Theory and simulation of electron-cloud instabilities - CiteSeerX

Theory and Simulation of the Electron Cloud Instability G. Rumolo and F. Zimmermann SL/AP CERN, Geneva, Switzerland Abstract

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A quasi-stationary electron cloud can drive single-bunch instabilities on a circulating beam through a head-tail coupling mechanism. The oscillation frequency of the electrons in the transverse field of the bunch, and how it can be expressed as a function of the bunch properties, is essential for the understanding of this instability. The process of the instability has been investigated via computer simulation. The model that we have applied makes use of two sets of macro-particles for the bunch particles and for the electrons, too, which interact at one or more locations along the beam orbit (kick approximation). The code has been employed to simulate possible instability mechanisms in the SPS as well as in the the LER of the KEK-B factory. The dependence of the induced emittance growth rates on chromaticity, bunch intensity and length, and electron cloud density has been studied and interpreted for both cases.

1

INTRODUCTION

Photoemission, or gas ionization, and electron multiplication due to the secondary emission process on the inner side of the beam pipe may cause the build up of an electron cloud, which can significantly degrade the performance of rings operating with closely spaced proton or positron bunches. One of the undesired effects of the presence of the electron cloud inside the beam pipe is the possible beam instability that it brings about. Coupled-bunch as well as single-bunch phenomena have been studied in the past and it was shown that the electron cloud can be the source of unconventional wake fields which affect the beam dynamics. A multi-bunch dipole-mode instability was observed at the KEK Photon Factory on positron beam operation [1], and it was subsequently explained by K. Ohmi as an effect of the variation in the electron-cloud centroid position, which can couple the motion of subsequent bunches [2]. More studies on this subject were carried out by an IHEP-KEK collaboration at the Bejing Electron Positron Collider [3], and at the PEP-II B factory [4, 5]. The effect on a bunch of the deformation in the electron cloud distribution caused by a preceding displaced bunch was therefore introduced as an option of study in the existing codes for the simulation of the electron cloud build up [2, 6, 7]. In the Electron Cloud Session of Chamonix X (2000), F. Zimmermann has shown that the multi-bunch instability is not a concern for LHC at top energy because the growth time, which scales with the effective bunch-to-bunch dipole wake field W 1,y (x) ac-

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τ≈

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is in the order of magnitude of 1 s even for maximum Secondary Emission Yields (SEY’s) around 2.0 [8]. Nevertheless, the multi-bunch instability can be a concern for LHC at injection energy and in the Super Proton Synchrotron (SPS), too. This issue is now under investigation, because with W1,x > W1,y the induced horizontal instability might be not negligible; this would explain why coupled bunch phenomena have recently been observed at the SPS in the horizontal plane [9]. A single-bunch instability of a proton or a positron bunch is different from this multi-bunch mechanism and it can be considered as a two-stream instability of the same type as studied in plasma physics. In this case, the head and the tail motions of a single bunch are coupled by the induced distortion of the electron cloud distribution, and the oscillations performed by the electrons around the bunch as it passes through the cloud, play a key role. A single-bunch instability of a positron bunch due to electrons created by ionization of the residual gas was first discussed for linear accelerators [10]. Later on, when a blow up of the vertical size was observed at the Low Energy Ring (LER) in the KEK B Factory, a plausible explanation was a similar single-bunch two-stream instability [11, 12]. In the simulation code presented in Ref. [12] written to study this phenomenon, the positron bunch was represented by a large number of micro-bunches with a fixed transverse size equal to the beam size. An improved version of the code using macro-particles both for the electrons and for the bunch particles was developed at CERN and a preliminary discussion was given in Ref. [13]. Throughout the present paper the attention will be solely focused on single-bunch phenomena. In the following, we will first describe how the single bunch instability sets in and what parameters characterize it (Sec. II). Then, we will introduce the macro-particle simulation code modelling the interaction between bunch and electrons and we will discuss its applications to the SPS at CERN and to the LER of the KEK B Factory (Sec. III). Finally, Sec. IV summarizes the results and draws an outline for future work and development.

