Outline: Motivation Comparisons with: > Thick wall formula > CST Thin inserts models Tests on the Mode Matching Method Webmeeting N.Biancacci, B.Salvant, V.G.Vaccaro Acknowledgement: A.Burov, F.Caspers, H.Day, E.Métral, N.Mounet, C.Zannini, M.Migliorati, A.Mostacci Conclusion and Outlook 1

Motivation We developed a Finite Length code to study the effect of finite length on simple geometries in order to investigate the effect for real devices. A list of benchmark has been done in order to validate the Mode Matching Method applied to simple geometries: 1.Comparison with the classical Thick wall impedance formula for different conductivities. 2.Comparison with CST for different conductivities of the filling material and length of the device. 3.Length dependence of impedance. 4.Application of the theory for thin insertion impedance and comparison with Shobuda-Chin-Takata model. length Filling material (ε’,ε’’,σ,μ) 2

Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=8 F/m σ variable 1-Thick Wall Formula: Test for high conductivity σ (1/1) 3 Thick wall formula: Varying conductivity

2-CST: Varying σ (1/4) 4 For high onductivity we can compare the impedance from the thick wall formula and CST. The model is a simple cavity in PEC. The wakefield is too low, this leads to numerical problem on the impedance. Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = 10 3 S/m σ=10 3

5 Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = 1 S/m 2-CST: Varying σ (2/4) σ=1

6 Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = S/m 2-CST: Varying σ (3/4) σ=10 -3

7 2-CST: Varying σ (4/4) Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = S/m σ=10 -4

cut off 8 2-CST: Varying Length (1/3) Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = S/m Varying length L=20 cm

9 Parameters: Inner radius=5cm Outer radius=30cm Length=60cm εr=1 F/m σ = S/m 2-CST: Varying Length (2/3) L=60 cm

10 Parameters: Inner radius=5cm Outer radius=30cm Length=100cm εr=1 F/m σ = S/m 2-CST: Varying Length (3/3) L=100 cm

3- Length dependence of impedance (1/2) 11 Parameters: Inner radius=7.7cm Outer radius=9.2cm Length=variable εr=9.4 F/m σ = S/m Longitudinal impedance for Alumina 96%. For Length >inner radius, longitudinal modes are well visible in case of low conductivity.

12 3- Length dependence of impedance (2/2) Parameters: Inner radius=2mm Outer radius=25mm Length=variable εr=1 F/m σ = 10 5 S/m We also studied the dependence of length for the case of ReWall impedance for multilayer beam pipes 1. In this case conductivity is high and the length does not play significant rule (all curves are overimposed). Discrepancy at low frequency is under investigation. 1.N.mounet, E.Metral, “Impedances of an Infinitely Long and Axisymmetric Multilayer Beam Pipe: Matrix Formalism and Multimode Analysis”

Small isolating insertions between beam pipe flanges in SPS could present impedance peaks at enough low frequency to overlap with the bunch spectrum. But Thin insertions Geometry is difficult to study with e.m. Simulators like CST. The thickness of the insertion is on the order of 200 um, the radius of the beam pipe 15 cm. insertion But.... ModeMatching Method (MMM) and Shobuda-Chin-Takata’s 2 (SCT) model could help. Courtesy of B.Salvant 2. COUPLING IMPEDANCES OF A SHORT INSERT IN THE VACUUM CHAMBER 13

Mode-Matching Method and Shobuda-Chin-Takata’s model for thin inserts Quick comparison..: MMM Provides a numerical-analytical way to compute the e.m. Fields excited inside cavity-like discontinuities in circular beam pipes. Based on cavity eigenmodes decomposition + field matching on the boundary and separation surfaces. The cavity can be filled with whatever material provided an analytical description (e’,e’’,σ). S2S2 S1S1 S3S3 SCT’s model Provides an analytical way to compute the e.m. Fields excited inside thick discontinuities in circular beam pipes. Fields are decomposed in sum of scatterd waves along the pipe, the insert, and outside in vacuum, no longitudinal variation is taken into account, no PEC boundary over the insert (radiation). 14

Thin insertion with MMM Thickness = 15 mm Gap width = 800 μm Pipe radius = 7.5 cm PEC bounded on the top Insert properties: Material properties: Beam parameters: CST – MMM comparisons (1/4) 15

CST – MMM comparisons (2/4) 16

1.67 GHz 4.8 GHz 8.0 GHz wakefield 20m CST – MMM comparisons (3/4) 17

1.67 GHz 4.8 GHz 8.0 GHz 30% discrepancy in magnitude... But CST peaks are still not satu- rated as my CPU memory. Wake length= mm Bunch length=15 mm TotalNumberOfMesh=10 5 CST – MMM comparisons (4/4) 18

Thin insertion with SCT’s model Thickness = 15 mm Gap width = 800 μm Pipe radius = 7.5 cm No boundary on the top, free space. Insert properties: Material properties: Beam parameters: CST – SCT comparisons (1/4) 19

CST – SCT comparisons (2/4) 20

The wake is shifted down. CST – SCT comparisons (3/4) 21

0 frequency peak: Thin insertion has very low conductivity, image current finds an open and accumulate charges on the insert extremities building up a static electric field. It is an effect of this simplified model. In reality currents finds closed loops that move this peack to low frequency. In SCTs model there is a restriction to pure transverse modes (the scattered field is always supposed constant longitudinally) here is a good approximation: the first longitudinal mode goes over hundreds of GHz. CST – SCT comparisons (4/4) 22

Conclusion and outlook CONCLUSIONS The Mode Matching Method has been succesfully applied to benchmark the Thick wall impedance formula, and different geometries simulated in CST particle Studio with different conductivity and length. Having a Finite-Length method is important in order to correctly model low conductivity/ high permeability materials, where, if the length is greater than the transverse dimension, longitudinal modes start to be relevant in comparison to 2D models. Thin inserts models of MMM and SCT have been succesfully benchmarked with CST showing the differences between the two methods in modelizing thin insert impedances. Little discepancies in magnitude of renonance peak have been shown when a complete convergence cannot be reached in the emsimulator CST. OUTLOOK These results show us the importance of having 3D simple models for impedance extimation. Further extension to the transverse impedance, quadrupolar and dipolar, is foreseen as well as further analysis and comparisons on 2D/3D difference and limitatons. 23

Thanks for your attention! 24

Backup slides Notes: 25

26 Parameters: Inner radius=5cm Outer radius=30cm Length=20cm εr=1 F/m σ = 10 4 S/m CST: Varying σ σ=10 4

27 Parameters: Inner radius=5cm Outer radius=30cm Length=40cm εr=1 F/m σ = S/m 2-CST: Varying Length (2/5)

28 Parameters: Inner radius=5cm Outer radius=30cm Length=80cm εr=1 F/m σ = S/m 2-CST: Varying Length (4/5)