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Nov 28, 2000 - During the passage of the LHC bunch train, an electron cloud will build ..... Electron Cloud Effect in the Arcs of the LHC”, CERN-LHC-Project-.
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-SL-Note-2000-057 AP

Electron-Cloud Energy and Angular Distributions A. Arauzo and F. Zimmermann

Abstract

We discuss simulated energy and angular spectra of electrons lost to the vacuum chamber wall during beam-induced multipacting in SPS and LHC. Our simulations demonstrate that electron energies and impact angles are strongly correlated, and that the angular distribution can be modified by the beam magnetic field. The energy spectrum is found to be remarkably sensitive to changes in the bunch length or, to a lesser extent, in the transverse beam size, which may open new possibilities for beam diagnostics. From our simulation results, we estimate the energy and angular acceptance required for future electron-cloud diagnostics as well as the electron impact rate.

Geneva, Switzerland November 28, 2000

1

Introduction

During the passage of the LHC bunch train, an electron cloud will build up inside the vacuum chamber of the SPS and the LHC [1, 2, 3, 4, 5, 6, 7]. This electron cloud arises as follows. Each bunch generates a certain number of primary electrons, either via photoemission, at the LHC, or via gas ionization, at the SPS. Subsequently, beam-induced multipacting can lead to an exponential growth in the number of electrons along the batch, until the electron-cloud density saturates under the influence of its own space-charge field. The growth rate of the electron cloud and the saturated electron density depend on the secondary emission yield of the chamber surface, the value of which will slowly decrease as a result of the electron bombardment [8]. In the SPS, the electron cloud build up has been observed with the LHC test beam since about two years [9, 10, 11]. Its primary effects are an increase of the vacuum pressure, by up to two orders of magnitude, an impairment of the pick-up signals for the transverse damper, and electron-cloud driven beam instabilities [12, 13, 14]. For the LHC, the heat load on the cold beam screen inside the superconducting magnets is also a concern. Electrons of the cloud acquire energy in the electromagnetic field of the charged proton beam, when proton bunches pass by. The energy transferred from the beam to the electrons is deposited on the beam screen (or beam pipe), and amounts to an extra heat load which must be absorbed by the cryogenics system. In order to determine precisely the energy transfer to the cloud electrons, we have performed a series of computer simulations, aimed at characterizing the energy spectrum of cloud electrons in the SPS and the LHC. For both accelerators, we have chosen beam and chamber parameters which give rise to an exponential growth in the number of electrons, or to the so-called ’multipacting’. In the simulation, we have recorded the energy and the angle of incidence of all primary or secondary electrons impinging on the beam-pipe wall. In calculations of the beam-electron interaction, the magnetic force of the beam is often neglected, since the electrons are much slower than the speed of light. In the simulations reported below, the effect of the beam magnetic field was included, using a modified version of the simulation program, since we suspected that the magnetic field might modify the longitudinal electron momenta and thus the angles of incidence. We will show that as expected the beam magnetic field has little influence on the energy spectrum, but that it can increase the rms angular spread of the electrons incident on the chamber wall. A difference between SPS and LHC is that in the LHC primary electrons are generated primarily by synchrotron radiation, whereas in the SPS they are created mainly by ionization of the residual gas. As a consequence, the primary photon emission rate in the LHC is taken to be almost four orders of magnitude higher than in the SPS; see also Table 1. For the present study, we consider a 1-m long region free of extrenal fields with 2.5 cm radius, and the nominal charge and bunch spacing of an LHC batch. We further assume a maximum secondary emission yield of δmax = 1.9. Simulations for this value have well reproduced last year’s observations in the SPS [6]. Similar secondary emission yields also are expected for the early LHC commissioning. With the above parameters, the simulation shows electron multipacting and electron-cloud build up for both SPS and LHC beam sizes. This paper describes the main characteristics of the energy spectra and the angular distribution of the lost electrons so obtained. Our motivation for this study was to quantify the electron-cloud energy spectra, both for comparison with future measurements and for deriving specifications on electron-cloud diagnostics which may be designed.

