Geometric beam coupling impedance of LHC secondary collimators

Nuclear Instruments and Methods in Physics Research A 810 (2016) 68–73

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Geometric beam coupling impedance of LHC secondary collimators Oscar Frasciello a,1,n, Sandro Tomassini a, Mikhail Zobov a, Benoit Salvant b, Alexej Grudiev b, Nicolas Mounet b,2 a b

Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Frascati, Italy CERN, CH-1211 Geneva 23, Switzerland

art ic l e i nf o

a b s t r a c t

Article history: Received 26 May 2015 Received in revised form 12 November 2015 Accepted 20 November 2015 Available online 11 December 2015

The High Luminosity LHC project is aimed at increasing the LHC luminosity by an order of magnitude. One of the key ingredients to achieve the luminosity goal is the beam intensity increase. In order to keep beam instabilities under control and to avoid excessive power losses a careful design of new vacuum chamber components and an improvement of the present LHC impedance model are required. Collimators are among the major impedance contributors. Measurements with beam have revealed that the betatron coherent tune shifts were higher by about a factor of 2 with respect to the theoretical predictions based on the LHC impedance model up to 2012. In that model the resistive wall impedance has been considered as the dominating impedance contribution for collimators. By carefully simulating also their geometric impedance we have contributed to the update of the LHC impedance model, reaching also a better agreement between the measured and simulated betatron tune shifts. During the just ended LHC Long Shutdown I (LSI), TCS/TCT collimators were replaced by new devices embedding BPMs and TT2-111R ferrite blocks. We present here preliminary estimations of their broad-band impedance, showing that an increase of about 20% is expected in the kick factors with respect to previous collimators without BPMs. & 2015 Elsevier B.V. All rights reserved.

Keywords: Wakefields Impedance LHC collimators GdfidL

1. Introduction The LHC collimators [1] are among the main beam coupling impedance contributors. This can be clearly seen in Fig. 1, where the detailed contributions, in percent, to the whole 2012 LHC vertical dipolar impedance model are reported [2,3]. Here, collimators play the major role ( 90%) over a wide frequency range, both for real and imaginary parts. At that time, in 2012, the model was essentially based on the resistive wall impedance of collimators, the resistive wall impedance of beam screens and warm vacuum pipe and a broad-band model including pumping holes, BPMs, bellows, vacuum valves and other beam instruments. The geometric impedance of collimators was approximated only by that of a round circular taper. However, several measurements were done in 2012 of the total single bunch tune shifts vs. intensity, both at injection and at 4 TeV. A specific measurement was also performed for several collimator families all at top energy, giving their tune shifts upon moving back and forth the jaws. The experimental results, in terms n

Corresponding author. Tel.: þ 39 06 9403 8114. E-mail address: [email protected] (O. Frasciello). 1 Also at University “La Sapienza”, Rome, Italy. 2 Currently at EPFL, Lousanne, Switzerland.

http://dx.doi.org/10.1016/j.nima.2015.11.139 0168-9002/& 2015 Elsevier B.V. All rights reserved.

of tune slope vs. intensity, were compared with the numerical simulation results, which used the wake fields from the LHC impedance model. The measured tune shifts are higher than predicted ones by a factor of 2 at top energy and of 3 at injection [4]; so the existing LHC impedance model accounted only for a fraction, 13 12, of the measured transverse coherent tune shifts. This fact led to the need for an LHC impedance model refining which, first of all, required a careful collimator geometric impedance calculation. For this purpose, we carried out numerical calculations of the geometric impedance of the LHC secondary collimator, as close as possible to its real design and evaluated the collimator contribution to the overall LHC impedance budget.

