Fast Instability Caused by Electron Cloud in Combined Function Magnets

One of the factors which may limit the intensity in the Fermilab Recycler is a fast transverse instability. It develops within a hundred turns and, in certain conditions, may lead to a beam loss. The high rate of the instability suggest that its cause is electron cloud. We studied the phenomena by observing the dynamics of stable and unstable beam, simulating numerically the build-up of the electron cloud, and developed an analytical model of an electron cloud driven instability with the electrons trapped in combined function dipoles. We found that beam motion can be stabilized by a clearing bunch, which confirms the electron cloud nature of the instability. The clearing suggest electron cloud trapping in Recycler combined function magnets. Numerical simulations show that up to 1% of the particles can be trapped by the magnetic field. Since the process of electron cloud build-up is exponential, once trapped this amount of electrons significantly increases the density of the cloud on the next revolution. In a Recycler combined function dipole this multi-turn accumulation allows the electron cloud reaching final intensities orders of magnitude greater than in a pure dipole. The estimated resulting instability growth time of about 30 revolutions and the mode frequency of 0.4 MHz are consistent with experimental observations and agree with the simulation in the PEI code. The created instability model allows investigating the beam stability for the future intensity upgrades.

FIG. 1. The first batch above the threshold intensity suffers the blow-up after injection into the ring [3]. The bunch intensities of six consecutive injections in the Recycler are shown on the left. The first bunch train (the top one) has the intensity below the threshold (dashed line), and the following injections -above the threshold. The vertical axis shows the injection position of the bunch trains in the ring. On the right -turn-by-turn measurements of beam center positions of the corresponding bunch trains during the first 100 turns after the injection. Only the first batch above the threshold intensity becomes unstable.

Electron cloud trapping
The most likely candidates for the location of electron cloud in Recycler are its combined function magnets. They occupy about 50% of the ring's circumference. In a combined function dipole the electrons of the cloud move along the vertical field lines, but the gradient of the field creates a condition for 'magnetic mirror' (Fig. 2) -an electron will reflect back at the point of maximum magnetic field Bmax if the angle between electron's velocity and the normal to the field lines is less than (for the derivation see, for example, [5,6]): Let us look at the process of electron cloud trapping in more detail. Let's assume that a proton bunch train has created some electron cloud and consider the last two bunches of the train. The first bunch kicks the electrons of the cloud, typically supplying them an energy of the order of a hundred eV. The electrons drift along the magnetic field in the vacuum chamber, finally reaching its walls and producing secondary electrons with the energies of a few eV [7]. In the absence of the beam these secondary electrons will reach the aperture and, perhaps after a few elastic reflections, will eventually get absorbed by the vacuum chamber. But the next proton bunch can stop a fraction of the secondaries, reducing their angle  to below the critical value (1). These electrons will remain trapped in the magnetic field after the beam is gone.
The presence of the second bunch is necessary to stop the particles, created by the first one, and therefore a single bunch cannot trap the cloud. Instead it clears the space, kicking the cloud to the physical aperture. The secondary electrons, created in the process, will eventually reach the vacuum chamber and be absorbed. This clearing bunch can be used to indicate the presence of the electron cloud [8] or to bring the electron cloud density below the threshold, stabilizing the beam.
The long-term confinement of the trapped electron cloud can be affected by two effects: longitudinal drift and scattering. The drift is caused by the horizontal derivative of the magnetic field  16 , where re -is the classical electron radius, ε -the electron energy, c -the speed of light, ΛC -the Coulomb logarithm, and χmin -the minimal scattering angle. χmin can be estimated as min max   , since the scattering does not lead to a particle loss if it stays within the trapping cone max    . Then for the electron energies~1 10   eV the cross-section 17 10 According to the experimental measurements [10], the scattering cross-section for many residual gases is of the order of 15 2 10 cm  at the energies 10   eV. Combining the two scattering effects we obtain a lifetime ~ 1 ms for the electron cloud density ne < 10 13 m -3 and the residual gas pressure P1 0 -8 Torr. Since all the loss mechanisms result in the lifetime much larger than the revolution period of 11 μs, all the trapped cloud will be present on the next turn. It will act as the new seed electrons, and can lead to a higher electron cloud density on the next revolution.

