Contributions to the impedance budget impedance of pumping holes and - - PowerPoint PPT Presentation

Contributions to the impedance budget → impedance of pumping holes and interconnects

Bernard Riemann FCC-hh impedance and beamscreen workshop 2017-03-30

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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

• rders of magnitude

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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)

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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.

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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 exist123 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.

• 1A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014
• 2G. Stupakov, Phys. Rev. E 51, 3515 (1995)

3S.S. Kurennoy, Part Acc. 39, 1 (1992)

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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, no dispersion (TEM, beam) 2d problem (TE/TM) z-dependent (TE/TM) frequency / a.u.

• long. wavenumber / a.u.

⊲ Group velocity dω/dk around the resonance condition is connected to the external quality factor of a large number of periods in a segment.

• 4T. Wangler, RF linear accelerators, 2nd ed. (Wiley-VCH, 2008)

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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

• 5A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014
• 6T. Wangler, RF Linear accelerators, 2nd ed. (Wiley-VCH, 2008)

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Periodicity of pumping holes

Mesh for one half of the geometry, rendered with COMSOL Multiphysics 5.2, COMSOL Inc., http://www.comsol.com/

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) ⊲ The fields for these resonance conditions are most important

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Periodicity of pumping holes

20 40 60 80 100 120

• long. wave number / (rad/m)

3 4 5 6 7 8 9 10 mode frequency / GHz

• mag. bound.
• el. bound.

ω = ck Dispersion diagram for the shown vacuum geometry

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20 40 60 80 100 120

• long. wave number / (rad/m)

10 11 12 13 14 15 16 mode frequency / GHz

• mag. bound.
• el. bound.

ω = ck

⊲ Proper seperation of modes is an issue (approach: Fourier decomposition of on-axis field) ⊲ Frequency limitation due to no of meshcells

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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 simplified FCC chamber geometry

with cylindrically orthogonal surfaces

⊲ Approach based on Fedotov and Gluckstern who computed a rectangular pipe hole with an outer enclosing pipe7 (B & C) ⊲ r < rA: evanescent TE/TM modes, r > rA: TEM/TE/TM (coaxial) ⊲ Neglect material size and sync.rad. ”mirrors”

7A.V. Fedotov and R.L. Gluckstern, Phys. Rev. E 56 (3) (1997)

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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 simplified 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)

• 8A. Mostacci, PhD thesis, University of Rome (2001), CERN-THESIS-2001-014

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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.

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Interconnects

2d problem, no dispersion (TEM, beam) 2d problem (TE/TM) 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

• aD. Ferrazza, bachelor’s thesis, University of Rome (2016)

bCOMSOL Multiphysics 5.2, COMSOL Inc., http://www.comsol.com/

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Interconnects

2 4 6 8 10 12 mode index 3.0 3.2 3.4 3.6 3.8 4.0 4.2 Frequency / GHz

electric-electric electric-magnetic

2 4 6 8 10 12 mode index 10-6 10-4 10-2 100 102 104 106 108 1010

Qext estimate Z |Ez|dz / V R / Q / Ω

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 = cks are relevant. ⊲ This means to handle interconnects using multicell RF approaches.

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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.

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Thank you for your atention!

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