Contributions to the impedance budget → impedance of pumping holes and interconnects Bernard Riemann FCC-hh impedance and beamscreen workshop 2017-03-30 1 / 17
Contributions to impedance budget A list of (passive) contributions ⊲ beam pipe crossings ⊲ RF cavity resonators ⊲ collimators Beam screen: ⊲ interconnects ⊲ vacuum pumping holes ⊲ surface roughness ⊲ coatings ⊲ resistive-wall impedance ⊲ Essentially, everything that changes the cross-section contributes to the impedance (with the addition of resistive-wall effects) ⊲ Characteristic (wave)length of the contributions span orders of magnitude 2 / 17
Pumping holes ⊲ High pumping efficiency needed to maintain low pressure in beam pipe ⊲ This goal for itself has a large technical margin for the sizes of pumping holes (e.g. hole length (z): 1 . 5 mm → 25 . 75 mm ) ⊲ The characteristic hole size lead to different approaches in impedance computation from: I. Bellafont, ”Studies on the beam-induced effects in the FCC-hh”, EuroCirCol meeting, Barcelona (2016-11) 3 / 17
Pumping hole impedance Computational challenges ⊲ How to compute the small field contributions of a part hidden behind a shield? ⊲ For a limited piece of vacuum pipe, the difference between 3d FEM/FIT models with/without hole ”drowns in numerical noise” ⊲ Similar to external-Q computation problem in SRF cavities ⊲ For small holes only: how to approximate small apertures in a much larger structure? ⊲ 3d FEM/FIT require ultra-fine meshes (number of mesh cells ∝ λ − d min ) ⊲ using 2d Poisson solvers or simple pipe geometries,one can treat a variety of small hole types as perturbations, ( → slide 6) ⊲ Where does the computational domain end? ⊲ Electromagnetic fields emanate through the holes and get reflected behind these holes by the respective materials. ⊲ A reasonable limit would be the ”next order” of holes, e.g., holes in the sourrounding material are neglected, and we assume the surrounding to be circular homogenic. 4 / 17
Periodicity of pumping holes ⊲ Effects of pumping holes are negligible on a local scale, but can constructively interfere and add up to a significant contribution due to periodicity ⊲ Analytical treatments exist 123 but in most of them holes necessarily are approximated as small-size perturbations. ⊲ The negligible local effect limits wakefield solvers even for larger holes. ⊲ For distributed structures, we need an impedance estimate per length. ⇒ Find a way to utilize structure periodicity to compute estimates. 1 A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014 2 G. Stupakov, Phys. Rev. E 51 , 3515 (1995) 3 S.S. Kurennoy, Part Acc. 39 , 1 (1992) 5 / 17
Dispersion in periodic structures ⊲ See also: B. Isbarn et al., WEPMA026, Proc. IPAC15 ⊲ In a 2d waveguide-like structure, phase velocity is superluminal for TE/TM modes and luminal for TEM (coaxial) modes ⊲ longitudinal variation allows for TE/TM phase velocities ≤ c . 4 ⊲ In this case, the resonance condition between particle beam and (mixed) TM/TE modes can be fulfilled. 2d problem (TE/TM) 2d problem, no dispersion (TEM, beam) z-dependent (TE/TM) frequency / a.u. long. wavenumber / a.u. ⊲ Group velocity d ω /dk around the resonance condition is connected to the external quality factor of a large number of periods in a segment. 4 T. Wangler, RF linear accelerators , 2nd ed. (Wiley-VCH, 2008) 6 / 17
Periodicity of pumping holes Simplified vacuum geometry for the FCC beam screen, rendered with COMSOL Multiphysics 5.2, COMSOL Inc., http://www.comsol.com/ ⊲ Compute long. dispersion diagram of the pipe numerically using Floquet-periodic boundaries in beam direction ⊲ Analytical dispersion diagrams used for concentric beam pipes. 5 ⊲ Numerical dispersion diagrams frequently used for multi-cell (S)RF. 6 5 A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014 6 T. Wangler, RF Linear accelerators, 2nd ed. (Wiley-VCH, 2008) 7 / 17
Periodicity of pumping holes Compute using electric or magnetic boundary conditions on the vertical half-plane. Expectations: ⊲ ”hidden” modes with very similar frequencies for both cases. ⊲ londitudinal geometry variation ⇒ band gaps (in dependence on hole influence on mode) ⊲ At very high frequencies: crossings of the ω = ck line (slow-down of waveguide phase velocity due to holes) Mesh for one half of the geometry, rendered with ⊲ The fields for these resonance COMSOL Multiphysics 5.2, COMSOL Inc., conditions are most important http://www.comsol.com/ 8 / 17
