This section is divided into two parts, presenting the results of damping and collider ring, respectively. For the damping ring simulations, we launch an initial set of electrons with a uniform distribution as primary seed, and omit electron generations due to synchrotron radiation, since we only wish to determine the onset of exponential amplification. By contrast, for the collider ring we consider photo-emitted (macro-)electrons launched at the chamber wall during each bunch passage, as primary electron source. The reason is the much lower electron-density threshold (see Table 1), where electrons from photoemission alone could already render the beam unstable.

Simulating the collider case is more challenging also for another reason: The transverse beam sizes in the collider arcs are much smaller than the radius of the vacuum pipe. Therefore, as a first step, the convergence of the simulations is confirmed by comparing the central and line electron densities for different discretization levels of the Finite Difference solver and the space charge Particle in Cell solver, also varying the time step and the initial macro particle size [26]. We note that PyECLOUD computes the central electron density by counting the (macro)-electrons in a circle around the centre of the beam pipe, whose radius can be arbitrarily chosen. Therefore an additional convergence study was performed. The radius of the optimum computational circle was finally determined to be about 7.5 mm.

FCC-ee DR injection and extraction

In this study, the only difference between DR injection and extraction is the different longitudinal and transverse beam sizes. For example, the vertical beam size decreases by up to factor ≈60 between injection and extraction. We consider a DR bunch spacing of 50 ns [4].

In our first numerical experiment, we scan SEY values for the smallest radius value ((r_{0})) of 10 mm, and the highest initial electron density (IeD) 1012 m−3. Figure 2 (upper picture) shows that for (mathrm{SEY}leq 1.5) the electrons ultimately vanish when the computational domain is uniformly loaded at time step zero with cold electrons. In addition, no increase of the electron density from the initial seed was observed up to (mathrm{SEY} =1.7) [30]. However, the (mathrm{SEY} =2.1) curve reveals a build-up-like behavior. We investigate this case in greater detail; in particular, we vary the initial electron density value and the beam pipe radius, with results shown in the right picture of Fig. 2. Here, we observe large oscillations in the electron density for the smallest beam pipe radius and for the largest initial density. In this case, the electron density value reaches maximum of (8times 10^{8}) per meter.

Figure 2
figure 2

Simulated electron-cloud evolution from an initial uniform seed for DR injection, using the Furman–Pivi model. 16 bunches with 50 ns spacing are followed by a gap. Upper figure: (r_{0} = 10text{ mm}) and IeD = 1012 m−3, lower figure: SEY =2.1

We note that the results for the DR at injection were obtained with PyECLOUD and considering the Furman–Pivi SEY model, which provides larger electron build-up values than the ECLOUD model and may be considered a pessimistic scenario. Consequently, the simulation results indicate that a serious electron build-up for the FCC-ee DR injection is not expected if the parameters are in the range of (mathrm{SEY}leq 1.9), (mathrm{IeD} leq 10^{12}text{ m}^{-3}) and (r_{0} geq 10text{ mm}).

For the case of DR extraction, we only observe small changes in the number of electrons in the saturation region as compared to injection curves [28]. We do not reproduce the corresponding electron build-up curves here, as they are fairly similar to the injection case.

Instead, we focus on the kinetic energy distributions of the electrons between two sequential bunch passes, specifically the time period from 150 ns to 200 ns, i.e., between the 3rd and 4th bunch passage after time zero. The first plot of the top row and the last plot of the second row of Fig. 3 show cold electrons in typical stripe formation at the short time interval of 62.5 ps before the next bunch passes. Energies of the electrons, which are accumulating especially around the center region of the vacuum chamber, increase up to 2.5 keV, when the positively charged bunch arrives at time 150 ns (second plot in top row). Afterwards, following the vertical field lines, the energized electrons reach the top and bottom sections of the chamber and generate new electrons though the secondary emission process. Significant electron motion continues until most of the primary electrons have lost their energy and lower-energy secondaries emitted from the chamber wall have penetrated into the chamber.

