1 Introduction. slhc Project Note Slim elliptical cavity at 800 MHz for local crab crossing

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1 slhc Project Note Slim elliptical cavity at 800 MHz for local crab crossing Luca Ficcadenti, Joachim Tuckmantel / BE-RF Keywords: crab cavity, crab crossing, LHC upgrade, LHC-HL, deflector cavity. Summary A slim highly eccentric elliptical Crab cavity with vertical deflection at 800 MHz, compatible to beam line distances everywhere in the LHC ring, was designed. It is a good fall-back solution in case of problems with new compact 400 MHz designs. Simulated RF characteristics of the deflecting mode, HOM spectra and damping, tuning and multipacting effects are presented. First the most simple HOM coupling system from a point of view of geometry, machining and cleaning was investigated. The rejection of the working mode of such coupler depend on very tight mechanical tollerances. To overcome this issue a notch filter was added to the design. Results of both cases will be presented. 1 Introduction slhc-project-note /09/2011 R. Palmer [1] first proposed the crab crossing scheme in 1988 as an idea to enable quasi headon collisions with a crossing angle in linear colliders. This scheme used transverse deflecting cavities, the cavities being phased such that head and tail of the bunch are deflected in opposite directions, causing a tilt of the bunch. Such cavities are called crab cavities. In 1989 K. Oide and K. Yokoya [2] proposed to apply this scheme also for circular colliders. The proposed LHC upgrade includes the installation of crab cavities to increase the luminosity at the interaction points [3, 4, 5, 6, 7]. In the global scheme [3] the crab system might be installed anywhere in the LHC. The most practical location would be in the IP4 region where the distance between the counter-rotating beams is increased with doglegs to 420 mm to house the 400 MHz main RF cavities. This is not the case for the local scheme [4] [5], where the crab cavities have to be installed close and on both sides of the detectors in IR1 and IR5 where there is a smaller beam spacing of 194 mm only. Allowing for the width of the other beam pipe and a small margin for the necessary metal sheets, the cavity cannot exceed 290 mm in width. To obtain sufficient linearity of the deflection along the LHC bunches the highest usable frequency is 800 MHz; presently 400 MHz is preferred, if feasible. The classical elliptical shape of superconducting (SC) cavities has been established over decades and is/was used in many machines already, proving its reliability. New designs of compact cavities at 400 MHz are now under study [4, 5] which fulfill the local dimension constraint but deviate from the classical proven shape and hence carry the risk of unforeseen problems. Therefore we have examined the realization of a modified elliptical cavity, following more closely the estabilished design principles, with consequently far lower risk. Even at IP4 a classical SC cavity with circular axial symmetry at 400 MHz does not fit. For this reasons several cavities at 800 MHz, fitting at IP4, have been designed [6] [7]. But these cavities are still too large to fit for the local option. Therefore we have studied the possibility of an oblong cavity at 800 MHz matching the local constraints anywhere in the LHC. This is an internal CERN publication and does not necessarily reflect the views of the CERN management. 1

2 In the present paper we will demonstrate an 800 MHz oblong cavity following the estabilished design rules for SC cavities that fits anywhere in the LHC ring. It would allow LHC crab operation in the preferred local option and represents a valid fallback solution if the 400 MHz compact designs encounter problems that cannot be solved in the foreseen time scale. 2 The Basic Concept Two dipole-mode polarizations exist in an axisymmetric deflecting cavities and both modes have the same frequency. When shrinking the cavity in one direction the mode deflecting in this plane shifts up in frequency; this shift essentially cannot be compensated in increasing the perpendicular dimension. In contrast to this the frequency of the mode deflecting perpendicular to the shrinking direction can be recovered by increasing the other cavity dimension. Therefore it is possible to design a cavity with vertical deflection that is as slim as necessary in the required horizontal direction while its vertical dimension is increased to keep the frequency at 800 MHz. The prize to pay is a lower pr{qq K of the working mode. But practically the field limitation is E peak and B peak while a lower pr{qq K can be tolerated with correspondingly inceased liquid helium (LHe) consumption. Since only very few crab cavities are required, this is not a principal obstacle. We start from a cylindrical cavity of 800 MHz, deforming the transverse dimensions to maintain the same resonant frequency by decreasing the horizontal dimension and increasing the vertical dimension. Fig. 1 shows the LHC pipe dimensions and separation up to scale and the process of deformation, finally obtaining an oblong cavity at 800 MHz consistent with the geometric constraints of the local scheme. Figure 1: Schematic of the LHC beam pipe separation (in the LHC beam lines) and the cavity deformation steps. A pure pillbox cavity which has the first deflecting resonant mode at 800 MHz has a radius of about mm with an pr{qq K of no more than 32 Ω. In Fig. 2 the pr{qq K of a horizontally squashed pillbox was calculated for different scale values of the horizontal semiaxis. A horizontal scale factor of approximately 0.6 corresponds to a cavity that meets the horizontal space limitations imposed by the machine; then for a vertical scale factor of approximately 1.5 the TM 110 -like mode is resonant at 800 MHz. This deformation separates the frequencies of the two polarities of dipole modes, which otherwise degenerate in a perfectly circular cavity. 2

3 Figure 2: pr{qq K for a pillbox without beam-pipes at 800 MHz for different horizontal scale factors. In all cavities, the pr{qq of the working mode should be maximized, e.g. by using a small beam pipe. In many cases it is further enhanced by making the cell shape re-entrant. In the superconducting case, the cavity cleaning by (high pressure) ultra-pure water is an essential fabrication step and a re-entrant cavity geometry is an obstacle to the water flow. Such a geometry also increases the pr{qq and with it the beam impedance of undesired other order modes (HOM). With the extremely low surface resistance (R s ) of superconducting cavities, even after taking into account the refrigerator efficiency, the dissipated power is no longer an overriding issue. Superconducting cavities thus can have larger beam holes and renounce on re-entrant geometry. The penality, a lower pr{qq for the working mode, is not serious. However the reduced pr{qq for HOM s and the accompanying reduced beam-cavity interaction pay a big dividend in a high current machine as LHC. 3 Slim CRAB Cavity parameters A 3D sketch of the slim elliptical crab cavity design is shown in Fig. 3. A long series of optimization simulations was carried out before reaching the final design. The final cavity design includes the working mode (WM) feed power coupler and the damping coupler optimization for all the other order modes (OOM). In chapter 4 we discuss the optimization of the transverse and longitudinal profiles of the cavity in terms of peak fields and pr{qq K and study the resonance frequency of the working mode as a function of local geometry parameter. In chapter 5 we analyze all the other order modes and the ways to mitigate their influence. As will be explained in detail later, a single coaxial coupler is sufficient to attenuate both the lower order and higher order modes. However, the coupling to the WM is not sufficiently low so that a coaxial stub filter was included in the design. In Table 1 one can see the values of the principal parameter as calculated with both MicroWave Studio [8] and HFSS [9] high frequency simulator codes. The agreement between the two simulators is excellent. The geometrical dimensions have been chosen after a careful process of optimization with main requirements respecting the dimensions suited to the local scheme while optimizing the 3

