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

M A Furman

Center for Beam Physics, Bldg. 71R0259, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley, CA 94720-8211, USA

Email: mafurman@lbl.gov

Received 27 March 2006
Published 28 November 2006

Contents

• 1. Introduction and summary
• 2. Electron sources
  • 2.1. Primary mechanisms
  • 2.2. Secondary electron emission
• 3. EC build-up
  • 3.1. General considerations
  • 3.2. Results for the MI
• 4. Discussion
• Acknowledgments
• References

1. Introduction and summary

An upgrade to the Main Injector (MI) storage ring 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 ×10¹⁰. Such an increase would place the MI in a regime in which a significant electron-cloud (EC) effect has been observed at other hadron machines [2]–[5].

In this paper, we present an examination of the EC at the MI by means of computer simulations with the code POSINST [6]–[9]. For the purposes of the present work, we fix two important parameters, namely the beam energy E_b 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 ×10¹¹, beyond which ρ_e grows exponentially in time with an e-folding time τ≊100 ns upon injection of the beam into an empty ring, and reaches a steady-state value that is 10⁴ 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 contribution Δν≊  + 0.05 to the space-charge tune shift. An assessment of possible dynamical effects on the beam from the EC, such as emittance growth and instabilities, falls outside the scope of this paper, as does a systematic sensitivity analysis of our results on various assumed input parameters, particularly δ_max.

2. Electron sources

2.1. 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, the number of electrons generated is proportional to N_b, hence 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, which we express in units of electrons per proton per metre, or (e/p)/m.

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 and the ionization cross-section [10]. Assuming parameter values listed in table 1, we obtain a contribution n '_e(i)~ 10^–7 (e/p)/m from this process. The contribution from stray protons striking the chamber walls is given by the product of the proton loss rate per unit length n '_pl (`pl' stands for `proton loss') and the effective electron yield per proton-wall collision η_eff. We focus here on the beam injection process, since the most significant fraction of beam loss (~ 1% of the beam) occurs during this time, which lasts for Δ t_inj  =  0.4 s. Assuming that the beam losses occur uniformly throughout the machine, and uniformly during Δ t_inj, we obtain n '_e(pl)~ 10^–8 (e/p)/m, where we have also assumed a typical value η_eff  =  100. Further details can be found in [11].

Parameter	Symbol (unit)	Value
Ring and beam parameters
Ring circumference	C (m)	3319.419
Beam energy	Eb (GeV)	8a
Relativistic beam factor	γb	8.526312
Revolution period	T0 (μs)	11.1493
Beam pipe cross-section	···	Elliptical
Beam pipe semi-axes	(a,b) (cm)	(6.15, 2.45)
Harmonic number	h	588
RF wavelength	λRF (m)	5.645270
No. bunches per beam	···	504
Bunch spacing	sb (m)	5.645270
Gap length	··· (buckets)	84
Bunch population	Nb	(0.6–3)× 1011
RMS bunch length	σz (m)	0.75
Longitudinal bunch profile	···	Gaussian
Transverse bunch profile	···	Gaussian
Average beta function	(m)	25
Normalized tr. emittance (95%)	εN (m-rad)	40π
RMS relative momentum spread	σp/p	10–3
Transverse RMS bunch sizes	(σx,σy) (mm)	(5,5)
Parameters for primary e– sources
Proton loss rate	n 'pl (p/m)	1 ×10–10
Proton–electron yield	ηeff	100
Residual gas pressure	P (ntorr)	20
Temperature	T (K)	305
Ionization cross-section	σi (Mbarns)	2
Proton-loss e– creation rate	n 'e(pl) [(e/p)/m]	1 ×10–8
Ionization e– creation rate	n 'e(i) [(e/p)/m]	1.27 ×10–7
Secondary e– parameters
Peak SEY	δmax≡δ(Emax)	1.3b,c
Energy at peak SEY	Emax (eV)	293b, 272c
SEY at 0 energy	δ(0)	0.32b, 0.38c
Backscattered component at Emax	δe(Emax) + δr(Emax)	0.53b, 0.13c
Simulation parameters
Simulated section	···	Drift or dipole magnet
Length of simulated region	L (m)	0.1
Dipole magnet field	B (T)	0.1
No. kicks/bunch	Nk	11
(Full bunch length)/(RMS bunch length)	Lb/σz	4
No. steps between bunches	Ng	9
No. primary macroelectrons/bunch	Me	10
Macroelectron charge at Nb  =  3×1011	Q/e	412
Time step size	Δ t (ns)	1

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 [10].