Figure 3
figure 3

Kinetic energy distributions of electrons inside the wiggler magnet chamber at particular time steps between two bunch arrivals as indicated (with (Delta t = 62.5text{ ps})) for FCC-ee DR extraction

FCC-ee collider arcs

The electron density at the center of the beam pipe is critically important, since these electrons can cause a vertical beam blow up, with an estimate for the threshold density value given in Eq. (1). For the collider the estimated threshold is extraordinarily low; see Table 1. Accordingly, in this section, simulations are mostly devoted to monitoring, and controlling, the electron density at the center of the beam pipe. We consider the two different SEY models discussed earlier, several possible bunch spacings, and also a few values for the bunch population, corresponding to the parameters of the Conceptual Design Report [1] and to the new baseline [24], respectively. The photoelectron generation rate and the secondary emission yield (SEY) parameters are varied over realistic ranges for the FCC-ee collider arcs. In addition, we determine a reference electron density level for the case (mathrm{SEY}approx 0), i.e., without any secondary emission. As a complementary information we also compute the electron line density, i.e., the electron density per unit length.

Our first numerical experiment for the collider arcs investigates the effect of choosing either the Furman–Pivi or the ECLOUD SEY model, considering various SEY and (n_{gamma}^{prime}) values; namely (mathrm{SEY}={1.1,1.2,1.3,1.4}) and (n_{gamma}^{prime }= {10^{-3}, 10^{-4}, 10^{-5}, 10^{-6} }text{ m}^{-1}). Since the electron distribution is fluctuating, we calculate the average of approximate minimum values prior to successive bunch arrivals in the saturation region. With this approach, we determine the dependence of the central electron density on the bunch spacing, starting from 10 ns up to 20 ns. The results are displayed in Fig. 4 (upper picture). In this figure, similarly colored curves belong to the same SEY value. The concept of the center density calculation is illustrated by the insert of the left-hand picture. Also in this left picture, the curves obtained using the Furman–Pivi model exhibit significantly larger electron density values than those obtained with the ECLOUD model. This result agrees well with the study presented in [14] for the sample LHC parameters. With either model, the density strongly depends on the bunch spacing. For the Furman–Pivi model, at a spacing of 10 ns, the central electron cloud density at the moment of bunch arrival is of order 1012 m−3, which is much higher than the threshold density of ({sim} 4times 10^{10}text{ m}^{-3}). For a bunch spacing of 20 ns, the density values approach more acceptable values. In the right picture of Fig. 4 (lower picture), we present the mininum and maximum central density values for the Furman–Pivi model. For a bunch spacing of 10 ns, we observe a order of magnitude difference between the minimum and maximum density. This variation reflects the strong “pinch” of the electron cloud [31] with much enhanced central density near the beam during each bunch passage. However, what matters for the instability is the initial (and minimum) density just prior to bunch arrival.

Figure 4
figure 4

Electron density at the center of vacuum chamber as a function of bunch spacing. Upper figure: Minimum central electron densities for the two SEY models, lower figure: Maximum and minimum central electron-cloud density in saturation, for the Furman–Pivi SEY model

The following simulations are performed to reveal whether the photoelectron generation rate or the secondary emission dominate the electron-cloud build up. We first fix the photoelectron generations with (n_{gamma}^{prime }= 10^{-6}text{ m}^{-1}) and scan the SEY value, as is shown in Fig. 5 (left picture). Then we hold (mathrm{SEY}=1.1) constant, and change the value of (n_{gamma}^{prime}), as is illustrated in Fig. 5 (right picture). By comparing the two pictures in Fig. 5, we notice that the variation of the photoemission rate does not affect the center electron densities as much as varying the SEY value. However, the effect of SEY decreases for increasing bunch spacing.

Figure 5
figure 5

Maximum central electron density for the Furman–Pivi model as a function of bunch spacing, varying either the SEY yield or the photoelectron emission rate. Left figure: (n_{gamma}^{prime }= 10^{-6}text{ m}^{-1}), right figure: SEY = 1.1

The photoelectron generation rate becomes more prominent, for both the Furman–Pivi and ECLOUD models, if we look at the minimum central density rather than the maximum, and at low values of SEY. Figure 6 shows results for (mathrm{SEY}=1.1) and 10−5 (essentially zero). In this figure, BS indicates bunch spacing and (N_{b}) is the bunch population. For SEY values larger than about 1.1, the influence of varying the photoelectron generation rate (n_{gamma}^{prime}) from 10−3 to 10−6 m−1 is negligible [28].

