1. LHC COLLIMATOR IMPEDANCE
E. Métral for the RLC team
Many thanks to all the people involved in the measurements!
 New results since my LTC talk on 08/12/04
Theory
Zotter2005’s formula + extension of its validity (in 2005)
GSI result for a LHC collimator (EPAC06 and after)
Effect of the finite length of the collimator (preliminary results in 06)
Measurements
Coherent tune shift from a LHC prototype at the SPS in 04 & 06
Bench measurement using 1 wire in 05 (by F. Caspers & T. Kroyer)
Coupled-bunch instability rise-time with 72 bunches in the SPS in 06
Effect of chromaticity and scans vs. gap, bunch spacing and bunch length
TCT and TCLI issue
Elias Métral, LTC, 06/12/06

2. ZOTTER2005’S THEORY FOR THE TRANSVERSE RESISTIVE-WALL IMPEDANCE
Zotter2004’s results revealed an impedance ~ 100 times higher at 8 kHz than the one from Burov-Lebedev (BL) / Vos / Tsutsui  This has been understood (error in the Mathematica Notebook)
The new (extended) Zotter’s theory is valid for a circular beam pipe
Any number of layers
Any beam velocity
Any frequency  Unification of 3 regimes (BL2002, “thick-wall” and Bane1991)
Any σ (conductivity), ε (permittivity) and μ (permeability)
… making the usual assumption of the infinite length
Elias Métral, LTC, 06/12/06

3. Elias Métral, LTC, 06/12/06

4. APPLICATION OF THE NEW ZOTTER2005’S FORMULA IN THE CASE OF A SPS MKE KICKER
Meas. by F. Caspers, T. Kroyer & E. Gaxiola
Theory plotted before the measurements!
Elias Métral, LTC, 06/12/06

5. GSI RESULT FOR A LHC COLLIMATOR EPAC2006 (26-30/06/06) On 13/10/06
d = 2.5 cm for the real LHC collimators
Elias Métral, LTC, 06/12/06

6. EFFECT OF THE FINITE LENGTH (Gluckstern and Zotter)
GEOMETRY OF THE PROBLEM
Elias Métral, LTC, 06/12/06

7. INFINITE LENGTH COMPUTATION FINITE LENGTH COMPUTATION
Application to the case of a LHC collimator
Length = g = 1 m
Half gap = b = 2 mm
Resistivity = 10 μΩm
Wall thickness = R = 2.5 cm
INFINITE LENGTH COMPUTATION
FINITE LENGTH COMPUTATION
(Preliminary…)
Elias Métral, LTC, 06/12/06

8. This nonlinear wake of FZ will not play an important role in the LHC
MEASUREMENT 1
Coherent tune shift from a LHC prototype collimator at the SPS (single bunch at 270 GeV/c) in 2004
Zimmermann et al., EPAC06
This nonlinear wake of FZ will not play an important role in the LHC
 This meas. can be fully explained but DOES NOT ADDRESS the issue of the inductive by-pass effect
Elias Métral, LTC, 06/12/06

9. b/s Nonlinear correction vs. gap (round beam)
LHC collimators settings  6 
Courtesy F. Zimmermann
b/s
Elias Métral, LTC, 06/12/06

10. MEASUREMENT 2
Transverse impedance from collimator bench measurements using 1 wire in 2005 (by F. Caspers & T. Kroyer)
From theory
 This meas. can be reasonably explained but DOES NOT ADDRESS the issue of the inductive by-pass effect as the frequency range is too high (from 57 MHz to 1.4 GHz)  Classical regime only!
Elias Métral, LTC, 06/12/06

11. The transverse beam sizes still need to be compared
MEASUREMENT 3
Coherent tune shift from a LHC prototype collimator at the SPS (single bunch at 270 GeV/c) in 2006
Doing this the classical thick-wall formula for the RW impedance is assumed
The transverse beam sizes still need to be compared
Courtesy R. Steinhagen
Elias Métral, LTC, 06/12/06

