Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295

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Nuclear Instruments and Methods inPhysics Research A

0168-90http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/nima

Simulations of the electron cloud buildup and its influenceon the microwave transmission measurement

Oliver Sebastian Haas a,n, Oliver Boine-Frankenheim a,b, Fedor Petrov b

a GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germanyb Technische Universität Darmstadt, Institut für Theorie Elektromagnetischer Felder, Schlossgartenstraße 8, 64289 Darmstadt, Germany

a r t i c l e i n f o

Article history:Received 30 May 2013Received in revised form5 July 2013Accepted 13 July 2013Available online 20 July 2013

Keywords:Electron cloud buildupMicrowave transmission methodWaveguide perturbation

02/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.nima.2013.07.051

esponding author. Tel.: +49 6159 71 2408.ail address: o.haas@gsi.de (O.S. Haas).

a b s t r a c t

An electron cloud density in an accelerator can be measured using the Microwave Transmission (MWT)method. The aim of our study is to evaluate the influence of a realistic, nonuniform electron cloud on theMWT. We conduct electron cloud buildup simulations for beam pipe geometries and bunch parametersresembling roughly the conditions in the CERN SPS. For different microwave waveguide modes the phaseshift induced by a known electron cloud density is obtained from three different approaches: 3D Particle-In-Cell (PIC) simulation of the electron response, a 2D eigenvalue solver for waveguide modes assuming adielectric response function for cold electrons, a perturbative method assuming a sufficiently smoothdensity profile. While several electron cloud parameters, such as temperature, result in minor errors inthe determined density, the transversely inhomogeneous density can introduce a large error in themeasured electron density. We show that the perturbative approach is sufficient to describe the phaseshift under realistic electron cloud conditions. Depending on the geometry of the beam pipe, the externalmagnetic field configuration and the used waveguide mode, the electron cloud density can beconcentrated at the beam pipe or near the beam pipe center, leading to a severe over- or underestimationof the electron density.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Electron clouds limit the intensity of hadron beams in modernhigh energy synchrotrons and colliders (see, e.g., Ref. [1]). In theCERN LHC effects related to electron clouds have been observed atdifferent bunch spacings (see Ref. [2]). The operation with 25 nsbunch trains is presently limited by electron cloud effects. Forelectron clouds the main observations are usually a pressure risein the warm regions and an increase of the beam screen tempera-ture in the cold sections due to the additional heat load. Besidesthere are the beam-based observations, like e.g. head–tail instabil-ities above a threshold bunch intensity or below a certain thresh-old bunch spacing. Usually the electron cloud density is inferredfrom beam observations or from the heat load. In additionretarding field analyzers can detect electrons close to the vacuumchamber walls. A method where the average electron clouddensity is measured over a longer section of the beam pipe isthe microwave transmission method (MWT). The MWT was firstintroduced at the CERN Super Proton Synchrotron (SPS) in byKroyer et al. (see Ref. [3]). The MWT has widely been used withgreat success for the detection of electron clouds. For example, in2011, this method was used for a relative comparison of the

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electron cloud density for different surface conditioned vacuumchambers by Federmann et al. [1]. So far a cold and homogeneouselectron density has been assumed for the interpretation of MWTmeasurements. The aim of the presented work is to evaluate theinfluence of realistic electron cloud density profile on the micro-wave transmission. For the same amount of electrons in the beampipe we will compare the phase shift obtained for realistic densityprofiles with the one obtained for an electron cloud fillinghomogeneously the pipe. The error due to the assumption of ahomogeneous cloud will be estimated.

In Section 2 we introduce the theoretical framework of theMWT as well as a simple perturbative model in order to accountfor transverse inhomogeneities of the electron cloud. Simulationresults are discussed in Section 3. First, buildup simulationsperformed with VORPAL (see Ref. [4]), are presented in Section3.1. Second, an analytic model for the MWT is compared toVORPAL simulations in Section 3.2. In addition the effect of aninhomogeneous cloud will be investigated using a 2D eigenpro-blem solver with a dielectric electron cloud model, and comparedwith the above mentioned perturbative approach.