2. Theoretical considerations In order to verify whether the geometric collimator impedance can give a noticeable contribution to the betatron tune shifts, we suggest to compare transverse kick factors due to the resistive wall impedance and the geometric one. First of all, as we show below, the tune shifts are directly proportional to the kick factors. Besides, this approach has several advantages: (1) it is a quite straightforward way to compare contributions from impedances having

O. Frasciello et al. / Nuclear Instruments and Methods in Physics Research A 810 (2016) 68–73

different frequency behaviour into the transverse tune shifts; (2) only calculations of the broad band wakes are necessary without the exact knowledge of the transverse impedance Z T ðωÞ; (3) both kick factors and broad band impedances are easily calculated by many numerical codes. The well known expression for the coherent mode tune shifts can be found in [5], P 1 I 2c p Z T ðωp Þhm ðωp ωξ Þ P Δωcm ¼ j ; ð1Þ 1 þ j mj 2ω0 Q ðE=eÞL p hm ωp ωξ where Δωcm is the shift of the angular frequency of the mth transverse coherent mode, with m being the azimuthal mode number, ω0 is the angular revolution frequency, Ic is the average bunch current, Q is the betatron tune, E is the machine energy, L is the full bunch length and ωξ ¼ ω0 ηξ is the “chromatic” angular frequency, with ξ being the chromaticity and η the slippage factor of a circular accelerator. For a given mode m, the bunch power spectrum for a Gaussian bunch, with rms bunch length σz, is given by ωσ 2j mj 2 z ; e ðωσ z =cÞ : ð2Þ hm ðωÞ ¼ c Given the sum in Eq. (1) being performed over the mode spectrum lines

ωp ¼ ðp þ ΔQ Þω0 þ mωs ;

1 o p o þ1

ð3Þ

where ΔQ is the fractional part of the betatron tune and ωs is the angular synchrotron frequency, it results that the tune shifts are real if the imaginary transverse impedance, IZ T ðωp Þ, differs from zero. For ξ ¼ 0 and dipole coherent mode (m ¼0) we get a proportionality relation: X 2 Δωc0 ¼ const I c ImfZ T ðωp Þge ðωσ z =cÞ : ð4Þ p

On the other hand, from the kick factor definition, we have Z 1 2 1 kT ¼ Im Z T ðωÞ e ðωσ z =cÞ dω 2π 1

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ð5Þ

so that, comparing Eq. (5) with Eq. (4), we find

Δωc0 p kT

ð6Þ

In what follows we use the impedance theory developed in [2], in order to evaluate the kick factors due to the wall resistivity.

3. Numerical calculations The geometric wakefields and impedance simulations were performed by means of the electromagnetic code GdfidL [6], assuming all metallic surfaces perfectly conducting. In order to simulate the collimator as close as possible to its real design, we used the collimator CAD drawing, shown in Fig. 2 (left), including all mechanical details (.stl file) as GdfidL input file. The image in Fig. 2 (right) shows the details of the collimator internal structure as reproduced by GdfidL. A very fine mesh (and, respectively, a huge CPU time) was necessary to reproduce in detail the long and complicated structure and to overcome numerical problems arising when simulating the long collimator tapers. In particular , we used several billions of mesh points to perform the wake field simulations for 2 mm long bunches that are to be used as a point-charge-like wake function in particle tracking codes. Our numerical studies have shown that the collimator broadband impedance is dominated by the impedance of the flat elliptical collimator tapers, the closest to the beam. We find it to be noteworthy to say that the difference in impedance between the round and the flat rectangular tapers can be very large, being proportional to the ratio of the horizontal to the vertical size [7].

Fig. 1. Real (left) and imaginary (right) parts of the 2012 LHC vertical dipolar impedance model.

Fig. 2. Collimator CAD design (left) and internal structure as reproduced by GdfidL (right).

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Fig. 3. GdfidL results for LHC collimator's low frequency broad-band transverse effective impedance (left) and kick factors' comparison (right).

