC, the quantity to examine is tune shift / ~current.
Slide 25
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U N C L A S S I F I E D
Tune Shift / ~Current as a Function of Turn
The units are correct, but conversion constants for the current monitor and BPM are needed for quantitative measurement.
It is clear that we can qualitatively measure something very small and relate it to the EC density.
We’ve taken into account the beam current, but we still observe a systematic trend in the tune shift / current.• Neutralization due to the EC?
Slide 26
~position
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U N C L A S S I F I E D
Tune Shift / ~Current as a Function of Turn (2)
Note: while time slice 325 and 375 (symmetric about the beam peak), the slope of slice 325 is greater than 375 indicating a great rate of neutralization.
Slide 27
~position
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High Charge Measurement
Slide 28
WM41 vd
~Position
Accumulated beam 1225 us injecting every other turn (CD 2), PW 290 ns.
Stored beam 150 us after tick, 420 turns.
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U N C L A S S I F I E D
Low Charge Measurement
Slide 29
~Position
WM41 vd
Accumulated beam for 625 us injecting every third turn (CD 3), PW 290 ns.
Stored beam 300 us after kick, 835 turns.
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U N C L A S S I F I E D
Low Charge Measurement (2)
Accumulated beam for 625 us injecting every third turn (CD 3), PW 50 ns.
Stored beam 300 us after kick, 835 turns.
Slide 30
~Position
WM41 vd
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U N C L A S S I F I E D
Bunched Coasting Beam
Slide 31
~Position
WM41 vd
Accumulated beam 625 us injecting every third turn (CD 3), PW 290 ns.
Stored beam 300 us after kick, 835 turns.
Observe 60-70 MHz frequency, signature of microwave instability
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U N C L A S S I F I E D
Bunched Coasting Beam (2)
Accumulated beam 625 us injecting every third turn (CD 3), PW 50 ns.
Stored beam 300 us after kick, 835 turns.
Observe 200 MHz frequency, signature of electron multipacting
Slide 32
~Position
WM41 vd
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U N C L A S S I F I E D
Summary
We have studied measurements of the coherent space charge tune shift.
We believe that we can measure small changes in the tune shift as a function of slices along the bunch and turns.
When we take into account the instantaneous current, we still observe a changing tune shift.• Could this be neutralization due to the EC?
Slide 33
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U N C L A S S I F I E D
Further Work
Change analysis from qualitative to quantitative by taking to account conversion constants for WM41.
Use the coherent space charge tune shift equation to better isolate the neutralization.
Take another set of measurement with high enough current to give good signal on EDs for an independent measurement of the EC.
Slide 34