1. New results on impedances, wake fields and electromagnetic fields in an axisymmetric beam pipe
N. Mounet and E. Métral
Acknowledgements: B. Salvant, B. Zotter, G. Rumolo, F. Caspers, F. Roncarolo.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

2. Context and general purpose
Impedance / wake fields (electromagnetic force on a particle due to a passing particle offsetted from the pipe center) are a source of instabilities / heat load.
In LHC, low revolution frequency and low conductivity material used in collimators → classic thick wall formula for the impedance is not valid:
we need the general formalism (B. Zotter / E. Métral) to compute impedances and wake fields,
the six electromagnetic field components are of interest to understand the global physical picture.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

3. Electromagnetic fields in a resistive cylindrical beam pipe
Analytic calculation from B. Zotter & E. Métral’s formalism
Position of the problem
General calculation of the field components in frequency domain from Maxwell’s equations
Field matching to get the constants
 multilayer case (matrix formalism)
Force (wake field) and impedance computation (summing all the azimuthal modes)
Approximations in the single-layer/conductor case
 impedance in the first frequency regime
Back to the time domain
Recovering Chao’s formulas for the second frequency regime
Wake field computation in the general case → new FT technique
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

4. Position of the problem
Beam circulating at speed v in a cylindrical pipe made of any multilayer linear medium.
The beam is a macroparticle (charge Q) offsetted from center.
In time domain:
In frequency domain:
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

5. Position of the problem
Azimuthal mode decomposition
 the point charge is replaced by a sum of ring-shaped charge densities.
Consider for the computation only one mode at a time (but we can sum all the modes in the end)
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

6. Multilayer field matching: Matrix formalism
The field components in each layer involve 4 constants, determined from field matching (continuity of the tangential field components) between the adjacent layers.
It is possible to relate the constants of a layer to those of the previous one using a matrix, finally getting between the first and last layer:
Constants (last layer) = M . constants (first layer)
 We get everything from a 4x4 matrix (M) which is the product of N-1 (relatively) simple 4x4 matrices (we do all of this analytically, under Mathematica).
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

7. Comparison with existing exact 2-layers formula and L
Comparison with existing exact 2-layers formula and L. Vos’s thin wall formula
For 2 layers we can compare with Elias’s formula, and the thin wall with inductive bypass one, when the pipe thickness is very small (with respect to its radius)
450 Gev, stainless steel, radius = 20 mm, thickness = 0.02mm
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

8. Comparison with existing multilayer code (from Benoît Salvant)
For 3 layers (the code is still Mathematica® in symbolic, the only difference is in the way to compute aTM)
Copper coated LHC graphite collimator (450 GeV)
Courtesy of B. Salvant (existing code)
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

9. Comparison with other multilayer formalisms
For 3 layers (copper coated LHC graphite collimator at 450 GeV)
Close agreement, except:
At very high frequency with BL
At low frequency (real part) with LV (at high freq., numerical pb – we did not use Mathematica)
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

10. Some multilayered examples
Calculation from 1 up to 5 layers (LHC graphite collimator at 450 GeV)
The biggest difference is between graphite alone and copper coated graphite
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

11. Matrix formalism: conclusion
We are no longer limited to 3 layers (we computed up to 5 layers).
Computation of the transverse impedance is about 130 faster than previously, in the case of three layers (computation time of a few hours is replaced by a few minutes).
We might find some useful approximations for some specific multilayer pipe, and derive simple analytical formula for the impedance (idea from S. Fartoukh).
Hahn and Ivanyan already thought about a similar matrix method (see H. Hahn, Matrix solution to longitudinal impedance of multi-layer circular structures, BNL, C-A/AP/#336, and M. Ivanyan et al, Multilayer tube impedance and external radiation, PRST-AB 2008), but not in the same formalism.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

12. Single-layer/conductor case: first result in the low frequency regime
LHC graphite collimator transverse impedance
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

13. Real part of the transverse wall impedance in the first frequency regime
From Elias Métral’s transverse impedance formula (developing the Bessel functions up to second order – the first order giving the inductive bypass limit) we get at low frequencies
d : skin depth
 proportional to -w ln(w), and to s ,
 what matters is d instead of b
F. Roncarolo et al, Phys. Rev. ST Accel. Beams 12, (2009)
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

14. Single-layer/conductor case: second frequency regime in time domain (m=1)
We recovered Chao’s formulas for the EM fields in vacuum in the second frequency regime (classic thick wall approximation), with additional dependencies (in b and in the permeability mr)
z is the distance behind the bunch, b the pipe radius, s its conductivity, P1 = Q a .
In that regime, the transverse electric field components are small with respect to the magnetic ones → validation of approximations made in measurements.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

15. Back to time domain: Wake fields from analytically computed wall impedances
In the general case (exact formulas), the wall impedance cannot be exactly Fourier transformed into wake fields
 A new Fourier transform technique to compute the wake
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

16. Summary of what has been done
Full derivation of the exact electromagnetic field components in frequency domain (multilayer, multimode) (a note will be published shortly)
A new and efficient matrix formalism in the multilayer case, to be linked with ZBase
In the single-layer case, with a pipe wall made of a conductive material
Impedance formula (real part) in the first frequency regime
EM fields inside the pipe, in the second frequency regime (in time domain as well) → we recover Chao’s formulas
For the general and “exact” time domain computation
Wake fields (and fields) computation using a new and accurate FT technique.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009

17. Future outlook
Electromagnetic fields computation in the single-layer/conductor case
Compute the field components in the first frequency regime (frequency domain then if possible time domain)
In the first and second regimes, we also want the fields in the pipe wall, to get useful physical insight.
From the exact impedance obtained, multibunch instabilities growth times in LHC have to be
analytically computed (Sacherer’s formula, and time domain analytical approach)
simulated using Headtail
→ the code needs to be extended.
BE/ABP/LIS section meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 07/09/2009