1. RF Quadrupole for Landau damping in HL-LHC
A. Grudiev, K. Li, K. Papke, M. Schenk
Acknowledgements
G. Arduini, H. Bartosik, X. Buffat, R. De Maria, A. Maillard, E. Métral, Y. Papaphilippou, G. Rumolo, E. Shaposhnikova, C. Zannini
72nd HiLumi WP2 Meeting 02. August 2016

2. Outline Introduction Numerical proof-of-principle HL-LHC study case
Synchro-betatron resonances
Potential location in HL-LHC
Summary and discussion
M. Schenk et al.

3. At second order: Detuning with longitudinal amplitude (action).
Introduction
Basic working principle of an RF quadrupole
Proposal by Alexej Grudiev[2,3]
An RF quadrupole introduces a betatron tune spread to ‘damp’ transverse collective instabilities via Landau damping.
Example: Pillbox cavity with TM210 mode.
Working principle
RF-modulated quadrupole kick
translates into a betatron detuning (assume ϕ0 = 0)
At second order: Detuning with longitudinal amplitude (action).
M. Schenk et al.

4. Introduction Betatron detuning with amplitude
Example: Landau octupoles (LO)
Detuning w. transverse amplitude
Detuning w. longitudinal amplitude
Examples: RF quadrupole, Q’’
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5. Introduction Qualitative comparison of stability diagrams
SD for RF quadrupole (2nd order)[5].
Size of SD depends on instability. (higher order modes have larger stable area)
Asymmetry of tune distribution is reflected.
One plane (here: H) shows more favourable stabilising behaviour than the other. (Note: Dipolar impedance induces Re(ΔQcoh) < 0)
‘Polarity’ / phase (ϕ0) of RF quadrupole is relevant.
LO and RF quadrupole benefit from the same underlying stabilising mechanism, but the stabilising behaviour is different.
There is a strong asymmetry between the two planes for the RF quadrupole as a consequence of the shape of the incoherent tune distributions.
SD for RF quadrupole is very approximate at the moment, making only qualitative statements possible.
Ongoing work and good progress by A. Maillard for more precise dispersion integral.
Typical stability diagram (SD) from LO.
Rather symmetric about Re(ΔQcoh) = 0.
M. Schenk et al.

6. Introduction Motivation for an RF quadrupoleLO
RF quad
Motivation for an RF quadrupole
LHC operation shows[16,17]
Landau damping is successfully used against instabilities.
LO are often operated close to their limit.
HL-LHC will have to handle beams of higher intensities and higher brightnesses[7]
Potentially leads to more violent instabilities.
LO less effective due to lower transverse beam size.
At 7 TeV, spread in Jz is times larger than that in Jx,y.
Even compact RF quadrupole can produce a large incoherent betatron tune spread.
RF quadrupole could be an ideal device to increase the margin for stable HL- LHC operation.
Additional advantage for potential future high energy accelerators
On the energy ramp, the detuning from LO is affected by both adiabatic damping and increased beam rigidity, i.e.
An RF quadrupole is only affected by the increased beam rigidity, i.e.
(given that the longitudinal emittance is blown up along the ramp – as done in (HL-)LHC)
M. Schenk et al.

7. Outline Introduction Numerical proof-of-principle HL-LHC study case
Synchro-betatron resonances
Potential location in HL-LHC
Summary and discussion
M. Schenk et al.

8. Numerical proof-of-principle[4]
LHC, 3.5 TeV study case (I)
If = -Id = -10 A
Instability observed during LHC commissioning, [18].
LHC at 3.5 TeV, single bunch, mode m = -1 (H) head-tail instability.
Cured with LO If = -Id at
A (experiment)
A (PyHEADTAIL)
A (PySSD).
LO(ΔQx)RMS ≈ (2.4 ± 0.3)∙ ≈ Qs
Experiment, simulations and theory show that Landau damping cures the instability.
Case ideally suited to test the RF quadrupole numerically.
E. Métral et al.[18,19]
Octupoles (PyHEADTAIL)
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9. Numerical proof-of-principle[4]
LHC, 3.5 TeV study case (II)
RF quadrupole (PyHEADTAIL)
b(2) = 0 Tm/m
b(2) = Tm/m
b(2) = Tm/m
b(2) = Tm/m
An RF quadrupole is equally able to cure the instability by introducing a large enough betatron tune spread.
RFQuad(ΔQx)RMS ≈ (3.5 ± 0.5)∙10-5 ≈ Qs
LO(ΔQx)RMS ≈ (2.4 ± 0.3)∙10-5 ≈ Qs
Stability diagram theory makes clear that the stabilising behaviour for magnetic octupoles and RF quadrupole is not the same.
RMS tune spread alone does not provide a complete picture.
M. Schenk et al.