Periodicity of pumping holes 10 mag. bound. el. bound. 9 ω = ck 8 mode frequency / GHz 7 6 5 4 3 0 20 40 60 80 100 120 long. wave number / (rad/m) Dispersion diagram for the shown vacuum geometry 9 / 17
16 mag. bound. el. bound. 15 ω = ck mode frequency / GHz 14 13 12 11 10 0 20 40 60 80 100 120 long. wave number / (rad/m) ⊲ Proper seperation of modes is an issue (approach: Fourier decomposition of on-axis field) ⊲ Frequency limitation due to no of meshcells 10 / 17
Pumping hole impedance: Field matching A inner beamscreen mantle with sync.rad. slots (2d) B outer beamscreen mantle with pumping holes (3d) C vacuum enclosure (2d) sketch of a simpli fi ed FCC chamber geometry with cylindrically orthogonal surfaces ⊲ Approach based on Fedotov and Gluckstern who computed a rectangular pipe hole with an outer enclosing pipe 7 (B & C) ⊲ r < r A : evanescent TE/TM modes, r > r A : TEM/TE/TM (coaxial) ⊲ Neglect material size and sync.rad. ”mirrors” 7 A.V. Fedotov and R.L. Gluckstern, Phys. Rev. E 56 (3) (1997) 11 / 17
Pumping hole impedance: Field matching A inner beamscreen mantle with sync.rad. slots (2d) B outer beamscreen mantle with pumping holes (3d) C vacuum enclosure (2d) sketch of a simpli fi ed FCC chamber geometry with cylindrically orthogonal surfaces ⊲ Can be viewed as continuation of semi-analytical impedance models, 8 but without dipole approximations ⊲ Can allow computation of wake impedance on a large frequency scale ⊲ Generalization of known cases (cross-checking possible) 8 A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014 12 / 17
Interconnects Three main approaches ⊲ Wakefield computation (CST only) ⊲ Compute distortion and resulting wake for a single interconnect ⊲ Eigenmodes of resonant structure (comparison) ⊲ Periodic boundary conditions ⊲ Only approach in list that includes coupling of interconnects within reasonable computation time. ⊲ Assuming the large number of interconnects in the ring, periodicity is a good approximation as a boundary condition. 13 / 17
Interconnects 2d problem (TE/TM) 2d problem, no dispersion (TEM, beam) z-dependent (TE/TM) frequency / a.u. long. wavenumber / a.u. Eigenmode comparison ⊲ Recompute results from a thesis on impedances of beamscreen interconnects which were done using CST. a ⊲ Trapped eigenmodes do only depend on the structure near the inner cavity. ⊲ Eigenmodes in the range up to ≈ 4 . 4 GHz could be confirmed. b a D. Ferrazza, bachelor’s thesis, University of Rome (2016) b COMSOL Multiphysics 5.2, COMSOL Inc., http://www.comsol.com/ 14 / 17
Interconnects 4.2 Q ext estimate 10 10 Z | E z | d z / V R / Q / Ω 4.0 10 8 10 6 3.8 10 4 Frequency / GHz 3.6 10 2 10 0 3.4 10 -2 3.2 10 -4 electric-electric electric-magnetic 10 -6 3.0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 mode index mode index Eigenmode comparison ⊲ For modes above 3 . 9 GHz mode coupling occurs (simplified model). ⊲ For a large number of interconnects, only the fields that match ω = ck s are relevant. ⊲ This means to handle interconnects using multicell RF approaches. 15 / 17
Summary ⊲ Two approaches for distributed impedances under investigation 1 FEM eigenmodes with Floquet-periodic boundaries. compute dispersion diagram and fields using periodic boundary conditions. Get R/Q from fields, Q from fields and group velocity. 2 Field matching on simplied geometry. modify approach by Fedotov and Gluckstern to include an inner sloted pipe (beamscreen). ⊲ Pumping holes: 1 Eigenmode approach seems reasonable, but may suffer from numerical instabilities and/or long computation time, especially for high frequencies. 2 Field matching : While effort is required to implement this, and not all details of the beamscreen shape can be considered, it can be cross-checked with existing results for special cases. ⊲ Interconnects: 1 Here, the eigenmode approach seems feasible. In the limit of frequencies with wavelengths much smaller than the interconnect dimensions, the ”tapering” of the resonator should become invisible. 2 Mode matching is possible for simplified pillbox-like resonator structure, but less useful. 16 / 17
Thank you for your atention! 17 / 17
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