A preliminary assessment of the electron cloud effect for the FNAL main injector upgrade

: We present results from a preliminary assessment, via computer simulations, of the electron cloud density for the FNAL main injector upgrade at injection energy. Assuming a peak value for secondary emission yield deltamax = 1.3, we find a threshold value of the bunch population, Nb,th~;=1.25x1011, beyond which the electron-cloud density rho_e reaches a steady-state level that is ~;104 times larger than for Nb We present results from a preliminary assessment, via computer simulations, of the electron-cloud density for the FNAL main injector upgrade at injection energy. Assuming a peak value for secondary emission yield δ max = 1 . 3, we ﬁnd a threshold value of the bunch population, N b, th ’ 1 . 25 × 10 11 , beyond which the electron-cloud density ρ e reaches a steady-state level that is ∼ 10 4 times larger than for N b < N b, th , essentially neutralizing the beam, and leading to a tune shift ∼ 0 . 05. Our investigation is limited to a ﬁeld-free region and to a dipole magnet region, both of which yield similar results for both N b, th and the steady-state value of ρ e . Possible dynamical eﬀects from the electron cloud on the beam, such as emittance growth and instabilities, remain to be investigated separately.


I. INTRODUCTION.
An upgrade to the main injector (MI) at FNAL is being considered [1] which would increase the bunch intensity N b by a factor of 5 from its present value of 6 × 10 10 . Such an increase would place the MI in a regime in which a significant electron-cloud effect has been observed at other hadron machines [2][3][4].
In this article we present an examination of the EC at the MI by means of computer simulations with the code POSINST [5][6][7][8]. For the purposes of the present work, we fix two important parameters, namely the beam energy E at its injection value, and the peak value δ max of the secondary emission yield (SEY) of the vacuum chamber at 1.3. Furthermore, we confine our attention to only two regions of the ring: a drift, and a dipole magnet of strength B = 0.1 T. More specifically, we compute the electron density ρ e as a function of N b , and we consider two models of the SEY that differ in the emitted-energy spectrum at fixed δ max . We find a threshold value for the bunch intensity, N b,th 1.25 × 10 11 , beyond which ρ e grows exponentially in time with an e-folding time τ 100 ns upon injection into an empty ring, and reaches a steadystate value that is 10 4 times larger than for N b < N b,th . In steady state, for N b > N b,th , the EC essentially neutralizes the beam and leads to a tune shift ∆ν ∼ 0.05. An assessment of possible dynamical effects on the beam from the EC, such as emittance growth and instabilities, falls outside the scope of this article, as does a systematic sensitivity analysis of our results on various assumed input parameters, particularly δ max . * Work supported by the US DOE under contract DE-AC03-76SF00098.

II. ELECTRON SOURCES.
A. Primary mechanisms.
In general, the build-up of the electron cloud (EC) is seeded by primary electrons from three main sources: photoelectrons, ionization of residual gas, and electrons produced by stray beam particles striking the chamber wall. Since these processes are essentially incoherent, it is customary to quantify them in terms of the number of primary electrons produced per beam particle per unit length of beam traversal, n e . For the MI, the contribution to the primary electron density from photoelectrons is wholly negligible. The contribution from residual gas ionization can be estimated from the gas density. The inverse of the mean free path for an ionization event by a particle traveling in a gas is given by ρ g σ i , where σ i is the ionization cross-section and ρ g the gas density. Expressing ρ g in terms of the pressure P and temperature T yields Implicit in this formula is the assumption that the ionization event yields a single electron. A typical value for σ i , which we assume in this note, is 2 Mbarns [9] (for simplicity, we neglect here the dependence of σ i on beam energy). Assuming P = 20 nT and T = 305 K, Eq. (1) yields n e(i) = 1.27 × 10 −7 (e/p)/m. The contribution from stray protons striking the chamber walls is given by n e(pl) = η eff n pl (2) where n pl is the number of lost protons per stored proton per unit length of beam traversal, and η eff is the effective electron yield per proton-wall collision ("pl" stands for "proton loss"). We focus on the beam injection process, since the most significant fraction of beam loss (f = 1.2% of the beam) occurs during this time, which lasts for ∆t inj = 0.4 s or N inj = 3.59 × 10 4 turns. Assuming that the beam losses occur uniformly throughout the machine circumference C, and uniformly during ∆t inj , we obtain The effective electron yield per proton-wall collision η eff has been estimated [10] at the PSR to be η eff 100−200.
