Notes on Multi-Layer Impedance

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-AB-2003-093 (ABP)

The Impedance of Multi-layer Vacuum Chambers

Luc Vos

Summary

Many components of the LHC vacuum chamber have multi-layered walls : the copper coated cold beam screen, the titanium coated ceramic chamber of the dump kickers, the ceramic chamber of the injection kickers coated with copper stripes, only to name a few. Theories and computer programs are available for some time already to evaluate the impedance of these elements. Nevertheless, the algorithm developed in this paper is more convenient in its application and has been used extensively in the design phase of multi-layer LHC vacuum chamber elements. It is based on classical transmission line theory. Closed expressions are derived for simple layer configurations, while beam pipes involving many layers demand a chain calculation. The algorithm has been tested with a number of published examples and was verified with experimental data as well.

Geneva, Switzerland October , 2003

1 Introduction

For the LHC it has been necessary to compute the impedance of mult-layered vacuum chambers since a major part of the beam pipe belongs to this family : the copper coated cold beam screen, the titanium coated ceramic chamber of the dump kickers, the ceramic chamber of the injection kickers coated with copper stripes, the copper coated µ-metal pipes of the septum magnets, the copper coated ceramic TDI injection collimators, the copper coated cold-warm transition pieces, the copper coated carbon collimators, the NEG coated warm vacuum pipes. The impedance of multi-layer vacuum chamber walls can be found by solving Maxwell’s equations taking into account the proper boundary conditions. The task is tedious and has been done before[1]. A physically transparent alternative exists which is easy to understand and simple to implement. The basic idea is as follows. A particle beam moving with velocity βc, is accompanied by transverse electric and magnetic fields. In free space these fields tend to infinity while they are simply cut by a perfectly conducting vacuum chamber. Both cases are identical from the point of view of the beam. The situation changes drastically when the vacuum chamber is no longer perfectly conducting or has other propagation properties than those of free space. Fields penetrate the inner surface of the vacuum chamber material. The tangential magnetic field is continuous across the inner boundary. The penetrating magnetic field generates a surface current density which requires a longitudinal electric field in the face of a non-zero surface impedance. This field pair (tangential magnetic field and longitudinal electric field) forms a transverse wave in the vacuum chamber material. The problem at hand consists in finding the longitudinal electric field. The transverse wave is launched in a sequence of layers with different but known characteristics. The situation is essentially that of a transmission line comprising different sections, each with a different but known intrinsic impedance and propagation constant. Hence, the input impedance of this structure, which is the longitudinal impedance, can be found in a straightforward manner. The theory of transmission lines implicitly takes into account Maxwell’s equations with the proper boundary conditions.

2 Basic transmission line parameters [2]

Each material can be characterised electro-magnetically by three parameters. They are the electric conductivity σm, the permeability µm and the dielectric constant εm. Both µm and εm can be complex. The intrinsic impedance of the material m is :

jωµ m Zm = . (1) jωε m +σ m

The other parameters that are required are the propagation constants. Zotter specifies without proof the radial propagation constant [3, page 158] ,[6]:

  2 2  ω  γ Tm =   + jωµ m (σ m + jωε m ). (2)  βc 

The derivation of this quantity goes as follows. Consider a beam travelling in vacuum with a speed βc parallel to a material boundary. The intrinsic propagation constants in vacuum and in the material are known quantities : γ0 and γm. In the most general case one can write the transverse propagation constants as :

γ 2 = γ 2 − h 2 T 0 s , (3) 2 2 2 γ Tm = γ m −hsm where γT is the transverse propagation constant and hs the longitudinal propagation constant. From the continuity of the magnetic field across the boundary it follows that hs = hsm. Moreover, there is no transverse propagation in the vacuum such that γT = 0 and γ0 = hs = hsm. Hence: 2  j  2 2 2  ω  γ Tm = γ m −γ 0 = jωµ m (σ m + jωεm )−  . (4)  βc 

3 Setting-up the transmission line relations

Consider a set of parallel layers numbered n = 1, 2 ... from the first vacuum-material boundary onwards. Each set is characterised by its intrinsic impedance Zn, valid for a unit area, and its transverse propagation constant γTn = γn (drop subscript T since only transverse waves are dealt with from this moment onwards) and its length or thickness. The last layer, which can be of any nature (vacuum, conductor, insulator) has infinite length. No reflections occur in this part of the transmission line, hence it loads the previous section with its characteristic impedance. This allows the computation in a piecewise fashion of the input impedance and transfer impedance of each preceding line section. The input impedance of the first section (inverse order, last in computation!) is simply the surface impedance experienced by the beam.

