1. N. Mounet and E. Métral - ICE meeting - 16/03/201 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.

2. N. Mounet and E. Métral - ICE meeting - 16/03/202 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).

3. N. Mounet and E. Métral - ICE meeting - 16/03/203 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).

4. N. Mounet and E. Métral - ICE meeting - 16/03/204 Multilayer cylindrical chamber (Zotter formalism) Chamber cross section  Source (in frequency domain, k=  /v) decomposed into azimuthal modes: whereis the wave number.

5. N. Mounet and E. Métral - ICE meeting - 16/03/205 Multilayer cylindrical chamber: Zotter formalism (CERN AB-2005-043)  For each azimuthal mode we write Maxwell equations in each layer (in frequency domain) where  c and  are general frequency dependent permittivity and permeability (including conductivity).  Taking the curl of the 3 rd equation and injecting the 1 st and the 2 nd ones:  Taking the curl of the 2 nd equation and injecting the 3 rd and the 4 th ones: with

6. N. Mounet and E. Métral - ICE meeting - 16/03/206 Multilayer cylindrical chamber: longitudinal components  Along the longitudinal axis → “simple” (uncoupled) Helmholtz equations:  For E s the equation is inhomogeneous (right-hand side is the driving term from the beam), but homogeneous for H s.  Outside ring-shaped source ρ m → homogeneous → separation of variables: → get harmonic differential equations for both Θ and S. and

7. N. Mounet and E. Métral - ICE meeting - 16/03/207 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 m e and m h are not necessarily equal to m.  Reinjecting those into the Helmholtz equations for E s and H s, we get Bessel’s equation (here written for E s ):

8. N. Mounet and E. Métral - ICE meeting - 16/03/208 Multilayer cylindrical chamber : longitudinal components  Introducing the radial propagation constant →are modified Bessel functions of order m e and m h. → There are 4 integration constants per layer: C Ie, C Ke, C Ih and C Kh.

9. N. Mounet and E. Métral - ICE meeting - 16/03/209 Multilayer cylindrical chamber : transverse components  In each layer, all the transverse components can be obtained from the longitudinal ones: reinjecting E s and H s into the 2 nd and 3 rd Maxwell equations and using again the invariance properties along the s axis: Superscript (p) indicates quantities taken in the cylindrical layer p.

10. N. Mounet and E. Métral - ICE meeting - 16/03/2010 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=a 1 (continuity of E s and H s at the ring-shaped source, and two additional relations for dE s /dr and dH s /dr by integration of the wave equations between r=a 1 -δa 1 and r=a 1 +δa 1 ).  Fields should stay finite at r=0 and r=∞ → take away constants C Ke and C Kh in the first layer, and C Ie and C Ih in the last one → only 4N unknowns. Except for p=0 (surface charge & current at the ring- shaped source) → 4(N-1) equations.

11. N. Mounet and E. Métral - ICE meeting - 16/03/2011 But … what about those azimuthal mode numbers ?  There are still those integers m e and m h 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 m e = m h = m  This has to do with the axisymmetry → if no axisymmetry, there would be some coupling between different azimuthal modes. 

12. N. Mounet and E. Métral - ICE meeting - 16/03/2012 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 M p+1,p is an explicit 4x4 matrix, and C Ig =   c C Ih, C Kg =   c C Kh “Source” term, due to the beam (from matching at r=a 1 )

13. N. Mounet and E. Métral - ICE meeting - 16/03/2013 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)  TM (m)= the only “wall” constant (frequency dependent) needed to compute the impedance. 1 if m=0, 0 otherwise

14. N. Mounet and E. Métral - ICE meeting - 16/03/2014 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   (m) are constants (still dependent on  ).  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 ka 1 /  << 1 and kr /  << 1.

15. N. Mounet and E. Métral - ICE meeting - 16/03/2015 Cylindrical chamber wall impedance  Total impedance: EM force on a test particle in (r=a 2,   ), 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 (x 1 = source coordinate, x 2 = test coordinate) New quadrupolar term

16. N. Mounet and E. Métral - ICE meeting - 16/03/2016 Cylindrical chamber EM fields results  Example: Fields 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 a 1 =10  m:

17. N. Mounet and E. Métral - ICE meeting - 16/03/2017 Cylindrical chamber wall impedance results For 3 layers (  m-copper coated round graphite collimator surrounded by stainless steel, at 450 GeV with b=2mm), dipolar and quadrupolar impedances (per unit length):  New quadrupolar impedance small except at very high frequencies.  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).

18. N. Mounet and E. Métral - ICE meeting - 16/03/2018 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).

19. N. Mounet and E. Métral - ICE meeting - 16/03/2019 Longitudinal impedance at low frequency (F. Caspers’ question) For 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  → Im(Z || )=0 is forced.  But the real part has to be symmetric.  Re(Z || )=0 means zero power loss at 

20. N. Mounet and E. Métral - ICE meeting - 16/03/2020 EM fields at low frequency (F. Caspers’ question) For a 1 layer LHC round graphite collimator (b=2mm) at injection, EM fields at the surface of the wall, vs. frequency (note: G=  0 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 symmetry considerations, in our 2D model E s 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)

21. N. Mounet and E. Métral - ICE meeting - 16/03/2021 Impedance at high frequency  Why is it that the beam-coupling impedance always go to zero at high frequency ?  Answer: there is 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 ~  / 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.

22. N. Mounet and E. Métral - ICE meeting - 16/03/2022 Multilayer flat chamber Chamber cross section (no a priori top-bottom symmetry)  Source (in frequency domain) decomposed using an horizontal Fourier transform: Source used

23. N. Mounet and E. Métral - ICE meeting - 16/03/2023 Multilayer flat chamber: outline of the theory  For each horizontal wave number k x, 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 k x (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 k x: Constants (depend on  and k x, obtained from field matching)

24. N. Mounet and E. Métral - ICE meeting - 16/03/2024 Multilayer flat chamber: integration over k x  Next step is to integrate over k x to get the total fields from the initial point-like source (equivalent of multimode summation in cylindrical)  The last term in E s 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 k x. We could get numerically the fields but computation would have to be done for each x, y, y 1 and  → really heavy.  Somehow, we would like to sort out the dependence in the source and test particles positions.

25. N. Mounet and E. Métral - ICE meeting - 16/03/2025 Multilayer flat chamber: integration over k x  The great thing is that we can transform into with  mn given by (numerically computable) integrals over k x of (complicated) frequency dependent quantities. To do so: - Transform (x,y) to cylindrical coordinate (r,  ), - Make a Fourier series decomposition, - Demonstrate and use - Use

26. N. Mounet and E. Métral - ICE meeting - 16/03/2026 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 ky 1 /  and kr /  << 1 (x 1 & y 1 and x 2 & y 2 = positions of the source and test particles): Quadrupolar terms not exactly opposite to one another (≠ A. Burov –V. Danilov, PRL 1999, ultrarelativistic case) + Constant term in vertical when no top-bottom symmetry:

27. N. Mounet and E. Métral - ICE meeting - 16/03/2027 Comparison to Tsutsui’s formalism For 3 layers (see parameters in previous figures), 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.

28. N. Mounet and E. Métral - ICE meeting - 16/03/2028 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.

29. N. Mounet and E. Métral - ICE meeting - 16/03/2029 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.