12. Instability rise-time with a batch of 72 bunches in the SPS in 2006
MEASUREMENT 4 (1/7)
Instability rise-time with a batch of 72 bunches in the SPS in 2006
Summary of the predicted rise-times (in SPS turns) for 1 batch of 72 bunches (1.151011 p/b) at 270 GeV/c (See APC 13/10/06)
Without the nonlinear terms of FZ, hoping to see the inductive bypass effect…
Factor ~ 2 difference
Elias Métral, LTC, 06/12/06

13. Relative chromaticity [×10]
MEASUREMENT 4 (2/7)
Case 1 (coast 5)
Collimator out ( 30 mm)
Intensity [1010 p/b]
i.e. ~ 1500 SPS turns
x  35 ms
y  50 ms
Relative chromaticity [×10]
Courtesy B. Salvant
Elias Métral, LTC, 06/12/06

14. Courtesy S. Redaelli & B. Salvant
MEASUREMENT 4 (3/7)
Courtesy S. Redaelli & B. Salvant
Elias Métral, LTC, 06/12/06

15. Relative chromaticity [×10]
MEASUREMENT 4 (4/7)
Case 5 (coast 12)
Collimator in (~ 2 mm)
x  [12, 32 ms]
Intensity [1010 p/b]
y  42 ms
Relative chromaticity [×10]
Courtesy B. Salvant
Elias Métral, LTC, 06/12/06

16. Courtesy S. Redaelli & B. Salvant
MEASUREMENT 4 (5/7)
Courtesy S. Redaelli & B. Salvant
Elias Métral, LTC, 06/12/06

17. Collimator  11.5 mm, and ξx = 0.04 (Anim_07h29)
MEASUREMENT 4 (6/7)
Collimator  11.5 mm, and ξx = 0.04 (Anim_07h29)
Courtesy F. Roncarolo
Elias Métral, LTC, 06/12/06

18. Collimator  3 mm, and ξx = 0.64 (Anim_07h33)
MEASUREMENT 4 (7/7)
Collimator  3 mm, and ξx = 0.64 (Anim_07h33)
Courtesy F. Roncarolo
 Coupled-bunch instability damped with a high chromaticity
Elias Métral, LTC, 06/12/06

19. TOP ENERGY Stability diagrams (Y-plane)
From Landau octupoles at max.
EFFECT OF CHROMATICITY
TOP ENERGY
Stability diagrams (Y-plane)
Mode 0
Mode 1
Mode 2
Mode 3
Elias Métral, LTC, 06/12/06

20. Elias Métral, LTC, 06/12/06

21. From nonlinearities + space charge (2D) Stability diagrams (X-plane)
INJECTION ENERGY
Stability diagrams (X-plane)
Mode 0
Mode 1
Mode 2
Mode 3
Elias Métral, LTC, 06/12/06

22. Elias Métral, LTC, 06/12/06

23. SCAN VS. GAP, BUNCH SPACING AND BUNCH LENGTH
1.1) For n most critical
Re (Q)  ~ 1/b2.5
Im (Q)  1/b2
1.2) For n = 3499
Im (Q)  ~ 1/b
Reminder: What is called the most critical (coupled-bunch) mode n is the (usual) one giving the largest imaginary part of the tune shift, but n = 3499 (= M - [Qx] - 1) gives the largest real part, and if we take the modulus it is the latter one which is the most critical
Elias Métral, LTC, 06/12/06

24. 2) SCAN VS. BUNCH SPACING (assuming a constant bunch intensity)
Re (Q) ~ constant (it is the "famous“ almost single-bunch effect due to the inductive bypass)
Im (Q)  ~ 1/bunch spacing
3) SCAN VS. BUNCH LENGTH
3.1) For n most critical
Re (Q) ~ 1/(bunch length)1/4
Im (Q) constant
3.2) For n = 3499
Re (Q) ~ 1/(bunch length)1/5
Elias Métral, LTC, 06/12/06

25. The trapped modes we will damped using ferrites
TCT AND TCLI ISSUE
The 2 TCTV at D1 (at points 2 and 8) and the TCLIB.6R2 (2-beam devices) have been simulated by A. Grudiev
For this 3 devices (2 used at top energy and 1 used at injection) the Broad-Band impedance is quite high  A minimum gap of 12 mm has been proposed
The trapped modes we will damped using ferrites
Elias Métral, LTC, 06/12/06