The classical formulae for the transfer and input impedances are :

Ztn = Zin+1 cosh(γ nsn )+ Zn sinh(γ nsn )

Ztn , (5) Zin = Zn Zn cosh()γ nsn + Zin+1 sinh()γ nsn where Zin, Ztn are the input and transfer impedances of section n and Zin+1, Ztn+1 are the input and transfer impedances of section n+1. The length of section n is sn. The penetration of the electro-magnetic fields through the different layers is defined by the transfer impedances and the input impedances as will be shown below.

Consider a flat beam travelling through vacuum between two planar perfect conductors. The electro-magnetic fields travel with a speed βc, no reflections occur. Interrupt the perfect conductors and insert a piece of material with known electro-magnetic and geometric properties and which length is unity. The primary fields that accompany the beam are not altered, namely the tangential magnetic field. Its interaction with the input impedance of the insertion generates a longitudinal electric field. The input impedance is the longitudinal impedance of the vacuum chamber insert. The equivalent transmission line diagram for three consecutive layers is shown in Fig 1. The last layer is always infinitely thick, in this case layer three, and serves as load impedance of the previous section. Since we work with fields, impedances are surface impedances. The magnetic field at the entry of the first transmission line section is given by Ampere’s law :

Hi = Hi1 = Jb , (6)

3 where Jb is the planar beam current density .

transverse transmission lines Hi1 Z1, γ1, Z2, γ2,

Ei1 Z3

Figure 1 : Equivalent transmission line circuit of a vacuum chamber consisting of three layers.

The transverse wave will be partially reflected and transmitted at each transmission line junction except at the last one since it is infinitely long. The effect of the partial reflections and transmissions is duly taken into account in the formulae of the input and transfer impedances (Eq.5). The computation proceeds backwards and yields the input impedance Zi1. When needed, the penetration of the magnetic field through the various layers can be found with the transfer impedances Ztn. Indeed,

Ei1 Hi1 = , Zi1

Ei1 Zi1 Hi2 = = Hi1 , (7) Zt1 Zt 1

Zi1 Zi 2 Hi3 = Hi1 . Zt1 Zt 2

4 Cylindrical geometry versus planar geometry

The input magnetic field in cylindrical geometry is given by :

i H = b , (8) i1 2πb where ib is beam intensity and b the radius of the pipe. The beam is located in the center. In free space the magnetic field decreases as r-1, where r is the increasing distance from the beam. Consider two consecutive layers. If the area of the first is unity,then the area of the second is unity increased by ∆1/b, ∆1 being the thickness (length of equivalent transmission line) of the first layer. The increased cross-section of the second layer increases in a similar way the inductance, the capacitance and the conductivity. Hence the characteristic impedance is not changed and the transmission constant increases with ∆1/b. For subsequent layers the transmission constant increases by ∆2/(b+∆1), etc. Obviously, this effect is only

4 significant when ∆1/b is not too small. In practice it turns out that the simple planar geometry is a good approximation for most cylindrical cases.

5 Special cases

A few cases with a small number of layers lead to relatively simple closed formulae.

5.1 A single metallic layer with infinite thickness

The input impedance is the intrinsic impedance of the metal :

jωρ Z1 = Z0 ≈ jωµρ , (9) jωρ + Z 0c where Z0 is the impedance of free space and ρ =1/σ. Eq. 9 is the classical skin depth formula.

The approximation is valid for radian frequencies ω c << Z0 ρ .