4 Figure 3: Perspective view of the cavity with couplers encircled (left); View in beam direction of the cavity with spatial constraints highlighted (right). surface peak fields for the same kick voltage V K as low as possible. In Appendix B it is possible to find all the instructions to draw the entire cavity in a CAD program for a mechanical drawing or in a simulation code for future developments. Table 1: Slim crab cavity parameters Input parameter Unit Cavity transverse dimensions ˆ 729 mm Cell length mm Beam pipe radius 56 mm Iris curvature radius 25 mm Cell rounding radius 90 mm WM (resulting) parameter HFSS CST-MWS frequency MHz pr{qq K Ω E K MV/m B K mt Generator frequency at MHz. f ă 1%. 4 Geometrical sensitivity studies The crab cavity cell design aims to achieve a high transverse pr{qq K for the operating mode, in order to provide the required deflecting voltage with a minimum of LHe consumption and the surface peak fields as low as possible, while meeting the LHC aperture requirements. In this chapter the sensitivity study of the four principal RF performance parameters pr{qq K, E peak, B peak and f res is presented for four different (small) perturbations of cavity shape parameters: pipe-length, pipe radius, cell rounding radius and wall inclination. Fig. 4 shows the effect 4

5 Figure 4: Lateral sketch of a quarter cavity in the plane spanned by beam-axis and long semiaxis. In each sketch the name of the varied parameter is underlined. Variation of cell length (top-left), pipe radius (top-right:), cell rounding radius (bottom-left), wall inclination (bottom-right). of the variation of these parameters of the sidewise projection (containing the long semiaxis) that allow defining uniquely this cavity projection while always keeping both semiaxis constant. Each was varied individually, with the others kept constant each time. In Figs. 5 to 8 the behaviour of the performance parameters is shown under variation of the geometrical parameters as mentioned above. R /Q [Ω] L [mm] AXIS Peak surface E (@2.5MV) [MV/m] L [mm] AXIS Peak surface B (@2.5MV) [mt] L [mm] AXIS Figure 5: Variation of L AXIS ; left: R K {Q; middle: E peak ; right: B peak. From Fig. 5 one concludes that the best choice concerning the parameter ( L AXIS ), that controls the cavity length on axis, is 0 mm while pr{qq K increases with increasing L AXIS. However, reducing the peak fields is more important so that the optimum remains at L AXIS 0 mm. To respect the mechanical beam aperture the pipe radius cannot be smaller than the LHC pipe radius of 42 mm. For reasons explained in more detail later, the inner radius of the coaxial 5

6 R /Q [Ω] Pipe radius [mm] Peak surface E (@2.5MV) [MV/m] Pipe radius [mm] Peak surface B (@2.5MV) [mt] Pipe radius [mm] Figure 6: Variation of the pipe radius; left: R K {Q; middle: E peak ; right: B peak. R /Q [Ω] Cell rounding radius [mm] Peak surface E (@2.5MV) [MV/m] Cell rounding radius [mm] Peak surface B (@2.5MV) [mt] Cell rounding radius [mm] Figure 7: Variation of the cell rounding radius; left: R K {Q; middle: E peak ; right: B peak. R /Q [Ω] Wall degrees [deg] Peak surface E (@2.5MV) [MV/m] Wall degrees [deg] Peak surface B (@2.5MV) [mt] Wall degrees [deg] Figure 8: Variation of the Wall inclination; left: R K {Q; middle: E peak ; right: B peak. 6

7 pipe hosting the OOM coupler has been chosen equal to the LHC pipe radius and for the outer radius 56 mm is used. To obtain the necessary coupling with the working mode and to have longitudinal symmetry of the cavity, the pipe radius on the working mode coupler side was also chosen to be 56 mm. The rounding radius has been independently increased to its maximum value, given its lower impact on the electromagnetic parameters, to mitigate any signs of multipacting electron avalanche (further explained in chapter 7). Regarding the slope of the wall, zero degree is used for all subsequent studies, but nothing precludes, in a future upgrade, to use e.g. 3 for the wall slope to increase the electromagnetic performance (Fig. 8) and to have a better ability to clean the cavity. On the other hand a too large wall inclination might cause problems for the operational tuning due to reduction of the longitudinal elastic range of the cavity. 4.1 Frequency tuning With the exception of the cell rounding radius we have studied the small perturbation effect within the linearity range on the resonant frequency of the cavity around the chosen operating point. The geometrical perturbations are shown in Fig. 9 while Table 2 shows the linear sensitivity coefficients in terms of frequency, pr{qq K, as well as electric and magnetic peak fields on the cavity surface. The resonant frequency sensitivity as function of the axial cell length is highlighted in Table 2 since it is foreseen as operational WM tuning by elastic deformation as also done for the accelerating superconducting cavities. Around our working point we have therefore a tuning sensitivity of 0.3 MHz/mm. Figure 9: Side view of the cavity with three different geometry perturbations. Table 2: Geometric sensitivity coefficients Geometry parameter Frequency (R/Q) E B Cell length a 0.3 MHz/mm -0.1 Ω/mm 0.04 (MV/m)/mm -3.6 mt/mm Pipe radius MHz/mm -0.1 Ω/mm 0.5 (MV/m)/mm 0 mt/mm b Wall degrees 11.5 MHz/deg -0.8 Ω/deg 1.5 (MV/m)/deg 14.8 mt/deg a Pratical frequency tuning parameter. b Very low value. 7

8 5 Lower Order and Higher Order Mode Properties In any cavity there exist a large variety of modes but only one of them the working mode (WM) is driven by an RF source for accelerating or deflecting purposes. In accelerating cavities generally the WM is the lowest monopole mode and all other modes are then called Higher Order Modes (HOMs). For a usual deflecting cavity the lowest, most efficient, deflecting mode has a higher frequency than the lowest accelerating mode. Then this lowest monopole is called Lower Order Mode (LOM). In a symmetric cavity (e.g. circular, quadratic) deflecting modes degenerate into two polarities with the same frequency. In this case the unwanted sister of the working mode with the same frequency is called Same Order Mode (SOM). One can then group all modes except the WM as Other Order Modes (OOMs). All OOMs can be excited by the beam, then possibly causing instabilities and leaking power from the beam. Therefore in high current machines such as the LHC and especially for superconducting cavities with intrinsically very high Q 0 for all modes, these modes have to be damped artificially. This poses the problem of providing a very good attenuation for the OOMs, but avoiding any damping of the WM. Separation can be realized by frequency, by field pattern or a combination of both. In our case of an oblong cavity we have no SOM since the mode degeneration does not exist here but there are a LOM monopole below the WM and a variety of HOMs. These constraints impose the strategy for damping the OOMs without damping the WM. When a beam passes through the cavity, these modes get excited due to the interaction with the charged particles, generating wake fields that act upon the trailing beam. The beam-induced voltage and power depends on the beam intensity, the coupling constant pr{qq and the natural decay time of each mode when no artificial damping is applied. The natural energy decay is given by the filling time constant τ Q n {ω n where ω n is the resonant frequency of mode n and Q n is its intrinsic quality factor. The modes excited by the leading particles may affect other particles in the same bunch or particles in bunches trailing behind 1, depending on the decay times of the wake fields. Short wake fields mostly act on the same bunch leading to single bunch effects, which depend more on the pr{qq as the decay times are relatively small. Multiple bunch effects need larger decay times and act on the trailing bunches as well. The longitudinal pr{qq for modes having an electric component along the beam axis and for particles with the speed of light c is given by (circuit Ω convention): ˆR Q ˇ ş`8 8 E zpx 0, y 0, zqe iωz ˇ c dz 2 ωu with the cavity stored energy U. The pr{qq K for modes with deflecting component can be calculated using two methods. First, using the definition of this deflecting force, we have: ˆR Q K ˇ ş`8 8 rýñ E K px 0, y 0, zq ` ip v ˆ ÝÑ B px 0, y 0, zqq K se iωz ˇ c dzˇ2. (2) 2 ωu Exploiting the Panofsky Wenzel Theorem (see Ref. [10]), for dipole modes with E z 9x close to the cavity axis this is identical to: ˆR Q K ˇ ş`8 1 including the exciting bunch itself in circular machines. 8 E zpx x 0, y 0, zqe ιωz ˇ c dz 2 pkx 0 q 2 ωu ˇ2 ˇ2 (1) (3) 8