2. Theoretical background

This section contains the basic principles of the MWT forelectron clouds. It is important to note, that full simulations of

O.S. Haas et al. / Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295 291

MWT measurements, i.e. simulations over several revolutionperiods, over a length of several meters and with reasonableaccuracy, are currently not feasible. Instead, the aim of this workis to evaluate the MWT by simplifying to several time independentsimulations, given approximations of slow varying longitudinalelectron density. On a surplus, this gives insight on severalpossible sources of error separately.

2.1. Principle of the microwave transmission method

In the MWT a beam pipe of length L acts as a waveguide. Ifthere is an electron cloud present in the beam pipe the dispersionrelation changes, which can be expressed as a phase shift relativeto the empty beam pipe. For a cold and homogeneous plasma onecan calculate the resulting phase shift just by the difference of thepropagation constants for the empty and the filled waveguide (see,e.g., Ref. [3])

ΔϕL

¼Δkz ¼ kz;empty�kz;ec ¼1c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2�ω2

c

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2�ω2

c�ω2p

q� �; ð1Þ

where ωc ¼ 2πf c is the cutoff frequency of the used mode of theempty waveguide, ω¼ 2πf is the injected frequency and c is thespeed of light.

The cutoff frequency usually belongs to the lowest mode,e.g. the TE10 mode in the case of a rectangular beam pipe. Thesingle mode operation is then ensured by choosing a microwavefrequency below the second lowest cutoff frequency, leading toexponential attenuation of all modes except the lowest mode. Ifsingle mode operation cannot be ensured and amplitude ratios ofthe exited modes are not known, it is not possible to calculate theelectron cloud density unambiguously. Simplified speaking thereare several modes with different cutoff frequencies in Eq. (1) andthe modes cannot be disentangled without further knowledge,i.e. just from measuring the phase shift. The plasma frequency

ωp ¼ffiffiffiffiffiffiffiffiffiffiffinee2

ϵ0me

s≈56:4 m3=2 s�1 � ffiffiffiffiffi

nep

; ð2Þ

where ne is the electron density, e is the elementary charge, ϵ0 isthe dielectric constant and me is the electron mass, can beunderstood as the cutoff frequency of the plasma. The plasmafrequency in the considered cases is in the range of tens of MHz, asthe electron density in order of magnitude 1012 m�3 to 1013 m�3.A cutoff frequency of tens of GHz is thus significantly larger thanthe plasma frequency, which is often used for a linearization ofEq. (1)

ΔϕL

¼ ω2p

2cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2�ω2

c

p : ð3Þ

It is important to note, that the electron cloud or plasma can beexpressed as a dielectric with relative permittivity (see, e.g., [5])

ϵr ¼ 1�ω2p

ω2 : ð4Þ

This relation can be used to model the electron cloud. One of themajor difficulties in the MWT measurement arise due to the verysmall value of the phase shift, which is often in the order of10 mrad only. Furthermore, the phase shift is time dependent,which leads to a phase modulation of the transmitted microwavesignal

sMWðtÞ ¼ a0 sin ðωt þmðtÞÞ; ð5Þwhere a0 is an arbitrary amplitude and m(t) is the phase modula-tion due to the electron cloud. There are few setups described inthe literature, which successfully demodulate the microwavesignal directly, e.g. in Ref. [6]. Often the maximum phase shiftΔϕmax is calculated from the modulation sidebands in the

spectrum of the transmitted microwave signal instead, as thedirect demodulation is technologically more demanding.The measurement in the spectrum is done, e.g., in Ref. [1]. Asthe demodulation is a more direct measurement, the presentedwork focuses and calculating modulation signals like they wouldbe obtained in such a measurement, and not the spectrum.

The temperature of a plasma in a plasma-filled waveguide hasbeen shown by Kalluri and Prasad [7] to have no influence on thedispersion relation in the case of TE modes and a homogeneousplasma. For TM modes, however, the temperature modifies thedispersion relation. One can estimate (Ref. [7]) that even for TMmodes and energies of the electron cloud up to 1 keV the effect ofa finite temperature on the phase shift are not measurable.