5.2 A single metallic layer with finite thickness

The impedance of the outer layer is Z0. The metal layer has resistivity ρ, thickness s1, and the usual free space value for permeability µ and dielectric constant ε. The intrinsic impedance and the propagation constant are for β = 1:

jωρ Z1 = Z0 jωρ + Z 0c . (10) jωµ γ = 1 ρ

This yields the following surface impedance :

jωρ  jωµ    1+ tanh s1  jωρ + Z0c  ρ  Z = Z . (11) i1 0 jωρ + Z c   0  jωµ  1+ tanh s1  jωρ  ρ 

For ω = 0 the impedance is real and corresponds to a parallel circuit formed by the impedance of vacuum (Z0) and the surface impedance of the metal wall ρ/s1. The impedance reduces to the previous case for frequencies where the skin depth is smaller than s1.

5.3 A dielectric layer followed by an infinitely thick conductor

The impedance of the outer layer is :

jωρ Z2 = Z0 . (12) jωρ + Z0c

5 The dielectric has a thickness s1, conductivity zero, permeability µ and dielectric constant ε1 = εε r where εr is the relative permeability. The intrinsic impedance and the propagation constant of this layer are:

jωµ Z 0 Z1 = = jωε 1 ε r . (13)  ω  2 ω γ =   −ω 2 µε = 1 −ε 1  c  1 c r

The input surface impedance is then :

jωρ + Z c  ω  1+ 0 tanh 1−ε s  jωρ jωρε  c r 1 Z = Z r . (14) i1 0 jωρ + Z c jωρε  ω  0 1+ r tanh 1−ε s   r 1 jωρ + Z0c c

ω π A resonance can be identified when ε −1s = , or f = c 4 ε −1s . Take for c r 1 2 res r 1 example a 5mm thick ceramic with εr = 4, hence fres = 8.66 GHz. For lower frequencies the impedance reduces to a pure surface inductance :

εr −1 Li1 = µ s1 . (15) ε r

5.4 A thin conductive layer followed by an infinitely thick conductor with different conductivity.

The thickness of the first layer is s1 and its resistivity is ρ1. The resistivity of the second conductor is ρ2 different from ρ1. The permeability and permittivity are those of free space. The intrinsic impedances are :

jωρ ω Z = Z 1 ≈ j Z ρ 1 0 jωρ + Z c c 0 1 1 0 , (16) ω Z = j Z ρ 2 c 0 2 where ω/c << Z0/ρ1,2. The propagation constant of the first layer is :

jωµ γ 1 = . (17) ρ1

If the skin depth in the thin layer is larger than s1 , then the input surface impedance is :

6 jωµ 1+ s1 Z2 coshγ 1s1 + Z1 sinhγ 1s1 jω ρ2 Zi = Z1 ≈ ρ 2Z 0 . (18) Z1 coshγ 1s1 + Z2 sinhγ 1s1 c ρ2 jωµ 1+ s1 ρ1 ρ2

If ρ1 > ρ2, that is a bad conductor on the inside of a good conductor, then :

jω  jωµ   ρ jωµ  jω  ρ     2   2  Zi = ρ2 Z0  1 + s1  1− s1 ≈ ρ2 Z0 + jωµs1 1 −  . (19) c  ρ2   ρ1 ρ2  c  ρ1 

Note that the first term is the impedance without the thin layer. Hence the thin layer adds an inductance to the impedance of the outer (good) conductor. If ρ1 >> ρ2 then the additional inductance is simply:

L = µs1 . (20)

If ρ1 < ρ2, that is a good conductor on the inside of a bad conductor, then the admittance is :

ρ2 jωµ 1+ s1 1 ρ1 ρ2 s1 −1 s1 δ 2 Yi = ≈ + Z2 = + (1+ j), (21) jω jωµ ρ1 ρ1 ρ2 ρ2 Z0 1+ s1 c ρ2 where δ2 is the skin depth in the bad conductor. The approximation is valid for 2 ω c < ρ2 (Z0 s1 ). The impedance is a parallel circuit of the thin good conducting layer (first term) and the intrinsic impedance of the (thick) bad conducting material behind.

