1. General wall impedance theory for 2D axisymmetric and flat multilayer structures
N. Mounet and E. Métral
Acknowledgements: N. Biancacci, F. Caspers, A. Koschik, G. Rumolo, B. Salvant, B. Zotter.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

2. Context and motivation
Beam-coupling impedances & wake fields (i.e. electromagnetic forces on a particle due to another passing particle) are a source of instabilities / heat load.
In the LHC, low revolution frequency and low conductivity material used in collimators → classic thick wall formula (discussed e.g. in Chao’s book) for the impedance not valid e.g. at the first unstable betatron line (~ 8kHz):
need a general formalism with less assumptions on the material and frequency range to compute impedances (also for e.g. ceramic collimator, ferrite kickers).
N. Mounet and E. Métral - ICE meeting - 23/03/2011

3. Two dimensional models
Ideas:
consider a longitudinally smooth element in the ring, of infinite length, with a point-like particle (source) travelling near its center, along its axis and with constant velocity v,
integrate the electromagnetic (EM) force experienced by a test particle with the same velocity as the source, over a finite length.
Neglect thus all edge effects → get only resistive effects (or effects coming from permittivity & permeability of the structure) as opposed to geometric effects (from edges, tapering, etc.).
Main advantage: for simple geometries, EM fields obtained (semi-) analytically without any other assumptions (frequency, velocity, material properties – except linearity, isotropy and homogeneity).
N. Mounet and E. Métral - ICE meeting - 23/03/2011

4. Multilayer cylindrical chamber (Zotter formalism)
Chamber cross section
 Source (in frequency domain, k=w /v) decomposed into azimuthal modes: 1
where
is the wave number.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

5. Multilayer cylindrical chamber: Zotter formalism (CERN AB-2005-043)
For each azimuthal mode we write Maxwell equations in each layer (in frequency domain)
where ec and m are general frequency dependent permittivity and permeability (including conductivity).
Taking the curl of the 3rd equation and injecting the 1st and the 2nd ones:
Taking the curl of the 2nd equation and injecting the 3rd and the 4th ones:
with
N. Mounet and E. Métral - ICE meeting - 23/03/2011

6. Multilayer cylindrical chamber: longitudinal components
Along the longitudinal axis → “simple” (uncoupled) Helmholtz equations:
For Es the equation is inhomogeneous (right-hand side is the driving term from the beam), but homogeneous for Hs.
Outside ring-shaped source ρm → homogeneous → separation of variables:
→ get harmonic differential equations for both Θ and S.
and
N. Mounet and E. Métral - ICE meeting - 23/03/2011

7. Multilayer cylindrical chamber : longitudinal components
From symmetry with respect to the θ=0 (mod π) plane, translation invariance of vector , and invariance w.r.t :
Up to now, no boundary condition have been used, and the integers me and mh are not necessarily equal to m.
Reinjecting those into the Helmholtz equations for Es and Hs, we get Bessel’s equation (here written for Es):
N. Mounet and E. Métral - ICE meeting - 23/03/2011

8. Multilayer cylindrical chamber : longitudinal components
Introducing the radial propagation constant
are modified Bessel functions of order me and mh.
There are 4 integration constants per layer: CIe , CKe , CIh and CKh.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

9. Multilayer cylindrical chamber : transverse components
In each layer, all the transverse components can be obtained from the longitudinal ones: reinjecting Es and Hs into the 2nd and 3rd Maxwell equations and using again the invariance properties along the s axis:
Superscript (p) indicates quantities taken in the cylindrical layer p.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

10. Multilayer field matching
4(N+1) integration constants to determine from field matching (continuity of the tangential field components) between adjacent layers:
4 more equations at r=a1 (continuity of Es and Hs at the ring-shaped source, and two additional relations for dEs /dr and dHs /dr by integration of the wave equations between r=a1 -δa1 and r=a1 +δa1).
Fields should stay finite at r=0 and r=∞ → take away constants CKe and CKh in the first layer, and CIe and CIh in the last one → only 4N unknowns.
Except for p=0 (surface charge & current at the ring-shaped source) → 4(N-1) equations.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

11. But … what about those azimuthal mode numbers ?
There are still those integers me and mh to determine in each layer !
To ”everybody” it looked like those had to be necessarily equal to the initial azimuthal mode number of the ring-shaped source, m (see R. Gluckstern, B. Zotter, etc.). They did not even mention that there is something to prove here…
Using the field matching relations of the previous slide, it is actually possible (and lengthy) to prove that in any layer
me = mh = m
This has to do with the axisymmetry → if no axisymmetry, there would be some coupling between different azimuthal modes.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