9 where k 2π{λ, λ is the wave length and x 0 is the offset from the beam axis. For dipole modes E z is about proportional to the x direction then pr{qq K is independent of the choice of x 0. pr{qq and Q n of each mode are properties of the cavity and in the case of Q n also its surface quality. Superconducting cavities have intrisically very high Q n. Therefore additional damping has to be supplied by HOM couplers with low Q ext. The total (loaded) Q L,n is then 1{Q L,n 1{Q n`1{q ext,n. Since Q n " Q ext,n it can be approximated by Q L,n Q ext,n. The corresponding filling time τ (the energy decay time) is then τ n Q ext,n {ω n, hence the field decay time τ F,n is given by τ n 2 Q ext,n {ω n The resultant longitudinal and transverse impedances (on resonance) are given by: ˆR Z,n Q Z K,n ω ˆR c Q Q ext,n rωs (4) Q ext,n r Ω K m s (5) More details can be found in the bibliography. It is important to determine pr{qq for all parasitic modes present in the cavity to determine the required damping. The longitudinal impedances are due to the longitudinal electric field component E z, the transverse impedances are (over the Panofski-Wenzel formula) due to the gradient of V z, the field on axis being zero for ideal multipole modes. These impedances were determinated for the 800 MHz slim cavity designs using CST Microwave Studio [8] and HFSS simulation codes [9]. Due to a strong squashing ratio and symmetry breaking couplers, it is not pratical to categorize the OOMs as monopole, dipole, etc. as in a conventional cavity with circular symmetry. The OOMs are identified by the main field orientation in the cavity along the beam axis, and are grouped in 5 main types of modes as summarized in Table 3 with field definitions shown in Fig. 10. Figure 10: (From left to right) pure accelerating, horizontal and vertical deflecting and not excited modes in a device having the two main transverse symmetries. (The missing mode s type in the figure are local modes in the coupler). The deflecting modes with the field E x, H y give a net deflection in the x direction (horizontal), the other polarity having the fields E y, H x with a net deflection in the y direction (vertical, as the working mode). The longitudinal magnetic field present does not contribute to excitation of any OOMs, the modes having E z on axis are the accelerating modes. As will be explained in chapters 5.2 and 5.3 a single damping coupler at 45 with respect to the vertical direction is 9

10 Table 3: Mode Types Essential field on beam axis Type of mode Chosen marker E z Accelerating E x, H y H-Deflecting E y, H x V-Deflecting H z Not excited Negligible local in coupler sufficient to damp both vertical and horizontal transverse HOMs and LOMs. However, since this design has not enough rejection for the working mode, it has to be foreseen with an additional notch filter F resonant [MHz] Mode number Figure 11: Resonant frequency of all cavity modes up to 2.0 GHz, the first green square marker is the WM. Since the structure has no principal symmetries, modes are no longer purely deflecting or accelerating. This means that those modes that are derived from purely accelerating modes may also have deflecting components and the deflecting modes may also have a small accelerating component. We refer to these impedences as parasitic impedances. For the 800 MHz slim cavity design including the HOM coupler all the modes up to 2.0 GHz, the longitudinal LHC pipe cut-off frequency, were calculated. The resonant frequencies of the cavity modes are plotted in Fig. 11. The meaning of the markers chosen is defined in Table 3. All black stars represent modes concentrated in the coupler(s) and not really beam-excited modes since they have most of the electromagnetic energy confined inside the input or damping couplers or at the outer equatorial volume of the cell, presenting very low axis fields. One should note the frequency separation between the working mode, resonating at 800 MHz, and the adjacents modes. The first dangerous mode is the acceleratig mode resonating 150 MHz below the working mode, the next mode is a deflecting mode resonating 50 MHz above the working mode. As we shall see in the subsequent chapters the latter frequency separation is enought to couple sufficently the OOMs while still causing a large damping for the WM. 10

11 The pr{qqs for all modes were computed for the final geometry including the 45 OOM coupler that breaks the symmetries of the device, so we have computed the shunt impedances for both transverse deflections and the longitudinal acceleration for all modes, to show how much the OOM asymmetry perturbs the purity of the originally accelerating, vertical deflecting and horizontal deflecting modes (R/Q) [Ohm] Mode frequency [MHz] Figure 12: Longitudinal pr{qq of all modes up to 2.0 GHz. included. Values below 10 2 were not Calculations were generally done for closed volumes, hence the coupler ports were assumed to be closed and no energy can leak out there. Then in the simulation one special class of modes appears which we have called local modes. Their field is essentially non-zero only inside the small volume of one of the couplers with hence low stored energy. Therefore their numerical (R/Q), proportional to the inverse of the stored energy, can become comparable to other classical modes. But since these local modes are sitting only in a coupler, in a real cavity with open ports their stored energy will readily leak out of the coupler corresponding to a very low Qext so that either no resonant mode would exist then or at least their impedances seen by the beam stays very small. Therefore these local modes were considered as not dangerous. In figures 12, 13 and 14 we display respectively the values of longitudinal pr{qq, horizontal pr{qq K and vertical pr{qq K calculated for all modes up to 2 GHz. We have not included the very low parasitic impedences values ((R/Q) below 10 2 Ω) of the modes that are not so much affected by the OOM coupler asymmetry and thus have a significant strength only along the field direction of the pure mode from which they derive. In Fig. 12 the longitudinal pr{qq of all modes are summarized. As expected, the modes having the bigger longitudinal impedances are the accelerating modes (blue circle), the most dangerous mode is the first accelerating mode resonating at 640 MHz with a longitudinal pr{qq of about 65 Ω. We see that due to the coupler asymmetries several deflecting modes also manifest a non-negligible longituinal impedance. In this figure the working mode at 800 MHz does not appear, it has a longitudinal pr{qq lower than 10 2 Ω thus, despite the OOM coupler breaking the symmetry, the working mode keeps excellent purity. In Figures 13 and 14 horizontal and vertical pr{qq K are shown. All dipole modes other than the working mode are higher order modes, the working mode has the highest vertical pr{qq K 11

12 10 2 Horizontal (R/Q) [Ohm] Mode frequency [MHz] Figure 13: Horizontal pr{qq K of all modes up to 2.0 GHz. Values below 10 2 were not included Vertical (R/Q) [Ohm] Mode frequency [MHz] Figure 14: Vertical pr{qq K of all modes up to 2.0 GHz. Values below 10 2 were not included. 12