6 More complicated cases

For more than two layers it is not impossible but cumbersome to write down the expression for the input surface impedance (equal to the longitudinal impedance) of the beam pipe. The cascade calculation based on Eq.5 for impedance and on Eq.7 if penetration is the issue, is simple to perform in the form of a notebook (MATHCAD or MATHEMATICA).

As an example consider a ceramic layer with a thin metallic coating on the inside, outside is air. Assume the same type of ceramic as in 5.3. The numerical notebook results show the same resonance at 8.66 GHz. The impedance for lower frequencies appears to be the impedance of a parallel circuit formed by the thin metallic layer (ρ/s1) and the outer layer (air) Z0.

7 Benchmarking with published data

7.1 Double metallic layer

In [4] Lambertson studies the case of a double walled beam tube at the PEP-II interaction point. The parameters used for the calculation are as follows. Two coaxial cylindrical Be

7 tubes with thickness 0.8 mm and 0.4 mm with a space of 1.6 mm between. A copper shield is placed at 2 mm from the outer Be tube. The inner radius of the assembly is 25 mm. The conductivity of Be is 3*107 Ω-1m-1 and that of copper 5.8*107 Ω-1m-1. The relativistic beam parameter is γ = 6070. It is desired to know the beam electro-magnetic field at the inside face of the copper shield where silicon detectors are installed. A field attenuation factor of 11.898 - j117.45 is found at the orbital frequency of 136 kHz ([4] Eq 29 page 9). The attenuation factor computed with the transmission line algorithm in a plane geometry turns out to be 10.861-j105.72.

7.2 Coated ceramic chamber in a kicker magnet

In [4] the longitudinal impedance is computed for a ceramic chamber with a thin metallic inside coating inserted in a ferrite kicker magnet. The parameters used in the calculation are as follows. The thin coating has a surface resistivity of 0.3 Ω, a conductivity of 3.5*105 Ω- 1m-1 and a relative permeability of 1000. The ceramic is 6.4 mm thick and has a relative permittivity of 9. The ferrite is 25 mm thick and has a relative permeability of 1300 and a permittivity of 10. The outermost layer is copper with a conductivity of 5.8*107 Ω-1m-1. The relativistic beam parameter is γ = 6070.

The result of the transmission line algorithm is shown in Figure 2. 2

Real

] 1.5 Ω

stance [ 1 i

Imaginary 0.5 Surface res

0 100 1 103 1 104 1 105 Frequency [Hz]

Figure 2 : Surface impedance of a coated ceramic chamber.

The agreement with Fig 1 on page 11 of [4] is very good.

7.3 Layered vacuum chambers

In [5] the transverse impedance of the LHC beam screen is computed with LAWAT, a computer program based on [6]. The parameters of the calculation are the following. A 2 mm thick stainless steel chamber with radius b = 0.019m is coated on the inside with 50 µm copper at cryogenic temperature. The conductivity of the copper layer is 5.5 109 Ω-1m-1. The conductivity of the steel is 2 106 Ω-1m-1. An outer shield with infinite conductivity is placed at 3.45 mm from the stainless steel chamber. The transverse impedance is computed from 1

8 to 1010Hz. The transmission line algorithm only yields the longitudinal surface impedance. In order to compare the results it was necessary to transform the longitudinal surface impedance into a longitudinal impedance by the standard integration operator (2c ωb 2 ) to obtain the transverse impedance. The result is shown in Figure 3. It should be noted that the ‘inductive bypass’ effect has been taken into account for the computation of the transverse impedance[7]. The image is identical to Fig 2 of [5] apart from a resonance like phenomenon on the reactive part of the impedance claimed to be provoked by the direct space charge.

1 109

1 108

1 107

6 1 10

1 105

4 1 10 1 10 100 1 103 1 104 1 105 1 106 1 107 1 108 1 109 1 1010

Figure 3 : Transverse impedance of a layered vacuum chamber.