The real taper is elliptical in the case under study (see, for example, Fig. 2 in [8]), with high ratio of the horizontal to the vertical size. That is why it is better modeled by the flat rectangular taper formula. In Fig. 33 (left image) we compare the low frequency transverse impedance provided by GdfidL as a function of the collimator half gap with the following analytical impedance formulas. For a round taper [9] the transverse impedance is given by ZT ¼ j

Z0 2π

Z 0 2 b dz; b

ð7Þ 0

where Z 0 ¼ 377 Ω, b is the variable taper radius and b its derivative with respect to z. For a flat rectangular taper [7] the transverse impedance is Z Z 0 w ðg 0 Þ2 dz; ð8Þ ZT ¼ j 4 g3 where w is the taper constant width in the transverse vertical direction, g the taper variable gap and g 0 its derivative with respect to z. In both Eqs. (7) and (8), the integration is performed over the taper length. The width of the jaws where the taper is carved is 74 mm to be compared with the typical collimator gap g of few millimeters in operating conditions. What we obtain is a kind of a broad-band impedance, since at some point we have to truncate the wake and perform its Fourier transform. This makes the “effective impedance” bunch length dependent. For the nominal LHC bunch length of 7.5 cm, the whole bunch spectrum stays below the beam pipe cutoff, so remaining inductive. That is why the analytical formula comes back to describe it finely. Since the jaws taper represent the closest discontinuity to a beam, we can expect that the taper would give the dominant impedance contribution. Due to the fact that the collimator taper has a very flat geometry, the available analytical formula for a low frequency transverse impedance of a rectangular taper, Eq. (8), should give a reasonable estimate of the impedance in this case. Indeed, as is seen on the left plot of Fig. 3 (upper curve), the calculated low frequency impedance of the complicated collimators’ structures is very well approximated by the rectangular taper formula Eq. (8). The analytical formula for a round taper Eq. (7), instead, used in the early LHC impedance model, drastically underestimates the taper impedance (lower curve on the left plot of Fig. 3). 3 In both plots of Fig. 3 solid and dotted smoothed lines join the five computed points for the five different half gaps.

On the right-hand side of Fig. 3, finally, the comparison between the geometric kick factors and the resistive wall ones, both for Carbon Fiber Composite (CFC) and Tungsten (W) made collimators’ jaws, is reported, always as a function of the half gap. As a result, for W collimators geometrical impedance contribution dominates in almost the whole range of half gaps (from 1.5 mm onward), while only from about 8 mm onward for CFC collimators. Even considering the partial contribution in the case of CFC, it is evident that the geometrical impedance is not negligible with respect to resistive wall one. Both the transverse and longitudinal impedances exhibit many resonant peaks at different frequencies. These higher order modes (HOM) are created in the collimator tank, trapped between the sliding contacts in the tapered transition, etc., with parameters depending very much on the collimator gap. As an example, Fig. 4 shows the longitudinal impedance for two different collimator gaps. We found out that the HOM parameters obtained in our simulations, such as HOM frequency patterns, their strength and dependence on the gap width are very similar to those obtained in earlier works with simpler collimator model [10–12]. The modes shunt impedances are relatively small compared to typical HOMs in RF cavities. However, possible additional RF losses and related collimator structure heating due to these modes, in the conditions of higher circulating currents in High Luminosity LHC, still deserve a deeper investigation. Finally, in order to allow beam dynamics studies by means of particle tracking codes, detailed longitudinal and transverse dipolar and quadrupolar wake field calculations were carried out for 5 different collimator jaws' half gaps, namely 1 mm, 3 mm, 5 mm, 11.5 mm and 20 mm. For this purpose the bunch length was just 2 mm, instead of the 7.5 cm long LHC nominal bunch length, and the wake field was traced over 1 m. These wake potentials can approximate “point-like” wake functions for multi-particle tracking codes. We report, for example, in Fig. 5 the longitudinal and dipolar transverse wake potentials, computed by GdfidL, for 1 mm and 20 mmm collimator jaws' half gaps.

4. LHC impedance model update Recently the LHC impedance model was updated taking into account several improvements, namely the evaluation of the geometric impedance of the collimators as explained above, a full revision of the resistive wall model of the beam screens and the warm vacuum pipe, a theoretical re-evaluation of the impedance of the pumping holes, the inclusion of details of the triplet region (tapers


and BPMs) and the broad-band and High Order Modes (HOMs) of the RF cavities, CMS, ALICE and LHCb experimental chambers. Comparing the old and the updated models, a quite common impedance behaviour at low frequency and a consistent increase ð 30%Þ at frequencies close to 1 GHz for the latter, can be assessed [3,13]. This should explain a part of the factor 2 discussed in the Introduction.