13 and does not present any appreciable horizontal pr{qq K. Some accelerating modes have, albeit it remains low, a transverse pr{qq K component. To keep the beam stable effective wakefield damping is crucial. All LOMs and HOMs need to be damped to meet the beam dynamics requirements, which are Z,n ă 2.4 MΩ for the accelerating modes and Z K,n ă 1.5 MΩ{m for the dipole HOMs [11]. In LHC there are two interaction points (ATLAS and CMS) where the beam has to be crabbed and each one needs one station to start the crabbing oscillation and one to stop it again after the detector, hence four stations in the ring. We assume that each station needs only one cavity, hence the impedance per cavity has to be one quarter of the total impedance. Should it reveal that one cavity cannot deliver the required kick strength, the impedance per cavity has to be lowered correspondingly. The goal of the damping coupler design is to lower the Q ext of each mode so that the impedance will not exceed these limits. Fig. 3 shows the proposed damping coupler positioning, effective for all modes. 5.1 The damping concept A special design is needed to have strong coupling (low Q ext ) to all parasitic modes but essentially not to couple to the working mode (very high Q ext ). We follow the method applied to the KEK-B crab cavity [7] [12]. All circular symmetric modes TE 01 and TM 01 can couple to the TEM mode in the coaxial line which can propagate at any frequency, while all multipole modes are attenuated up to the corresponding cut-off frequency. For dipole modes the cut-off frequency is there where the circumference of the line corresponds to about a free space wave length. The dimensional requirements of the beam pipe in LHC at the crab cavity locations allow to dimension the coaxial line so that the working (dipole) mode is still under cut-off, the next higher (parasitic) dipole mode is only slightly under cut-off while all other (higher frequency) dipole modes are propagating. The rejection of the working dipole mode relies on the perfect symmetry of the coaxial line with respect to the working mode field. A (small) eccentricity will then appear as a weak monopole component that propagates at any frequency along this line. In section 5.5 the effect of this asymmetry is studied and even exploited. The coaxial line in our case is shorter at least five times than the one of the KEK-B crab cavity and it is hoped that in this case sufficiently alignment precision can be realized and kept. The OOM power is coupled out by a perpendicular standard coaxial antenna rotated by 45 o against the horizontal and at the same time against the vertical axis so that both polarities are coupled with the same strength. The main coaxial line is closed forming a stub at a well tuned position to optimize damping to all OOMs (see Fig. 15, left). This WM rejection scheme relies on the attenuation along the main coaxial line. As shown later, it permits to get enough OOM damping for all dangerous modes. However, with a mechanically perfectly centered inner conductor the Q ext of the WM is 10 7 only, allowing too much WM power leaking into the damping system threatening to overload components and charging the WM transmitter. The couplers break the up-down and right-left symmetry of the structure, hence it is not surprising that the electrical central axis is not precisely identical to the mechanical axis of the bare cavity body. We have tried to find this electrical central axis in displacing the coupler transversely by a small amount. As shown later, in fact we find a position where the coupling to the WM decreases sensitively without too much loss in coupling for the OOMs. However, the position of the inner conductor is then quite close to the outer conductor in one direction and precision problems might appear. Therefore we have examined a second scheme where a choke for the WM is introduced and as output coupling resistive material in the coaxial line, but practically far away from the cavity for thermal reasons, was studied. The choke has its entry close to the cavity and is bent away from the cavity so that the WM is clipped off close to the cavity. Should the magnitude of the internal fields of the choke cause problems, one could 13

14 also invert the direction of the choke so that its entry is on the farther end, with in the coaxial line more attenuated WM fields, the attenuating effect being the same. This solution has the problem that the coaxial line must be long (for thermal reasons) so that alinement problems as in the KEK cavity might appear. In this case one could replace the resistive material by a then much closer lateral antenna, a hybrid between both schemes. However, we did not study the details of this hybrid solution. 5.2 Coupling to the LOM The data in fig. 12 show several monopole modes (blue circles) with a high pr{qq that have to be well damped. The mode with the highest value is the first accelerating (fundamental) mode. The damping coupler is designed to maximize the coupling to this mode. Fig. 15 shows the configuration of the damping couplers for both the investigated solutions to damp the fundamental mode. It contains also the values of Q ext, the resonant frequency of the fundamental mode and the qualitative trend of the surface magnetic field amplitude in false colors. The azimuthal orientation of the damping coupler is not relevant for monopole modes. In the lateral antenna case, the most important geometric parameter for tuning the effective damping of the accelerating mode is the length of the choked coaxial pipe. Figure 15: Damping scheme for the first accelerating monopolar mode. Surface magnetic field strength is shown colour coded. It was possible to damp all LOMs and all monopole HOMs using a simple coaxial to coaxial coupler as shown in Fig. 15 (left). All accelerating monopole modes are propagating without cut off along the coaxial to coaxial coupler with different coupling factors strongly depending on the longitudinal position of the intersection between the coaxial line and the outer length of the coaxial pipe. In the notch filter case the OOM power is not taken out by a lateral antenna but by resistive absorbers down the main coaxial line. Calculations were done without representing the absorber but assuming that the wave at the end of the represented section is fully absorbed (Fig. 15 (right)). 5.3 Coupling to the horizontal and vertical HOMs In Figure 16 we compare the coupling to the WM with a high Q ext (rejection) and to the first dangerous higher order dipole mode with a low Q ext (good coupling) for the antenna scheme. Fig. 17 shows the same for the notch filter scheme. 14

15 Figure 16: Left: working mode interacting with the OOM coupler - essentially no coupled; Right: damping of the nearest dipolar mode - sufficiently good coupling. With 50 mm and 56 mm for the inner and outer radii respectively of the coaxial pipe, the OOM coupler is also able to attenuate sufficiently the first horizontal dipole mode at 850 MHz. To assure the maximum rejection of the working mode while keeping the first parasitic dipole mode sufficiently damped, the best compromise for the chosen cut-off frequency is approximately 900 MHz, as schematically shown in Fig. 18. Figure 17: Left: working mode interacting with the OOM coupler - essentially no coupled; Right: damping of the nearest dipolar mode - sufficiently good coupling. Without using the notch filter the maximum rejection of the WM in terms of Q ext is 10 7, as the left part of the Fig. 16 shows. This Q ext is much too low and has to be improved by at least two orders of magnitude. One way would be to displace the inner conductor of the main coaxial line; as studied in Sec. 5.5, Q ext for the WM can be made sufficiently high with this method. Another possibility is to introduce the notch filter as shown in Fig. 17. The designed notch filter is tuned to be λ g {4 long at 800 MHz and acts also on the other modes of the coaxial pipe. The short circuit at the right end is transformed into an open circuit in a lambda quarter resonator, producing a theoretically infinite impedance at the intersection with the pipe (stop filter). This notch filter increases the rejection of the WM by a factor of at least ; the at the same time happening increase by about 30% of Q ext for the most critical higher order dipole mode resonating at 850 Mhz presents not a real problem as shown in the following paragraphs. 15

16 Figure 18: Trasmission coefficient of TEM and TE 11 modes of the coaxial pipe. 5.4 The achieved damping In the following we report on the damping results as calculated with MicroWave Studio [8]. To fullfil the beam impedance constraints in the LHC ring for each HOM, we checked the right coupler dimensions to optimize damping for all parasitic modes. The final damping results are shown in Fig. 19, where one can see the Q ext values obtained with both versions of the OOM couplers for all modes up to 2 GHz WM (Shifted coaxial line) WM (Centered coaxial line) 10 8 Q ext 10 6 Q ext Mode frequency [MHz] Mode frequency [MHz] Figure 19: External quality factor Q ext as calculated from MicroWave Studio. The local modes were not considered. Left: The 45 antenna design, in which the hollow markers represent the case with the coaxial pipe inner conductor shifted by 2.2 mm (see the Fig. 24). The rejection is much better than obtained for the centered inner conductor. Right: The notch filter design with very high rejection of the WM, larger than (out of scale). On the left are the Q ext for the antenna coupler without using a notch filter. The working mode resonating at 800 MHz shows a Q ext,oom slightly larger than At the nominal V K 2.5MV the leakage power would be «18 kw, cleary too high. This rejection Q ext should be 16