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2

Analytical Considerations

If an electron is accelerated from rest in the field of a passing bunch, the relation between longitudinal and radial momentum transfer is [15], v⊥ /c ∆pz ≈ r , 2 ∆p⊥ v⊥ 2 1 − c2

(1)

∆pz where v⊥ is the electron velocity after the bunch passage. The ratio | ∆p | is the expected ⊥ impact angle on the surface. For example, final electron energies of 200, 400 or 1000 eV would correspond to impact angles of 0.014, 0.020 or 0.031 radians (or to 0.08, 1.15 and 1.78◦ ). The critical radius defined by Berg [16] is the starting amplitude at which an electron performs a quarter oscillation in the beam potential during a bunch passage. For a round beam this radius is approximately given by [16]

s

rc = 2

Nb re σz πλmax

(2)

where Nb denotes the bunch population, re the classical electron radius, σx the rms bunch length, and λmax a form factor, which depends on the longitudinal distribution. The maximum energy gain, estimated as the energy acquired by a particle launched at rc using the so-called ‘autonomous approximation’ (constant beam density), reads ∆Emax =

me c2 2Nb re rc λmax log , σz c0 σ⊥

(3)

with σ⊥ = σx,y the rms transverse beam size,√and a round beam is assumed. For a Gaussian longitudinal √ γ/2 distribution one finds λmax = 1/ 2π, and for a transverse Gaussian distribution ≈ 1.06 (γ here is Euler’s constant). The critical radius, Eq. (2), is 1.68 cm for c0 = 2e the SPS and 0.84 cm for the LHC. The approximate maximum energy gain according to Eq. (3) evaluates to 707 eV for the SPS and 5217 eV for the LHC. However, though Eq. (3) is quite accurate for a rectangular distribution, it widely overestimates the maximum energy for a Gaussian longitudinal profile. To obtain a better estimate, we can use a symplectic integration routine, also written by Berg [17] (in the following referred to as ‘Berg program’), which considers an electron that starts at rest and computes its energy gain for various radial starting amplitudes. Typical results of this program for SPS transverse beam sizes and two different bunch lengths are shown in Fig. 1. The numerical integration results suggest that the maximum energy gain is about 400 eV for the nominal SPS parameters (solid line) and 1750 eV for the LHC. These numbers are a factor 2–3 smaller than those predicted by Eq. (3).

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Simulation Approach

The electron-cloud simulation splits the bunches and interbunch gaps into many slices, typically 50. During the passage of each bunch slice a number of primary macro-electrons are generated on the chamber wall, each representing a certain number of real electrons. At each time step the motion of the macroelectrons is tracked under the influence of the beam electric field, the electron space-charge field, as well as beam and electron image charges. Whenever a macroelectron is lost to the wall, it is reemitted as a secondary electron, with different charge and initial energy of a few eV. Details of the simulation model can be found in Refs. [2, 4, 6, 7]. 3

Figure 1: Electron energy gain during the bunch passage as a function of initial electron position, for SPS beam sizes and two different bunch lengths. The curves are obtained by 4th order symplectic integration of the electron motion starting at rest [17].

3.1

Magnetic field of the beam

Hitherto, we have always neglected the beam magnetic field in our simulation. Note however that external magnetic fields were frequently included, and these could have a strong effect on the electron motion and on the multipacting. For a round beam in a round chamber, the magnetic field of the beam is related to the electric field via ~ ~ = 1 ~v × E (4) B c2 where ~v (|~v| ≈ c) is the velocity of the proton beam. The magnetic force on an electron reads ~ F~m = −e~ve × B

(5)

with ~ve denoting the electron velocity and (−e) the electron charge. Only considering the transverse electric field components Ex and Ex (the longitudinal component Ez is relativistically suppressed), we can rewrite this as 





ve,z ve,y ve,z ve,x Ex ~ux + Ey ~uy − Ex + Ey ~uz F~m = e c c c c



(6)

where ~ux , ~uy , and ~uz form a right-handed system of orthonormal unit vectors. In the code the magnetic field was implemented by modifying the transverse kicks from the beam field

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applied at each tracking step, and introducing a longitudinal deflection as follows: 

∆p(em) e,y ∆p(em) e,z



ve,z c   v e,z = ∆p(e) e,y 1 − c ve,x (e) ve,y (e) ∆pe,x + ∆pe,y . = c c

= ∆p(e) ∆p(em) e,x e,x 1 −

(7) (8) (9)

Since |ve |  c, the transverse magnetic force is a small correction to the electric force. A more important effect of the magnetic force is that it imparts longitudinal momentum to the electrons, which the electric force does not.

