Electron Cloud Cyclotron Resonances in the Presence of a Short-bunch-length Relativistic Beam

: Computer simulations using the 2D code "POSINST" were used to study the formation of the electron cloud in the wiggler section of the positron damping ring of the International Linear Collider. In order to simulate an x-y slice of the wiggler (i.e., a slice perpendicular to the beam velocity), each simulation assumed a constant vertical magnetic field. At values of the magnetic field where the cyclotron frequency was an integral multiple of the bunch frequency, and where the field strength was less than approximately 0.6 T, equilibrium average electron densities were up to three times the density found at other neighboring field values. Effects of this resonance between the bunch and cyclotron frequency are expected to be non-negligible when the beam bunch length is much less than the product of the electron cyclotron period and the beam velocity, for a beam moving at v~;;c. Details of the dynamics of the resonance are described. Abstract Computer simulations using the 2D code “POSINST” were used to study the formation of the electron cloud in the wiggler section of the positron damping ring of the International Linear Collider. In order to simulate an x-y slice of the wiggler (i.e., a slice perpendicular to the beam velocity), each simulation assumed a constant vertical magnetic field. At values of the magnetic field where the cyclotron frequency was an integral multiple of the bunch frequency, and where the field strength was less than approximately 0.6 T, equilibrium average electron densities were up to three times the density found at other neighboring field values. Effects of this resonance between the bunch and cyclotron frequency are expected to be non-negligible when the beam bunch length is much less than the product of the electron cyclotron period and the beam velocity, for a beam moving at v ≈ c. Details of the dynamics of the resonance are described.


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In order to investigate this issue, a series of computer simulations was performed for the parameters of the wigglers of the ILC positron damping ring using the two-dimensional computer code POSINST [3][4][5]. In each simulation the field of an ideal dipole was imposed (i.e., a static uniform magnetic field in the vertical (+y) direction, where the beam is assumed to be traveling in the +z direction). Each simulation thus was a simplified representation of an x-y slice of the wiggler. The calculations (which simulate an interval of 500 bunch passages) covered the "buildup" phase--i.e., the electron cloud formation--which takes place over a time interval of a few microseconds (~200 bunch passages) for present ILC parameters. In this time the density of the electrons in the vacuum chamber first increases, then plateaus at what will be referred to here as its "equilibrium" value, since this value represents an equilibrium between electron production and loss. POSINST assumes the distribution function of the beam to be unaffected by the electron cloud, which is a reasonable assumption for this timescale.
The cloud evolution was calculated for the range of magnetic field values expected in the wiggler, i.e., B = 0 -1.6 T.
Results of the simulations showed very little dependence of the equilibrium cloud average density (i.e., the electron density at the density plateau averaged over the vacuum chamber volume) on B for B greater than approximately 0.6 T. But below this value there were magnetic field values for which the average cloud density was enhanced by factors of up to 3 over the density at nearby magnetic field values. The magnetic fields at which this enhancement occurred have been identified as fields at which the bunch LBNL-1002E CBP-795 spacing (i.e., the time between the arrival of one beam bunch and the next) is an integral multiple of the electron cyclotron frequency. At these fields the resonance condition is fulfilled, where τ b is the bunch spacing, m e is the electron rest mass, -e is the electron charge, with n equal to a positive integer. This is a plausible situation for resonant behavior, since each time a beam bunch appears electrons will be at the same position in their cyclotron orbit if relativistic mass increase can be neglected. Electrons are born at the walls at very low energy and this nonrelativistic approximation is therefore true for some part of their lifetime in the system. As will be shown below, almost all electrons leave the system before any appreciable mass increase occurs. It is also to be noted that all of the electrons in the system for which the relativistic mass increase is negligible are in resonance, since the magnetic field is constant throughout the cross section. Thus the resonance can have a significant effect on the whole cloud.
There are two somewhat similar resonances to those we have found that have been mentioned in the literature. Rumolo and Zimmerman in reference [6] showed a very similar phenomenon that was a resonance between ion cyclotron motion and the bunch passage. Because of the large ion gyroradius and two-species nature of the problem, the dynamics in their case were in detail somewhat different from what is discussed here, but the resonance is similar. In the other instance, Cai et. al. in reference [7] discussed an increase in the electron cloud density for a case where the vacuum chamber was surrounded by a solenoid. The increase occurred when the bunch period was equal to LBNL-1002E CBP-795 half the electron cyclotron period. The dynamics they discuss, however, were quite different from the dynamics described in this paper. The resonance with which they were dealing was the multipacting resonance. Because of the solenoidal field, the distance the electron traveled from birth to the time it hit the wall again was not the width of the vacuum chamber as in the usual case, but rather the length of a cyclotron orbit. Therefore the cyclotron period appeared in the multipacting resonance condition. In contrast, in the ILC wiggler case we examine here, the electron stays in resonance with the bunch through several bunch passages and the increase in the cloud density is due to an increase in electron perpendicular energy due to synchronization of the bunch appearance with the cyclotron orbit.
In this report we will discuss the cause of the high electron cloud density at the magnetic fields at which the resonance condition is fulfilled. We will also describe the dynamics of the electron cloud formation there, and propose a reason that such an effect is not seen at higher field or for long bunches. In Section II the computational model and parameters for the simulations are described. Section III describes the simulation results. In Section IV the results of a tracking code for individual particle dynamics are used to elucidate the dynamics of the resonance process. Section V compares the simulation and single particle results and contains further discussion of the dynamics of electrons in resonance.
In Section VI special features at high and low magnetic field are explained. Results of an analytical model are given in Section VII. Both the analytical model and the single particle tracking have simplifying assumptions and do not reflect the full dynamics calculated in POSINST. But they add considerable insight, and each complements the LBNL-1002E   CBP-795   other, since the single-particle tracking reflects the situation well at low B and has no   constraint on electron phase space, whereas the analytical model is good for all B and thus can explain important features at high B, but is restrictive in the phase space treated.
Conclusions are discussed in Section VIII.

