Build up of Electron Cloud in the PEP-II Particle Accelerator in the Presence of a Solenoid Field and with Different Bunch Pattern

We have augmented the code POSINST to include solenoid fields, and used it to simulate the build up of electron cloud due in the PEP-II positron ring. We find that the distribution of electrons is strongly affected by the resonances associated with the cyclotron period and bunch spacing. In addition, we discover a threshold beyond which the electron density grows exponentially until it reaches the space charge limit. The threshold does not depend on the bunch spacing but does depend on the positron bunch population. (Submitted to Physical Review Special Topics: Accelerators and Beams) ∗Work supported by the Department of Energy under Contract No. DE-AC03-76SF00515 and DEAC03-76SF00098. Build up of Electron Cloud in the PEP-II Particle Accelerator in the Presence of a Solenoid Field and with Different Bunch Pattern Y. Cai and M. Pivi∗ SLAC, Stanford University, 2575 Sand Hill Road, Menlo Park, CA 94025. M. A. Furman† Center for Beam Physics Accelerator and Fusion Research Division Lawrence Berkeley National Laboratory, MS 71-259 Berkeley, CA 94720 (Dated: DRAFT: September 10, 2003) We have augmented the code POSINST to include solenoid fields, and used it to simulate the build up of electron cloud due in the PEP-II positron ring. We find that the distribution of electrons is strongly affected by the resonances associated with the cyclotron period and bunch spacing. In addition, we discover a threshold beyond which the electron density grows exponentially until it reaches the space charge limit. The threshold does not depend on the bunch spacing but does depend on the positron bunch population. PACS numbers: 29.27.Bd,79.20.Hx,29.20.Dh


I. INTRODUCTION
It is well established by many experimental evidences [1,2] at KEKB and PEP-II that the instabilities caused by electron impose a severe limitation upon the luminosity in e + e − storage rings.Based on the experiments [1] at KEKB, there exists a current threshold beyond which the vertical beam size at the interaction point starts to grow like σ * y ∝ N 2 p /S b , where N p is the bunch population and S b is the spacing between two sequential bunches.Since N p is normally set at the limit allowed by the beam-beam interaction, this observation implies that S b cannot be too small otherwise the vertical blow-up degrades the luminosity.As a result, both B-factories are currently operated S b ≈ 2 m, which is larger than its design value.
Experimentally, the solenoid field raises the threshold of the blow-up and therefore allows the increase of luminosity.On the other hand, we know from the simulation performed by Zimmermann [3] that longitudinal solenoid field B s confines the electrons near the wall of the vacuum chamber and therefore reduces the cloud density near the positron beam.All this indicates that both S b and B s play vital roles in the physics of electron cloud instability.In this paper, we will study the dynamics between the positron beam and electron cloud with different S b and B s to reveal the physics indicated from the simulations and experiments in the PEP-II low energy ring (LER).
The main sources of electrons in the PEP-II are given by: residual gas ionization, photoemission from synchrotron radiation and secondary emission from electrons hitting the walls.In the present simulations we initially generate a certain large number of electrons uniformly (in azimuth) at the chamber wall and let the electron cloud develops by the secondary emission process until an equilibrium (saturation) density is reached.This approach is valid in the limit that small number of electrons are generated at each bunch passage compared to the equilibrium level.

B. Secondary Emission Process
The secondary electron yield (SEY) δ(E 0 ) and the corresponding emitted-electron energy spectrum dδ/dE (E 0 =incident electron energy, E = emitted secondary energy) are represented by a detailed model described elsewhere [4].The parameters have been obtained from detailed fits to the measured SEY of various materials [5].The main SEY parameters are the energy E max at which δ(E 0 ) is maximum and the peak value δ max = δ(E max ), see Table I and Fig 1 .To be consistent with our previous simulations we have used a value for δ max ∼2.0 and E max =300 eV.Furthermore, for the results shown below, we do take into account the elastic backscattered and rediffused components of the secondary emitted-electron energy spectrum dδ/dE at E 0 0. The backscattered component typically becomes more important at low incident electron energies.To account for this behavior we have used a fit extrapolated data for copper measured at CERN [6] and assumed a δ(0) 0.5 corresponding to a 50% reflectivity for electrons impinging the wall with an energy close to 0 eV.