12. Multilayer field matching: Matrix formalism
In the initial formalism, solves “with brute force” the full system of equations (4N eqs., 4N unknowns)  computationally heavy for more than 2 layers.
But we can relate constants between adjacent layers with 4x4 matrices:
In the end, with the conditions in the first and last layers:
where Mp+1,p is an explicit 4x4 matrix, and CIg=m0 c CIh , CKg=m0 c CKh
“Source” term, due to the beam (from matching at r = a1)
N. Mounet and E. Métral - ICE meeting - 23/03/2011

13. Multilayer field matching: Matrix formalism
The matrix between adjacent layers are explicitly found as:
with
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14. Multilayer field matching: Matrix formalism
To solve the full problem, only need to multiply (N -1) 4x4 matrices and invert explicitly a 2x2 matrix: in the vacuum region
Still some numerical accuracy problems, so need to do this with high precision real numbers (35 digits, typically).
Note: other similar matrix formalisms developed independently in H. Hahn, PRSTAB 13 (2010), M. Ivanyan et al, PRSTAB 11 (2008), N. Wang et al, PRSTAB 10 (2007)
aTM(m)= the only “wall” constant (frequency dependent) needed to compute the impedance.
1 if m=0, 0 otherwise
N. Mounet and E. Métral - ICE meeting - 23/03/2011

15. Total fields: multimode extension of Zotter’s formalism
Up to now we obtained the EM fields of one single azimuthal mode m.
Sum all the modes to get the total fields due to the point-like source:
and aTM(m) are constants (still dependent on w).
First term = direct space-charge → get the direct space-charge for point- like particles (fully analytical).
Infinite sum = “wall” part (due to the chamber). Reduces to its first two terms in the linear region where ka1 /g << 1 and kr /g << 1.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

16. Cylindrical chamber wall impedance
Total impedance: EM force on a test particle in (r=a2, q = q2), in frequency domain, integrated over some length L (length of the element), normalized by the source and test charges (+ some sign / phase):
Taking the linear terms only, the “wall” impedances are then (x1 = source coordinate, x2 = test coordinate) L a2 q2
New quadrupolar term
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17. Cylindrical chamber EM fields results
Example: Fields (including direct space-charge) in a two layers round graphite collimator (b=2mm) surrounded by stainless steel, created by the mode m=1 of a 1C charge (energy 450 GeV) with a1=10mm:
N. Mounet and E. Métral - ICE meeting - 23/03/2011

18. Cylindrical chamber wall impedance results
For 3 layers (10mm-copper coated round graphite collimator surrounded by stainless steel, at 450 GeV with b=2mm), dipolar and quadrupolar impedances (per unit length):
 Importance of the wall impedance (= resistive-wall + indirect space-charge) at low frequencies, where perfect conductor part cancels out with magnetic images (F. Roncarolo et al, PRSTAB 2009).
 New quadrupolar impedance small except at very high frequencies.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

19. Comparison with other formalisms
In the single-layer and two-layer case, some comparisons done in E. Métral, B. Zotter and B. Salvant, PAC’07 and in E. Métral, PAC’05.
For 3 layers (see previous slide), comparison with Burov-Lebedev formalism (EPAC’02, p. 1452) for the resistive-wall dipolar impedance (per unit length):
Close agreement, except:
at very high frequency (expected from BL theory),
at very low frequency (need to be checked).
N. Mounet and E. Métral - ICE meeting - 23/03/2011

20. Longitudinal impedance at low frequency (F. Caspers’ question)
For e.g. a 1 layer LHC round graphite collimator (b=2mm), longitudinal impedance per unit length goes to zero at low frequency:
The imaginary part has to be antisymmetric with respect to w=0 → Im(Z|| )=0 is forced.
But the real part has to be symmetric → Re(Z|| )=0 not forced at zero frequency.
Re(Z|| )=0 means zero power loss at w=0.
In our 2D model, Re(Z|| ) always goes to zero at zero frequency.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

21. EM fields at low frequency (F. Caspers’ question)
For a 2 layers LHC round graphite collimator (b=2mm, thickness 25mm, vacuum around) at injection, EM fields at the outer surface of the wall vs. frequency (note: G=m0 c H):
All the electric field components go to zero in the wall, at low frequencies → no heat load since no current density (from Ohm’s law),
From translation invariance of the potential along s (2D model) Es has to be zero at DC → zero longitudinal impedance.
If it’s non zero “in real life”, it has to come from 3D (e.g. EM fields trapped when beam enters a structure)
N. Mounet and E. Métral - ICE meeting - 23/03/2011

22. Impedance at high frequency
Why is it that the beam-coupling impedance always goes to zero at high frequency ?
Answer: there are no more induced currents in the wall at high frequency, because the EM fields from the beam (=direct space-charge, i.e. the fields present if no boundary was there) decays before reaching the wall:
→ decays in a length ~g / k , so angular frequency cutoff around
(b = wall radius)
Frequency cutoff typically around GHz for low energy machines, and 10THz for LHC collimators of 2mm radius, at injection.
This cutoff DISAPPEARS if g = ∞ → fundamentally non-ultrarelativistic effect !
N. Mounet and E. Métral - ICE meeting - 23/03/2011