II. The Computational Model and Parameters
The model used by POSINST is described in several publications [3][4][5]. The system we have simulated is an x-y slice of the accelerator in the plane perpendicular to the direction of beam propagation (+z). A constant uniform magnetic field is applied in the vertical (+y) direction. Table I gives numerical and physical parameters used for all the simulations described in this report.
The beam bunch density is assumed to be Gaussian in x, y, and z, with the bunch centroid centered transversely in the vacuum chamber. The beam distribution function does not evolve with time in these calculations. We assume that this is a reasonable approximation for the few microseconds required for the electron cloud buildup. The bunch centroid passes through the center of the vacuum chamber-no attempt is made here to include the details of the centroid orbit in the wiggler.
The beam magnetic field is neglected in POSINST since its force is negligible compared to the beam electric field because the electrons are at low velocity. As can be seen LBNL-1002E CBP-795 below, even though the energy of electrons increases due to the resonance, essentially all have energies small enough that the nonrelativistic approximation remains valid.
POSINST uses a particle-in-cell (PIC) algorithm to calculate the time-evolving electric field of the electrons and to apply this field to their motion. The photons emitted by the beam are assumed to travel concurrently with the beam, so that the flux hitting the wall as a given slice of the beam passes through the system is proportional to the line charge density of that beam slice. The number of photons emitted per bunch particle per meter is specified as input to the code. This is multiplied by an The secondary electron emission model is described in ref. [5]. It is a heuristic model based on fits to experimental data. The coefficients needed by the model were obtained from bench measurements [5] and extrapolation from data from the CERN Super Proton Synchrotron for a stainless steel surface [8], with the peak of the secondary yield vs.
energy for normal incidence chosen to be at E max =195 eV and the peak secondary emission at normal incidence set to 1.4. These assumed values are reasonable for typical surfaces. While quantitative details of the phenomenon discussed in this article may depend on the precise parameter values of the SEY model, the essential physics does not.
Numerous simulations were performed in order to find the PIC mesh resolution and time step necessary for accuracy in these runs. This was done by increasing the numerical resolution in space or time until no further change was seen in the time history of the electron cloud buildup. The values chosen can be found in Table 1.
The number of electrons generated is large, and continually increases both the computational time per time step and the memory needed as the simulation progresses.
This numerical challenge was handled by "culling" the electron macroparticle population noise that the equilibrium density value could be accurately ascertained, so the runs here were done with that threshold.
The vacuum vessel was assumed to be a perfectly conducting circular chamber with radius a=2.3 cm. An antechamber of full height 1 cm was located at the +x midplane.
Any electrons hitting this part of the surface were assumed to be lost and produced no secondaries. Note that the ILC damping ring wiggler chamber would have an antechamber on each side of the chamber, but these runs, because they began as a benchmarking exercise, have only a single antechamber on the +x side. or real dynamics produced significant fluctuations during this interval, the interval was widened enough so that the value was affected by the fluctuations by no more than a few percent.