The effects of external magnetic fields on the microwave arenot considered in this work, as either the magnetic field strengthor the polarization of the microwave are normally chosen suchthat this does not affect the plasma wave (see, e.g., Ref. [1]).

2.2. Effect of the electron cloud inhomogeneity: longitudinal

The MWT measures an integrated electron density over alonger section of the beam pipe. The permittivity in Eq. (4) isclose to unity and varies smoothly with the electron cloud density.Thus no additional reflections occur due to the electron cloud. Ingeneral a significant amount of reflections of the microwave,e.g. due to geometry variations of the beam pipe, can obliteratethe MWT measurement. But as the MWT has been used success-fully, we assume that this is not an irresolvable problem for mostsetups. We thus assume an idealized setup with no reflections.

Considering a longitudinally inhomogeneous electron cloud ina geometrically homogeneous waveguide, the problem can thus bedescribed by individual slices of an infinitesimal length dz withlongitudinally homogeneous electron density and a propagationconstant for each slice. In the following sections the time depen-dent propagation constant kz;ecðtÞ for one slice will be calculated. Itcan be mapped to all other slices due to the symmetry of theproblem as described below. The resulting phase shift or modula-tion signal of the waveguide can then be obtained by integrationover the phase shift of these slices.

For the mapping to be done correctly it is necessary to considerthe different velocities of the microwave and the electron cloud.The resulting modulation signal can be written as

mðtÞ ¼ZLðkz;empty�kz;ecðt�zðv�1

g �v�1b ÞÞÞ dz ð6Þ

where vb is the velocity of the beam and vg is the group velocity ofthe microwave. The longitudinal electron density profile followsthe beam, thus travels with the beam velocity. The later is close tothe speed of light in most accelerators of interest. The microwave,however, has in most cases a significantly lower group velocity vgthan the speed of light, usually somewhere in between 50% and90% of the speed of light, depending on cutoff and microwavefrequency. In the integration in Eq. (6) the group velocity vgdepends in general on the electron density, but as the electroncloud varies slowly in longitudinal direction this effect is negli-gible. The group velocity can thus be assumed to be the groupvelocity of an empty waveguide. Note that in this context thegroup velocity just expresses where and which exact part of theelectron cloud interacts with the microwave, the actual phase shiftcaused by the varying phase velocity is included in the propaga-tion constants. The signs of the velocities in Eq. (6) are chosen suchthat the microwave travels in beam direction. While this propaga-tion behavior theoretically has an influence on the measurement,e.g. the microwave propagating in or against beam direction, theeffect is negligible for typical waveguide lengths and groupvelocities.

2.5

3.0

3.5

2.32.01.7

O.S. Haas et al. / Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295292

2.3. Effect of the electron cloud inhomogeneity: transversal

The effect of transverse electron density inhomogeneities canbe treated to a limited extent within perturbation theory. Similarto the cavity perturbation technique (see, e.g., Ref. [8]) one can usethe electric field solution of the homogeneous waveguide E0 tocalculate an effective electron cloud density ne;eff for use in Eq. (1).The effective electron cloud density for a waveguide with cross-section A is given by

ne;eff ¼1R

AjE0j2 dA

ZAne x; yð Þ E0j2 dA:

�� ð7Þ

The effective electron cloud density in Eq. (7) can be understood asa weighted average of the electron cloud density: in regions wherethe electric field amplitude of the used mode is higher, the densitygets weighted more compared to regions where the electric field islow. The TE10 mode of a rectangular beam pipe for example ismore sensitive in the center of the beam pipe in x-direction, whilein y-direction there is no change in sensitivity (see Fig. 1). Thiseffect could lead to an over- or underestimation of the electrondensity. In order to calculate the effective electron density for theMWT it is necessary to first obtain the actual electron densityprofile from detailed electron cloud buildup simulations. Toillustrate the effect of the transversal profile it is reasonable todefine the ratio of the measured average electron density – underthe assumption of a homogenous electron cloud – and the realaverage electron cloud density. This ratio corresponds to