23. Multilayer flat chamber
Chamber cross section (no a priori top-bottom symmetry)
 Source (in frequency domain) decomposed using an horizontal Fourier transform:
Source used
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24. Multilayer flat chamber: outline of the theory
For each horizontal wave number kx, solve Maxwell equations in a similar way as what was done in the cylindrical case, in cartesian coordinates (with source = ) → separation of variables, harmonic differential equations in each layer.
Same kind of considerations for the horizontal wave number kx (all equal in the layers) + field matching with matrix formalism (two 4x4 matrices in the end, one for the upper layers and one for the lower layers).
Longitudinal electric field component in vacuum, for a given kx:
Constants (depend on w and kx, obtained from field matching)
N. Mounet and E. Métral - ICE meeting - 23/03/2011

25. Multilayer flat chamber: integration over kx
Next step is to integrate over kx to get the total fields from the initial point-like source (equivalent of multimode summation in cylindrical)
The last term in Es can be integrated exactly: in both layers 1 and -1
→ this is the direct-space charge (see cylindrical).
The other two terms (“wall” part of the fields) are a much bigger problem: the integration constants are highly complicated functions of kx
We could get numerically the fields but computation would have to be done for each x, y, y1 and w → really heavy.
Somehow, we would like to sort out the dependence in the source and test particles positions.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

26. Multilayer flat chamber: integration over kx
Idea: in the end we should get a formula similar to the axisymmetric case in the vicinity of the beam (we could always solve the problem in some artificial cylinder inside the chamber)
So to get something similar, let’s first move to cylindrical coordinate (slightly modified: (r,f)=(r,q-p/2), better since invariance f →-f), then make a Fourier series decomposition :
Fortunately, there is a formula for the last integral:
… but I had to redemonstrate it (do not blindly trust Math books…)
with
i.e.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

27. Multilayer flat chamber: integration over kx
We have isolated the dependence in the test particle coordinates. For impedance considerations we also would like to isolate the dependence in the source particle coordinates. For this we need first to explicit the dependence on y1 (source vertical offset) of the integration constants:
Then, with the change of variable , , the formula
with amn given by
Analytical functions of w and kx , obtained from field matching
, and plugging everything back in, we get finally:
→ numerically computable integrals over kx of frequency dependent quantities (but only frequency dependent).
N. Mounet and E. Métral - ICE meeting - 23/03/2011

28. Flat chamber wall impedance
Direct space-charge impedances are the same as in the cylindrical case (as expected).
From wall part (infinite sums) → get wall impedance in linear region where ky1 /g << 1 and kr /g << 1 (x1 & y1 and x2 & y2 = positions of the source and test particles):
+ Constant term in vertical when no top-bottom symmetry:
Quadrupolar terms not exactly opposite to one another (≠ A. Burov –V. Danilov, PRL 1999, ultrarelativistic case)
N. Mounet and E. Métral - ICE meeting - 23/03/2011

29. Comparison to Tsutsui’s formalism
For 3 layers (LHC copper-coated graphite collimator, see slide 18), comparison with Tsuitsui’s model (LHC project note 318) on a rectangular geometry, the two other sides being taken far enough apart:
 Very good agreement between the two approaches.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

30. Form factors between flat and cylindrical wall impedances
Ratio of flat chamber impedances w.r.t longitudinal and transverse dipolar cylindrical ones → generalize Yokoya factors (Part. Acc., 1993, p. 511). In the case of a single-layer ceramic (hBN) at 450 GeV:
 Obtain frequency dependent form factors quite ≠ from the Yokoya factors.
N. Mounet and E. Métral - ICE meeting - 23/03/2011

31. Conclusion
For multilayer cylindrical chambers, Zotter formalism has been extended to all azimuthal modes, and its implementation improved thanks to the matrix formalism for the field matching.
The number of layers is no longer an issue.
For multilayer flat chambers, a new theory similar to Zotter’s has been derived, giving also impedances without any assumptions on the materials conductivity, on the frequency or on the beam velocity (but don’t consider anomalous skin effect / magnetoresistance).
Both these theories were benchmarked, but more is certainly to be done (e.g. vs. Piwinski and Burov-Lebedev, for flat chambers).
New form factors between flat and cylindrical geometries were obtained, that can be quite different from Yokoya factors, as was first observed with other means by B. Salvant et al (IPAC’10, p. 2054).
Other 2D geometries could be investigated as well.
N. Mounet and E. Métral - ICE meeting - 23/03/2011