III. Simulation Results -Electron Cloud Density vs. B
Results of the simulations can be seen in Fig. 2a, where the equilibrium average density for each simulation is graphed vs. the ratio of the bunch spacing to the cyclotron period, which we refer to as "n", following Eq. (1). The "cyclotron period" referred to is the cyclotron period of a non-relativistic electron (γ=1). The abscissa is thus proportional to B. As mentioned above, it is evident from the figure that at the higher fields there is no  Vacuum chamber is circular and fits exactly within the plot.

IV. Dynamics of the Resonance
In this section we will show the details of the resonance dynamics using the simple case of a system without the electron space charge.
As the beam bunch passes, all of the electrons in the system that are not at the center of the chamber experience a force directed toward the beam. Since the bunch is short and traveling at the speed of light, we will for the moment consider this force to be an instantaneous kick. This approximation will be examined in Section VI. Because the beam is traveling at relativistic speed, the direction of the force is in the x-y plane to a very good approximation. The y component of the force accelerates the electron along LBNL-1002E CBP-795 the dipole field lines. The x component affects its cyclotron motion, which is in the x-z plane. As can be seen in fig. 4, the effect of the x kick is to rotate v ⊥ toward a direction parallel to the x axis and pointing toward the y axis (i.e., toward the -x direction for an electron with x>0, and +x direction for electron with x<0). If the resonance condition given in Eq. (1) is fulfilled, when the next beam bunch arrives the electron will be at the same position in the x-z plane, and its v ⊥ will be rotated again, so that if the electron remains in the system long enough, its v ⊥ will end up at 180° to the x axis if the electron is at x>0, and 0° if its x<0. For the beam intensity and chamber size used here, since the electrons are born at the wall with just a few eV of energy, the beam kick is large compared to the initial energy and quickly (in a few bunch passages) should rotate v ⊥ into this position. When v ⊥ is at 180° to the x axis, the electron is at the bottom (lowest z position) of its cycle, where it has completed 270° of its cyclotron orbit as measured from the x axis. We will call this latter polar angle describing the electron position the "cyclotron phase angle". If the magnetic field is not close to a resonant value, the electron will be at different positions as sequential bunches arrive, with the velocity vector therefore changing its direction between kicks, and the effect of the kicks on the cyclotron motion will therefore tend to average to zero over time.
For the resonant case, each beam kick will increase the magnitude of v ⊥ if its x component is in the same direction as the beam force-i.e., if v ⊥x < 0 for x>0 or v ⊥x > 0 for x<0. This orientation of v ⊥ can occur either due to initial conditions or because the beam kicks have rotated it to this position. The change in v ⊥ will also change the LBNL-1002E CBP-795 cyclotron radius, as shown in fig. 4. Since the effect of the force of the beam is to rotate the v ⊥ into a direction where it will increase in magnitude, the average v ⊥ / v || will increase, and therefore the average cosine of the angle of the electron wall impacts with respect to the normal will decrease, assuming that the electrons stay in the system long enough for the beam to have a significant effect on them. A significant decrease in the average cosine is indeed seen in the POSINST simulations at resonant magnetic fields.
The change made in the angle and magnitude of the velocity will result in a change in the position of the gyrocenter (again, see fig. 4). This movement of the gyrocenter is the well-known "ExB guiding center drift".
An individual-particle tracking code was written to elucidate the dynamics described above, which in On the left is n=12.0 (resonant case); on the right is n=11.5.
A fully relativistic tracking calculation shows a somewhat more complicated picture. As the mass of a given electron increases, it detunes from the resonance. Again there is a population of electrons that, as described above, tends to oscillate in y, staying in the system for a few tenths of microseconds before they hit the wall. But in the relativistically correct calculation these have shorter lifetimes (~0.1-0.2 µs, rather thañ 0.5 µs, for the n=12 case). Thus these electrons, which stay in the system long enough    of the electrons is changed at resonance to one which raises the effective SEY. Since v ⊥ is increased, the electrons will also, on the average, hit the wall at shallower angles, which also increases the effective SEY.