26.  “Very” confident in our impedance model!
CONCLUSION (1/3)
Zotter2005’s formula has been compared to other approaches from Burov-Lebedev2002, Tsutsui2003 (theory and HFSS simulations), Vos2003 and Al-khateeb et al. (GSI)2006  Similar results obtained in the new (Burov-Lebedev2002) low-frequency regime
The new (extended) Zotter2005’s formula has been used to compute the transverse impedance of a SPS MKE kicker and a good agreement has been obtained with 2-wire measurements from F. Caspers and T. Kroyer
 “Very” confident in our impedance model!
Theoretical work in progress
Finalize the work on the finite collimator length
Multi-bunch or “long” bunch  Wave velocity  Beam velocity
Elias Métral, LTC, 06/12/06

27. The 2006 measurements still need to be analyzed in detail
CONCLUSION (2/3)
All the measurements performed (and analyzed) so far are in agreement with our theoretical predictions but DO NOT ADDRESS the issue of the inductive by-pass effect
The 2006 measurements still need to be analyzed in detail
Coupled-bunch instability in the LHC induced by the collimators
At injection  Should be damped by a feedback (which should be able to damp instabilities with rise-times of few tens of ms)
At top energy  Should be damped by octupoles + (controlled) chromaticity (but then other problems may happen…)
Estimated max. intensity?  Few tens of % of the nominal one…
A good control of the tunes and chromaticities will be needed to increase the intensity
Elias Métral, LTC, 06/12/06

28.  Uses PS batches of 48 bunches in 2.4 s
CONCLUSION (3/3)
An “alternative” bunch filling scheme for the LHC is being proposed (in case of problems or as a possible step on the way to 72 bunches)
 Uses PS batches of 48 bunches in 2.4 s
More robust through the injector chain (less losses…)
Less bunches (2592 instead of 2808) and more gaps in the LHC  Better for the coupled-bunch instability induced by the collimators
8 % less bunches  8 % less luminosity
Filling time shorter by 5.5 % …
Elias Métral, LTC, 06/12/06

29. APPENDIX 1: A new physical regime for LHC  “Inductive by-pass”
First unstable betatron line
Skin depth for graphite (ρ = 10 μΩm)
Collimator thickness 
 One could think that the classical “thick-wall” formula would be about right
Elias Métral, LTC, 06/12/06

30.  A new physical regime was revealed by the LHC collimators
In fact it is not  The resistive impedance is ~ 2 orders of magnitude lower at ~ 8 kHz !
 A new physical regime was revealed by the LHC collimators
Usual regime :
New regime :
 This inductive by-pass effect is therefore observed even with a single layer extending up to infinity!
Elias Métral, LTC, 06/12/06

31. APPENDIX 2: New (extended) Zotter2005’s theory
Zotter2005’s formula for the transverse RW impedance is more precise than the one of BL in 1 aspect
 He considers both TE and TM modes, whereas BL considers only TM mode
Using Zotter’s formalism his formula was extended in 2 aspects. The new formula is now valid
Without making the “low-frequency” approximation
Without assuming (necessarily) a good conductor for the first layer
 This new Zotter2005’s formula is therefore more “precise” than the one from BL in 3 aspects
Elias Métral, LTC, 06/12/06

32. Elias Métral, LTC, 06/12/06

33. Elias Métral, LTC, 06/12/06

34. Graphite (2.5 cm, 10 μm) with vacuum outside
APPENDIX 3: Transverse resistive-wall impedance and coupled-bunch tune shift (at 7 TeV) for the case of a circular 1 m long collimator with half gap 2 mm and at average beta
Graphite (2.5 cm, 10 μm) with vacuum outside
Copper (2.5 cm, 17 nm) with vacuum outside
Graphite (2.5 cm, 10 μm) with vacuum outside and a copper coating (5 μm, 17 nm) inside
Elias Métral, LTC, 06/12/06