Numerical simulation
We simulated electron cloud build-up over multiple revolutions in a Recycler dipole using the PEI code [11]. The code simulates the build-up and 2D transverse coupled motion of the electron cloud and the beam. The electron cloud is represented by an ensemble of macroparticles of a constant weight, and the beam -by a series of rigid bunches with Gaussian transverse shape. The beam-cloud interaction is calculated using the Basetti-Erskine model [12]. The ring was modelled as a linear transfer matrix with one interaction point, representing a combined function magnet. The input parameters of the simulation are summarized in Table I.
The main source of primary electrons in Recycler is the collisional ionization of residual gas by the beam. To simulate it we put the seed electrons at the beam center with their linear density following where Nb is the number of protons in a bunch [13].
The model of secondary emission includes true secondary and elastically reflected electrons and assumes normal incidence [14]. In a dipole field, however, an electron hits the wall of a vacuum chamber at an angle. That increases the time the electron spends near the wall surface and consequently the SEY. Experimental data on angular dependence of SEY fits an empirical formula: where a1 and a2 are material specific parameters and SEY0 -the yield measured at normal incidence [15]. For a pure dipole field, the cloud rapidly builds up during the passage of the bunch train and then decays back to the initial ionization electron density in about 300 RF buckets, or ~ 6 μs (Fig. 4). When the field gradient is added, up to 1% of the electron cloud stays trapped, increasing the initial density on the next revolution. The final density, which the cloud reaches after ~ 10 revolutions, can be as high as two orders of magnitude greater than in the pure dipole case (Fig. 4). The resulting cloud distribution is a stripe along the magnetic field lines, with higher particle density being closer to the walls of the vacuum chamber ( Fig. 5).
At lower densities ~10 -2 of particles are trapped, which agrees with an estimate from Eq. (1); as the density of electron cloud increases the trapping ratio goes down to ~10 -3 , probably due to the space charge of electron cloud.
FIG. 4. In a combined function magnet the electron cloud accumulates over many revolutions, reaching much higher line density, than in a pure dipole. A clearing bunch destroys the trapped cloud, preventing the accumulation. Results of a numerical simulation using the PEI code.
FIG. 5. Electron cloud forms a stripe inside the vacuum chamber; beam center and its 2-σ boundary are shown in white. Results of a numerical simulation using the PEI code.
As mentioned above, a trapped cloud can be cleared by a single bunch following the beam at a sufficient distance. In Fig. 4 a bunch of 10

10 
protons, added 120 RF buckets after the main batch, destroys the trapped cloud, preventing the multi-turn build-up. First, one can see a small increase in the cloud density as the clearing bunch kick the cloud and it reaches the vacuum chamber, producing the secondary electrons. Then, the density rapidly drops as these secondaries reach the aperture. The multi-turn electron cloud accumulation due to the trapping mechanism might play an important role in a proton ring, where the density of the primary ionization electrons is relatively low. For a positron machine of a similar energy the amount of primary electrons is much greater due to the photoemission by synchrotron radiation. Because of the large number of primary electrons, the cloud can reach a saturation density during the passage of on bunch train. Then the presence of trapping would only slightly affect the overall picture, shifting the saturation towards the head of the batch. The recent studies at CESR show that, the cloud in its combined function magnets reaches a saturation density during the passage of one positron bunch train [16].