4 4.1

Simulation Results Parameters

Table 1 summarizes the relevant parameters assumed in our simulations. The only differences between SPS and LHC are the bunch length, the transverse beam sizes, and the primary electron creation rate. The primary number of electrons chosen for LHC corresponds to the average photon flux in the arcs and a photoemission yield of 5%, the SPS number to residualgas ionization at 50 nTorr carbon-monoxide pressure. The initial energy distribution of the secondary electrons follows a half-Gaussian (centered at 0) with rms spread 5 eV, that of the photoelectrons a truncated Gaussian centered at 7 eV, also with 5 eV rms value. The initial angular distribution of the photoelectrons is uniform in the two spherical coordinates θ and φ. The emission angles of the secondary electrons are distributed according to dN/dθ = cos θ sin θ, or dN/dΩ = cos θ, where Ω is the solid angle and θ the angle with respect to the surface normal. Table 1: SPS and LHC parameters assumed in the simulation. variable symbol bunch number nb bunch population Nb bunch spacing Lsep rms bunch length σz primary photoelectron rate per proton Ype reflectivity R max. secondary emission yield δmax energy of max. sec. em. yield max rms horizontal beam size σx rms vertical beam size σy radial half aperture hr

4.2

SPS 80 1.05 × 1011 7.48 m 30 cm 2.5 × 10−7 m−1 100% 1.9 450 eV 2.7 mm 2.7 mm 2.5 cm

LHC 80 1.05 × 1011 7.48 m 7.5 cm 1.25 × 10−3 m−1 100% 1.9 450 eV 303 µm 303 µm 2.5 cm

Electron Dose

In the LHC simulation, during the passage of an 80-bunch train a total of about 2 × 1011 electrons are lost to the wall. Dividing by the length of the train and the surface area, this 5

translates into 1.3 × 1018 e− /m2 /s. For the SPS, the numbers are about two times smaller, namely 9 × 1010 electrons hit the wall, which corresponds to a dose of 5.7 × 1017 e− /m2 /s. The factor of two difference arises, because, due the much smaller number of primary electrons in the SPS, about half of the bunch train has passed by before the electron cloud reaches an appreciable density [6]. 4.3

Energy spectrum

Figure 2 shows the energy distribution obtained for the electron cloud with nominal LHC conditions. The total number of electrons has been normalized to an one and the energy spectrum intensity along the vertical axis is given in these ‘arbitrary’ units. The maximum energy limit considered in the program was 3.5 keV. No electrons were found at larger energies. Figure 2 bottom provides a close-up view of the spectrum. There is a high peak near zero energy values and a second main peak at 200–300 eV. The intensity decreases for higher energies, and nonzero values extend up to about 2 keV.

Figure 2: Energy spectrum of electrons hitting the chamber wall in the LHC. Top: full spectrum showing the two main peaks. Bottom: enlarged view revealing details of the second main peak. For comparison, the energy spectrum obtained for the SPS beam parameters is displayed in Fig. 3. Also here we observe a large peak at very low energies. The detailed higher-energy 6

spectrum (Fig. 3 bottom) shows a richer structure than for the LHC. The intensity drops to zero for energies beyond about 400 eV.

Figure 3: Energy spectrum of electrons hitting the chamber wall in the SPS. Top: full spectrum revealing the peak at very low energies. Bottom: enlarged scale illustrating details of the spectrum at higher energies. The energy spectra indicate that there is a maximum electron energy. Such limit is most clearly seen in the SPS spectrum of Fig. 3 It is not unexpected since the momentum transfer to an electron must assume a maximum value as a function of the beam potential and the initial position of the electron. We can compare our simulation result with the energy gain during the passage of a single bunch predicted by the 4th order symplectic integration of electron motion [17] mentioned in Section 2. To do so, using the Berg program we have computed the energy gain of electrons due to the passage of a bunch for the SPS and LHC beam parameters as a function of their initial position. An example of such calculation was shown in Fig. 1. From this and assuming a uniform radial distribution, we can obtain a rough image of the predicted energy spectrum, by histogramming the energy values (the vertical coordinate vector). Note that in reality the distribution of the electron cloud may not be exactly uniform in radius r, but perhaps it might be closer to uniform in cartesian coordinates. 7

Figure 4: Comparison of simulated electron-cloud energy spectrum (top) and the prediction of Berg’s program (bottom) for LHC beam conditions.