V. Comparison of Simulation Results to Single Particle Tracking
The complete POSINST simulations include the space charge force from the electrons and the effect of a finite-length beam, both of which are not present in the single particle results of the last section. Therefore in this section we use the results of the last section to identify quantities of interest in the POSINST simulations and examine the agreement between the results of the full simulations and the simple single-particle model. Figure 9 shows the energy spectrum for n=11.5 and n=12 from POSINST simulations for electrons striking the vacuum wall during time interval 0-0.2 µs, i.e., during the cloud buildup. The spectrum is quite similar to that shown in Fig. 7. From the resonance concept and also the single particle tracking results we expect this to be due to an increase in perpendicular momentum for the resonance case. Indeed, inspection of the LBNL-1002E CBP-795 spectra of v y for various x for these electrons shows very little difference between the resonance and non-resonance cases. But as shown in Fig. 10, the perpendicular velocity is enhanced in the resonance case, and the cosine of the angle of impact is less, which also increases the effective SEY.  v ⊥ is quite wide in x, which is consistent with the density distribution of Fig. 3. Knowing the workings of the resonance mechanism, this distribution can now be explained. We can approximate the x component of the force of the beam on the electrons, F x , as that of For x≤2 mm Fig. 11 shows no growth in v ⊥ for the resonance above what is seen in the non-resonant case. We attribute that to the fact that, as seen in Fig. 3, for the resonance case there are few electrons in this region except at high y, where the x component of the beam force is small.

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CBP-795 Both the single particle tracking calculations and POSINST simulations show that almost all electrons remain in the simulations for less than 7 bunch passages and never reach high v/c.
We conclude that the dynamical features shown by the single particle tracking are represented in the full simulations, giving further credence to the resonance explanation and to the details of the electron dynamics shown in Section IV.

VI. Discussion of Features at High and Low B
It remains to explain special features of the equilibrium density vs. B dependence, such as why no such effects are seen at the higher magnetic fields, e.g., above 0.6 T in fig. 2, and why there are "double" peaks, with a local minimum near the resonance, for low B. In this section we attempt to understand these aspects of the problem.
We first discuss the absence of peaks at high magnetic field. We have assumed in the discussions above that the force of the beam on the electrons is an instantaneous kick.
Given this assumption, at resonance each time a bunch appears the electron is at the same LBNL-1002E CBP-795 position, and the dynamics are as we have described, until the relativistic mass increase gradually detunes the electron from resonance. The assumption of an instantaneous beam kick is good when the time for the bunch to pass an electron is much less than the smallest relevant timescale in the problem, the cyclotron period. Thus the assumption is good when where l b is the bunch length and c is the speed of light. As B increases, the cyclotron period decreases until equation (2) is no longer valid. Instead the beam force during one bunch passage is integrated over a significant fraction of the cyclotron rotation. The concept of a "resonance" becomes invalid. The effect of the beam force is added to a velocity whose direction is changing with time, and it will therefore add less to the magnitude of the velocity. For high enough B the net effect will be negligible. For l b ~ 4 σ z the condition of Eq. (2) fails in the same range where the amplitude of the peaks in fig.   2 decreases to zero. To our knowledge the resonance we discuss here has not been discussed in the literature of the electron cloud before. That is presumably because in the parameter regimes of the accelerators that have been studied the bunch length was longer, and the magnetic fields studied were perhaps higher, so that Eq. (2) did not hold.
The energies of electrons in the simulations were for the most part less than about 600 eV, so that the electrons are essentially nonrelativistic. For large enough B, however, the small change in γ, and therefore in the cyclotron frequency, can still lead to a nonnegligible detuning from resonance, because there will be many cyclotron orbits