Although η eff is a function of the beam energy (E = 1.7 GeV for the PSR vs. 8.9 GeV for the case considered here), we set η eff = 100 for definiteness, hence Eq. (2) yields n e(pl) = 10 −8 (e/p)/m. The total primary-electron production rate, therefore, is n e = n e(pl) + n e(i) = 1.37 × 10 −7 (e/p)/m (4) and the number of primary electrons generated by one bunch passage through a section of length L is given by where the numerical value corresponds to the choices L = 0.1 m and N b = 3 × 10 11 . As for the time dependence of n e , the fact that the primary electron-generation processes are incoherent implies that n e (t) ∝ I b (t) where I b (t) is the instantaneous beam current at the ring location under investigation [12].

B. Secondary electron emission.
The secondary emission yield (SEY) function δ(E 0 , θ 0 ) is the average number of electrons emitted when an electron of kinetic energy E 0 impinges on a surface at an incident angle θ 0 (conventionally measured relative to the normal to the surface). The SEY reaches a peak value δ max (conventionally specified at normal incidence) at an energy E 0 = E max . A fairly detailed phenomenological probabilistic description of the secondary emission process is presented in Refs. [7,8], upon which we base the analysis in this article.
Closely related to δ is the emitted-energy spectrum of the secondary electrons, dδ/dE at given incident energy E 0 , where E is the emitted electron energy. The spectrum covers the region 0 ≤ E E 0 , and it exhibits three fairly distinct main components: elastically reflected electrons (δ e ), rediffused (δ r ), and true secondaries (δ ts ). The SEY is given by δ = δ e + δ r + δ ts . The three components are emitted with qualitatively different energy spectra. Depending upon various features of the storage ring considered, the three components can contribute differently to various aspects of the ECE.
Since we do not have data for the SEY of the MI vacuum chamber, for the discussions in this note we adopt two models, 1 which we call "K" and "H," that may be considered representative of the possible range of SEY parameters for the MI. As seen in Fig. 1, the SEY functions δ(E 0 ) are essentially the same for the two models, but the emitted energy spectra are not: the SEY for model K has a larger backscattered component (composed of elastic plus rediffused electrons) than model H (see Table I). When these two models are applied to the estimate of the EC power deposition in the LHC arc dipoles, for example, one finds significantly different results [12], underscoring the importance of the emission spectrum.

III. ELECTRON-CLOUD BUILD-UP.
A. General considerations.
A convenient phenomenological parameter to characterize the EC build-up (and decay) is the effective SEY, δ eff , defined as an average over a time window of the convolution of δ(E 0 , θ 0 ) with the energy-angle electron-wall collision spectrum (normalized to unity) dN/dE 0 dθ 0 , The spectrum dN/dE 0 dθ 0 is a function of many variables such as the bunch intensity and fill pattern, the vacuum chamber geometry, etc. This spectrum is not known a priori, and hence neither is δ eff . Nevertheless, in general, δ eff has a monotonic dependence on δ max . In effect, the integral in Eq. (6) is evaluated during the simulation process: δ eff is obtained by dividing the number of emitted electrons by the number of incident electrons during any given time window. When δ eff < 1 the chamber walls act as net absorbers of electrons, and the EC build-up is dominated by the production of primary electrons. Since the beam, on average, produces a fixed number of primary electrons per unit time, the EC line density at a given location in the ring, λ e , grows linearly in time t following injection of the beam into an empty chamber according to whereλ b = eN b /s b is the average beam line density anḋ n e is the number of primary electrons generated per beam particle per unit time,ṅ e = n e v b , where v b is the beam velocity. After a typical time τ , the EC line density reaches saturation when the number of primary electrons generated per unit time equals the number of electrons absorbed by the walls per unit time. The saturated value of λ e and the time constant τ are given by [11] λ e =λ bṅe τ , where ∆t tr is the characteristic traversal time of the electrons across the chamber under the action of the beam. This situation typically happens when N b and/or δ max are low, 2 although it can also happen when N b is very large owing to the fall-off of δ(E 0 ) at very high E 0 . If the production of primary electrons ceases (for example, when the beam is extracted, or during a gap in the bunch train), the EC density decays exponentially in time if the space-charge forces are negligible [12]. 3 If, on the other hand, δ eff > 1, the EC build-up is dominated by secondary electron emission quickly following injection of the beam into an empty chamber on account of the inherently compound effect of secondary emission: the more electrons are present, the more are generated. In this case the average EC density grows exponentially in time until a saturation is reached when the spacecharge forces from the EC suppress further secondary emission from the walls. The saturation level reached by the EC density is insensitive to n e . It does not grow indefinitely as δ eff → 1 − , as Eq. (8) might imply, but rather reaches a limit comparable to the beam neutralization level. This situation happens when N b and/or δ max are sufficiently high. In the exponential growth regime, the growth time τ of the EC density is related to δ eff and ∆t tr by [12] δ eff = e ∆ttr/τ (δ eff > 1).