Actual values for all parameters used here, including those pertaining to the primary electrons, are listed in table 1. As mentioned above, and as illustrated below, the primary electrons act primarily as seeds in the formation of the EC when the beam current is above threshold. In this regime, secondary electron emission typically contributes several orders of magnitude more electrons to the EC density, hence the precise values of the primary electron parameters have little impact on the EC in steady state. For this reason, we have not attempted to accurately pin them down.

2.2. Secondary electron emission

The SEY function δ(E₀,θ₀) is the average number of electrons emitted when an electron of kinetic energy E₀ impinges on a surface at an incident angle θ₀ (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 E₀  =  E_max. A fairly detailed phenomenological probabilistic description of the secondary emission process is presented in [8, 9], upon which we base the analysis in this paper.

Closely related to δ is the emitted-energy spectrum of the secondary electrons, dδ/dE at given incident energy E₀, where E is the emitted electron energy. The spectrum covers the region 0≤ E    E₀, 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 in various degrees of importance to various EC effects.

Since we do not have data for the SEY of the MI vacuum chamber, for the discussions in this note, we adopt two models, which we call `K' and `H,' that may be considered representative of the possible range of SEY parameters for the MI. These models correspond, respectively, to the fits to stainless steel and copper data in [8, 9], except that in the present paper we scale all three components of δ by a common factor so that δ_max  =  1.3 instead of the original value 2.05. This scaling has the consequence that δ(0) becomes proportional to δ_max. Since we do not know the precise value of δ(0), this scaling is intended only as a practical step in the parameter exploration, and is not meant to reflect the phenomenology of the secondary emission process.

As seen in figure 1, the SEY functions δ(E₀) 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 1). When these two models are applied to the estimate of the EC power deposition in the Large Hadron Collider (LHC) arc dipoles, for example, one finds significantly different results [10, 12], underscoring the importance of the emission spectrum.

The choice δ_max  =  1.3 used here is meant as a first step of a more complete assessment that is yet to be carried out. It is likely that N_b,th is sensitive to δ_max and to other variables. In practice, the value of δ_max is a function of the conditioning state of the material, as it decreases monotonically with electron bombardment. Vacuum chambers made of copper or stainless steel have δ_max values in the range ~ 1.5–2.5, or even higher, in the `as-received' condition. For aluminium, the values are typically higher than this. Bench experiments show that, if the material is bombarded in vacuum with a steady flow of electrons, δ_max decreases to ~ 1.1 after a dose ~1 C cm^–2  [13]–[16]. The MI vacuum chamber is made of stainless steel; our choice δ_max  =  1.3 is generally believed to correspond to a more or less well-conditioned state of this material. However, the sensitivity of our results to δ_max is an important issue that remains to be investigated.

3. EC build-up

3.1. 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(E₀,θ₀) with the energy-angle electron-wall collision spectrum (normalized to unity) dN/dE₀d θ₀,

The spectrum dN/dE₀d θ₀ 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 (1) 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, , grows linearly in time t following injection of the beam into an empty chamber according to

where is the average beam line density and is the number of primary electrons generated per beam particle per unit time, , where v_b is the beam velocity. After a growth 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 growth time τ and the saturated value of λ_e are given by [17]

where Δ 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 sufficiently low, although it can also happen when N_b is very high because, as N_b increases, typical values of E₀ can exceed E_max, hence δ(E₀) decreases hence so does δ_eff. 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 [10].^Note1 

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 space-charge 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 (3a) 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 Δ t_tr by [10]

The traversal time Δ 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. Δ 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.

3.2. Results for the MI

For the studies presented in this paper, we have used the simulation code POSINST [6]–[9]. 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_b  =  8 GeV.^Note2  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 1. 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 × 10¹⁰≤ N_b ≤ 3 ×10¹¹, while we fix δ_max  =  1.3. We carry out simulations for one revolution period (T₀  =  11.15 μs) for an 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 section 4.