2 MECHANISM OF THE SINGLE BUNCH INSTABILITY INDUCED BY ELECTRON CLOUD An electron cloud uniformly distributed all inside the beam pipe can act as a short-range wake-field and be responsible for single-bunch instabilities. The mechanism

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that causes the beam to go unstable under the action of the electron cloud wake-field is schematically shown in Fig. 1. Assuming that the bunch enters the electron cloud with its head slightly offset (upper part of Fig. 1), there will result a global net force acting on the electrons towards the head centroid position and consequently an accumulation of electrons in that region. The newly reconfigured electron distribution will thus kick the following bunch particles toward the higher density region. The motion of the head will therefore be transmitted and amplified at the tail of the bunch. The tail deflection increases over successive turns (lower part of Fig. 1). This simple intuitive picture applies if the inverse of the synchrotron tune of the bunch is much smaller than the number of turns which the instability would take to appear. Otherwise, the bunch particles will in general mix longitudinally. This prevents the coherent dipole motion from growing to high values, but the instability will still manifest itself as a head-tail Transverse Mode Coupling Instability (TMCI) and as an emittance growth distributed all along the bunch.

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equation of motion (for example for the vertical y direction) in the simple form d2 y 2λ(t)re c2 + [y− < y >bunch (t)] = 0 , dt2 (σx + σy )σy

(1)

where After a few passages through the electron cloud

Figure 1: Schematic of the single-bunch instability caused by electron cloud. When the bunch passes through the electron cloud, the electrons oscillate in the electric potential of the positron bunch. At first the oscillation is incoherent, but gradually a coherent oscillation of both electrons and bunch particles develops along the bunch. The coherent oscillation grows from any small initial perturbation of the bunch (or electron) distribution. The oscillation of the electrons is not purely harmonic because of the nonlinear nature of the forces and because of the nonuniform longitudinal bunch profile, as well. Nevertheless, if we consider only the electrons transversely very close to the bunch, within the range of linearity of the bunch field, we can write down their

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 Npr (ct − 2σz )2 λ(t) = √ exp − 2σz2 2πσz

−2σz < ct < 2σz ;

σx , σy and σz are the rms beam sizes in x, y and z; Npr is the number of protons contained in the bunch; < y >bunch (t) is the beam centroid vertical position along the bunch. Fig. 2 shows the y(t) and the phase space trajectories of five test electrons starting from initial conditions (y0 , y˙ 0 ) = (iσy , 0) with i = −2, ..., 2. We have used the SPS parameters summarized in Table I 1 in order to solve Eq. (1) numerically, choosing < y > bunch (t) as a random number in the interval (−σ y /10, σy /10) at each different time step. We can deduce from Fig. 2 that in this case the electrons perform less than one full oscillation over the 1 By mistake, in most of the simulations discussed hereafter the value of α was the one reported in this table, which is actually the slip factor η of the SPS for protons at 26 GeV. This explains why these simulations were for a bunch below transition energy, and a resonance is observed when carrying out the scan of the positive chromaticities. We also report some results obtained using the correct value α = 1.8 × 10−3 when already available. The full analysis has not yet been repeated

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Table 1: SPS parameters used in the simulations. Circumference 6900 m Relativistic γ 28.7 Bunch population ≈ 10 11 protons 2 Emitt. (x,y = σx,y /β) 6.25/2.27(×10−7) m Tunes (Qx,y,s) 26.632/26.586/0.006 Rms-beam-sizes (σx,y,z ) 5.0/3.0/300 mm Average beta functions β x,y = 40 Rms-energy-spread 0.001 Mom. compaction factor α = 5.78 × 10 −4 Chromaticities (ξx,y ) up to 0.4 in both planes y (x)

The quantity n osc is important because it determines whether the instability appears (if it’s too low, no instability is expected to set in because the electrons don’t

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As we can see,  this electron oscillation frequency is proportional to Npr /σz . We can evaluate the number of oscillations that the electron performs when the bunch passes, simply multiplying the frequency ω ey (x) /(2π) by the bunch length expressed in units of time ∆t bunch = 4σz /c:  ωey (x) 2Npr σz re 1 y (x) ∆tbunch = (3) nosc = 2π π σy (x) (σx + σy )

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move enough during the bunch passage) as well as the profile of the instability along the bunch. Typical values are: ωey = 2π × 0.22 GHz and nyosc = 0.85 for the SPS; ωey = 2π × 30.33 GHz and nyosc = 1.62 for the KEKB-LER.