Witness bunch experiment
We used a clearing bunch technique, similar to that used at Cornell [8] to check whether the instability is caused by trapped electron cloud. If a trapped electron cloud is present in the machine, a single bunch of high enough charge following the main batch, will kick it and clear the aperture. This clearing of electron cloud then can be noted by observing a change in beam dynamics. p is injected in the machine before the high-intensity batch, the later remains stable (Fig. 6, bottom). The position of the clearing bunch does not change the picture -it can be as far as half of the ring (or ~ 5 μs) apart from the batch, suggesting that there is a portion of the electron cloud that survives over one revolution, and it can be removed with a clearing bunch. This agrees qualitatively with the simulation of electron cloud build-up and trapping in Recycler dipoles (Fig. 4).  (Fig. 7). The detector has a 1 GHz bandwidth and a linear response within 75% of the physical aperture of 110 mm [17].
FIG. 7. Cross section of Recycler stripline detector; dimensions in mm [17]. Figure 8 depicts the betatron tune shift within the 80-bunch train with respect to the first bunch, measured over 600 revolutions with a stripline detector, with the dampers off during the measurement. The positive horizontal tune shift is a clear signature of the presence of a negative charge at the beam center. The vertical tune shift is negative, indicating that the maximum density of the cloud is outside the beam, which agrees qualitatively with the simulated distribution. Figure 9 shows the simulated potential of the electron cloud, seen by the last bunch of the train (the disctribution of the cloud density is depicted in Fig. 5). Its "saddle" shape near the beam center leads to horizontal focusing and vertical defocusing. When a clearing bunch is added, the tune shifts decrease, indicating a reduction of electron cloud density, which agrees with the simulation (Fig. 4). The remaining linear slope in the vertical tune shift is likely to be due to the resistive wall impedance. According to the recent measurements, in Recycler the vertical impedance is about five times larger than the horizontal [18]. ppb. The absolute value of the potential is plotted as a function of displacement from the beam center. The cloud potential decreases for small horizontal displacements and increases for small vertical ones.

Simple analytical model of beam-cloud interaction
As a reactive medium, the electron cloud interacts with the beam similarly to a low-Q impedance [19][20][21]. One of the first attempts to describe the resulting instability belongs to Burov and Dikansky in 1997 [22], who studied the cases of a field-free drift section and a bending magnet. Later Channell solved the coupled e-p motion in the presence of a quadrupole magnetic field [23]. Balbekov analysed the electron cloud instability and nonlinear effects, considering the cloud in the strong dipole field as an immobile 'snake' [24]. In this chapter we study the coupled beam-cloud motion in a strong dipole field, modelling the motion of the cloud 'stripe' as the mobility term, similar to the work [25]. This approach does not require making initial assumptions about the form of the electron cloud wake or its impedance.
First, consider a round coasting proton beam travelling in a ring, uniformly filled with electron cloud. Let us denote the position of the beam centroid at an azimuthal angle θ at time t as Xp(t, θ). Further, assume that the beam travels at a constant azimuthal velocity around the ring ω0 and use a smooth focusing approximation with a betatron frequency ωβ.
For simplicity, one can represent the electron cloud by a cylinder of a uniform charge density ne and the same radius as the proton beam, located at a horizontal position Xe. Let us further assume that the total number of electrons does not change in time. Because of the vertical dipole field, the individual electrons of the cloud cannot drift horizontally, but the position Xe can change as some regions build up and others are depleted, following the transverse motion of the proton beam. The characteristic rate of this slow motion of the electron cloud  is then the rate of its build-up: For small oscillation amplitudes we can assume the electron-proton interaction force to be linear in displacement. Then the coupled collective motion of the beam and the electron cloud is described by the following system of equations: where Γ is the rate of Landau damping. The coupling frequency ωp is approximated as where rp is the classical proton radius and γ -the relativistic factor. The linear damping term Γ in Eq. (5) arises from the spread in betatron frequencies for particles oscillating with different amplitudes. The characteristic rate of the Landau damping can be estimated as~, where Qx is the horizontal tune and ΔQx is its rms spread.