10. Outline Introduction Numerical proof-of-principle HL-LHC study case
Synchro-betatron resonances
Potential location in HL-LHC
Summary and discussion
M. Schenk et al.

11. RF quadrupole for HL-LHC
Simulation setup and chromaticity scan
Single bunch at 7 TeV with HL-LHC design parameters[7].
Stabilising systems
LHC transverse damper, idealised.
LHC magnetic octupoles.
Superconducting RF quadrupole at βx,y = 200 m (conservative)[1-3].
LHC operation shows that Q’ = 10 can be a potential working point.
With transverse feedback system, we observe a head-tail mode (0, 2).
Similar instability experimentally observed in LHC[16].
Without RF quadrupole, an LO current of If = (170 ± 10) A is required for stabilisation. (only accounting for impedance)
PyHEADTAIL
Q’ = 10
What is the effect of an RF quadrupole on the required LO current If?
Study the dependence of required LO current on RF quadrupole strength b(2) for two different RF quadrupole frequencies – 800 MHz and 1.2 GHz.
Sensitivity too low (# turns)
Head-tail mode
(0, 2)
radial
azimuthal
M. Schenk et al.

12. RF quadrupole for HL-LHC
800 MHz cavity
If, thr w/o RF
quadrupole
Tunespreads for stabilisation with RF quadrupole and LO alone
RFQuad(ΔQx)RMS ≈ (6.9 ± 0.4)∙10-5 RFQuad(ΔQy)RMS ≈ (4.5 ± 0.4)∙10-5
LO(ΔQx,y)RMS ≈ (3.6 ± 0.2)∙10-5
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13. RF quadrupole for HL-LHC
1.2 GHz cavity
If, thr w/o RF
quadrupole
Tunespreads for stabilisation with RF quadrupole and LO alone
RFQuad(ΔQx)RMS ≈ (1.3 ± 0.1)∙10-4 RFQuad(ΔQy)RMS ≈ (3.3 ± 0.4)∙10-5
LO(ΔQx,y)RMS ≈ (3.6 ± 0.2)∙10-5
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14. RF quadrupole for HL-LHC
Summary 800 MHz vs. 1.2 GHz
b(2) needed to stabilise with RF quadrupole alone.
b(2) needed to halve the stabilising LO current.
The symbiosis between LO and RF quadrupole is successful.
The goal is not to replace LO, but to support them.
800 MHz cavity performs better than 1.2 GHz – better match with HL-LHC bunch length.
The focus lies on studies and design of the 800 MHz cavity.
Asymmetry in stabilising threshold between planes is visible and can be significant in particular cases.
M. Schenk et al.

15. Outline Introduction Numerical proof-of-principle HL-LHC study case
Synchro-betatron resonances
Potential location in HL-LHC
Summary and discussion
M. Schenk et al.

16. Excitation of synchro-betatron resonances
Introduction: Evaluation of RF quadrupole prototype for SPS
Coherent losses
(head-tail mode 0)
Stable beam
Incoherent losses (resonances)
Evaluate RF quadrupole prototype for proof-of-principle experiment in SPS, using mode 0 head-tail instability.
By means of an aperture (beam pipe), particle losses are quantified.
Allows to identify three regimes.
Hypothesis: RF quadrupole can excite synchro-betatron resonances at sufficiently high strengths.
Losses after 105 turns
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17. Excitation of synchro-betatron resonances
Theory
Since an RF quadrupole gives a kick in H (V) as a function of a particle’s x (y) and z positions, it can excite synchro-betatron resonances (SBR)
(note that to first order, the RF quadrupole does not couple x and y)*
SBR condition is given by the relation[11] m
k = 1
k = 1
m = -8
If we set e.g. l = 0*, the resonance lines in the (Qz, Qx) space are given by
M. Schenk et al.

18. Excitation of synchro-betatron resonances
Method and model
Model:
Simulations are made with PyHEADTAIL (first approach).
Model uses linear transfer maps both in transverse and longitudinal planes.
Excitation can be generated by the RF quadrupole which is modelled as a localised kick with b(2) = 0.5 Tm/m.
Method:
Initialise single particle of fixed Jz for every horizontal tune setting and track over 106 turns.
Measure Jx evolution and fit it with an exponential function to obtain the action growth rate.
Analyse the effective horizontal tune with Sussix.
Resonance condition is given by
where the strongest resonance is observed for k = 1.
M. Schenk et al.

