Since the design of the vacuum chamber and of the machine devices is still under development, the FCC-ee impedance model is constantly updated. In particular, in previous studies, such as [15] and [4], only the longitudinal impedance was included in the evaluation of collective effects. Just in [3], the transverse resistive wall impedance has been used for the study of the transverse mode coupling and coupled bunch instabilities.

The impedance revision that we are constantly carrying out has brought to a new transverse impedance model which includes now the same sources as the longitudinal one. Additionally, some elements, such as the bellows and the resistive wall of the vacuum chamber, which represent, so far, the main impedance sources, have been revised by considering more realistic models.

In the following subsections we first discuss these two revised models and then, at the end of the section, we present the total impedance that has been obtained in the three planes, longitudinal (*z*) and transverse *x* and *y*. For the beam dynamics studies, only the dipolar contribution has been taken into account so far.

### Resistive wall

In [3] it was shown that the resistive wall impedance of a two-layer system with a thin coating has an additional imaginary term, with respect to a single layer, proportional to the coating thickness. As a consequence, in order to reduce the resistive wall contribution, which represents the most important impedance source for FCC-ee, the NEG layer, foreseen for mitigating the electron cloud build-up in the positron machine and for pumping reasons in both rings, should be as thin as possible.

The experimental activity discussed in [4] has shown that a NEG thickness between 100 and 200 nm is a good compromise with respect to the balancing between the limitations of activation and the coupling impedance. So far, the beam dynamics studies used a thickness value of 100 nm (see, e.g. [15]). However, according to the vacuum experts [17], in order to guarantee a uniform coating all along the beam pipe, a value of 150 nm should be considered.

The increase in the longitudinal and transverse impedance budget with respect to the 100 nm case is shown in Fig. 1, where the code IW2D [18] has been used with four layers in circular geometry, as already discussed in ref. [15]. As can be seen from the figure, and also according to Eqs. (8) and (9) of [3], only the imaginary part of the impedance is affected by the increased thickness, and the loss factor does not change. Moreover it is possible to demonstrate that, in our frequency range of interest, only the first two layers (copper and NEG) are important.

The increased imaginary part of the impedance of course has an influence on beam instability thresholds, as it will be discussed in Sects. 3 and 4.

In addition to the increased coating thickness, we have also investigated the additional effect to the impedance due to the two lateral winglets of a more realistic vacuum chamber as shown in Fig. 2. The winglets are needed to place synchrotron radiation absorbers ‘hidden’ to the beam.

The figure has been obtained with CST Microwave Studio [19]. Indeed the code IW2D, used so far, gives the impedance and wakefield only for circular and flat geometries. On the other hand, CST is not the most suited code to investigate the resistive wall impedance of a multi-layer system. For such a problem, by using the material type ‘lossy metal’, the code simulates the structure by using a one-dimensional surface impedance model. However, if the electromagnetic field penetrates through the layer, as is the case of the NEG coating foreseen in FCC-ee, one should use the ‘normal type’ material for each layer. This requires a very high number of mesh cells with too heavy computational resources.

In order to overcome this problem and have an estimate of the effects of the winglets, we have still used CST, but in an ‘indirect’ way as follows.

We have first evaluated the impedance of a single thick wall of a material with a relatively low conductivity (here we used (sigma _{c}=10^{5}) S/m). The use of this conductivity was necessary due to the fact that with copper and a reasonable length of the beam pipe (half a meter for mesh cells reasons), the impedance would have been too small and of the same order of the numerical noise. We observe, in any case, that for this chosen conductivity we are in a good conductor regime, characterised by (sigma _{c}gg omega varepsilon _{0}), with (varepsilon _{0}) the vacuum permittivity, up to frequencies much higher than those of our interest. Indeed we remember that, for a Gaussian bunch, the cut-off angular frequency is (omega _{c}=c/(2pi sigma _{z})) with *c* the speed of light and (sigma _{z}) the bunch length. For the minimum bunch length of FCC-ee, equal to 4.32 mm, this corresponds to a frequency of about 11 GHz.