κ¼ ne;eff

ne; ð8Þ

where ne is the average electron density. The average electrondensity should not be confused with the weighted or effectiveelectron density ne;eff . κ can be understood as a correction factor. IfEq. (7) holds, there are theoretical limits given for this weighting:assuming the worst case for this weighting – a point-like electroncloud – the effective electron cloud density is just the electrondensity in this point times the normalized amplitude of jE0j2. Oneexample for the aforementioned theoretical limit would be in thecase of a rectangular beam pipe with ne;eff∈½0;2 � ne�, independenton waveguide mode. The worst possible outcome of a measure-ment would thus be the lack of detection of the electron cloud.

Finally it should be noted, that in the general transversallyinhomogeneous case the continuity conditions of the electric andmagnetic fields cannot be fulfilled by TE modes, thus leading to asolution with hybrid modes. The perturbative approach assumesthat the electric field distribution is not affected by the electroncloud, and that hybrid modes do not occur. In the present study anumerical approach – a 2D eigenproblem solver – will be used aswell to analyze those effects. The deviations of the perturbativeapproach and the 2D eigenproblem solver can then be attributedto the modification of the electric field distribution and mode.

Fig. 1. Weighting of the electron cloud density in a rectangular beam pipe.Especially strongly localized electron clouds, e.g. as depicted due to the presenceof an external magnetic field, affect the phase shift measurement. The local electrondensity neðx; yÞ gets weighted by the electric field of the used mode jE0j2.

3. Simulation results

The presented simulations consist basically of two differentparts: buildup and microwave transmission method. As an exam-ple, CERN SPS parameters was chosen, i.e. a high intensity,ultrarelativistic bunch train with short individual bunches.

3.1. Buildup

The main tool for the buildup simulations was VORPAL (see Ref.[4]), a commercial finite difference plasma physics code. InVORPAL the probabilistic Furman-model [9] for modeling second-ary emission is implemented. The beam parameters are basicallythose of the LHC-type 25 ns beam in the SPS. The beam is assumedto have the velocity vb ¼ c, radius (1 s) rb ¼ 1 mm, protons perbunch Nppb ¼ 1:15� 1011, bunch length (1s) sb ¼ 0:33 ns, bunchspacing Tb ¼ 25 ns and revolution period Tr ¼ 23 μs. The beamconsists of NB ¼ 4 batches with Nb ¼ 72 bunches each and NB�B ¼ 8empty bunches in between batches. The magnetic field isB¼ By ¼ 0:12 T if present. The beam was assumed to be rigid andultra-relativistic. The magnetic fields generated by the electronsand the beam were assumed to be negligible, as the velocities ofthe electrons are rather low. Thus a 2D electrostatic approach wasused, though a constant external magnetic field can be imprintedfor the particle push of the electrons. The secondary emission yield(SEY) depending on incident energy of the electrons δðEiÞ wascalculated as proposed by Cimino et al. (see Ref. [10]). Themaximum SEY δmax of the true secondaries (often shortly referredto as SEY) was assumed to lie in between 1.7 and 2.3, as the exactSEY is usually not known. As source for the seed electrons residualgas ionization as well as desorption due to lost beam particleshave been implemented as mentioned by Arduini [11].

The vacuum chamber of the considered standard stainless steelmain bend (MBB) SPS dipole has a width of wx ¼ 121:8 mm and aheight of 48.5 mm with a rounded rectangular cross-section.

A density ne ¼ 107 m�3 of background electrons at the start ofthe simulation has been assumed. The profile of this backgrounddensity was determined by simulating the buildup until saturationwas ensured, i.e. over hundreds of bunches, and then continuingthe simulation without beam (decay phase) over the rest of therevolution period.

Typically the results of buildup simulations are visualized asdepicted in Fig. 2, where the line electron density λe depending ontime is given. Note that the line density is equivalent to the overthe cross-section averaged electron density ne.

0 1 2 3 40.0

0.5

1.0

1.5

2.0

Fig. 2. Electron line density of the simulated electron cloud buildup in the SPSsetup. Note that usually just two batches were simulated if the electron cloudreaches saturation during the first batch, as the following batches are thusidentical.