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Since Eq. (1) for the coherent mode tune shift uses the impedance convolution over the bunch spectrum (all frequencies starting from 0), the contribution to the tune shift is expected to be smaller. The new DELPHI simulations performed according to the new data for collimators [14], indeed, show an increase of at most 15–20%, as an effect of geometric impedance. However, the impedance near 1 GHz may more strongly affect other beam

Fig. 4. Real and imaginary parts of the longitudinal impedance for (left) 1 mm and (right) 20 mm half gaps.

Fig. 5. Longitudinal (up) and dipolar transverse (down) wake potentials for (left) 1 mm and (right) 20 mm half gaps, calculated for 2 mm bunch length. The wake is traced over 1 m length, as reported in the text, but here we focus the plots on the first 5 cm in order to highlight the main differences in wake trends.

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Fig. 6. The updated LHC dipolar vertical impedance model real (left) and imaginary (right) parts.

Fig. 7. The new BPM embedded TCS/TCT collimator design for LHC Run II and its elaboration into GdfidL.

Table 1 Geometric transverse kick factors due to the two TCSG/TCT geometries, calculated at different half gap values. Half gaps (mm)

w/BPM cavity kT ðV=cmÞ

w/o BPM cavity kT ðV=cmÞ

1

3:921 1014

3:340 1014

3

6:271 10

13

13

5

2:457 1013

5:322 10

2:124 1013

dynamics aspects, e.g. TMCI threshold, since the different modes in the spectrum probe different frequencies in a different manner. The impact of the geometric impedance on the whole transverse dipolar impedance LHC model is clearly visible in Fig. 6, where the various percentile contributions are shown.

5. The new LHC Run II TCS/TCT collimators' design In view of LHC Run II and future high luminosity LHC upgrades, 2 old TCS CFC and 16 W TCT collimators were replaced, during the LSI, by new devices embedding BPMs in the jaws' tapering region and TT2-111R ferrite blocks in place of old transverse RF fingers (Fig. 7) [15]. In the spirit of the approach discussed in Section 2, we again performed calculations of the geometric broad band wakes of the old and new collimators. We aimed at gaining a preliminary estimation of the new design contribution to impedance. Values listed in Table 1, calculated for three different jaws' half gaps, clearly show that the geometric transverse effective impedance of the new collimators is expected to increase of about 20% with

respect to the old design, mainly due to the first tapers with a steeper angle.

6. Conclusions Calculations of wake fields and beam coupling impedance have been performed for the LHC secondary collimators, for five different gaps, by means of GdfidL electromagnetic code. From the comparison of the transverse kick factors, it has been shown that the geometric impedance contributions is not negligible with respect to the resistive wall one. In particular, for CFC made collimator, the geometrical kick starts to be comparable to resistive wall one at about 8 mm half gap. In turn, for W made collimators, the geometrical kick dominates almost for all the collimator gaps. The present study has contributed to the refinement of the LHC impedance model. It has also been shown that the geometrical collimator impedance accounts for approximately 30% of the total LHC impedance budget, at frequencies close to 1 GHz. LHC LS1 saw the old TCSG/TCT collimators replaced by new ones with a quite more complicated structure, embedding BPMs and ferrite blocks. The latter, in particular, are believed to be very effective in damping HOMs trapped in the collimators’ structure and deserve a dedicated careful study to be described in further papers. Till now we have performed preliminary calculations of the geometric transverse impedance of this new design, already showing that it contributes to an increase of about 20% in the transverse kicks, with respect to the old one.


Acknowledgements Work supported by HiLumi LHC Design Study, which is included in the High Luminosity LHC project and is partly funded by the European Commission within the Framework Programme 7 Capacities Specific Programme, Grant Agreement 284404. We would like to acknowledge the CERN LHC collimation team and Luca Gentini for their helpful collaboration.

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