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bunch passage, and that the random distribution of the local centres of attraction appears low-pass filtered into a kind of coherent lower frequency motion of the electrons. This feature is clearer yet in Fig. 3, where the centroid of the electron distribution is tracked along the bunch passage. Since the wake field left behind by the oscillating electrons is in first approximation proportional to the average transverse position of their distribution, the profile in Fig 3 gives an idea of the longitudinal shape of the wake field, too. The oscillation frequency of the electrons can be approximated by assuming that the bunch is uniformly distributed in the longitudinal direction, and this yields:  Npr re c2 . (2) ωey (x) = 2σy (x) σz (σx + σy )

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Figure 4: Snapshots of the horizontal and vertical phase space distributions of the electrons at four times during the bunch passage. Figures 4 and 5 show snapshots of the horizontal and vertical phase space and ordinary space distributions for the electrons at four times during the bunch passage. At the start of the interaction between bunch and electron cloud, the electrons are uniformly distributed in the x and y directions and their velocities are all set to zero. The sizes of the cloud are chosen to be 10 times the beam rms-sizes. At subsequent times we can observe how the electrons start rotating in the phase space: those closer to the bunch execute synchronous oscillations around the bunch centroid, whereas the ones further away are simply drawn towards the centre and progressively undergo nonlinear oscillations. This gives rise to an accumulation of electrons in the central region of the cloud, which peaks after 1/4 oscillation and then becomes wider again but keeps on growing due to the increasing number of electrons that move towards the centre. At the end of the process, the electron distributions

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show a higher density around their centres, with maximum values that are more than twice higher than the initial uniform values. This is essentially the deformation of the electron cloud distribution that acts back on the incoming part of the bunch and causes it to go unstable.

3 SIMULATION OF ELECTRON-CLOUD INDUCED SINGLE-BUNCH INSTABILITIES Model

When we simulate the single-bunch effects driven by the electron cloud, only perturbations of the cloud induced by the passing bunch are considered. All the relevant bunch and lattice parameters, as well as the average equilibrium density of the electron cloud along the ring, are basic input parameters for the simulation of the coupled motion between bunch and cloud electrons. For simplicity, the cloud is assumed to be localised at one, or more, definite sections along the ring, s = n× s el with n = 0, 1, .., (Nint − 1), and the bunch particles are kicked by the electrons when they go through these sections. Both the cloud and the bunch are

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Figure 5: Snapshots of the horizontal and vertical ordinary space distributions of the electrons at four times during the bunch passage.

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modelled as ensembles of macro-particles (with N p bunch macro-particles and N e macro-electrons in the cloud). The bunch is also divided into N sl slices, which interact with the electron cloud one after the other and cause the distortion of the initially uniform cloud distribution that can significantly affect the part of the bunch coming later. The principle of the simulation is schematically illustrated in Fig. 6. The interaction between bunch particles and cloud electrons is expressed by the equations of motion: d2 xp,i (s) 2rp (e) Neme · + K(s)xp,i (s) = − 2 ds γ ·

Nel N int −1  j=1

,

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F (xp,i (s) − xe,j ; σme )δ(s − nsel )

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Nsl  d2 xe,j 2 = −(2r c ) Npk F (xe,j − xoffk ; σk ) , e dt2

(5)

k=1

where the positions of electrons and bunch particles are represented by the vectors x e ≡ (xe , ye ) and xp (s) ≡ (xp , yp , zp ), z = s − ct being a co-moving longitudinal coordinate; K(s) is the transfer matrix with the focusing strengths between two beam-electrons interaction points; Neme and Npk represent the number of electrons in one macro-electron and the number of particles in the k-th bunch slice, respectively; x offk and σk are the transverse offset and the rms-size of the k-th slice; σ me is the rms-size of a macro-electron, which is assumed to have finite size (typically a tenth of the beam size); F is expressed by the Bassetti-Erskine formula [14]. The technique of using macro-particles with finite size is not new in the frames