Figure 2 shows the time evolution of the EC line density. The above-mentioned behaviours are clearly seen. For N_b  =  6×10¹⁰, the EC reaches an average line density for a drift and for a dipole, while for N_b  =  3 ×10¹¹, the EC density saturates at for both cases. This latter value should compared with the average beam line density, implying an average beam neutralization factor The exponential growth of the EC density for N_b  =  3 ×10¹¹ is clearly seen over four orders of magnitude in density during the first ~1.5 μs, with a growth time τ≊110 ns for the drift and τ≊90 ns for the dipole.

Figure 2. Average EC line density versus time. (a) Nb  =  6 ×1010; (b) Nb  =  3 ×1011. Note that the vertical scale for (a) is linear, while that for (b) is logarithmic. The exponential growth of the density for case (b) during the first ~1.5 μs has an e-folding time τ≊ 110 ns for a drift and τ≊ 90 ns for a dipole. The saturation level is for both cases. For case (b) the horizontal green line represents the average beam line density,

Figure 3 shows the time- and space-averaged electron-wall collision energy spectrum. For N_b  =  6 ×10¹⁰, the spectra are sharply cut off at E₀    200 eV and yield an average electron-wall collision energy ~ 50–100 eV, while for N_b  =  3 ×10¹¹ the spectra exhibit a high-energy tail up to ~500 eV, with an average ~100–150 eV. Referring to figure 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 equations (2)–(4), we consider the results for a drift, specifically the EC build-up in figure 2(a). For N_b  =  6 ×10¹⁰, the values for δ_eff, τ and Δ t_tr obtained directly from the simulation are ~0.85, ~140 and ~21 ns respectively, which satisfy (3a) well. Furthermore, using from the figure, we obtain from (3b) in good agreement with the direct result from the simulation. The above value of Δ t_tr, in turn, implies a typical electron energy ~45 eV, in agreement with the direct results from the simulation shown in figure 3(a). For N_b  =  3 ×10¹¹, we obtain δ_eff≊1.15 and τ≊110 ns during the exponential growth regime. Equation (4) implies Δ t_tr  =  15 ns, which implies an electron energy ~90 eV. This value is lower by a factor ~2 than what is independently deduced from the simulation (e.g., figure 3(b)), 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 Δν owing to the focusing effect of the electrons on the beam. Assuming that the EC density distribution is round in the transverse plane, the EC-induced tune shift per unit length of beam traversal through the cloud, Δν/L, is given by [18]

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 centre of the bunch, and β is the usual lattice beta function. For N_b  =  3 ×10¹¹, the steady-state value translates into a density ρ_e≊7.5 ×10¹² m^–3. Assuming a value of 25 m for the average beta function, we obtain

To get an idea of the magnitude of Δν, we replace L by the circumference C, yielding Δν  =  0.056. For N_b < N_b,th, the electron density is ~10⁸ m^–3, hence Δν~5 ×10^–6, a wholly negligible tune shift.

4. 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 ×10¹¹, 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 figure 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 behaviour 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 figure 2(a) for N_b  =  6 ×10¹⁰).

Although the assessment presented in this paper is of limited scope, 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.

It seems interesting to compare the EC buildup in the MI with other storage rings. The sudden onset of a significant EC signal as a function of N_b, and the actual value of N_b,th, are related to a combination of vacuum chamber parameters (both physical and electronic), bunch length and bunch spacing. Simulations for the EC buildup in the LHC arc dipole magnets, for example, show a gradual (essentially linear) dependence of the EC power deposition as a function of (N_b–N_b,th), where N_b,th~ 2 ×10¹⁰ [12, 19]. This behaviour appears to be qualitatively different from the MI; it is likely that the much longer bunch spacing in the LHC plays an essential role in explaining the difference. More research is needed to clarify these issues.

As mentioned in subsection 2.2, the choice δ_max  =  1.3 in this preliminary assessment is meant as a first step in a more complete analysis. We have chosen this value for δ_max because it is believed to correspond to more or less well conditioned stainless steel. The EC effect is a self-conditioning phenomenon in the sense that the very same electrons from the cloud condition the vacuum chamber as they strike its surface during normal machine operation, leading to a gradual decrease of δ_max and hence to a diminished EC effect [20, 21]. The electron dose required to reach an innocuous EC effect, is, roughly speaking, ~0.1–1 C cm^–2. For the MI conditions studied in this paper, the average electron flux at the walls (see figure 3 caption) is ~10^–10 A cm^–2 for N_b  =  6 ×10¹⁰, implying a self-conditioning time of hundreds of years. On the other hand, at N_b  =  3 ×10¹¹, the electron flux at the walls is six orders of magnitude larger, implying a self-conditioning time of hours. Of course, this analysis is very simplistic, as many other factors affect the conditioning time; nevertheless, the electron flux gives a rough estimate of the relevant timescales.

