Coherent Space Charge Tune Shift Measurements in the Los Alamos Proton Storage Ring (PSR)

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

2. Outline • Coherent Tune Shift Theory • Motivation • Cornell Electron Cloud (EC) Tune Shift Study • Goals of PSR Experiment • Measurement • Analysis • Further Work

3. 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 x s, t

4. 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) x s, t 1 2

5. 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 - Log of the analysis of beam response to 1-turn kick Oct 9, 2006 - R. Macek, 1/14/08, 1/15/08 draft

6. WM41 vd Motivation • 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? WC41 Data taken by R. Macek in Oct. 2006;1225 μs accumulation, 200 μs store.

7. Motivation (2) • 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? WM41 int vd Simulation preformed by R. Macek based on the Oct. 2006 data; 1225 μs accumulation, 200 μs store.

8. 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?

9. 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 G. Dugan, M. Billing, et. al., Proceedings of IPAC2012, New Orleans, Louisiana, USA, WEYA02.

10. 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.

11. 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.

12. 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?

13. 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

14. 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. 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.

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.

16. 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

17. 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)

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

19. 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. WM41 vd ~position

20. 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. WM41 vd ~position

21. 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. SRWM41 ~position

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

23. WM41 vd Tune as a Function of Turn (2) • Tune fit for all slices Single-turn kick ~position • At some point signal becomes too small to reliably fit the tune.

24. 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 ~position

25. 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.

26. 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? ~position

27. 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. ~position

28. ~Position High Charge Measurement • Accumulated beam 1225 us injecting every other turn (CD 2), PW 290 ns. • Stored beam 150 us after tick, 420 turns. WM41 vd

29. ~Position Low Charge Measurement • Accumulated beam for 625 us injecting every third turn (CD 3), PW 290 ns. • Stored beam 300 us after kick, 835 turns. WM41 vd

30. ~Position 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. WM41 vd

31. ~Position Bunched Coasting Beam • 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 WM41 vd

32. ~Position 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 WM41 vd

33. 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?

34. 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.