O.S. Haas et al. / Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295 293

One can see that the line density increases exponentially untilit saturates, i.e. the generation of secondary electrons and theelimination is in equilibrium. The rise time varies just slightly withthe SEY δmax41:8, but for lower SEY it is possible that the electroncloud does not reach saturation during the first batch.

The transversal density profile is most important in thesaturation region, thus the electron density in the cross-sectionis shown in Fig. 3 (top) for an SEY δmax ¼ 2:3. Due to the externalmagnetic dipole field the motion of the electrons is restrictedmostly parallel to the magnetic field lines, i.e. in y-direction. Theoccurring density profile is often described as two characteristicstripes. This has been measured and simulated in the past, e.g. inRef. [12]. The detailed structure of the saturated electron densityprofile in a strong dipole field depends on the SEY maximum atenergies of roughly 300 eV and on the space charge of theelectrons.

While the electrons obviously cross the beam pipe transversallydue to the kick by the bunch, the cloud at saturation is most of thetime more dense near the chamber walls due to the space chargeof the electrons and the grounded beam pipe potential. Thus theelectron density varies from practically zero off center, 1012 m�3 atthe center and 1014 m�3 near the beam pipe walls. The averagedensity is roughly 4:4� 1012 m�3. This is important for the MWTas well as for the effects on the beam (see Ref. [13]). Directly afterthe kick the electrons can have energies up to a couple of keV andthe average kinetic energy is approximately 85 eV. The secondaryelectrons have on average a lower energy than the incidentelectrons, thus the cloud cools down to an average kinetic energyof 10 eV.

For comparison in Fig. 3 (bottom) the profile of an almostidentical setup – just without external magnetic field – is shown.Here the electron cloud is not as localized as in the case withexternal magnetic field, the density varies from 1011 m�3 to1013 m�3. In general one can observe that asymmetric beam pipes,e.g. rectangular or elliptical, lead to more inhomogeneous trans-versal density profiles than symmetric beam pipes, as the strengthof the kick on the electrons by the bunch is not symmetric in x-and y-direction. For example in the simulated case as seen in Fig. 3most of the electrons get a larger kick in y-direction and thus more

Fig. 3. Electron density in the cross-section of the simulated buildup in saturationright before a bunch passing with (top) and without (bottom) magnetic dipole field.For the normalization nen ¼ 1012 m�3 was used. While the buildup is a highlydynamic process, the electron density profile is most of the time qualitativelysimilar to the depicted case, i.e. more dense in the center regarding the longer axisand more dense near the beam pipe walls regarding the shorter axis. The SEY in thedepicted case is 2.3, but it should be noted that the electron density profile insaturation is visually identical for the SEY from 1.7 to 2.3.

secondary electrons are produced on the top and the bottom of thebeam pipe. Furthermore the electron cloud is significantly morelocalized if there is an external magnetic field present due to therestriction on the motion of the electrons.

3.2. Microwave transmission

As a first step the analytical model in Eq. (1) of a cold andhomogeneous electron cloud is compared with 3D PIC simulationsin VORPAL under the same assumptions. As mentioned the chosenexample is close to the SPS setup (see Ref. [1]), where the cross-section is approximately rectangular. Therefore the nomenclatureof the waveguide modes of a rectangular waveguide will be used,e.g. TE10 and TE20. Furthermore it should be noted that the SPSsetup uses the TE20 mode, with cutoff frequency f c;TE20 ¼ 2:54 GHz.The phase shift can be calculated by simulating the transmission ofan empty, as well as a filled beam pipe and calculating the phaseshift from the S-Matrix parameter S21 (see, e.g., Ref. [14]). Anidealized excitation – a wave port – has been used in thesimulation, as the this work focuses on the study of the effectsof the electron cloud inhomogeneity. After the transmission at theend of the waveguide the wave gets absorbed. As seen in Fig. 4, thesimulation results coincide very well with the phase shift obtainedanalytically.