The traversal time ∆t tr is also an "effective" quantity in the same sense that δ eff is, namely it is an average of the traversal time of all electrons crossing the chamber over their energy and angles. ∆t tr is a function of the beam intensity and fill pattern, external magnetic fields, etc. As discussed below, both situations (δ eff < 1 and δ eff > 1) can be realized in the MI, depending upon the value of N b .

B. Results for the Main Injector.
For the studies presented in this note we have used the simulation code POSINST [5][6][7][8]. We consider only two regions of the MI: a drift, and a dipole magnet of field B = 0.1 T, and we fix the beam energy at its injection value, E = 8 GeV. 4 Since the longitudinal motion of the electrons is negligible over the time scales of interest, we perform separate simulations for these two sections. The simulation is restricted to the dynamics of the EC under the action of successive passages of bunches during one machine revolution. The beam is represented by a prescribed function of space and time, hence it is not dynamical. Therefore, aside from the tune shift estimate discussed below, all dynamical effects from the EC on the beam, including single-bunch and multi-bunch instabilities, emittance growth, etc., remain to be addressed.
Simulation parameters for the MI used here are listed in Table I. For the above-stated reasons, the length of the simulated region has negligible impact on our results, so we fix it at 0.1 m for definiteness. For the purposes of a first exploration of parameter space, we choose the bunch population N b in the range 6×10 10 ≤ N b ≤ 3×10 11 while we fix δ max = 1.3. We carry out simulations for one revolution period (T 0 = 11.15 µs) for a MI beam consisting of 504 full buckets followed by a gap of 84 buckets. A brief discussion on the SEY model dependence is presented in Sec. IV. Figure 2 shows the time evolution of the EC line density. The above-mentioned behaviors are clearly seen. For N b = 6 × 10 10 , the EC reaches an average line densityλ e 1 × 10 −5 nC/m for a drift andλ e 2 × 10 −5 nC/m for a dipole, while for N b = 3 × 10 11 , the EC density saturates atλ e 5.5 nC/m for both cases. This latter value should compared with the average beam line density,λ b = 8.5 nC/m, implying an average beam neutralization factorλ e /λ b 0.65. The exponential growth of the EC density for N b = 3 × 10 11 is clearly seen over 4 orders of magnitude in density during the first ∼1. ns for the dipole. Figure 3 shows the time-and space-averaged electronwall collision energy spectrum. For N b = 6 × 10 10 , the spectra are sharply cut off at E 0 200 eV and yield an average electron-wall collision energy ∼ 50 − 100 eV, while for N b = 3 × 10 11 the spectra exhibit a high-energy tail up to ∼ 500 eV, with an average ∼ 100 − 150 eV. Referring to Fig. 1, these averages explain qualitatively why δ eff < 1 in the first case while δ eff > 1 in the second.
To assess the simple model embodied by Eqs. (7-9), we consider the results for a drift. For N b = 6 × 10 10 the effective SEY at saturation, obtained directly from the simulation, is δ eff ∼ 0.85. The behavior of the build-up in Fig. 2a is in agreement with the simple model, which implies τ 140 ns and ∆ tr 21 ns. This value of ∆ tr , in turn, implies a typical electron energy ∼ 45 eV, in agreement with independent results (see Fig. 3a). For N b = 3 × 10 11 we obtain δ eff 1.15 and τ 110 ns during the exponential growth regime. Equation (9) implies ∆t tr = 15 ns, which implies an electron energy ∼ 90 eV. . The spectrum is averaged over time during one revolution and over the entire surface of the chamber section being simulated, and integrated over incident angles θ0. The spectrum is normalized so that its integral over E0 yields the incident-electron flux at the wall, J. For case (a), J 130 pA/cm 2 for a drift, and J 220 pA/cm 2 for a dipole magnet, while for case (b), the corresponding values are J 100 µA/cm 2 and J = 130 µA/cm 2 , respectively. This value is lower by a factor ∼ 2 than what is is independently observed in the simulation (eg., Fig. 3b), presumably owing to the excessive simplicity of the model. The results for a dipole are in qualitative agreement with the above results for a drift.