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ωR = ω

Z  t  , ωR ω − 1 + iQ ω ωR

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and the values of ω r , Zt and Q depend on the ring under consideration. For instance, the values recently measured for the SPS vertical impedance are ω R = 2π × 1.3 GHz, Zt = 108 MΩ/m and Q = 3.6 [18]. In our simulations, we have used 10 4 macro-electrons, and the same number of macro-particles for the bunch. The bunch has been divided into 20 to 30 slices. The interaction starts from the slice containing particles with the largest positive values of z. The model that we have used allows us to resolve the head-tail motion of the bunch, as well as the horizontal and vertical emittance and rms-sizes evolutions, both for individual slices and averaged over the full bunch. Similar simulations were reported in [12], where the beam was represented by an ensemble of micro-bunches with constant transverse size. Table 2: KEKB LER parameters used in our simulations. Circumference 3016 m Relativistic γ 7000 Bunch population 3.3 × 10 10 e+ Emittances (x,y ) 1.8 × 10−8 /3.6 × 10−10 m Tunes (Qx,y,s) 45.53/44.11/0.015 Rms-beam-sizes (σx,y,z ) 0.42/0.06/4 mm Average beta functions β x,y = 10 m Rms-energy-spread 0.0007 Mom. compaction factor α = 1.8 × 10 −4 Chromaticities (ξx,y ) up to 0.11/0.23

Vertical bunch offset - ybunch (mm)

of plasma physics [15, 16] and beam-beam simulations for linear colliders [17], where its purpose is to avoid singularities which may invalidate the simulations while retaining the long range behaviour of the fields that is responsible for collective effects. We have chosen to use elliptical macroelectrons, and the rms-sizes of each have been calculated from its “natural” transverse area (namely, the area transversely occupied by the total ensemble divided by the overall number of macro-electrons) assuming that their aspect ratio equals that of the bunch. These numbers have been further scaled down by an appropriate factor in order to account for Gaussian distributions. Adjusting this factor we ran the code for different macro-electron sizes, and verified that the influence of this parameter on the results of the simulations is in fact negligible, as long as the electrons stay much smaller than the beam transverse size but big enough that the probability of large kicks remains also low. The interaction between bunch and electron cloud is simulated following the steps that are shown in Fig. 6. The macro-electrons have initially a uniform distribution which extends transversely over a region 10 to 20 times larger than the bunch rms-sizes. The initial velocities of the macroelectrons are set to zero. The particles in the bunch have initial Gaussian distributions in each coordinate of their 6dimensional phase space, and the bunch is subsequently sliced. The bunch slicing is actually repeated each time the interaction starts, because when synchrotron motion is taken into account, particles mix longitudinally. As the bunch slice k interacts with the electron cloud concentrated at the kick point, the particles therein contained and the electrons receive a mutual kick; the kick on the electrons is determined from a soft-Gaussian approximation using the instantaneous mean position and rms-sizes of the k-th slice. The perturbed electron cloud then acts on the particles in the next bunch slice k+1, whereas the slice k will be newly kicked by the electron cloud at the next interaction point (after all the bunch particles have been propagated using a linear transport matrix, which can optionally account for the synchrotron motion and a chromatic rotation). Without synchrotron motion, the mechanism does not affect the head of the bunch, which always feels a zero total force from the electrons, but it can drive its tail unstable as for conventional beam break up. The electron cloud configuration is uniformly re-generated at the beginning of each interaction. Recently a new option has been introduced in the code to simulate the effect of the short-range wake field corresponding to a broad band impedance, as well. On each particle of a bunch slice, the kick due to the superposition of the trailing fields excited by all previous slices is applied in addition to the electron cloud kick and to the chromatic rotation. With this tool, it is possible to investigate the combined effect of the electron cloud and a broad band impedance. The impedance we consider is

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Figure 7: Bunch centroid vertical position in the SPS versus time when the contribution of a broad-band impedance is taken into account (black line), and when it is not (red line). For this run we used a bunch population of 10 11 protons, a cloud density of 10 12 m−3 , a bunch length of 30 cm and zero chromaticities.