C. Simulation Model
For the purpose of these present studies we simulate the passage of a train consisting of PEP-II bunches with a bunch population of 1×10 11 and having a 4 or 2 bucket spacing, corresponding to 8.4 and 4.2 nsec.The stainless steel vacuum chamber is assumed to be a cylindrical perfectly-conducting round pipe with a 45 mm radius.
Typically, the electrons are simulated by macroparticles, each one representing a defined number of electrons and carrying a fixed charge as described in [7].The secondary electron emission mechanism adds to these a variable number of macro-particles, generated according to the SEY model mentioned above.The bunch is divided up into N k slices (51) and the inter-bunch gap into N g (250) intermediate steps.The image and space charge forces are computed and applied at each slice in the bunch and each step in the gap.Typical beam and vacuum chamber parameters are listed in Table I.
For simplicity, we assume that the bunch travels at the speed of light and since the beam electric field is effectively two-dimensional, it is convenient to use complex notation to represent it [7,8].

D. Electron Motion
Our work starts with implementing longitudinal solenoid field in the code POSINST [7].For simplicity, we assume that B is a constant and ignore any end effects of the solenoid.For a relativistic electron, the equation of motion can be written where ω = e B/γmc is the cyclotron frequency of the electron.The solution of Eq. ( 1) is a helical orbit with the axis of the helix parallel to the magnetic field and the Larmor radius r = v ⊥ /ω.Along the field, electron moves in a constant speed v .This solution is programmed in the code to compute the motion of the electrons where the solenoid field is at presence.The parameters used in the simulation is tabulated in Table I. S b has to be a multiple of the RF spacing S RF .N p corresponds to the value at the peak of a typical fill in the recent operation.

A. Bunch Train
Our simulation focuses on the electrons accumulated through the secondary emission from the beam pipe in the straight sections where not many primary electrons should be generated because of lack of synchrotron radiation.
The bunch pattern used in the simulation consists of a short train, long abort gap, and a long train.The density of electron cloud is clearly building up along the long train after the gap as shown in Fig. 2. Without solenoid field, the average density grows extremely fast along the train but saturates quickly near twice the neutralization density ρ e = N p /πr 2 b S b due to the balance between the space charge and secondary yield.As the solenoid field increases, both the growth rate and the equilibrium level decrease.At B s = 15 G, we see a very gradual growth of the density along the train of 600 bunches.Assuming that the cloud density is proportional to the vertical beam blow-up, this simulation may be used to explain the observation of the very slow blow-up along the train after the initial installation of the solenoids at KEKB [1].As B s reaches 25 G, the average electron density does not grow and is kept below 5% of ρ e .That is near the density at which the head-tail instability occurs in the LER [9].Fortunately, the density near the beam drops even more since the solenoid field restraints the electrons near the wall.

B. Cyclotron Resonance
As B s increases further, we find at 40 G that the equilibrium density along the bunch train actually become larger than the density without solenoid as shown in Fig. 3 for a bunch spacing S b = 2S RF and in Fig. 4 for a bunch spacing S b = 3S RF .However, we observed that most of the electrons are confined in the vicinity of the wall as shown in Fig. 5.It is clearly seen from Figs 3and 4 that this phenomenon appears as a multipacting resonance.
Indeed, the result can be explained by a resonance of multipacting associated with the cyclotron frequency ω and the bunch spacing S b .Given the lowenergy nature of the secondary electrons (100 eV), the radius of cyclotron motions is much smaller than the radius of beam pipe.The time of flight of an electron being emitted at the wall, bending back by the magnetic field and finally hitting the wall is nearly half of the cyclotron period T c = 2π/ω.The resonance occurs when the time of flight coincides with the time interval between two consecutive bunches, namely Since γ ≈ 1 for typical secondary electrons, this resonance is almost independent of the velocities of electron and therefore much stronger than the resonance occurred in drift space [10].The condition of cyclotron resonance is given by Given S b = 1.26 m, we have B c s = 40 G.That agrees with the simulation.In addition, we can see from Fig. 3 that a minimum density occurs at s is reduced to 30 G, 20 G according to Eq. ( 3), as shown in Fig. 4 for the case S b = 3S RF .Indeed, that is well confirmed by simulations and by a different approach using Vlasov equation in [11].Moreover, we find that the characters of dynamics are essentially the same if we keep the product of S b and B s as a constant.
FIG. 5: Snapshot of the x-y phase space for a solenoid field of 30 Gauss, 4 RF bucket bunch spacing configuration.