35. Transverse RW impedance
Elias Métral, LTC, 06/12/06

36. Coupled-bunch instability growth-rates vs. coupled-bunch mode n
COPPER
Most critical mode
Elias Métral, LTC, 06/12/06

37. Coupled-bunch instability growth-rates vs. coupled-bunch mode n
GRAPHITE
Most critical mode
Elias Métral, LTC, 06/12/06

38. Coupled-bunch instability growth-rates vs. coupled-bunch mode n
COPPER COATED GRAPHITE
Most critical mode
Elias Métral, LTC, 06/12/06

39. Stability diagram with the 6 previous tune shifts
(CuC,3489)
(C,3489)
(Cu,3499)
(C,3499)
(Cu,3489)
(CuC,3499)
Elias Métral, LTC, 06/12/06

40. APPENDIX 4: LHC tune shifts at 7 TeV vs. chromaticity
Single-bunch
Y-plane
Coupled-bunch
Elias Métral, LTC, 06/12/06

41. APPENDIX 5: TCT and TCLI issue
Per beam, there are
1 TCTH and 1 TCTV at each interaction point  4 TCTH and 4 TCTV
The 4 TCTH are all of the same type (1-beam pipe)
2 TCTV at D2 (i.e. where the 2 beams are well separated  1-beam pipe) at points 1 and 5
2 TCTV at D1 (i.e. where the 2 beams are not separated  2-beam pipe and large cavity created) at points 2 and 8
In addition there is also 1 TCLI (TCLIA.4R2.B1) used only at injection (1-beam pipe) and 1 TCLI (TCLIB.6R2) used also only at injection (but 2-beam pipe)
The 4 TCTH, the 2 TCTV at D2 and the TCLIA.4R2.B1 are all (the 7) of the same type (= 1-beam TCS-type device), and of the same type of the collimator prototype used in the SPS in For this we have measurements of the trapped modes by F. Caspers & T. Kroyer, and simulations by A. Grudiev  OK
Elias Métral, LTC, 06/12/06

42. The new result is for the 2 TCTV at D1 and for the TCLIB
The new result is for the 2 TCTV at D1 and for the TCLIB.6R2 (2-beam devices), which have been followed-up by AG. And this is for this 3 devices (2 used at top energy and 1 used at injection) that the Broad-Band impedance is quite high. For the trapped modes we think we will be able to damp them with ferrites (opening the RF bypass for the ferrite to be effective or keeping the RF bypass but creating a small cavity to put the ferrite)
Elias Métral, LTC, 06/12/06

43. Conclusion: A (full) gap of 12 mm is needed for 2 devices and 30 mm in case of more than 2 devices
Elias Métral, LTC, 06/12/06

44.  In this case, the impedance will increase by a factor 10
 12 mm is chosen to have a BB impedance per device which is about 1% of the total BB impedance of the design report (j 2.67 MΩ/m at top energy), i.e. ~ j 0.02 MΩ/m in the previous plot
However this result is obtained for the nominal case where β  βav (~ 70 m)
If, as discussed with RA, the beam is squeezed in 2 and 8, the β function will increase to ~ 660 m, i.e. by a factor 10
 In this case, the impedance will increase by a factor 10
 The impedance has to be reduced by a factor 10
 The full gap has to be increased from 12 mm to 30 mm (see previous plot). In this case the BB impedance is similar to the 1-beam TCS-type device
 This limits the squeeze in 2 and 8*
* Reminder: The TCTV primary function is shadowing the triplet, which half aperture is ~ 10 mm  It should be at ~ 8 mm (half gap) maximum to play is role
Elias Métral, LTC, 06/12/06

45. The TCLI is used only at injection and to protect the arc
The TDI alone protects until ~ 50% of the nominal intensity
For higher intensities, the TCLI (graphite) is absolutely needed
A full gap of 12 mm is also chosen for the TCLIB.6R2 to have a BB impedance which is the same percentage of the total BB impedance of the design report (j 1.34 MΩ/m at injection) as for the 2 TCTV at D1 at top energy
Elias Métral, LTC, 06/12/06