17 increased to «10 9 which is possible with a slight shift of the inner conductor transverse position as the unfilled green marker show. In Sec. 5.5 this method to improve the rejection of the WM without using a notch filter for all the parasitic modes will be investigated. On the right of the figure there are the equivalent calculations made on the cavity with the tuned notch filter. The very high Q ext of the WM is out of scale, the results shows that the notch intruduction does not seriously perturb the coupling with the other dangerouse modes. Multiplying the computed Q ext by the computed longitudinal or transverse pr{qq using the formulas 4 and 5 we obtained the following results for the beam impedence, shown in figures 20, 21 and 22 for the case of the antenna coupler without notch filter. The dashed lines are the impedance limit for the LHC crab cavities for the worst case beam energy, hence covering the worst case. With the proposed coupling scheme all the unwanted modes are damped to stay below these limits MΩ 10 4 Z [Ohm] Mode frequency [MHz] Figure 20: Lateral antenna coupler: longitudinal beam impedance Z of all modes below 2.0 GHz. The unfilled markers represent the case with the coaxial pipe inner conductor shifted by p 1, 2q mm, see Fig. 24. Finally in Fig. 23 the equivalent results for the coaxial coupler with the notch filter are shown. As the plot shows some HOMs (especially for those above 1.3 GHz) are affected by the presence of the notch filter. However,we have to consider that the threshold of acceptable impedance increases exponentially with frequency above its minimum, represented by the dashed line. So, for the time being we conclude that further studies are needed to determine if these impedances can be really dangerous for the beam. The design with notch filter was born to correct the relatively low rejection of the antenna coupler for the WM. A careful analysis of this design requires a more detailed study on individual dangerous HOMs. Part of the work is already started, it will be soon published a note dedicated to the solution already presented here. 17

18 MΩ/m Horizontal Z [Ohm/m] Mode frequency [MHz] Figure 21: Lateral antenna coupler: horizontal beam impedance of all modes below 2.0 GHz MΩ/m Vertical Z [Ohm/m] Mode frequency [MHz] Figure 22: Lateral antenna coupler: vertical beam impedance of all modes below 2.0 GHz. 18

19 Z [Ohm/m], Z [Ohm] MΩ MΩ/m Mode frequency [MHz] Figure 23: Notch filter coupler: longitudinal and vertical beam impedance of all modes below 2.0 GHz. 5.5 OOM coupler working mode s rejection quality The main aim was to design an efficient damping system using a geometry as simple as possible. Figure 24: Effects of a transverse displacement of the inner conductor of the damping coupler on Q ext of the working mode. There have been efforts to find a configuration with a single HOMs coupler such that it could damp all dangerous HOMs and reject the WM at the same time. To prevent working mode power leakage through the OOM coupler it is very important to maintain the corresponding Q ext at a level largely above the Q ext of the feed power coupler. Unfortunately, as shown in the previous sections and in particular in Fig.s 19 and 20, although a geometry able to sufficiently dampen the HOMs was identified, the highest value obtained for 19

20 the WM Q ext,oom is not sufficient. The effect of a transverse displacement of the inner conductor of the coaxial line on the working mode and on the HOMs Q ext has been studied, observing an improvement of the rejection of the WM while not degrading the OOM damping achieved. Results are shown in Fig. 24. The transverse position of the inner pipe in the directions of the coaxial antenna ( u) and perpendicular to it ( v) have been varied. One can see that in contrast to a perfectly symmetric structure a small change in the position of the coaxial inner pipe does not always lead to decrease of Q ext. Transverse shifts of up to 4 mm have been explored. Fig. 24 shows the possibility of increasing the WM Q ext,oom up to acceptable value of about Unluckly, if this geometric modification produces good results with regard to the rejection of the WM, as shown in Fig. 20 the inevitable decoupling with some HOMs leads to an non-acceptable increase of the corresponding beam impedance. Fig. 20 shows the modified impedance of the first three accelerating modes (hollow blue markers). 5.6 Quadrupole and higher axial symmetry modes Till now we have only considered monopole and dipole modes, neglecting modes of higher axial symmetry which do not pass the coaxial line of the OOM coupler. However, deflecting forces of a quadrupole mode scale with the 1 st power, deflection strength with the 2 nd power of the particle distance from the ideal beam axis. For even higher symmetry modes these powers increase by 1 for each step in symmetry, e.g. for a septupole mode excitation scales with the 2 nd power and deflection with the 3 rd power. Then a beam transversely as small as the LHC beam gets much less excitation and deflection by these modes, corresponding to a lower impedance. And finally these modes couple at least weakly to the WM power coupler so that they do not have extremely high Q ext. 6 Working mode power coupler studies An ideal crab cavity mode has no longitudinal E-field on axis, hence no coupling to the beam. Then the most efficient feed power coupler would have a Q ext,p C Q 0 and one could operate a superconducting crab cavity with only a few Watts of RF power. However in reality the beam is never precisely on axis (and for our cavity the mechanical axis does not correspond to the electrical axis with V z =0) and hence the beam takes or gives power, forcing an operation with a much lower Q ext,p C. Allowing a certain maximum beam-excursion but taking also machine protection asking for a high Q ext,p C with slow field changes and microphonics asking for a low Q ext,p C with large band width into account, a reasonable compromise is a Q ext,p C of about To avoid too much (relative) power leaking from the WM to the OOM coupler, the corresponding coupling should stay significantly higher than Q ext,p C. A baseline design for LHC crab cavity is shown in Fig. 3. It incorporates the OOM coupler discussed in previous sections as well as the feed power coupler. The feed power coupler consists of a simple coaxial antenna, as also used e.g. for the LHC main accelerating RF cavities, placed on the cut-off tube opposite to the OOM coupler. To obtain a value of not more than 10 6 for the Q ext,p C of the working mode several simulations were carried out. The desired coupling could be obtained with an enlargement at the end of the inner conductor, making the probe mushroomshaped. The optimization process led us to choose a radius of the cut-off tube containing the power coupler equal to 56 mm, significantly larger than the LHC beam pipe radius. In this way the field of the working mode is less attenuated there and can therefore propagate better towards the power coupler. The geometrical dimensions of the input coupler are presented in Appendix B. The most important geometrical parameters to tune and optimize the coupler are the distance from the cavity end and the length of the antenna probe. 20

21 Figure 25: Coupling scheme for the working mode feed coupler. The details of the mushroomshaped antenna is shown magnified. In Fig. 25 one can see the longitudinal section of the latter device and the electric field trend on the surface when the cavity is fed from this feed power coupler. Fig. 26 shows the values of Q ext as obtained for different antenna positions. The probe position is set to 0 mm when the end of the probe coincides with the cut-off radius. One can see that a wide range of coupling strengths can be obtained, including the design value of about x Q ext Input power probe lenght [mm] Figure 26: Feed power coupler Q ext for the working mode at different probe positions. 0 mm corresponds to the end of the probe coinciding with the cut-off radius, still considerably larger than the 45 mm of the LHC beam pipe. 21