19. Excitation of synchro-betatron resonances
Comparison of SPS and HL-LHC
SBR can be observed in SPS (26 GeV) simulations. qx,SPS = 0.13, Qs,SPS ≈
(ΔQx)RMS ≈9.2∙10-3 ≈ 0.54 Qs,SPS
The lowest order resonance closest to the working point is m = -8 for k = 1.
Good agreement between tracking and analytical formula.
SPS
Due to the simplicity of our model, the results are not yet conclusive.
Use SixTrack to study higher orders in more detail to give a definite statement about the excitation of resonances by an RF quadrupole in HL-LHC.
In HL-LHC (7 TeV, collision tunes), qx,HL = 0.31 and Qs,HL ≈
(ΔQx)RMS ≈ 1.3∙10-4 ≈ 0.06 Qs,HL
Since qx,SPS << qx,HL and Qs,SPS >> Qs,HL, SBR in HL-LHC are less likely to be an issue due to the high orders in m (m = -147 for k = 1).
In addition, required tune spread << Qs,HL.
With our simple model SBR cannot be observed around the HL-LHC working point.
HL-LHC
No resonances visible
M. Schenk et al.

20. Outline Introduction Numerical proof-of-principle HL-LHC study case
Synchro-betatron resonances
Potential location in HL-LHC
Summary and discussion
M. Schenk et al.

21. Potential location in HL-LHC
Point 4
RF infrastructure available.
Beta functions ≈ 200 m, but could in principle be increased by factor up to ≈10 if needed (R. De Maria).
Would reduce cavity number (linearly), thus also impedance and cost.
LHC, point 4
M. Schenk et al.

22. Summary
We study an RF quadrupole to enhance beam stability in the HL-LHC.
We developed a numerical model and showed its validity for an example case with Landau octupoles installed in the LHC.
Using this model, we demonstrated the stabilising effects from an RF quadrupole together with Landau octupoles for an operational scenario in the HL-LHC.
With the present HL-LHC impedance model, the LO current for single bunch stabilisation at flat-top and Q’x,y = 10 is If, thr = (170 ± 10) A – roughly one third of the maximum possible value.
By adding only about 2 RF quadrupole cavities[2]*, the Landau octupole stabilising currents can be reduced to 0 A and the margin for stable operation thus increases significantly.
*Active length of about 1 m. Integrated in one few metres long cryostat[2].
RF cavity design is ongoing. New designs may have larger cavity strength, leading to smaller number of cavities and hence reduced impedance and cost.
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23. Summary Is an RF quadrupole system required for HL-LHC?
The present impedance model suggests that the means for stabilisation available in LHC (i.e. Landau octupoles, transverse damper) are sufficient.
But, LHC operation reveals additional destabilising effects throughout the cycle, which are not captured by this model.
A definite answer whether HL-LHC needs an RF quadrupole cannot be given at presence. Further studies are required.
However:
It is clear that with higher intensity and higher brightness, instabilities are more likely to become a limiting factor.
Should it turn out that they do limit or exclude nominal HL-LHC performance, it will be vital to have a cure at hand.
To be prepared for that case, we encourage that RF quadrupole studies and R&D be continued. This would ideally include a proof-of-principle experiment with a prototype cavity, e.g. in the SPS.
In addition, the use of such a device for higher energy accelerators (HE-LHC) may be great as it introduces tune spread as a function of longitudinal action, whose spread is typically held constant during the energy ramp.
M. Schenk et al.

24. Open questions and future studies
Several effects are not yet (or cannot be) included in the model
Crab cavities impedance
Impedance model uncertainties
Unknown impedance contributions …
Modelling of e-cloud effects is currently under detailed investigation. Once scenarios are defined, we can assess the effectiveness of the RF quadrupole also in presence of e-cloud.
The same applies concerning coupled-bunch instabilities.
They may shrink the margin for stable operation.
Continue studies of excitation of (synchro-betatron) resonances at the HL-LHC working point by means of SixTrack.
What would be the requirements for a proof-of-principle experiment in the SPS?
For discussion: Potential location for the RF quadrupole in the HL-LHC.
M. Schenk et al.