3.2

Application to SPS and LER of the KEK-B Factory

Using the model previously introduced, hereafter we present computer simulations of beam instability for the CERN SPS and for the LER of the KEK B factory. Typical parameters that have been used for this study are

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were kept constant at a fixed value. The emittance growth observed in all cases was always linear (see, for example, Fig. 8) and therefore the growth rate of the instability could be quantified in terms of emittance growth over a time of 4.5 ms (this also gives the slope of the linear emittance increase). For an electron cloud density of 10 12 m−3 and zero chromaticities, Figs. 9 and 10 show the dependence of the instability on bunch population (bunch rms-length was 30 cm) and on bunch length (bunch population was 10 11 protons). As expected, the instability gets worse as these two parameters increase, though one can clearly see a saturation for values of the bunch population bigger than 10 11 protons. 12

E-cloud density = 10 m Horizontal emittance Emittance growth over 4.5 ms (in percent)

εx (µm)

summarized in Tables 1 and 2. Different values for the average electron cloud density along the ring (n el ) have been chosen, ranging from 2 × 10 11 m −3 to 1013 m −3 (as results from the simulations of electron-cloud build up [8, 11, 12]). All simulations made for SPS show that no significant single-bunch rigid-dipole mode instability is driven by the electron cloud alone (at least over the first 4.5 ms of evolution), but an increase of the bunch centroid vertical oscillation is observed when the contribution of the broad band impedance is also taken into account (Fig. 7). A more systematic study of the combined effect of electron cloud and broad band impedance will be thus conducted in the future. As the runs in these conditions are very time-consuming, a thorough analysis with the short-range wake field contribution has not been yet completed. All results reported in the following refer to simulations done without the contribution of the broad band impedance.

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We wanted to check the dependence of the instability on critical parameters like the cloud density, the bunch population and length, and chromaticity. To this purpose, we made a series of runs in which one of these parameters was varied within a reasonable span, whereas the others

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Figure 8: Simulated SPS emittance growth over 4.5 ms for a case with Npr = 1011 protons, σz = 30 cm, cloud density 1012 m−3 , and zero chromaticities. The upper picture shows the horizontal emittance growth, and the lower picture the vertical one.

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Figure 10: Emittance growth versus bunch length for SPS bunches (Table I). The reason why the instability levels off for higher values of the bunch population is that for these values the electrons perform about one full oscillation during the bunch passage. For a larger number of oscillations, the interaction with the m = 1 head-tail mode number becomes weaker, and higher order modes may be excited [19]. As the insta-

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Figure 13: Emittance growth versus electron cloud density for SPS bunches when the simulations are carried out using the correct value α = 1.8 × 10 −3 .

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Figure 12: Emittance growth versus electron cloud density for SPS bunches (Table I). In Fig. 12 the emittance growth as a function of the electron cloud density is plotted for zero chromaticities, a bunch rms-length of 30 cm and a bunch population of 1011 protons for the parameters of Table I. Figure 13 shows instead the emittance growth versus the electron cloud density for the same set of parameters, but when the α is correctly chosen as 1.8 × 10 −3 ; no major difference with respect to the previous case is observed. As expected, the higher the density of the electron cloud is, the worse the instability degrades the beam.

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bility is actually dependent on the number of plasma oscillations of the electrons over one bunch passage, Eq. (3), we can summarize the two dependencies above in the only dependence of the instability on this number: Fig. 11 clearly shows how the emittance growth increases for small values of ∆tbunch · ωe but then saturates when the number of oscillations exceeds 1. This appears consistent with an analytical estimate for a 2-particle model [12].

Figure 14 contains finally the dependence of the instability on chromaticity when the cloud density is 10 12 m−3 , the bunch length 30 cm and the bunch population 10 11 protons. For this high value of electron cloud density no strong dependence of the unstable evolution on chromaticity is to be observed, except for the one point with normalized chromaticities of around 0.15, where even a dipole mode vertical instability appears and the emittance growth exhibits a larger value. Checking the chromatic frequency shift corresponding to this value, one obtains ωξ = ∆Qω0 /η = 0.23 GHz= ωe . Therefore, we interpret this point as follows: for this value of chromaticity a resonance occurs due to the chromatic shift that pushes the bunch spectrum toward the oscillation frequency of the electrons (most probable candidate point where an “electron-cloud impedance” may be peaked). Figure 15 shows the dependence of the instability on chromaticity when the correct value of the momentum compaction factor is considered and the bunch is therefore above