Looking for solutions of Eq. (5) in a form
It can be solved perturbatively, under the assumption that 0 , , , Solving Eq. (8) for each integer wave number n one gets: where the small complex tune shift | The imaginary tune shift in Eq. (11) consists of two parts with the first being the Landau damping term. The most unstable mode, for which Im( )   is the greatest, is max    and its wave number max n is and the growth rate of this mode is The threshold electron cloud density ne,thr can be found from the condition γmax = 0, which yields Since, in general, the strength of Landau damping Γ depends on the density of the electron cloud, this equation might have one, many, or no solutions at all [21]. In this simple model we do not consider the electron cloud's contribution to Landau damping, which may arise from the nonlinear spread of the betatron tunes, created by the cloud.
In an experiment an observer will see the most unstable mode as it masks the others thanks to its higher exponential growth rate. Thus Knowing the complex frequency shift   we can find the impedance of the cloud as (see for example [26] Eq. (6.262)): where N is the number of protons in the ring and Z0 is the vacuum impedance. Because the electron cloud shifts both the coherent and the incoherent frequencies, we subtracted here the incoherent tune shift.
Knowing the impedance one can compute the wake functions using the formula (2.72) from [26]: In the case of a bunched beam, in a rigid bunch approximation, one needs to compute W(z) only in a discreet set of bunch positions k rf z kc  , where rf  is the RF period.
Finally, from the impedance of the most unstable mode one can estimate the instability growth rate of a bunched beam as [27]: where C is the ring circumference and L is the total length of the magnets. For the Recycler / 1 / 2 L C  .

Instability in Recycler
In order to use the model and estimate the parameters of the fast instability in Recycler one needs to know the density of the electron cloud and the rate of its build-up. We obtain these quantitative parameters by measuring the betatron frequency shift and comparing it with the build-up simulations.
We injected one batch of 80 proton bunches of The most unstable mode has the frequency of about 0.4 MHz, its impedance, calculated using Eq. (16), is 20 MΩ/m (Fig. 11). Figure 12 depicts the corresponding wake function W(n) as a function of bunch number n. W(n) fits an exponential decay curve The estimate of the mode frequency qualitatively agrees with the simulation in the PEI code and the stripline measurement. PEI simulated the ring, completely filled with 588 bunches of 10 5 10  p.
The resulting frequency is about 0.7 MHz (Fig. 13). In the stripline measurement one batch of 80 bunches of the same charge was injected. The measured frequency was about 0.9 MHz. Both simulated and measured frequencies agree on the order of magnitude with each other and the estimate.
Using the calculated value of the real part of the impedance we can now estimate the growth rate using Eq. (13). We obtain the growth rate of ,max 0.

Conclusion
A fast transverse instability in the Fermilab Recycler might create a challenge for PIP-II intensities.
The PIP-II upgrade goal is nearly doubling the beam intensity and reaching 10 8 10  protons per bunch [4]. Since the fast instability has been already observed at the intensities as low as 10 4 5 10   ppb, understanding its nature is important for making predictions about the machine performance at higher intensities.
We have observed that the fast instability can be mitigated by injection of a single low intensity clearing bunch. This finding suggests that the instability is caused by electron cloud and the cloud is trapped in Recycler magnets. Bunch-by-bunch measurements of the betatron tune have shown its shift towards the end of the bunch train. The tune shift decreases after the addition of the clearing bunch, which is also consistent with the electron cloud picture.
There is practically no doubt that the source of trapping is the combined function magnets, occupying around 50% of Recycler circumference. Combined function magnets are widely used in contemporary accelerators and are a technology of choice for some future machines. According to numerical simulations in PEI,~10 -2 -10 -3 of particles are trapped by the magnetic field of those magnets. That allows the electron cloud to gradually build up over multiple turns, reaching final intensities orders of magnitude greater than in a pure dipole. The results of electron cloud build-up simulation in the Recycler combined function dipoles agree qualitatively with the observed stabilization of the beam by a clearing bunch and quantitatively with the measurements of betatron tune shift. According to the simulations, the estimated cloud density is 11 6 10  m -3 on the beam axis and the characteristic times of its build-up and decay are 40 and 10 RF periods respectively.
We have created a simple analytical model of the transverse multibunch instability, driven by the electron cloud trapped inside the combined function magnets. The model allows an estimation of the instability threshold, the frequency of the most unstable mode and its growth rate. For the current parameters of the Recycler beam we find the mode with a frequency of 0.4 MHz and a growth time of 30 revolutions, which is consistent with the observations of the fast instability and the simulations in PEI. The model allows the prediction of the rate of the instability for higher intensities of the proton beam, given an estimate of the electron cloud density, which can be obtained from numerical simulations.