Figure 6
figure 6

Minimum central electron cloud density versus time, comparing SEY models and defining the reference electron formation level for (text{SEY}approx 0). Left figure: (n_{gamma}^{prime }= 10^{-6}), BS = 32 ns, (N_{b}=2.8 times 10^{11}), right figure: Zoomed view of the left picture at the moment of a bunch passage

Now we investigate the central electron density that could be reached in the ideal case of approximately zero SEY and for the lowest possible photoelectron generation rate of our parameter scan range. We can consider this an important reference value for the electron density. Figure 6 reveals that the reference level is approximately (5 times 10^{7} e^{-}/text{m}^{3}), and the same for both SEY models (as the secondary emission contribution is irrelevant here).

Next, if we choose a larger bunch spacing of 32 ns and a bunch population as (2.8 times 10^{11}), both values consistent with the new parameter baseline [24], the central electron density obtained with the two SEY models agree quite well even for nonzero SEY yields [26]. However, the overall electron density per unit length is not necessarily equal [26].

We now examine more closely the effect of the bunch spacing and the number of positrons in the bunch trains on the electron center density level. For this, we decrease both the bunch spacing and the bunch population to half their original values. It is a common understanding that decreasing the bunch spacing increases the electron cloud density while lowering the bunch population can either enhance or attenuate the electron cloud build up, depending on the initial value and other parameters such as the beam pipe radius.

We perform the numerical experiment for the FCC-ee collider arc dipole parameters, choosing the Furman–Pivi model with (mathrm{SEY} = 1.1) and (mathrm{SEY} = 1.4). The simulation results are shown in Fig. 7. We can immediately conclude that the larger SEY drastically increases the initial speed of the electron-cloud build-up. Furthermore, decreasing the bunch population and halving the bunch spacing from 10 to 5 ns has a beneficial effect that might permit injecting closely spaced bunches of lower intensity as well as avoiding an exponential electron growth in the vacuum chamber.

Figure 7
figure 7

Time evolution of the central electron density for bunch spacings of 10 and 5 ns, with associated bunch population (N_{b}=1.7times 10^{11}text{ m}^{-3}) and (8.5times 10^{10}text{ m}^{-3}), respectively, at a high electron generation rate of (n_{gamma}^{prime}=10^{-3}text{ m}^{-1}) (BS:bunch spacing, (N_{b}): bunch population). left figure: SEY = 1.1, right figure: SEY = 1.4

Our last numerical example employs the updated FCC-ee arc dipole/drift parameters for a new 91-km layout with 4 interaction points [24]. In this case, the bunch population, the bunch spacing and bunch length (without beamstrahlung) are increased, compared with the FCC CDR [1], namely to (N_{b}=2.76 times 10^{11}), (L_{mathrm{sep}}=30) or 32 ns, and (sigma _{z}=4.32text{ mm}), respectively. Additionally, also the horizontal and vertical beam sizes are increased, by a factor ({approx} sqrt {3}), as can be seen in Table 1.

After substantial simulations (≈450 hours computer run time), we are in a position to infer certain combinations of photoelectron rate and peak secondary emission yield that result in central electron density values below the estimated threshold, (rho _{mathrm{thr}}approx 4 times 10^{10}text{ m}^{-3}), from (1); also see Table 1. As a result, we find that photoelectron generation rates (n_{gamma}^{prime}) of 10−4 m−1, 10−5 m−1, or 10−6 m−1 for the dipoles and 10−5 m−1 or 10−6 m−1 for field-free drift regions, combined with SEY values in the range from 1.1 to 1.4, lead to simulated central electron densities lower than the estimated threshold, for both the ECLOUD and Furman–Pivi SEY model, and considering either 30 ns or 32 ns bunch spacing, as is illustrated in Fig. 8.

Figure 8
figure 8

Simulated electron densities at the center of the beam pipe for the FCC-ee arc dipole & drift regions using the new baseline beam parameters for four IPs [24]

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