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transition. We observe that in this case the resonant point around 0.15 disappears, because now both ξ x,y and η are positive, and the chromatic shift has the right sign not to make the bunch spectrum exactly overlap with the impedance spectrum. The instability tends to get worse for values of chromaticity higher than 0.5: here a change of the working point could help to cure this effect. The results from similar simulation scans for the LER of the KEK B factory are presented in Figs. 16,17,18. In these diagrams, only the vertical emittance blow up is shown, because no unstable evolution in the horizontal plane appeared from the simulations. In good agreement with the previous results, we observe here a faint dependence on chromaticity (Fig. 16), and an instability strength increasing with the electron cloud density (Fig. 17) and with the bunch population (Fig. 18).

Emittance growth over 5 ms (in percent)

Vertical emittance 50 45 40 35 30 25 20 15 10 5 0 0.5

11

Bunch population =3.3 10 E-cloud density = 10 1

1.5

2

12

2.5

m

protons

-3

3

3.5

4

4.5

5

5.5

Chromaticities as multiples of (Q, x ,Q, y)=(1,2)

Figure 16: Emittance growth versus vertical chromaticity for LER bunches. All this appears to be consistent with Ohmi’s simula-

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10

11

11

-3

m )

Figure 17: Emittance growth versus electron cloud density for LER bunches.

Vertical emittance Emittance growth over 5 ms (in percent)

Figure 15: Emittance growth versus (horizontal or vertical) chromaticity for SPS bunches when the simulations are carried out using the correct value α = 1.8 × 10 −3 and the bunch is therefore above transition.

9

Cloud density (10

Normalized chromaticity ξx,y

100 90 80

Chromaticities = (0,0) E-cloud density = 10

12

m

-3

70 60 50 40 30 20 10 0

1

2

3

4

5

6

7

8

Bunch population (10

9 10

10

positrons)

Figure 18: Emittance growth versus bunch population for LER bunches. tions [12], though a stronger dependence of the instability on chromaticity should have been found if we had studied the unstable evolution for lower values of the electron cloud density (below 7×10 11, as was suggested in Ref. [12]). Ohmi’s new simulations running with a Particle In Cell (PIC) scheme for the calculation of the electron cloud electric field acting on the bunch particles [20] have been compared with the outcomes of our code for the case of a LER run without synchrotron motion. Fig. 19 shows how the bunch centroid vertical amplitude and the bunch vertical rms-size along the bunch are predicted to be after 300 turns. The agreement between the two codes looks satisfactory. In Fig. 20 the electron vertical distribution after interaction with the bunch is plotted, and we can see that the agreement between the two codes is still good, though Ohmi’s code predicts a double shoulder structure which is smoothed out in our simulation.

4 CONCLUSIONS In this paper we have given a physical picture of the singlebunch instability induced by electron cloud, and we have

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Distribution of the electron cloud

Dipole amplitude of bunch (after 300 turns) 0.0002

8000

0.00015

7000

6000

0.0001 Bunch vertical rms-size

N (arb. un.)

y (m)

0

-5e-05

4000

3000

Bunch centroid vertical amplitude

-0.0001

2000

-0.00015

1000

0 -0.0004

-0.0002 -2

-1.5

-1

-0.5

0 z/sigz

0.5

1

1.5

2

0.2

dNe/dy (x 107 m-1)

y-offset and σy (mm)

Ohmi’s vertical distribution

5000

5e-05

0.15 0.1

Bunch vertical rms-size

0.05

-0.0002

-0.0001

0 y (m)

0.0001

0.0002

0.0003

0.0004

14

Vertical distribution obtained with our code

12 10

0

8 6 4

Bunch centroid vertical amplitude

-0.15 -0.2 -2

-0.0003

16

-0.05 -0.1

Cloud

2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Bunch (in z/σz)

Figure 19: Bunch centroid vertical amplitude and bunch vertical rms-size along the bunch after 300 turns (case without synchrotron motion). The upper picture comes from Ohmi’s PIC solver [20], whereas the lower one is the outcome of our simulation.