Once the CST impedance of this model was determined, we divided it by its surface, one-layer, impedance (Z_{s1}) and multiplied it by the surface impedance of a double layer given by [20]

$$ Z_{s2}(omega )=( 1+j)sqrt{frac{omega Z_{0}}{2sigma _{c} c}} frac{alpha tanh [frac{1+j}{delta _{1}}Delta ] +1}{alpha +tanh [frac{1+j}{delta _{1}}Delta ] }, $$

(1)

where *ω* is the angular frequency, (Z_{0}) the vacuum impedance, (sigma _{c}) the coating conductivity, (delta _{1}) the skin depth of the coating, Δ its corresponding thickness, and, for a good conductor, (alpha simeq delta _{1}/delta _{2}), with (delta _{2}) the skin depth of the substrate, which is supposed to be of infinite thickness.

The final impedances, for both the longitudinal and transverse planes, are represented in Fig. 3, in the left-hand and right-hand side, respectively. As a check of the validity of this method, we have compared the results with those of IW2D of circular geometry, four layers, and multiplied by a form factor of 1.1 in both planes.

The results of CST have been obtained by considering a Gaussian bunch of 2 mm bunch length, thus having a cut-off frequency of about 24 GHz, and this is the reason why, in this comparison, we have arrived up to 30 GHz. In this frequency range there is a good agreement between the two results.

The slightly different behaviour of the real part of the transverse impedance of CST at small frequencies is due to the lossy metal material type that has been used. Indeed, the frequency dependent skin depth of the fields must be much smaller than the thickness of the metal solid around the beam pipe (that we have chosen in CST to be 2 mm), but at such low frequencies this hypothesis is not verified by CST.

However, since the goal of this comparison was not to check the frequency behaviour given by IW2D, but to understand possible corrections necessary in the results due to the winglets, from the results we can conclude that, as expected, the winglets produce a very small perturbation to the resistive wall impedance of the order of 10%. We can take into account this effect by multiplying the impedance of the circular pipe given by IW2D by a factor 1.1. This factor can be seen as an approximated numerical ‘Yokoya form factor’ [21] valid in the relativistic case and for the geometry with the winglets.

Finally, we observe that the winglets give also a contribution to the quadrupolar impedance due to the breaking of the cylindrical symmetry. However, by using the method just illustrated, we have evaluated a factor of about 20 less that that of the the dipolar impedance, and, in this paper, we neglect this quadrupolar contribution.

### Bellows

The presence of the winglets, on the other hand, plays an important role in the impedance of the bellows. Indeed, the early stage of the design featured a simplified model that considered the beam vacuum chamber with a circular profile without the winglets, as shown in Fig. 4, left-hand side. The main contribution to the broadband impedance, both in the longitudinal and transverse planes, is due to the modes trapped between the beam pipe and the bellows, a space that, from the electromagnetic point of view, can be thought as a cavity coupled with the beam through the apertures due to the RF fingers. The presence of the winglets changes this space, thus modifying the parameters of the modes and their contribution to the broadband impedance.

The study of the bellows has been performed in three different steps, each one considering a design more realistic with respect to the previous one. The corresponding models are shown in Fig. 4, left (simplified circular shape), centre (circular shape with the addition of the winglets), and right-hand side (mechanical design provided by the CERN vacuum group [22]).

A crucial component of the device is the RF shielding with comb-type fingers and small electric fingers to ensure the electric contact between the two sides of the shielding. These fingers are shown in the lower part of the figure, at centre (for the first two models) and at the right-hand side (for the realistic model). The contribution of the shielding is fundamental to suppress the low frequency resonances due to the bellows which otherwise would lead to a high impedance contribution.

The realistic model on the right side of the figure consists of a vacuum chamber designed with a mechanical CAD, with different comb-type fingers that best suite the bellow’s shielding, and a realistic version of the bellows with squared profile, as shown in the top right part of the same figure.