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CBP-795 between bunch passages. This will be another contributing factor to the decrease in peak amplitude at high B. However most electrons exit the system in 3-4 bunch passages.
Thus the effect will only be significant for electrons of several hundreds of eV.
We turn now to the question of why the low-n peaks in the density vs. B curve have a minimum at resonance. To investigate this we have extensively compared POSINST simulation data for n=2 with results at the nearby off-resonance peak (n=1.93). The data show that at resonance the electrons do attain higher v ⊥ than in the off-resonance case, but this happens because they remain in the system longer, building up energy with each bunch passage. As a result, while their average SEY is slightly higher than for the electrons in the off-resonance case (1% higher during the cloud buildup phase), they hit the wall a lot less often (rate of wall bombardment is 23% lower than for the nonresonance case), therefore producing fewer secondaries. As n increases, this difference in the bombardment rate decreases. For n=5 and the off-resonance peak at n=4.96, for instance, where the minimum at resonance is a lot shallower than at the n=2 resonance, the difference in bombardment rate is only 0.5%. Thus the longevity of the electrons at resonance at low n accounts for the minimum in cloud density there.
Before the simulation data mentioned above were analyzed, one possible explanation for the low-n minima at resonance seemed to be that the action of the resonance might cause electron energy to increase past E max , resulting in electrons with a smaller effective SEY than for the case just off resonance. This did not prove to be the case-the SEY for the high-v ⊥ electrons at resonance was still near the peak, partly because their impact angle LBNL-1002E

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was lowered by the increase in perpendicular energy (see Fig. 13 for SEY vs. energy and impact angle).
The divergence between the cloud density vs. time curves for n=2 and n=1.93 occurs after space charge forces becomes important. Therefore a detailed quantitative explanation of the variation of the longevity difference with n, including its behavior in the vicinity of a resonance, would be likely to require solution of the full dynamics of the electron motion in the presence of spatially nonuniform and time-varying space charge.
We have not attempted to solve this problem in this report, and leave it for future work.

Fig. 13 SEY vs. Impact Energy for our parameters
In considering the peaks at low B it should also be mentioned that at lower order resonances (i.e., low n) the dynamics can be influenced by finite gyroradius effects. Although these approximations restrict parameter values (e.g., B cannot be too small), and the electron's phase space and time histories, they allow for a qualitative explanation of the resonance behavior at small B and its suppression at high B (or for long bunches).
The line charge density of the positron beam at time t and longitudinal location z is given where the summation is over successive bunches. We choose the origin of time such that the center of the first bunch (k=0) crosses z=0 at t=0. At this instant, the centers of bunches k=0,1,2,3,... are located at z = −kcτ b . While in a real accelerator the number of bunches is finite, the infinite upper limit in the above summation does not affect the results in the analysis that follows.
The electric field generated by the beam is where G(x,y) , with dimensions of 1/length, is the 2-dimensional Bassetti-Erskine field.
Under our stated assumptions, the equations of motion for a single electron (charge -e and rest mass m e ) are where p ≅ m e v is its momentum. The magnetic field B has an external component B=B This effect is not present in the POSINST calculations or in this analysis, and will be explored in future work.
Eq. (5) then becomes We first neglect the field from the beam. The equations become where ω=eB/m e . The solution in the x-z plane is where the subscript "f" stands for "free" (i.e., in the absence of the beam). The gyroradius ρ 0 is determined by the initial condition, ωρ 0 = v ⊥0 , where v ⊥ 0 is the initial speed in the xz plane; φ 0 is the initial phase. The solutions for x and z are: where (z c , /ω) is the gyrocenter in the (x-z) plane expressed in terms of the components of the velocity at t=0. As for the motion in y, it is free-particle motion: v y =v y0 and y=y 0 +v y0 t.
Now taking d/dt of the 3rd component of Eq. (6) and combining it with the 1st yields Under our stated assumptions we can derive an approximation to this equation by setting x, y and z in the right-hand side to their values at t=0, keeping only the essential time dependence in the exponential factor. Without any loss of generality we choose z 0 =0, which is the equation for a driven harmonic oscillator. We readily where the free part, v f , is given by Eq. (8) and the driven part is given by where θ(t) is the conventional step function, To complete the derivation we note that the 3rd component of Eq. (6) is and where the driven parts are given by The 1st-order equation for motion in y can be obtained in similar fashion, but it does not add much useful information to the discussion. One finds that those electrons for which the x component of the gyrocenter, x c , is comparable to or slightly smaller than the chamber radius a oscillate harmonically about y=0 with an angular frequency but those electrons whose gyrocenter x c is within ~a/2 of the pipe center (x=0) are very unstable under the action of the beam, and are driven to the wall of the chamber within one to a few bunch passages (for the parameter values in Table 1).
Equations (12), (14) and (15)  (14)) is uniquely determined by the beam driving force. One readily finds from Eqs. (12-14) that the phase advance relative to the x axis, from the time a bunch passes until v z becomes aligned with the beam motion is ωΔt = (3/4)×2π for x c >0 and ωΔt = (1/4)×2π for x c <0, in agreement with the single-particle tracking analysis presented in Sec. IV, where Δt is the corresponding time interval. To reach the above conclusion, one needs to recall that the Bassetti-Erskine field component G x has the same sign as x. For integer n, the electron energy therefore grows like |A| 2~t2 , explaining why in this case the electronwall collision energy is larger than for non-integer n. Finally, if the bunch length is too large, or the B field too strong, the resonant growth of the amplitude is suppressed by the phase-averaging factor € κ = e -(ωσ t ) 2 / 2 [11].