22 7 Multipacting analysis Multipacting (or multipactor) is an electron phenomenon where resonant electron trajectories cause an exponential growth of the number of electrons by Secondary Emission Effect, SEE (more details can be found in the bibliography). For the first superconducting cavities multipacting was a major performance limitation. Eventually it was found that in β 1 cavities multipacting can be overcome by a proper choice of an elliptic cavity geometry. But multipacting remains a permanent danger when deviating from this classical shape and is often present in ancillaries as the power or the HOM couplers. Therefore multipacting simulations in RF cavities have received a lot of attention. When such a resonance is found in simulation, one can have reasonable confidence that it exists in reality. But when no resonances were found, there is always a small doubt that one might have missed something. Simulations of multipacting in the slim crab resonator have been carried out using CST Particle Studio (CST-PS, [8]). Since CST-PS is probably the first commercial code that can simulate realistic electron multiplication, the result of this study could also be useful as code benchmarking. Before reaching valid multipacting results we had to calibrate the simulation code settings. Through the analysis of many simulations we confirmed that the emitted particles dynamics and all related effects are strongly dependent on the number of spatial points inside the cavity volume, on the initial energy of the emitted electrons, on the surface chosen to put the particle source and on the time step chosen for the tracking calculations. The well-known parameter to describe the Secondary Electron Emission (SEE) is the Secondary Electron Yield (SEY) or coefficient, which is defined as the ratio of the incident and the secondary or emitted current δpeq I s {I 0 peq, for a given incident electron energy E. As the detailed physical process is very complex, the SEE model used in CST Particle studio [14] to calculate the SEY, is a statistical model based on random numbers (Monte Carlo) for the emission angle and the kind of emission. One can find a detailed article [13] about the emission model used in CST-PS. There are three basic types of secondary electrons: the elastic reflected, the rediffused and the true secondary ones. The implemented SEE model is based on a probabilistic, mathematically self-consistent model developed by Furman [13] and includes the three kinds of secondary electron mentioned above. CST-PS has a Particle in Cell (PIC) solver and a Particle Tracking (PT) solver that allow the simulation of charged particle interaction with RF electromagnetic fields to be carried out. For multipacting studies, due to the large number of particles involved, we used the PT solver which excludes space charge effects and hence is computationally less demanding. Multipacting is a very complex phenomenon. To predict the onset of an electron avalanche resonating with the RF cavity fields, one must take a great care in choosing the settings of the simulation code. An important requirement for a reliable multipacting analysis is that the electromagnetic fields are known accurately, especially close to the surface, since even small errors in the RF field may largely falsify the electron trajectory calculations. We have three diagnostic elements to identify multipacting using CST-PS code. First, we can observe the trend of the number of electrons in time, for a limited number of RF periods. In CST-PS we can obtain these diagnostic values in post processing, this is automatically given as a post processing plot at the end of the simulation. The second diagnostic consists in comparing the SEY curve used in the simulation with the averaged xseyy and the average impact energy, xe i y. This can be obtained from the collision information table at the end of the simulation. Finally it is very useful to observe the electron trajectories in order to understand and visualize the kinematics of the multipacting. 22

23 7.1 Multipacting (MP) simulation results for slim crab cavity The results of MP simulations in the slim crab cavity are shown below. The simulations were aimed to search for any signs of an electron avalanche resonant with the electromagnetic fields in the cavity volume. The simulations with the PT were applied on the geometry obtained after the optimization process described in the chapters above. PT requires the introduction of a metallic coating around the cavity, the only modification made on the initial geometry. The material chosen for all simulations described below is a perfect conductor (PEC) with the properties of secondary emission significantly worse than those found experimentally on niobium samples (see [24] for more details). Fig. 27 shows the Secondary Emission Yield for the chosen material. Figure 27: The total SEY (magenta) used in MWS-PS simulations obtained as a superposition of the elastic (red), rediffused (green) and true secondary (blue) components. In all simulations carried out and reported below the particle source was located all around the iris near the OOM damping coupler. This area is one of the most critical cavity region as well as close to the coaxial coupler with the highest field strength zone inside the cavity. In our simulations we set the seed electrons by using a fixed emission, the simplest model of the tracking solver. This allows us to select on one hand the initial mean electron energy and energy spread, on the other hand a fixed current or a fixed current density to be defined. The current value has to be obligatorily defined when one considers the space charge effect, not considered here. After setting up the particle source properties the secondary emission properties have to be defined. For our simulations we chose the Furman-Pivi secondary emission model as mentioned above; at this stage we can select the maximum number of generations and maximum number for secondaries per impact. The maximum number of secondaries per impact was set to 10. Before running a simulation it is necessary to define what RF field will be used durinig a particle tracking computation. One can choose if the fields have to be calculated in the same project or will be imported from an other project. In the latter case one could use different 23

24 meshing for the tracking solver independently from the RF field meshing. We preferred the import method enabling us to use a different meshing. The RF field amplitude (in terms of field level) and phase can be parametrized to perform parametric simulations, a series of independent simulations with a varying parameter were performed. Since essentially only the working mode at 800 MHz is excited (all OOMs are sufficiently damped) we performed multipactor simulations only for this mode. The working mode derives from the pure TM 110 in a pilbox; since it is a dipole (m=1) mode, the field-strength along one azimuthal turn around the beam axis varies (about) as cos φ. Neglecting the small asymmetry introduced by the couplers, the dynamics of particles emerging from the top half of the iris is identical to that of the particles emerging from the bottom half. For this reason it was only necessary to vary the field phase within 180 and not 360 while the emission area was set on the entire iris to cover 360. Before obtaining the results presented in subsequent sections, different simulations were performed to optimize the settings. Driven by suggestions from other expert designers ([26] [27] [28] [29] and [30]), we performed simulations in order to increase the accuracy of the results. These optimizations are described in the Appendix A. Table 5 shows the main simulator parameters selected after the optimization process for the following results. 1.3 Multipactor bands 10 6 <SEY> Impact Energy <Ei> [ev] Deflecting Voltage [MV] Figure 28: Average ăseyą and average impact energy as a function of total deflecting voltage. In this paper the averaged secondary emission yield per impact xseyy ([27], [29]) is used as an indicator of multipactor. All runs were performed using the complete Furman-Pivi secondary emission model to include elastic and inelastic reflections. This implies an inevitable spread and overlap of the multipactor band. For the slim crab cavity at 800 MHz the field level was scanned from tens of kv up to 3 MV deflecting voltage, at each field level all the RF phases and at least 50 RF cycles were simulated to obtain the parameters of possible resonant trajectory. In Fig. 28 the averaged SEY and the average impact energy are shown as functions of the deflecting voltage. These are the results calculated from simulations running at peak performance 24

25 of the computer system used. In the same figure two broad multipactor bands are visible, the first starting at low field stretching up to 400 kv with an average impact energy below 2 kev, and the second band starts above 2 MV. There are two types of trajectories that are considered for potential multipacting. The first type involves particles with trajectories resonant with the RF. These particles will impact the surface at the same locations with constant energy. If the impact energies are within the range for SEY bigger than unity the trajectories could generate an electron avalanche. Some particle tracking codes extract these events individually and determine the multipacting type in terms of order and number of impact points (order: number of RF cycles to return to the original site, points: number of sites per multipacting cycle). CST particle tracking code does not distinguish the different orders, but with a manual analysis of the 3D trajectories it is possible to determine the order of trajectories. The second type of multipacting involves particles with run-away resonances. These particles start resonant but slowly drift out of the resonant region and the RF phase. The particles impact the surface numerous times with energies that could produce larger than unity yield before leaving the right RF phase. Run-away resonances with high number of particles may cause observable multipacting activies during the RF processing. In the following we will present simulation results on different regions of the cavity that may support resonant trajectories Multipacting in the cavity region Fig. 28 shows two multipactor bands, the first at low field up to 400 kv deflecting voltage and the second at high field above 2 MV. Run-away resonant trajectories were found at several locations in the cavity and within the low field band as shown in four time instances in Fig. 29. In the same figure the number of particle versus time up to more than thousand RF periods are shown. At low field, in the range of the deflecting voltage between 50 kv and 400 kv, three different effects were found. Particles starting to resonate with the RF in the equatorial area could produce run-away resonant trajectories in the direction of the small radius equatorial area, extinguished in less than 400 RF periods, or could produce longer run-away resonant trajectories having a more stable position around the large radius equatorial area; also this resonance disappears after 1 µs as Fig. 29 shows. The third effect visible in the illustration is the so-called ponderomotive or Gaponov-Miller effect [32], which tends to push charged particles towards regions of low field amplitude; this can have a substantial effect on the multipactor regions. Some trajectories at 100 kv deflecting voltage are plotted in Fig. 29 as example. Other low voltage trajectories in the low field band have very similar behaviour. Stable resonant trajectories were found around the large radius equatorial area within the high field band, for example in Fig. 30 it is possible to see some resonant trajectories and the number of particle versus time after 10 RF periods. As the trajectories show, after the first ten periods an electron avalanche is established indicating the onset of multipactor. Trajectories also show a predominance of first order multipacting with two stable points crossing the cavity equator, often called Ω MP. Here one should note that the stable resonant trajectories within the cavity volume have been found only at voltages above the working voltage and using a material with a much worse SEY than that measured on niobium samples [24]. One-point multipacting was common in cavities of older design, such as pill-box or muffintin cavities. This problem was eliminated with the invention of the spherical/elliptical cavity shape [21]. However, there were indications that two-point multipacting still occurred on some occasions. First observations of such occurrences along the cavity equator were made at CERN with LEP cavities ([22]). In many cases, such a multipacting barrier can be surmounted by RF processing [23]. This is done by allowing multipacting to progress for several minutes, while 25