Preliminary simulations and most of the convergence studies related to the impedance behaviour have been carried out using the simplified CST models, firstly with the circular beam chamber and then including the winglets, since the realistic model required heavy computational resources. Afterwards, simulations on the realistic model were carried out to verify the results obtained with the simplified versions.

For all the considered models, longitudinal and transverse simulations have been performed with the wakefield solver of CST, and they were preliminary focused on the numerical convergence of the results, which turned out to be the most critical and challenging part for the electromagnetic characterisation of the device.

Actually, the complexity of the simulations deriving from the small mesh size, required to proper model the tiny fingers of the shielding, led to time consuming simulations and to the need of important computational resources. The situation was even more cumbersome in performing transverse simulations due to the limitation in the use of symmetries to reduce the number of mesh cells. In this scenario, the achievement of the numerical convergence turned out to be essential to allow the correct study of the problem under consideration and in many situations this required to use a large number of mesh cells, up to one billion.

One check that was performed after the convergence studies, was the possibility to reconstruct the wake potentials of a longer Gaussian bunch, for example of 3.5 mm, with that of 0.4 mm used as pseudo-Green function. The comparison between the direct result of CST and the reconstruction by means of the convolution integral is shown in Fig. 5 for both the longitudinal (left-hand side) and transverse (right-hand side) planes.

The results of the longitudinal wakefields for 3.5 mm Gaussian bunch and the impedances of all the three studied models by considering a single bellow are reported in Fig. 6 for comparison in the left and right-hand side, respectively.

The most obvious aspect is the absence of the first resonance of the circular model around 11 GHz in both the other two models with the winglets. This behaviour has been carefully studied and it turned out that the first resonance is related to the coupling between the beam vacuum chamber and the cavity of the bellow. When the radius of the beam chamber approaches the radius of the bellow cavity, the resonance decreases its amplitude more and more until it is suppressed. According to these results, it is not surprising that the resonance is almost suppressed in the case of the chamber with winglets, since the horizontal aperture of the chamber approaches the radius of the bellow cavity. As a result, the realistic vacuum chamber has a better performance from the impedance point of view.

In the transverse plane we can draw similar conclusions: for the realistic model the low frequency dipolar impedance is negligible, almost null and the first resonance appears above 20 GHz. A similar situation occurs for the quadrupolar term of the impedance.

It is important to remind that for the total wakefield and impedance of the bellows we have to multiply these results by their number. The exact number is still unknown, however, if we consider 2900 dipole arcs 24 m long with bellows every 8 m [23] plus 2900 quadrupoles/sextupoles arcs as in the CDR [2], we have a total of 11,600 bellows. In addition to these, we need to take into account the bellows for the RF system, injection system, collimation, etc. Overall, to be conservative, we have overestimated them by using a total number of 20,000 bellows.

### Total impedance

The impedance model evaluated so far takes into account also the beam position monitors and the RF system which includes the tapers connecting the cryo-modules. The total imaginary and real part of the broadband longitudinal and transverse impedances, together with the different contributions, are shown in Fig. 7 and Fig. 8

It is important to note that, except for the resistive wall and the bellows, all the other devices give a small contribution (just the beam position monitors show a small peak around 5 GHz). Indeed, also the photon stoppers, or synchrotron radiation absorbers, have been evaluated in the past [4, 24] resulting in a negligible contribution. A system that could be more critical, in particular in the transverse plane, is represented by the collimators. We are still missing this impedance and the study is in progress. We have to remind, however, that we almost doubled the impedance of the bellows in order to take into account, at this preliminary stage, other important and yet unknown possible impedance sources.

In addition to the update of the resistive wall and bellows, other accelerator systems are also under development and may change in the future. For example, an alternative RF system is under study. It considers to use, instead of 52 single cell 400 MHz cavities grouped in 13 cryo-modules, each one having a double taper, as in the CDR [2], a system using the so called two-cell 600 MHz Slotted Waveguide ELLipltical (SWELL) cavity [25]. The longitudinal broadband impedance for a single cavity is shown in Fig. 9 in comparison with the 400 MHz one.

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