In the PIC simulation of the MWT through a homogeneouselectron density we gave the electrons an average kinetic energyof the earlier mentioned Ekin ¼ 100 eV and observed no change inthe phase shift. This observation agrees with analytical resultscalculated with Ref. [7] as mentioned in Section 2.3. Hereinafter itis thus assumed, that the temperature of the electron cloud itselfhas no effect on the MWT.

Already the relatively simple simulation of the MWT througha homogeneous cloud requires a very high mesh refinement anda high number of macro particles to reach a sufficient accuracy.A combined simulation of electron cloud buildup and the MWTwithin a 3D EM PIC scheme was thus not feasible. In the followingthe approach described in Sections 2.2 and 2.3 will be used.

To evaluate the influence of the inhomogeneity of a realisticelectron density on the MWT, the density profiles were obtainedfrom the buildup simulation, as described in Section 2.3. Theimportant remaining step is the calculation of the propagationconstants kz;ec for the thin slices of length dz from neðt; x; yÞ. Thepropagation is then emulated by integration as shown in Eq. (6). Inaddition to the already described methods – the perturbativeapproach and PIC 3D simulations – a solver for the 2D eigenwaveproblem has been implemented (see, e.g., Ref. [15]). A dielectric

2.602.713.50

Plotmarkers: VORPAL PIC 3DCurves: Analytical

f in GHz

0 1 2 3 4 5 6 7 8 9 100

5

10

Fig. 4. Phase shift of simulated MWT for a cold and homogeneous density. Therelative errors of the simulations compared to the analytic model in Eq. (3) areabout 4%.

0 1 2 3 40

10

20

30

40

50HomogeneousInhomogeneous TE20Inhomogeneous TE10

Fig. 6. Phase shift signal of the example setups. Inhomogeneous electron clouddistributions can lead to modified phase shift signals in the MWT measurement.This modification depends on the waveguide mode used in the transmission.Possible results are the over- or underestimation of the electron density. Themicrowave frequencies in the depicted examples have been chosen such that thephase shift for a homogeneous electron cloud is identical, i.e. f c;TE10

¼ 1:30 GHz,f TE10 ¼ 1:61 GHz, f c;TE20 ¼ 2:54 GHz and f TE20 ¼ 2:71 GHz. The SEY in the correspond-ing buildup simulation is 2.0 and the magnetic field 0.12 T. For further buildupsimulation parameters refer to Section 3.1.

O.S. Haas et al. / Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295294

model (see Eq. (4)) and thus frozen electron density profile wasused. The solver calculates the propagation constant kz for atransversally inhomogeneous, but longitudinally homogeneouscloud in a beam pipe. This propagation constant is equivalent tothe propagation constant of a thin slice kz;ec as described in Section2.2. In both cases – the perturbative approach and the 2Deigenproblem solver – the phase shift signal can then be calculatedfrom the propagation constants of the thin slices with Eq. (6).

The main goal was to compare the 2D eigenvalue solver to theperturbative approach, i.e. compare ΔϕPert and ΔϕEig, and estimatethe range of validity of the latter. This is visualized in Fig. 5 for theexample with external magnetic dipole field. Here it was assumed,that the profile of the saturated electron cloud density staysapproximately the same for increasing average density. One canobserve, that for the given example, the perturbative approach isvalid up to ne≈1:6� 1014 m�3 with a 5% error. While this result isnot universally valid, it is reasonable to assume that for realisticsetups all parameters are in the same order of magnitude, e.g. themicrowave frequency fi and cutoff frequency f c are in the range ofGHz, thus the results should be in the same order of magnitude aswell. Furthermore the presented example is a rather extreme caseof an inhomogeneous electron cloud. As electron densities of up to1013 m�3 are usually the limit, one can conclude that the pertur-bative approach should be valid for most realistic setups.

It should be noted that the results of the 2D eigenproblem solverhave been compared to 3D PIC simulations with stationary electronclouds in VORPAL as well. No significant deviations have beenobserved, despite the relative errors of the 3D PIC method mentionedabove. Note that a time-dependent electron cloud was computation-ally not feasible in 3D PIC simulations as mentioned above.