0 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

y (mm)

Figure 20: Electron vertical distribution after interaction with the bunch. The upper picture comes from Ohmi’s PIC solver and was taken after the 300th beam-cloud interaction [20], whereas the lower one is the outcome of our simulation after just one interaction.

5 ACKNOWLEDGEMENTS discussed a macro-particle model for the description of the electron-cloud single-bunch interaction in order to study numerically this phenomenon. The oscillation frequency of the electrons during the bunch passage is an essential parameter for the understanding of the blow up and for the interpretation of the instability evolution in different conditions. From the application of our code to SPS, we have learnt that the instability quickly saturates when the number of oscillations performed by the electrons over one bunch passage becomes greater than unity. Beside that, for a high electron cloud density, the instability does not depend much on chromaticity, except for a resonance when the chromatic frequency shift exactly equals the oscillation frequency of the electrons. For positive chromaticities, this can only happen below transition. We presented preliminary evidence that the combined effect of electron cloud and broad band impedance is likely to cause a dipole-mode instability with consequent beam loss, but the issue has to be investigated further. In order to reduce the computing times, the implementation of a PIC or Cloud In Cell (CIC) module is planned as future work.

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The authors would like to express their thankfulness to G. Arduini, K. Cornelis, K. Ohmi, F. Ruggiero and D. Schulte for helpful discussion and/or information.

6

REFERENCES

[1] M. Isawa, Y. Sato and T. Toyomasu, Phys. Rev. Lett. 74, 5044 (1995). [2] K. Ohmi, Phys. Rev. Lett. 75, 1526 (1995). [3] Z. Y. Guo et al., KEK-PREPRINT-98-23, in Proc. 1st Asian Particle Accelerator Conference (APAC 98), Tsukuba, Japan (1998). [4] M. Furman and G. R. Lambertson, talk for PEP-II Machine Advisory Committee and updates, January 1997 (1997). [5] S. A. Heifets, “Transverse Instability Driven by Trapped Electrons”, SLAC/AP-95-101 (1995). [6] M. A. Furman and G. R. Lambertson, in Proceedings of the International Workshop on Multibunch Instabilities in Future Electron and Positron Accelerators, Tsukuba, KEK 1997 edited by Y. H. Chin (KEK Proceedings 97-17, 1997) p. 170.

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[7] F. Zimmermann, ‘A simulation study of electron-cloud instability and beam-induced multipacting in the LHC”, CERN-LHC-PROJECT-REPORT-95, and SLAC-PUB7425 (1997). [8] F. Zimmermann, “Electron-cloud simulations for SPS and LHC”, in Proceedings of the Workshop on LEP-SPS performance -Chamonix X-, Chamonix, 2000 edited by P. Le Roux, J. Poole and M. Truchet (CERN-SL-2000-007 DI) p. 136. [9] K. Cornelis, G. Arduini et al. Elsewhere in these proceedings. [10] T. O. Raubenheimer and F. Zimmermann, Phys. Rev. E52, 5487 (1995). [11] F. Zimmermann, “Electron-Cloud Studies for the LowEnergy Ring of KEKB”, CERN SL-Note-2000-004 AP (2000). [12] K. Ohmi and F. Zimmermann, Phys. Rev. Lett. 85, 3821 (2000). [13] G. Rumolo, F. Ruggiero and F. Zimmermann, Phys. Rev. ST Accel. Beams, 4, 012801 (2001). [14] M. Bassetti and G. A. Erskine, “Closed Expression For The Electrical Field Of A Two-Dimensional Gaussian Charge”, CERN-ISR-TH/80-06 (1980). [15] J. M. Dawson, Rev. Mod. Phys. 55, 403, (1983). [16] R. W. Hockney and S. W Eastwood, “Computer Simulation Using Particles”, Hilger Bristol U.K. (1988). [17] D. Schulte, (private communication) and PhD Thesis TESLA-97-08 (1996). [18] G. Arduini et al, “Measurements of coherent tune shift and head-tail growth rates at the SPS”, CERN SL-Note-99-059 MD (1999). [19] A. W. Chao, “Physics of Collective Beam Instabilities in High Energy Accelerators”, Wiley (1993). [20] K. Ohmi, (private communication).

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