As seen in table 1, the backscattered component of the SEY at E₀  =  E_max is (δ_e(E_max) + δ_r(E_max))/δ(E_max)  =  0.41 for model K and 0.10 for model H. In the regime of interest to the MI this implies that, in SEY model K, the 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 figures 4–6 for a more complete set of results).

A set of delicate measurements of δ(E₀) and dδ/dE for copper samples at low temperature (T≊9 K) carried out by Cimino and Collins [22] at the European Organization for Nuclear Research (CERN) exhibits an upturn in δ(E₀) as E₀ decreases below ~ 20 eV, reaching δ(0)≊1 (an indication of a similar upturn is apparent in another set of measurements: see [23], figure 5). The Cimino–Collins data exhibit the usual conditioning effect whereby δ_max gradually decreases with electron bombardment. However, the data also exhibit the novel feature that δ(E₀) is insensitive to electron bombardment for E₀    10–20 eV. Measurements of the spectrum dδ/dE for several values of E₀ allowed the extraction of δ_e(E₀) and δ_r(E₀) + δ_ts(E₀), which showed that δ_e(E₀)→ 1 in the limit E₀→ 0 regardless of the state of conditioning of the sample, while δ_r(E₀) + δ_ts(E₀)→ 0 in the same limit. Since δ_ts(E₀)→ 0 in this limit, these measurements imply δ_r(0)≊0. By contrast, in the models used here for the MI simulations, δ(E₀) decreases monotonically as E₀→0, and its three components have the following values: model K: δ_e(0)  =  0.32,δ_r(0)  =  δ_ts(0)  =  0; model H: δ_e(0)  =  0.31, δ_r(0)  =  0.07, δ_ts(0)  =  0. EC buildup simulations showed that the upturn in the Cimino–Collins data leads to a substantially larger EC signal relative to the more conventional model in which δ(E₀) decreases monotonically as E₀→0 [24]. This relatively large effect of the low-energy details of the SEY can very likely be attributed to the long survival time in the vacuum chamber of the backscattered electrons when the bunch spacing is sufficiently large, as in the LHC [12]. For the case of the MI, the much shorter bunch spacing diminishes the relatively large importance of the backscattered electrons. A simulation spot check of the EC buildup for N_b  =  3 ×10¹¹ (results not shown) with an SEY model corresponding to the Cimino–Collins data showed that the exponential growth time τ is somewhat larger than the results in subsection 3.2, although in general there were no qualitative differences.

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_b  =  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 E₀ 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 saturated value of the EC density is probably always comparable to the beam neutralization level, which is independent of beam energy. Therefore, above threshold, the EC tune shift is expected to follow the rather simple scaling Δν~ 1/E_b, leading to the estimate Δν≊3 ×10^–3 at E_b  =  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 dependences 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 ten equal time steps, (i.e., 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 Δ 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 primary electrons generated per bunch passage in the section of ring being simulated are represented by M_e  =  10 macroparticles of charge Q/e≊ 400. The rather low value of M_e accounts for the noisiness of the EC line density for N_b  =  6 ×10¹⁰ (figure 2(a)) 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.

Acknowledgments

I am grateful to A Chen, W Chou, K Y Ng, J-F Ostiguy, P Yoon and X Zhang for valuable discussions. Work was supported by the US DOE under contract DE-AC02-05CH11231.

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CrossRef

Notes

Note1  The simpler arguments used in [10] lead to τ  =  –Δ t_tr/ln δ_eff, which agrees with (3a) only when δ_eff≊1. A fuller discussion will be presented in [17].

Note2  Owing to a misunderstanding, we erroneously chose 8 GeV in our simulations instead of the actual value of 8.9 GeV. The slightly lower value has a negligible effect on our simulation results, except possibly that it leads to an overestimate of the tune shift (5) by ~10%.