A straightforward consequence of the EC density is a tune shift ∆ν owing to the focusing effect of the electrons on the beam. Assuming that the EC density distribution is round in the transverse plane, the tune shift per unit length of beam traversal through the cloud, ∆ν/L, is given by [13] ∆ν/L = r p βρ e 2γ b where r p = 1.535×10 −18 m is the classical proton radius, γ b is the relativistic factor of the beam, ρ e is the EC density (with dimensions of volume −1 ) seen by the center To get an idea of the magnitude of ∆ν, we replace L by the circumference C, yielding ∆ν = 0.056. For N b < N b,th , the electron density is ∼ 10 8 m −3 , hence ∆ν ∼ 5 × 10 −6 .
IV. DISCUSSION. Figure 4 summarizes the results for the electron density at saturation as a function of N b . A threshold value for N b , N b,th 1.25 × 10 11 , is strongly indicated both for drifts and dipoles, which seems fairly insensitive to the SEY model. The saturated value of ρ e , on the other hand, shows a sensitivity to the SEY model on the level of a factor of ∼2. Figure 5 shows the growth time τ of the EC density upon injection into an empty chamber, and Fig. 6 the effective SEY δ eff . As is the case for ρ e , τ and δ eff show some sensitivity to the model, but N b,th does not (the non-smooth behavior in the dipole cases in these three Figures for low N b is probably due to the fact that the EC has not quite reached steady state after one turn, as is apparent in Fig. 2a for N Although the assessment presented in this article is limited, this threshold dependence is the most striking conclusion. Above threshold, the EC density is high enough to lead to a tune shift ∼ 0.05. However, owing to the intrinsic limitations of the simulation technique used, we cannot assess the dynamical effects upon the beam. As seen in Table I, the backscattered component of the SEY at E 0 = E max is (δ e (E max ) + δ r (E max ))/δ(E max ) = 0.41 for model K and 0. 10  electrons are emitted with higher average energy than in model H. The higher energy implies a faster traversal across the chamber, and an effectively higher yield in subsequent electron-wall collisions, which helps to explain why ρ e , δ eff and 1/τ are higher in the former model than in the latter (see Figs. 4, 5 and 6 for a more complete set of results). The essential parameters that determine N b,th are almost certainly δ max , E max and δ(0). It seems imperative, therefore, to determine N b,th as a function of these three quantities. In addition, the beam energy may play an important, but indirect, role primarily through the bunch length σ z . At top energy, E = 120 GeV, σ z is shorter by a factor of 5 relative to injection energy. This shorter bunch length probably leads to longer high-energy tails in the E 0 spectrum, and therefore to a possibly higher value of δ eff relative to the injection-energy case. The dependence of N b,th on σ z should, therefore, also be established. However, once threshold is exceeded, the sat-urated value of the EC density is probably always comparable to the beam neutralization level, which is independent of beam energy. Therefore, above threshold, the tune shift follows the rather simple scaling ∆ν ∼ 1/E, which leads to the estimate ∆ν 3 × 10 −3 at E = 120 GeV.
For simplicity, we have assumed a tri-gaussian density distribution for the bunch, with round aspect ratio in the transverse plane. In reality, the bunch has an elliptical aspect ratio owing to the variation of the β function, while the longitudinal profile is probably not quite gaussian. The dependence of our results on deviations from these simplifying approximations should be quantified, and an assessment of the EC density in other magnets, especially quadrupoles, should be investigated.
In addition to the above-mentioned possible dependencies on physical parameters, the simulation parameters should also be checked for numerical stability. In the cases presented here, we have taken bunch length effects into consideration by dividing the full bunch length into 10 equal time steps, (ie., N k = 11 kicks), and the inter-bunch spacing into N g = 9 steps. Given the beam parameters, this slicing leads to time steps of size ∆t 1 ns both within the bunch and in between bunches. The EC space-charge forces are computed and applied at every time step by means of a 2D grid of size 5 mm×5 mm. The N e primary electrons, Eq. (5), are represented by M e = 10 macroparticles of charge Q/e = N e /M e = 411.6. The rather low value of M e accounts for the noisiness of the EC line density for N b = 6 × 10 10 , Fig. 2a, but it is practically inconsequential above threshold. From our experience with EC simulations for other storage rings, it appears that these simulation parameters provide approximately stable results, although methodical tests remain to be carried out.