1 ramic

hind ce 0.1 e Piwinski d b l etic fie n 0.01 tive mag rela

1 10 3 4 3 1 10 1 10 0.01 0.1 1 10

thickness coating / resistivity Figure 4 : Penetration of magnetic field through a coated ceramic chamber. Comparison between the results obtained with the transmission line algorithm(blue) and those obtained by A. Piwinski in [8] (red).

9 In [8] A. Piwinski computes the penetration of the beam fields through a coated ceramic chamber coaxial with a good conducting pipe. The ceramic has a thickness of 6 mm and a relative permittivity εr = 9. Piwinski computes the magnitude of the magnetic field behind the coated ceramic for four different wavelengths λ (0.01, 0.1, 1, 10m) as function of the inverse surface resistance of the coating. The inner radius of the ceramic is 0.037m. Fig 4 of [8] is recomputed with its formula (20) and compared in Figure 4 with the results obtained with the transmission line algorithm. The wavelengths λ = 0.01 and 0.1m are not included for the simple reason that a resonance occurs in the ceramic at λ = 0.07m . This phenomenon is not taken into account in Piwinski’s treatment making a comparison impossible in this frequency range.

8 Experimental benchmark

In [9] the results obtained in a shielding experiment are reported. The basic experimental set-up consisted of a 5 mm thick ceramic chamber coated on the inside with a thin layer of titanium. The magnetic field was measured on the outside of the ceramic for several configurations. The penetration was measured without and with an additional outer shield made of brass. The penetration in the first case (a simple one layer case) was extremely small as expected. The second case is more interesting and consists of three layers with different properties. The outer shield was connected to the up and downstream vacuum chamber in two cases and in a third case it was isolated. The connecting shield was mounted tight on the ceramic in the first case and kept at a much larger distance in the second case. In the second and third case the edge effects are very important (reflections and an important inductive insert) and can be accounted for. However, in the first case the edge effect is negligible and allows a straightforward check of the calculation. Indeed the computed penetrating current (Figure 5) and the measured one agree very well (top of Fig 8 in [9]) apart from the tail signal which is somewhat larger than calculated.

4 p bunch current am 3 ENT /

R 2

LDED CUR 0

1 0 5 10 15 20 25 30 NON SHIE TIME/ ns Figure 5 : Bunch current (red) and computed current (black) behind ceramic coated with a resistive layer but inside an outer conductor. This result is to be compared with top of Fig 8 in [9].

9 Conclusion

An easy to implement algorithm based on classical transmission line theory has been presented that allows the calculation of the longitudinal impedance of multi-layered vacuum chambers of arbitrary but known properties and an arbitrary number of layers. The transverse impedance can be derived from the longitudinal one with the standard transformation. The results have been verified with published data (both longitudinal and transverse) and with experiment. The algorithm has been used to compute the impedance of numerous elements of the LHC beam pipe.

10 References

[1] B. Zotter, Longitudinal Instabilities of Charged Particle Beams inside Cylindrical Walls of Finite Thickness, Particle Accelerators, 1970, Vol1. [2] C. Jordan, K, Balmain, Electromagnetic Waves and Radiating Systems, Prentice- Hall, Inc. 1968. [3] B. Zotter, S. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators, World Scientific, Singapore. 1998. [4] G. Lambertson, Fields in Multilayer Beam Tubes, LBNL-44454, 1999. [5] E. Keil, O. Meincke, B. Zotter, The Impedance of layered vacuum chambers, EPAC 1998. [6] B. Zotter, Transverse Oscillations of a Relativistic Particle Beam in a Laminated Vacuum Chamber, CERN 69-15 Rev, 1969. [7] L. Vos, The Transverse Impedance of a Cylindrical Pipe with Arbitrary Surface Impedance, CERN-AB-2003-005 ABP, 2003. [8] A. Piwinski, Penetration of the Field of a Bunched Beam Through a Ceramic Vacuum Chamber with Metallic Coating, IEEE, Vol. NS 24, 1977. [9] D. Brandt, e.a., Penetration of Electro-Magnetic Fields through a Thin Resistive Layer, AB-Note-2003-002 MD, 2003.

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