Figure 4 compares the simulation results with those from Berg’s program for the LHC beam parameters. The main features of both spectra agree well: the shape and position of the main peak at about 200–300 eV and the peak near zero energy values. The maximum energy level for the Berg calculation appears at 1750 eV, where — we can hardly see it — in the simulated spectrum one observes a small broad peak. Figure 5 shows the equivalent results for the SPS. The agreement is again good. The main features of the simulation are reproduced by the Berg calculation and the maximum limits in energy are equally sharp. We can thus identify the maximum energy limit of the simulation with the maximum of the energy gain curve (as a function of radius), visible in Fig. 1. The similarity of the different computations gives us some confidence in the reliability of the results. It is worth pointing out that the calculations made by the Berg program consider the electric field of only one bunch, whereas in the simulation, depending on the beam parameters and the initial position of an electron, the latter can receive the kick from a second bunch, and, in addition, is also subjected to image-charge and space-charge forces.

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Figure 5: Comparison of simulated electron energy spectrum (top) and Berg prediction (bottom) for SPS beam conditions.

4.3.1

Dependence of energy on bunch length

Apart from the primary electron generation rate, the LHC and the SPS cases only differ in the three beam sizes. Nevertheless the energy spectra are quite different. To explore the transition from the SPS to the LHC beam conditions, we have simulated the energy spectra for different bunch lengths in the SPS. Figure 6 illustrates the strong dependence of the energy gain on the bunch length. The spectrum in black corresponds to the SPS nominal conditions. As the bunch length is shortened, the energy spectrum starts to resemble that in the LHC (σz = 7.5 cm). The most sensitive part of the spectrum is the peak at high energy which shifts to even higher energies as the bunch length decreases. 4.3.2

Dependence of energy on transverse beam size

A dependence of the electron-cloud energy spectrum on the beam size is observed as well. An example is shown in Fig. 7, which compares the SPS electron energy spectra for different beam sizes. Again, as the beam radius decreases. the maximum energy limit shifts to higher energies. This dependence, however, is not as strong as that on the bunch length in Fig. 6. 9

Figure 6: Simulated electron-cloud energy spectrum for different bunch lengths in the SPS. 4.4

Angular distribution. Beam magnetic field effect

In order to design a spectrometer device for measuring the electron cloud energy spectra, as considered, e.g., in Refs. [18, 19, 20], it is useful to study the angular distribution of electrons impinging on the beam pipe. This angular distribution will give us an indication of the angular acceptance required for a detector. We define the angle θ as the angle of the electron momentum with respect to the surface normal. The angular distribution has been simulated for SPS as well as for LHC beam parameters, both with and without the beam magnetic field. Though the presence of the magnetic field has no effect on the energy spectra, as we will see, for the LHC it does modify the angular distribution. 4.4.1

LHC beam parameters

The angular distribution obtained for LHC beam parameters is shown in Fig. 8 with and without the beam magnetic field. The distribution exhibits a main peak at low impact angles, between 0.0 and 0.2 rad, and a broad component at large angles. The main peak is slightly shifted to higher angle values due to the extra kick given by the beam magnetic field, whereas 10

Figure 7: Electron energy spectrum for different transverse beam sizes in the SPS, for a constant bunch length of σz = 30 cm.

the signal at larger angles stays unchanged. In order to gain more insight, we have examined the correlation between energy and impact angle. Figure 8 shows a scatter plot of the number of electrons hitting the beam screen as a function of the impact angle and their energy. Only intensities greater than zero are plotted. It can be seen that most parts of the energy spectra correspond to small incident angles and only the high peak near zero energies is associated with large impact angles. In particular, the broad signal at large impact angles in Fig. 8 corresponds to low energy electrons. The apparent ‘resonant’ structure at low energies is an artifact of the binning in the code, where each point represents the charge of all electrons in an energy window of 7 eV and in an angle window of 0.18◦. Electrons with large angles will likely be out of range for any kind of detection system. These low energy electrons are secondaries generated between bunches with large emission angles, which are lost again to the beam screen before the following bunch passes by. Such losses occur naturally, since in 25 ns a 5-eV an electron travels 3.3 cm, and a 1-eV electron 1.5 cm. These lengths are comparable to the chamber dimensions. The energy distribution of these electrons indeed resembles the energy distribution assumed for the secondary emission 11

Figure 8: Angular distribution of electrons hitting the vacuum pipe. Red continuous line corresponds to simulations with the magnetic field from the beam. Black discontinuous line is the result without the effect of the beam magnetic field.

