Electromagnetic Modeling of Open Cell Conductive Foams for High Synchrotron Radiation Rings

Proceedings of IPAC2014, Dresden, Germany TUPRI047 ELECTROMAGNETIC MODELING OF OPEN CELL CONDUCTIVE FOAMS FOR HIGH SYNCHROTRON RADIATION RINGS S. Petracca∗ , A. Stabile, University of Sannio, Benevento, INFN, Salerno, Italy Abstract Electrical Properties of Conducting Foams Open cell conductive foams (OCMF) have been recently Electromagnetic modeling of OCMFs has been studied suggested as a possible alternative to perforated metal by several Authors during the last decade. A numerical ap- patches for efficiently handling gas desorption from the proach based on Weiland finite integration technique (FIT, beam pipe wall in high synchrotron radiation machines, in [4]) has been proposed by Zhang et al. [5] to compute the view of its superior performance in terms of residual gas (frequency, thickness and angle of incidence dependent) re- concentration and beam shielding. Here we discuss the flection coefficient of SiC foam. In the quasi-static limit λ 0, the conductivity of OCMFs can be computed us- electromagnetic properties (characteristiuc impedance and −→ propagation constant) of OCMFs and how they affect the ing effective medium theory (EMT), for which several for- beam coupling impedance. mulations exist giving comparable results (see, [6],[7] for a review), e.g., ν σeff = σ0(1 p) , (1) OPEN CELL METAL FOAMS − where σ0 is the bulk metal conductivity, p the porosity, and ν a morphology-dependent factor. Open cell metal foams (OCMF) are highly gas- Measurements of the microwave electromagnetic shield- permeable reticular materials, consisting of a 3D web of ing efficiency of OCMF panels [8] indicate that a simple thin conducting ligaments (Fig. 1), whose typical struc- Drude model ture is shown in Figure 1. OCMF have remarkable struc- 2 ω ǫ0 σ(ω) = p (2) ı ω + ν provides a good description of the frequency-dependent conductance of metallic foams. Typically, the relevant plasma and collision frequencies, ωp and ν are of the order of a few tens of GHz and a few tens of KHz, respectively, much smaller than their solid metal counterparts (typically in the PHz and GHz range, respectively [9]). OCMF Impedance and Skin Depth Throughout a typical beam current frequency spectrum, OCMF and bulk metals behave differently1. In bulk metals , ω ν ω so that the characteristic impedance Z ≪ ≪ p m and (complex) propagation constant k˜m can be written Figure 1: A typical OCMF at two different viewing scales. 1 + ı ων 1/2 1 ı ω ˜ (3) Zm 2 Z0, km − k0(ωp) ∼ √2 ωp ∼ √2 r ν tural properties (low density and weight, high (tensile and 1/2 shear)-strength to weight ratio, nearly isotropic load re- with Y0 = (ǫ0/µ0) = 1/Z0 the vacuum admittance, and ˜ sponse, low coefficient of thermal expansion), which qual- k0(ω) = ω/c. In metallic conductors, both Zm and km are ified them among the most interesting new materials for ω1/2. ∝ aerospace applications [1]. The key morphological param- In OCMFs ν ω ωp throughout the beam current ≪ ≪ eters of OCMF are the ”pore” size, and the porosity (vol- spectrum, so that the material exhibits a plasmonic behav- ume fraction of pores). Pore sizes in the range from 10−6 ior. The OCMF wall (characteristic) impedance Zf and −3 to 10 m and porosities in the range 0.7 - 0.99 are typ- (complex) propagation constant k˜f are thus given (to low- ical. These parameters determine the gas-permeability of est order in the small quantities ν/ω and ω/ωp) by the material, and, together with the electrical properties of ν ω ν the metal matrix, its electrical characteristics. OCMF may Z Z + ı , k˜ k (ω ) ı (4) f ∼ 0 2ω ω f ∼ 0 p 2ω − replace perforated metal patches in high synchrotron radi- p p ation accelerators [2], in view of their potentially superior 1In the case of LHC, the beam current power spectrum consists of lines performance in terms of outgassing and beam shielding [3]. at integer multiples of f0 = c/δb, δb being the bunch spacing, with an envelope approximately ∝ cos2. The −20 dB bandwidth is ∼ 1 GHz, 5 −1 roughly 10 times the circulation frequency ωR, and 10 times the cut- ∗ [email protected] off frequency of the lowest beam pipe waveguide mode [10]. 05 Beam Dynamics and Electromagnetic Fields ISBN 978-3-95450-132-8 D05 Instabilities - Processes, Impedances, Countermeasures 1671 Copyright © 2014 CC-BY-3.0 and by the respective authors TUPRI047 Proceedings of IPAC2014, Dresden, Germany Hence, the OCMF characteristic resistance Rf = are simply proportional to the average wall impedance, as- Re[Zf ] and skin depth δf are both frequency inde- sumed piecewise constant pendent, and, e.g., for the case of high-grade (ρ 5.510−10 ohm cm−1 at 20K) Copper foam with p = 0.