26 Figure 29: Electron trajectories at 4 different time and at low deflecting voltage (100 kv), and the number of particle along time. Figure 30: Left: electron trajectories after 10 RF periods at high deflecting voltage (2.8 MV). Right: the number of particle along time. 26

27 slowly raising the RF power. In general, once multipacting barriers have been processed, they do not reappear easily, provided the cavity is kept under vacuum and cold MP in OOM coupler Among various high power RF components used to drive RF accelerators, coaxial power couplers are known to be prone to multipacting, and several schemes have been proposed to control and possibly suppress it, complicating its technical design and eventually lowering its reliability [33], [26]. Figure 31: Left: electron trajectories after 15 RF periods at low deflecting voltage (100 kv). Right: the number of particle along time. Stable resonant trajectories were found at the location close to the end of the inner coaxial pipe near the iris in the OOM side, at low field level as shown in Fig. 31. No resonant trajectories were identified in the OOM coupler area and in the coaxial region at high deflecting voltage within the second multipactor band (see Fig. 28). The MP bands in smilar coaxial couplers have been observed but often it can be processed away by RF processing. We would not expect the MP in the coaxial damping coupler to present a truly significant barrier. 8 Conclusions The electromagnetic design of an 800 MHz crab cavity sufficiently compact to fit anywhere in the LHC ring has been realized. It allows operating the LHC for the High Luminosity upgrade (HL-LHC) with the local crab option, providing a peak voltage of 2.5 MV, powered by a dedicated high power RF system. The proposed crab cavity is based on a single-cell working on the vertical polarity of the TM 110 -like mode loaded by a special coaxial damper integrated on one of the cavity beam tube. Two different damping system were investigated. The first system, having the simplest geometry, was the lateral antenna coupler placed at 45 with respect to the vertical direction. Although this damping system works well with respect the HOMs damping, the WM needs stronger rejection (as extensively shown in section 5). Modes different from the operational one are well damped. The second system explored was aimed to increase such relatively low rejection of the WM by the OOM coupler. The intruduction of a dedicated notch filter between the cavity body and the OOM power output has provided the expected preliminary results by increasing the Q ext of 27

28 the WM to acceptable values. Further studies on the effect of the notch filter on the HOMs are necessary and are already started. The cavity geometry has been optimized to achieve the lowest peak surface fields keeping the pr{qq K as high as possible. The classic ellipsoidal shape of the structure allows easy cleaning of the interior surfaces as required for the preparation of superconducting cavities. Particle tracking simulations were also performed to predict or identify any signs of multipacting in a range of transverse voltage from tens of kv up to 3 MV, well above the rated voltage. The preliminary results show no signs of multipacting inside the cavity itself, but emergence of a multipacting phenomenon between the inner and outer conductor of the coaxial pipes for low values of the deflecting voltage. The multipacting phenomenon is analitically predicted and is well known to the accelerator community; it is hoped that it can be processed away. 9 Possible future steps Studies of the effect of the notch filter on the HOMs. Increase the understanding of multipacting at low and high voltage hard bands, using a real SEY curve measured on niobium material. Design grooves inside the OOM coupler if some hard multipacting bands were found using realistic niobium as measured. Creation of a technical drawing for the realization of a prototype. Acknowledgements The authors would like to thank Elena Shaposhnikova, Ed Ciapala and Erk Jensen for carefully reading the paper and suggesting improvements as well as Ed Ciapala and Erk Jensen for their support for this work. References [1] R.B. Palmer, SLAC PUB-4707 (1988). [2] K. Oide, K. Yokoya, Beam-beam collision scheme for storage-ring colliders, Phys. Rev. A 40, 315 (1989). [3] Y.-P. Sun, et al., Study with one global crab cavity at IR4 for LHC, PAC09. [4] LHC-CC10, 4th LHC Crab Cavity Workshop, December 2010 CERN, [5] R. Calaga, S. Myers, F. Zimmermann, Summary of the 4th LHC crab cavity workshop LHC-CC10, CERN-ATS [6] L. Xiao, Z. Li, et. al., 800 MHz crab cavity conceptual design for the LHC upgrade, SLAC- PUB-13648, May [7] k. Akai, J. Kirchgessner, et.al., Development of Crab Cavity for CESR-B, Cornell University, Ithaca, NY. (1993). [8] Microwave Studio, CST, Darmstadt, [9] High Field Solver Simulator, Ansoft, 28

29 [10] W. K. H. Panofsky and W. A. Wenzel, Rev.Sci. Instrum. 27, 967 (1956). [11] E. Shaposhnikova, Impedance effects during injection, energy ramp & store, LHC-CC10, CERN, Dec [12] K. Hosoyama, et.al., Development of the KEK-B superconducting crab cavity, EPAC08, Genoa, Italy. [13] M.A. Furman and M.T.F. Pivi, Probabilistic model for the simulation of secondary electron emission, PRSTAB, Vol. 5, (2002). [14] F. Hamme, U. Becker, P. Hammes, Simulation of Secondary Electron Emission with CST PARTICLE STUDIO, ICAP 2006, Chamonix, France. [15] S. U. De Silva, J. R. Delayen, Multipacting Analysis of the Superconducting Parallel-Bar Cavity, PAC11, NY, USA. [16] J. R. Delayen, H. Wang, New compact TEM-type deflecting and crabbing RF structure, PRSTAB, 12, , (2009). [17] Z. Li, L. Xiao, C. Ng, T. Markiewicz, Compact 400MHz Half-Wave Spoke Resonator CRAB cavity for the LHC upgrade, SLAC, Menlo Park, [18] L. Ge, Z. Li, Multipacting Analysis for the Half-Wave Spoke Resonator CRAB cavity for LHC, PAC11, New York, USA. [19] B. Hall, G. Burt, et. al., Novel Geometries for the LHC CRAB Cavity, IPAC10, Kyoto, Japan. [20] N. Kota, KEK compact crab cavity, Crab Cavity WebEx meeting 5, [21] U. Kelin and D. Proch, in Proceedings of the Conference on Future Possibilities for Electron Accelerators, volume N117, Charlottesville, [22] W. Weingarten, Electron loading, in 2nd Workshop on RF Superconductivity, edited by H. Lengeler, pages , CERN, Geneva, Switzerland, [23] J. Knobloch, W. Hartung, and H. Padamsee, Multipacting in 1.5-GHz supercunducting niobium cavities of the CEBAF shape, Workshop on RF Superconductivity, Albano Terme (Padova), Italy, [24] N. Hilleret, C. Scheuerlein, M. Taborelli, The secondary electron yield of air exposed metal surfaces at the example of niobium,cern EST/ (SM). [25] CST staff, private comunication. [26] G. Romanov, Update on Multipactor in Coaxial Waveguides using CST Particle Studio, PAC11, New York, USA. [27] G. Burt, R. G. Carter, et.al., Benchmarking Simulations of Multipactor in Rectangular Waveguide using CST Particle Studio, SRF2009, Berlin, Germany. [28] F. Hamme, U. Becker, P. Hammes, Simulation of Secondary Emission with CST Particle Studio, ICAP2006, Chamonix, France. [29] G. Romanov, Simulation of Multipacting in Hins Accelerating Structures with CST Particle Studio, LINAC08, Victoria, BC, Canada. 29