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process, which is a Gaussian centered at 0 eV with 5 eV rms. We are primarily interested in the low-angle peak of the angular distribution, whose main contributions come from electrons hitting the beam screen with an angle between 0.01 and 0.1 rad (about 0.6◦ and 5.7◦ ).

Figure 9: Scatter plot for LHC beam conditions. Electrons hitting the vacuum pipe as a function of their energy in eV and angle of incidence in radians.

4.4.2

SPS beam parameters

The simulated angular distribution for the SPS is shown in Fig. 10. There is almost no difference between the distributions obtained with and without the beam magnetic field. According to Eq. (1), the longitudinal momentum gained in the magnetic field increases with the electron energy. Therefore, for SPS parameters the effect of the magnetic field is smaller than for the LHC. Although small, the beam magnetic field contribution is included in all the spectra and distributions presented in the following. 13

Figure 10: Angular distribution for e hitting the vacuum pipe. Red continuous line corresponds to simulations with the magnetic field from the beam. Black discontinuous line is the result without the effect of the beam magnetic field.

In Fig. 10, we again find a main peak at low angle values and a broad signal at high angle values. The origin is the same as in the LHC case. For the main peak, the dominant contribution comes from electrons with hitting angles between 0.02 and 0.12 rad (1.1◦ to 6.9◦ ). The scatter plot of Fig. 11 shows a similar correlation between energy and angle as Fig. 9 for the LHC. 4.4.3

Energy dependence

To study in more detail the distribution of the angles of incidence as a function of the electron energy, various angular distributions are plotted in Fig. 12 for energy intervals of 7 eV. We observe again that a strong correlation between energy spectrum and impact angle. For increasing electron energies the angular distribution becomes narrower and shifts to lower values. Secondary electrons are emitted with a broad angular distribution, but they are accelerated mainly transversally in the radial direction (assuming a round beam). The incident angle with the beam pipe is reduced as the (radial) energy gain is higher. The intensity of the different angular distributions in Fig. 12 mirrors the shape of the energy spectrum. 14

Some examples of low energy electron distributions for the SPS are shown in Fig. 13. In this case, the energy bin in the electron cloud simulations was chosen equal to 3 eV. The various angular distributions represent energies between 0 eV and 12 eV. The angular distribution is very broad and of negligible intensity at small angles. This confirms that the low-energy electrons give rise to the broad signal observed in the angular spectra for both LHC or SPS. The same electrons are also the source of the very high peak near zero energy values seen in the energy spectra of Figs. 2 and 3. The number of these electrons is large, but they are confined to a small range of energies at low values. These low-energy electrons are important neither from the point of view of energy deposition nor regarding energy spectrum measurements. 4.4.4

Angular dependence

In the previous section we have studied how the angular distribution changes with the electron energy. Conversely, here we look at the variation of the energy spectra with impact angle. The angle bin in the simulations is 0.18◦ . Figure 14 presents the spectra of electrons hiting the beam pipe at different angles, for the LHC. The main contribution to the energy spectrum (see Fig. 2 bottom) corresponds to angles of incidence between 2◦ and 5◦ . If one only considers electrons near normal incidence (0.5◦ ) the energy spectrum at medium and high energies would be overemphasized. On the other hand, for large angles (30.1◦ ), the energy spectrum only contains a low-energy component. Figure 15 shows energy spectra obtained at various angles of incidence for the SPS. The energy spectrum for the SPS is quite different from the LHC case. But also here we observe a large variation in the energy spectrum. Again, the spectra obtained for angles in the range from 2◦ to 5◦ agree quite well with the general energy spectrum in Fig. 3. At this point we can conclude that the angular distribution of the collected electrons should extend at least up to 2◦ to 3◦ to guarantee a reliable and meaningful measurement of the electron cloud energy spectrum.

5

Discussion

Since the energy spectrum of electrons incident on the wall strongly depends on the range of impact angles considered, for any future electron-energy diagnostics a minimum angular acceptance of 2–4◦ appears to be advisable. The relevant energy ranges for SPS and LHC differ by a factor of about 4, since the electron energies vary with bunch length, and, to a lesser extent, with the transverse beam size. Moreover, the electron energies will also scale between linearly and quadratically with the bunch population. All this suggests that electroncloud monitors should cover a large dynamic range of electron energies. The high sensitivity of the energy spectrum to bunch length and beam size opens the exciting possibility of utilising the electron-cloud phenomenon for monitoring these beam parameters, similar in spirit to gas ionization monitors. The effect of the beam magnetic field increases the angular spread of incident electrons by about 70% at the LHC, but it is insignificant for the longer bunches in the SPS. In neither case does the magnetic field noticably affect the electron energies.