To illustrate the effect of the transversal weighting in the MWTmeasurement calculated modulation signals for an SEY of 2.0 ofthe example setup with external magnetic dipole field is shown inFig. 6. It should be noted, that the results with the perturbativeapproach as well as the 2D solver are visually identical, i.e. therelative error is below 0.5%, and the former where thus omitted inFig. 6. Compared to a calculated modulation signal of an homo-geneous density with the same average density, one can clearlysee the modified amplitudes as well as shapes. Interestingly it canbe observed, that the modulation signal for the TE20 mode has arounded-off shape during the first batch. This can be explained bythe stripe profile of the electron density: due to the low spacecharge of the electron cloud in the beginning, just a single stripe in

10 20 50 100 200 5000.75

0.80

0.85

0.90

0.95

1.00

Fig. 5. Ratio of the phase shift in the perturbative approach ΔϕPert and the 2Deigenwave solver ΔϕEig of the example setup with external magnetic dipole field.With increasing electron density the error of the perturbative approach increases,as the electric field distribution gets deformed. The results shown were calculatedfor the TE20 mode with microwave frequency f ¼ 2:71 GHz and f c2 ¼ 2:54 GHz,although results for the TE10 mode are visually identical.

the center emerges. With increasing electron density two stripesappear.

Assuming a SEY between 1.7 and 2.3 one can determine thecorrection factor κ (see Eq. (8)) for the example setup withexternal magnetic dipole field as 1:7170:05 (TE10) and0:4570:06 (TE20). For the more homogeneous electron cloud inthe example with no external magnetic field, the correction factorsare more moderate with 1:2670:03 and 0:8170:02 for the TE10and TE20 mode, respectively. Note that in principle the correctionfactor varies over time and does not directly correspond to thechange of the amplitude of the modulation signal as seen in Fig. 6,e.g. due to the longitudinal integration (see Eq. (6)). However, forall realistic applications it is a very good approximation to justcalculate the electron density with Eq. (3) and correct by the abovementioned correction factor.

It should be noted that the chosen geometry is a case withlarger corrections as e.g. a circular beam pipe. For many setups –

especially without external magnetic fields – these correctionsshould thus be negligible. Nevertheless, the presented perturba-tive approach offers a simple way to quantify these corrections ifan electron density distribution is available from buildupsimulations.

4. Conclusions

The influence of realistic electron cloud density profiles onmicrowave transmission (MWT) measurement has been investi-gated. Between bunches the saturated electron cloud densityprofile is dominated by space charge and by the external magneticfield. The transverse density profiles are strongly inhomogeneous.The electrons can be localized at the beam pipe wall or near thebunch center in the form of well separated stripes. The detailedshape of the electron clouds will affect the results of MWTmeasurements. This is demonstrated for beam pipe and bunchparameters close to the conditions in the CERN SPS. Our buildupsimulations indicate that the local electron density varies frompractically zero to 1014 m�3. While other parameters, like thetemperature of the electrons, do not affect the MWT measure-ment, transverse inhomogeneities do. The actual change in thephase shift depends on the chosen waveguide mode. The totalphase shift can be obtained from a longitudinal integration over

O.S. Haas et al. / Nuclear Instruments and Methods in Physics Research A 729 (2013) 290–295 295

2D slices, assuming that electron cloud permittivity varies onlyslightly along the beam pipe. The important effect of the trans-verse inhomogeneity can be quantified using a simple perturbativeapproach, using the electron density from separate buildupsimulations. The perturbative approach has been shown to bevalid up to an average densities exceeding 1014 m�3 for the chosenexample parameters. From the perturbative approach we find thatthe necessary correction for the phase shift can add up to factors of2, over- or underestimation, for the chosen parameters.

Acknowledgments

The work was supported by the BMBF under contract no.05H12RD7. We would like to thank Fritz Caspers and MichaelHolz for fruitful discussions and insight on the MWT measure-ments at the SPS at CERN, Silke Federmann for the helpfulcorrespondence on the same measurement setup and MauroTaborelli for information on SEY data.

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