IV. ELECTRON-CLOUD INTENSITY THRESHOLD
A threshold for the electron-cloud was observed [2] between 700 mA to 900 mA with 692 bunches spaced 4-RF buckets at PEP-II.The measurement was carried out without or with the solenoid field of 30 G. To understand the threshold mechanism, we run simulations with the similar parameters as in the experiments.The re- Above threshold, the saturated density is proportional to the line density of the beam N p /S b indicating it is limited by the space charge.Since the peak beam current at PEP-II is already operated well above the threshold, the simulation predicts that a two-fold increase of the electrons when the bunch spacing is shorten from 4S F R to 2S RF even if the solenoid field is doubled.
The threshold in the simulation is about 6.5 × 10 10 compared with (4.6 − 6.0) × 10 10 in the observation.One should take into account that the effective peak of the secondary electron yield at the vacuum chamber surface may be smaller than 2 as assumed in the present simulation.
A possible explanation of the intensity threshold is given by considering the details of the secondary electron yield as shown in Fig 7 .The SEY exceeds unity in our model for a 25 eV primary electron energy.Then, inspecting Fig 8,which shows that the average collision energy of the electrons reaches 25 eV for a bunch population of ∼6.5×10 10 , one may explain the intensity threshold as observed in Fig. 6.
Without solenoid, however, the simulation disagrees with the observation because the absence of the nearby threshold in the simulation.
For an electron near the wall, the momentum kick due to a bunch is given by [7] ∆p and thus the energy received from the bunch is where r e is the classic radius of electron.If the electron reaches the wall before the next bunch arrives (B s > B c s ), multipacting of electrons occurs if ∆E ≥ E δ=1 (6) where E δ=1 (typically 20-100 eV) is the energy above which the secondary yield δ exceeds unity.This yields the threshold of bunch population where E 0 = mc 2 .In this simulation, we have E δ=1 30 eV.Using Eq. ( 7), we obtain N th p 8.8 × 10 10 compared with 6.5 × 10 10 found in the simulation.Besides reducing the secondary yield, enlarging the radius of beam pipe may be more effective way to increase the threshold as indicated in Eq. (7).
We compute the average electron density in the vacuum beam pipe compared with the electron density within an ellipse centered on the beam axis, with and without solenoid field as shown in Fig 9 .The area of the beam-ellipse is 20 σ x x 20 σ y = 14 mm x 4.6 mm.In particular, for a for a bunch spacing of S B = 3 S RF , the electron density near the beam reaches 7 ×10 4 e/cm 3 when the solenoid field is set at 30 Gauss.