≈9, Z = ξ Z(i) (10) h walli i wall both fairly small: Xi ξ being the surface fraction covered by patches with wall Z0 ν −5 c −4 i Rf 1.4 10 ohm, δf 6 10 m. (i) ∼ 2 ωp ≈ · ∼ ωp ≈ · impedance Zwall. (5) The OCMF characteristic reactance OCMF Impedance Budget ω Equations (7) to (10) allow to evaluate the impedance X = Im[Z ] ıZ (6) budget of an OCMF patched beam pipe. A straightforward f f ∼ 0 ω p solution, which is not included for brevity, of the electro- on the other hand, is large compared to that of bulk metal, magnetic boundary value problem for a (relativistic, van- and grows linearly with ω. For the case, e.g., of high- ishingly thin) axial beam in a circular conducting liner, 4 with radius a and thickness ∆ surrounded by a co-axial grade Copper foam with p = 0.9, Xf 0.5 ohm at 10 Hz. ≈ (infinitely thick) conducting circular tube (the cold bore) with radius b > a + ∆, shows that if ∆ > δf in eq. (5) BEAM COUPLING IMPEDANCES across the whole beam current spectrum, the Leontovich´ impedance of the cold bore backed liner wall OCMF patch The following relationship exists (to 1st order in the wall is fairly well approximated by the intrinsic impedance of impedance) between the longitudinal impedance per unit the OCMF. ¯ length Zk of a patched-wall beam liner and the (known) The contribution of the surface roughness of the foam to ¯(0) the OCMF wall impedance can be estimated as [14] longitudinal impedance per unit length Zk of the same pipe with a perfectly conducting wall [11] (rough) π h ω Zf ı Z0 h, (11) (0) ǫ0Y0 2 ≈ r32 L c Z¯ =Z¯ + Zw(s) E0n(s) ds, (7) k k cΛ2 I | | ∂S h and L being the r.m.s. height and correlation length of where Zw is the (local) Leontovich´ impedance [12] of the the surface roughness, respectively. patched wall, ∂S is the pipe cross-section contour, Λ the The numerical values of the above impedance compo- (0) nents, normalized to the mode number (i.e., multiplied by beam linear charge density, and En the (known) field component normal to the pipe wall in the perfectly con- the (ωR/ω) factor), have been collected in Table 1, where ducting pipe. A similar formula, not included for brevity, the solid metal values are also shown for comparison. Here we assume high-grade Copper (ρ 5.510−10 ohm cm−1 exists for the (dyadic) transverse impedance [11] as well. ≈ Beam coupling impedances embody a synthetic descrip- at 20K) for the solid and foamed material. For this lat- tion of beam stability. The absolute value and imaginary ter we assume a pore diameter 1 mm, and a ligament ∼ 10 −1 ¯ size 0.1 mm, yielding ωp 7.93 10 rad sec and part of Zk are, e.g., inversely proportional to the thresh- ν ∼3.66 104 Hz in the Drude≈ model· of Section , consis- old currents for (single-bunch) microwave instability and ≈ · ¯ tent with typical measured values of the static conductance, Landau damping suppression [13]). The real part of Zk de- termines the parasitic loss (energy lost by the beam per unit and a r.m.s. roughness scale h .125 mm with correlation length L 0.25 mm. ≈ pipe length), via [13] ≈ +∞ 1 2 ¯ Table 1: Zwall of Solid and OCMF High-Grade Cu ∆ = I(ω) e [Zk(ω)]dω, (8) E 2π Z−∞ | | ℜ [ohm] Solid OCMF −6 −5 where I(ω) and Zk(ω) ie the beam-current spectrum. (ωR/ω)Rwall 4.910 ωR/ω 1.410 (ωR/ω) −6 −5 According to (7), perforated/slotted or OCMF patches, (ωR/ω)Xwall 4.910 pωR/ω 5.510 (0) p placed where the normal field component En is minimum, have minimum impact on the longitudinal As seen from Table 1, the Copper wall resistance is 5 impedance. larger than that of the OCMF up to ω = 10 ωR. On the (0) The circular beam liner with radius s, where En is uni- other hand, as anticipated in Section , the OCMF wall re- form along ∂S, represents a worst case, where [13] the lon- actance is relatively large . gitudinal and transverse impedances per unit length It exceeds that of a perfectly conducting slotted by roughly one order of magnitude, at a 10% escape probabil- Z Z ity level). as well as that of solid Copper. Indeed, while the ¯ wall ¯ wall (9) Zk = h i, Z⊥ = h 3i 2πa 2πk0a typical escape probability of an OCMF wall is roughly one ISBN 978-3-95450-132-8 05 Beam Dynamics and Electromagnetic Fields Copyright ©1672 2014 CC-BY-3.0 and by the respective authors D05 Instabilities - Processes, Impedances, Countermeasures Proceedings of IPAC2014, Dresden, Germany TUPRI047 order of magnitude larger than the limiting one of a slotted [13] B.W.

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