25. References I
[1] A. W. Chao, Physics of collective beam instabilities in high energy accelerators, Wiley, [2] A. Grudiev, Radio frequency quadrupole for Landau damping in accelerators, Phys. Rev. ST Accelerators and Beams 17, , [3] A. Grudiev, RF quadrupole for Landau damping, Talk at ICE meeting, [4] A. Grudiev, K. Li, and M. Schenk, Radio Frequency Quadrupole for Landau Damping in Accelerators: Analytical and Numerical Study, Proceedings, HB2014 (USA), [5] J. Scott Berg and F. Ruggiero, Stability Diagrams for Landau Damping, LHC Project Report 121, [6] K. Papke, A. Grudiev, Design of RF quadrupole resonator, Slides, [7] O. Brüning and L. Rossi (Edts.), The High Luminosity Large Hadron Collider, Advanced Series on Directions in High Energy Physics Vol. 24, [8] C. Zannini et al., Benchmarking the SPS transverse impedance model: headtail growth rates, Talk at SPSU meeting, [9] H. Bartosik, Beam dynamics and optics studies for the LHC injectors upgrade, PhD thesis, 2013.
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26. References II
[10] M. Schenk et al., Use of RF Quadrupole Structures to Enhance Stability in Accelerator Rings, Proceedings of HB2016 (Sweden), [11] A. Piwinski, Synchro-Betatron Resonances in Circular Accelerators, DESY, [12] H. Bartosik et al., Improved methods for the measurement and simulation of the CERN SPS non-linear optics, Proceedings of IPAC16 (Korea), [13] C. Vaccarezza et al., Preliminary results on Daphne operation with octupoles, Proceedings, EPAC2002 (France), [14] V. V. Danilov, Increasing the transverse mode coupling instability threshold by RF quadrupole, Phys. Rev. ST Accel. Beams 1, , [15] E. A. Perevedentsev and A. A. Valishev, Synchrobetatron dynamics with a radio-frequency quadrupole, Proceedings of EPAC2002, Paris (France), [16] L. R. Carver et al., Current status of instability threshold measurements in the LHC at 6.5 TeV, Proceedings of IPAC16, Busan (Korea), [17] E. Métral et al., Measurement and interpretation of transverse beam instabilities in the CERN Large Hadron Collider (LHC) and extrapolations to HL- LHC, Proceedings of HB2016 (Sweden), [18] E. Métral et al., Status of the LHC Instabilities, 1st ICE Meeting, [19] E. Métral, B. Salvant, N. Mounet, Stabilization of the LHC single-bunch transverse instability at high-energy by Landau octupoles,
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27. Backup slides

28. RF quadrupole models in PyHEADTAIL
Detuner
Localised kicks
Opposite signs in H and V explain the difference in stabilisation behaviour in the two planes (see asymmetry in tune footprints and stability diagrams).
RF quadrupole stabilising effect could be optimised e.g. by installing one cavity with ϕ0 = 0 in low βx, high βy and another one with ϕ0 = π in low βy, high βx regions.
M. Schenk et al.

29. Evaluation of RF quadrupole prototype test in SPS
Identify a weak head-tail instability in SPS that can be Landau damped.
Focus lies on mode 0 head-tail instability in the vertical plane (Q’y < 0):
It is very clear and well-reproducible both in experiment and simulations (reliable impedance model);
Good agreement between measurements and PyHEADTAIL simulations;
Higher order modes (Q’y > 0) cannot be easily observed experimentally.
Can it be stabilised by an RF quadrupole and at what strength b(2)?
M. Schenk et al.

30. Excitation of synchro-betatron resonances
HL-LHC at working point 62.01
For a test, we scanned for SBR around the tune Qx = 62.01, keeping the same model and parameters as before.
At this low fractional tune, lower orders m of SBR become visible.
This is observed also in tracking simulations.
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31. Stability diagram theory
Transverse Landau damping with longitudinal amplitude
From Scott Berg and Ruggiero[5], the dispersion integral is given by
In the absence of tune spread, i.e.
M. Schenk et al.

32. Stability diagram theory
Transverse Landau damping with longitudinal amplitude
To obtain the stability diagram, we solve the equation
Assuming that
all the tunes are measured with respect to
Jz is in units of RMS longitudinal emittance.
The synchrotron tune is constant.
The dispersion integral in [5] includes several approximations
“… only correct under the assumption that the frequency of the impedance is small compared to the frequencies in the bunch spectrum.”[5]
“… ignore a term giving the longitudinal force due to the transverse wake…”[5]
Radial mode?
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33. Dependence on RF quadrupole phase
RF quadrupole for HL-LHC
M. Schenk et al.