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Figure 11: Scatter plot for SPS beam conditions. Electrons hitting the vacuum pipe as a function of their energy and angle of incidence.

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Figure 12: Angular distribution for different energy values and LHC beam parameters.

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Figure 13: Angular distribution for very low energy values and SPS beam parameters.

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Figure 14: Energy spectra of electrons hitting the beam pipe at different angles for LHC beam conditions

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Figure 15: Energy spectra for e-cloud electrons hitting the beam pipe at different angles, for the SPS beam conditions.

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6

Conclusions

By means of computer simulation, we have studied the energy and angular spectra of electrons hitting the chamber wall during beam-induced multipacting in SPS and LHC. The energy spectrum changes strongly with longitudinal and, to a lesser extent, with transverse beam dimensions. The electron impact energies are correlated with the angle of incidence. In case of the LHC, the angular spectrum is modified by the beam magnetic field. Future diagnostics should aim for a large dynamic range and a minimum angular acceptance of a few degrees.

References [1] O. Gr¨obner, “Technological Problems related to the Cold Vacuum System of the LHC”, Vacuum, vol. 47, pp. 591-595 (1996). [2] F. Zimmermann, “A Simulation Study of Electron-Cloud Instability and Beam-Induced Multipacting in the LHC”, CERN-LHC-Project-Report 95 (1997). [3] O. Gr¨obner, “Beam Induced Multipacting”, CERN-LHC-Project-Report-127 (1997). [4] O. Br¨ uning, “Simulations for the Beam Induced Electron Cloud in the LHC Beam Screen with Magnetic Field and Image Charges”, CERN-LHC-Project-Report-158 (1997). [5] M. Furman, “The Electron Cloud Effect in the Arcs of the LHC”, CERN-LHC-ProjectReport-180 (1998). [6] F. Zimmermann, “Electron-Cloud Simulations for SPS and LHC”, Proc. Chamonix 10, CERN-SL-2000-007-DI. [7] G. Rumolo, F. Ruggiero, F. Zimmermann, “Simulation of the Electron-Cloud Build Up and Its Consequences on Heat Load, Beam Stability and Diagnostics”, presented at ICAP 2000, Darmstadt (2000). [8] N. Hilleret, “Ingredients for the Understanding and the Simulation of Multipacting”, Proc. Chamonix 10, CERN-SL-2000-007-DI. [9] W. H¨ofle, “Observations of the Electron Cloud Effect on Pick-Up Signals in the SPS”, Proc. Chamonix 10, CERN-SL-2000-007-DI. [10] G. Arduini, “Observations in the SPS: Beam Emittance, Instabilities”, Proc. Chamonix 10, CERN-SL-2000-007-DI. [11] J. Jimenez, “Electron Cloud: SPS Vacuum Observations with LHC Type Beams”, Proc. Chamonix 10, CERN-SL-2000-007-DI. [12] M. Izawa, Y. Sato, T. Toyomasu, Phys. Rev. Lett. 74, 5044 (1995). [13] K. Ohmi, Phys. Rev. Lett. 75, 1526 (1995). [14] K. Ohmi and F. Zimmermann, “Head-Tail Instability caused by Electron Cloud in Positron Storage Rings”, CERN-SL-Report-2000-015 (AP). [15] J. Buon, F. Couchot, J. Jeanjean, F. Le Diberder, V. Lepeltier, H. Nguyen Ngoc, J. Perez-y-Jorba, P. Chen, “A Beam Size Monitor for the Final Focus Test Beam,” NIM A 306, p. 93 (1991). [16] S. Berg, “Energy Gain in an Electron Cloud During the Passage of a Bunch”, CERN LHC Project Note 97 (1997). [17] S. Berg, “Program for Computing Energy Gain of Photoelectrons”, (1997). 21

[18] K. Harkay, “Measurements of the Electron Cloud in the APS Storage Ring”, Proc. PAC99, New York, p. 1641 (1999). [19] A. Arauzo, C. Bovet, J. Buon, P. Puzo, “Proposal of a Gas-Ionization BSM for the LHC”, LHC-Project-Note-198 (1999). [20] M. Pivi, A. Variola, “Energy Spectrum Measurement of the Multipacting Electrons in the SPS”, CERN SL-Note-2000-040 BI (2000).

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