VIII. Conclusions
We have described a resonance when the beam bunch frequency is an integral multiple of the cyclotron frequency that causes an enhancement of the electron cloud density. Single particle tracking shows the effect clearly, and though the dynamics are more complicated when electron space charge is taken into account, POSINST simulations, which include space charge, show an increased cloud density at magnetic field values where the resonance condition holds, and perpendicular velocity and impact angle increase consistent with the resonance. One would expect to see the effects of this resonance in real systems in low-field dipoles, in the fringe fields of dipoles, and in the low-field areas LBNL-1002E CBP-795 of wigglers for accelerators, storage rings, and damping rings with positively-charged beams and short bunch length. The resonance has not been described previously in the electron cloud literature, probably because it occurs only for relatively low magnetic field values and bunch lengths sufficiently short that the time it takes for the beam to pass an electron is short compared to the cyclotron period.
The resonance has a very big effect on the spatial distribution of electrons in the vacuum chamber. At resonance the electrons are much more distributed throughout the chamber, not confined to the narrow stripes seen in the off-resonance case. This should be taken into account when locating electron cloud diagnostics, which are often placed with the characteristic stripes pattern in mind. Placing the diagnostics in the narrow x range appropriate for the off-resonance case will lead to a large underestimate of the cloud density and misunderstanding of its distribution and therefore its effect on the beam.
Different ranges of v ⊥ also tend to be found in different ranges of x for the resonant case, which implies a different distribution of wall heating due to the electron cloud.
The importance of the resonance, and the resultant density enhancement, is hard to gauge without the results of complete three-dimensional simulations. In 3D the ExB drift would push electrons in z, so that they would move in and out of resonance. This would probably decrease the density enhancement, but could possibly increase it by bringing more electrons into resonance. It should be noted that because the resonances can be quite narrow in B, 3D simulations of this effect must have enough z resolution to resolve well the gradient in the magnetic field in z. In a more realistic simulation there will also LBNL-1002E CBP-795 be some variation of B across the chamber, which will decrease the number of electrons in resonance. If the resonance effect is not decreased significantly in 3D, it could be an important factor in beam dynamics in cases where it produces an electron density that changes with a regular periodicity in z. In the wiggler case, for instance, the electron density would have the same periodicity as the wiggle of the beam centroid, and could cause a resonant deflection of the centroid. The effect of the resonance on cloud buildup was studied here. Its effect on the beam will be deferred for future work.