A. Calculation of the Vertical Dipole Wake Field Induced by the Electron-cloud
For the calculation of the wake function we follow [12] as reported in [7].After the electron-cloud have reached an equilibrium density a single perturbing bunch is displaced from the central orbit by an amount ∆y.In these simulations, we displace the 40th bunch vertically by ∆y=5 mm.The electron cloud is perturbed dynamically causing a dipole wake which affects the subsequent bunches.Let the ∆p y be the momentum kick experienced by the subsequent bunches as they traverse a section of length L. Assuming that there are N sections in the ring, the dipole wake field is computed as where F y = ∆p y /∆t is the force acting on the subsequent bunch during the traversal of the section, ∆p y is the actual momentum kick computed in the simulation and ∆t = L/c.By extracting the dipole wake function we compute the multi-bunch oscillation frequency in the first order approximation given in [12].Considering the ring filled with M equally-spaced bunches, we compute the coherent dipole frequency Ω µ corresponding to the dipole oscillation mode µ by W (ks B )e 2πik(µ+ν β )/M (9) where ω β = ω 0 ν β is the betatron angular frequency, ν β is the horizontal or vertical tune, the collective mode oscillation number is given by µ = 0, 1, 2...M − 1, E is the beam energy, and the overall summation is extended to the wake computed for the first subsequent bunches, here nw=10.Since the amplitude of the oscillation for each mode µ is proportional to exp(−iΩ µ t), the mode is unstable when ImΩ µ is negative and damped when positive.Simulation results show that the wake is short range and is significant for few trailing bunches following the perturbing bunch.Thus with good approximation, the instability growth rate τ −1 0 is given by the first k=1 term as the growth rate as a function of the solenoid field are shown in Fig. 11 and in Table II.we make here the approximation of a constant electron density along the circumference C of the ring.Finally noting that the k=0 mode is real and independent of the mode number µ, it is interpreted as the overall coherent tune shift with the analytical expression given by FIG. 9: Average electron density in the vacuum beam pipe compared with the electron density within an ellipse centered on the beam axis, with and without solenoid field.The area of the beam-ellipse is 20 σx x 20 σy = 14mm x 4.6 mm.The electron density near the beam reaches 7 ×10 4 e/cm 3 when the solenoid field is set at 30 Gauss.

B. Effect of the Solenoid Field on the Wake Field
As the solenoid field increases, the electrons are gradually confined within the vicinity of the wall.However, under the condition of a cyclotron resonance, there exist even more electrons than without solenoid.Since the electrons are confined far away from the beam axis, it is not clear if these electrons could cause any instability.To answer this question, we compute the long-range  II.
As the solenoid field increases, one may notice that the computed long-range wake function and the vertical tune shift are changing sign, from positive to negative.
Without solenoid field the tune shift generated by the electrons is positive because the electron-cloud has a focusing effect on the positron beam.As the solenoid field increases the electrons are confined near the wall resulting in a change of sign of the tune shift similarly to a conventional impedance.A neutral behavior at 20 Gauss is expected in the by 3RF bucket spacing.

VI. DISCUSSION
Based on the simulation, we find that the cyclotron motion of electrons plays important role in generating and accumulating secondary electrons.When the resonance condition is satisfied, we see huge amount electrons near the wall.Although they are far away from the positron beam, they still create the long-range wake strong enough to cause multi-bunch instability.
In addition, if the solenoid field is strong enough B s > B c s , we find that there exists a threshold for the electron-cloud under which there is no accumulated electrons.This discovery may provide us a method to completely eliminate the electron cloud with larger enough beam pipe and lower enough secondary yield.

FIG. 2 :
FIG. 2: Density of electron cloud as a function of time when a bunch train passes through a stainless-steel beam pipe and longitudinal solenoid at different settings.The bunch spacing S b = 2SRF .

FIG. 3 :FIG. 4 :
FIG. 3: Equilibrium electron-cloud density along the bunch train as a function of solenoid field.The bunch spacing S b = 2SRF .
FIG.6: Saturated density as a function of the bunch population.The circles represent the case of 4-RF spacing and 30 G solenoid field.The crosses represent 2-RF spacing and 60 G field.

FIG. 10 :
FIG. 10: Long-range wake function due to electron cloud for a bunch spacing S b = 3 SRF .

FIG. 11 :
FIG. 11: Multi-bunch instability growth rate as a function of the the solenoid field.The multipacting resonance condition at 30 Gauss is shown in figure for a bunch spacing S b = 3 SRF .

TABLE I :
Simulation parameters for the PEP-II LER

TABLE II :
Multi-bunch instability (MBI) simulations.Electron cloud induced vertical tune shift ∆ν β and instability growth time as a function of the solenoid field for a bunch spacing S b = 3SRF .FIG. 12: Vertical tune shift induced by the electron-cloud for a bunch spacing S b = 3 SRF .