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### LHC crab cavity impedance - 1st and 2nd thoughts

Frank Zimmermann

3rd LHC-ILC crab synergy meeting

5 April 2008

thanks to: Rama Calaga, Zenghai Li, Andrei Seryi

longitudinal impedance requirements – 1st thoughts:

- single and coupled-bunch modes should be Landau damped by frequency spread of synchrotron oscillations ; modes above 1 GHz are less important, and modes with frequencies above 2 GHz can most likely be discarded in view of the LHC bunch spectrum; concerning the narrow-band impedances of the crab cavities, from Ref. [2] we conclude that a longitudinal mode shunt impedance of 1-2 kOhm and Q values smaller than 1000 would be tolerable
- inductive low-frequency & broadband impedance should add no more than ~10% to the total LHC broadband & low-frequency impedance, to avoid bunch lengthening, longitudinal microwave instability and other harmful phenomena; from Ref. [1] we infer that the longitudinal low-frequency impedance of the crab cavity should be less than Z/n=j 10 mOhm

transverse impedance requirements – 1st thoughts:

- single and coupled-bunch modes should be Landau damped by natural frequency spread or frequency spread from Landau octupole magnets (complex tune shifts of a few 1e-4 can be suppressed),by positive chromaticity (head-tail damping) and via the transverse bunch-by-bunch damper (with a damping time of ~50 turns at injection); modes above 1 GHz are less important, and modes with frequencies above 2 GHz can most likely be discarded in view of the LHC bunch spectrum; from Ref. [2] we conclude that a transverse shunt impedance of 10 kOhm/m and Q values of order 100-1000 would suffice;
- in the global scheme the beta function at the crab cavity in IR4 would be between 300 and 800 m, which is up to 10 times larger than the average LHC beta function (~70 m). At local crab cavities the beta function can be 4-5 km (for beta*~0.25 m) [4]; even with these beta function values, the above narrow-band impedance numbers should be acceptable
- the beta-weighted transverse broadband impedance (beta x Z) should be smaller than the total LHC transverse broadband impedance and preferably less than 1 MOhm, where Z can be treated as effective impedance (the LHC rms bunch length at injection is 11.24 cm)

comments on 1st thoughts from Andrei and Zenghai

Andrei, 3 April 2008:

“In Deepa\'s paper ZAP code was used which performed an analytical evaluation of all the modes or sources of impedance, sort the more dangerous modes, etc. The question is -- how accurate is the ZAP calculation?For quick evaluation of particular modes we could use simple analytical estimation (should be just couple tens of lines of Matlab code). “

Zenghai, 2 & 4 April 2008:

“In the 800MHz design, one of the monopole modes has a R/Q of around 200 Ohm (for point charge), the Q would be lower than 10 to satisfy the 1-2kOhm requirement, which would be difficult…”

“Here are the R/Q numbers for the LOM, SOM, and the operating modes (→ next slide). These numbers are based on the first preliminary design. They will change some as design evolves. Should be a good starting point for impedance studies. We need the Qext requirement for the LOM and SOM modes for the damping coupler design. Let me know if you need more info. (If needed, the frequencies of the LOM and SOM can also be adjusted to avoid resonances).”

slide from Zenghai

Frank’s question:

should units of dipole R/Q not be Ohm/meter?

*(Based on Liling-1/20/08 talk. R/Q will change slightly as design evolves)

refined estimate of impedance tolerances

- “2nd thoughts”

longitudinal criteria

1. narrow-band impedance – threshold of coupled-bunch instability [5,6]

G~1.5

F~0.3

I0~0.58 A

b~1

f0~11.245 kHz

→ Rsh,L< 137 kW (inj.)

Rsh,L <196 kW (top)

→ Q < 100-200 should be fine for SLAC 800-MHz cavity

longitudinal criteria

2. broad-band impedance – loss of Landau damping [5,6,7]

G~1.5

F~0.3

Ib~0.21 mA

b~1

f0~11.245 kHz

→ Im(Z/n) < 0.24 W (inj.)

Im(Z/n) < 0.15 W (top)

transverse criteria

stability limit from Landau octupoles at 7 TeV

(Berg-Ruggiero [8], Resta-Lopez [9])

for m=0 modes

the two lines refer to

opposite or equal polarity

of octupole families

roughly speaking: stability if Re (DQ)< 3x10-4, Im (DQ)<10-4;

on the next page we will estimate DQ and deduce the resulting

limit on the transverse shunt impedance

coherent coupled-bunch head-tail tune shifts “DQ”

F. Sacherer, Transverse Bunched Beam Instabilities, 9th HEACC, Stanford, Springfield 1975, 347

A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, 1993

conservatively assume hm~1 and that sampling frequency falls onto resonance;

note that sampling interval is ~40 MHz (25 ns spacing) so that for Q~100

mainly one single line contributes to a multibunch mode (~1GHz/(40MHz)~25);

consider the rigid dipole mode m=0 and, slightly pessimistically, b~5 km

→ Rsh,T<< 2 MW/m (top)

Rsh,L< 100 kW/#cavities

(Q < 100-200 OK)

Im(Z/n) << 0.10 W/#cavities

Rsh,T<< 2 MW/m/#cavities

[1] F. Ruggiero, “Single-Beam Collective Effects in the LHC,” CERN SL/95-09 (AP) 1995.

[2] D. Angal-Kalinin, “Review of Coupled-Bunch Instabilities in the LHC,” LHC Project-Report 595, http://doc.cern.ch/archive/electronic/cern/preprints/lhc/lhc-project-report-595.pdf

[3] LHC Design Report, Vol. 1, Chapter 5, CERN-2004-003-V-1 (2004).

[4] R. Tomas, “Optics for Crab Cavities in the LHC,” LHC-CC08, BNL, March 2008

[5] E. Shaposhnikova, “Longitudinal Beam Parameters During Acceleration in the LHC,” LHC Project Note 242 (2000).

[6] F.J. Sacherer, “A Longitudinal Stability Criterion for Bunched Beams”, IEEE Tr. Nucl.Sci. NS-20 , 825 (1973).

[7] V.I.Balbekov,S.V.Ivanov, “Thresholds of Longitudinal Instability of Bunched Beam in the Presence of Dominant Inductive Impedance,” IHEP Preprint 91-14, Protvino (1991).

[8] J.S. Berg, F. Ruggiero, “Landau Damping with Two-Dimensional Betatron Tune Spread,” CERN-SL-AP-96-071-AP (1996).

[9] J. Resta Lopez, “Design and Performance Evaluation of Nonlinear Collimation Systems for CLIC and LHC,” PhD Thesis, U. Valencia (2007).

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