30 [30] P. H. Stoltz, et.al., Bwnchmarking Multipacting Simulations for FEL Components, FEL2010, Malmo, Sweden. [31] [32] A. V. Gaponov, M. A. Miller, Sov. Phys. JETP (1958) 168. [33] R. L. Geng, Multipacting simulations for superconducting cavities and RF coupler waveguide, PAC Bibliography H. Padamsee, J. Knobloch, T. Hays, RF superconductivity for accelerators (Wiley series in beam physics and accelerator technology, Cornell University Ithaca - New York, 2008). H. Padamsee, RF superconductivity (Wile-VCH, Cornell University Ithaca - New York, 2009). S. Turner, CAS - Superconductivity in Particle Accelerators (Haus Risse - Hamburg - Germany, 1995). John C. Slater, Microwave Electronics (D. VAN NOSTRAND COMPANY, Inc., 1990). 30

31 A Multipacting simulation studies The parameters chosen in the simulator code as well as the methods that led us to chose these parameters and preliminary MP results will be discussed in this section. Initially we had to optimize the code s running parameters in order to simulate a credible case while keeping the simulation runtime reasonable. Since CST-PS has a number of parameters we can specify when running the code, we investigated the effect of the variation of mesh density on the particle tracking simulation results. Other main parameters are: time sampling rate, spatial sampling rate, maximum number of secondaries per impact, the maximum number of generations which a source primary electron can produce, initial electron energy, electron emission model, secondary electron emission model, the number of initial electron and the area to put the initial electrons. In the following some of these simulations are shown, in which the amplitude was varied to explore a range for the deflecting voltage from a few kv to 3 MV sufficiently above the nominal voltage (2.5 MV). At each amplitude level, 18 equally spaced starting phases have been explored over one RF turn. Table 4: Preliminary simulations parameters MWS parameter Value Eigenmode meshing - LPW 40 (1.1M cells) Tracking meshing - LPW 40 (0.6M cells) Electron source area OOM side iris Number of initial electrons 60 Emission model Fixed emission Emission energy - [ev] 1 Number of secondary per hit 10 Maximum generation 20 Concerning the preliminary particle tracking simulations we used the setting parameters as shown in Table 4. Initially we performed simulations fast enough to have a starting point from which to optimize the various parameters of the tracking solver. It is possible to vary the meshing number of cells by acting on the number of lines per wavelength (LPW). The working mode field was calculated using the eigenmode solver at different LPW depending on the cases, as explained below. In Fig. 32 the number of particles during the first 800 RF periods are shown. These are the preliminary results of parametric simulations in which the deflecting voltage (V K ) was varied from 90 kv to 400 kv with steps of about 90 kv and at any amplitude change 9 phase points in half an RF turn have been considered as mentioned above. These data do not show a clear multipacting phenomenon with a resulting localized electron avalanche. There is a strong electron activity but not because of multipactor. Unfortunally, looking in more detail at some of these curves for some of the investigated parameters, one can observe the abnormal behavior as shown in Fig. 33. It shows the case with a voltage above 175 kv and the field phase fixed at 80, the particle emerging from the source surface sees a retarding field at different amplitude. It is a non-physical phenomenon as up to particles (in the case of V K =350 kv) are emitted before the first half RF period. This can be undestood as follows: CST-PS uses a small gap between the location where electrons are emitted and the true cavity boundaries. At high retarding fields this virtual gap may be large enough for electron to gain enough energy to produce apparently true secondaries. This leads to a very rapid but unphysical growth in the number of electrons. It is very important therefore to have a mesh dense enough to minimise this auxiliary gap size. For this reason we first 31

32 Figure 32: Number of electrons versus time for V K in the range from 90 kv to 400 kv, for each RF field the initial phases are spaced by 20 one from another. Figure 33: Unphysical behaviour due to an inadequate meshing. Number of particles versus time for all explored V K above 175 kv and field phase at 80 (retarding RF fields). 32

33 performed a series of simulations to parametrize both LPW parameters for the electromagnetic solver and for the tracking solver. A.1 Meshing scanning studies To obtain credible and reproducible results a careful study of the meshing of the structure is necessary. In MWS-PS it is possible to use two different, independent meshes for the RF field computation and the particle trajectory computation. Then the RF fields are calculated separately with their proper meshing and subsequently imported into the Particle tracking simulation. MWS supports two meshing types, tetrahedral and hexahedral. MWS staff suggested to use only hexahedral meshing for both the RF field solver and the Particle Tracking solver [25] because the tetrahedral solver is not fully debugged. It has been decided to use a hexahedral meshing with a number of independent cells for the two solvers. The particle tracking simulation with a deflecting voltage of 175 kv and a retarding phase of 80 shows the lowest non-physical peak in the number of particles, about 300 particles (Fig. 33). To study a more accurate meshing while keeping the simulation time low, the above setting has been chosen, instead of the other deflecting voltages with more unphysical particles. Figure 34: Electron number vs time during the first RF period for different meshing densities and initial electron number. Four simulation series were performed as explained in the text. The unphysical peak is extinguished rapidly with increased mesh density. Below the results for all meshing studies are shown. The following parameter variations were performed: 33

34 1. LPW(RF field solver) LPW(tracking solver) from 20 to 100 lines - 60 initial electrons; 2. LPW(RF field solver)=20; LPW(tracking solver) from 20 to 100 lines - 60 initial electrons; 3. LPW(RF field solver)=20; LPW(tracking solver) from 20 to 100 lines initial electrons; 4. LPW(RF field solver)=100; LPW(tracking solver) from 100 to 150 lines initial electrons; First we decided to vary both the mesh density increasing the lines per wavelength (LPW) for both solvers at the same time. Initially we set the field solver and the tracking solver mesh to 20 LPW. We increased the mesh in both solvers to calculate xseyy and xe i y, the number of particles in time and the electron trajectory. At this stage we used 60 initial particles. The same parameters were applied by varying only the tracking solver meshing and using more initial particles. Driven by a clear improvement of results in terms of consistency and the high performance of the computing system used, we got 150 lines per wavelength in the tracking meshing and 100 lines in the RF field meshing, with 900 initial particles. As shown in Fig. 34 every improvement of the meshing leads to a rapid decrease of the non-physical peak in the number of particles in time. In the same figure only some RF periods are shown as we were interested to observe the unphysical peak behavior while increasing the mesh cell number. 1.4 LPW MWS = LPW TK Number of particles: LPW MWS = 20 Number of particles: <SEY> 1 10 <E i > <SEY> 1 10 <E i > Lines Per Wavelength LPW MWS = 20 Number of particles: Lines Per Wavelength Particle Tracking LPW MWS = 100 Number of particles: <SEY> 1 10 <E i > <SEY> 1 10 <E i > Lines Per Wavelength Particle Tracking Lines Per Wavelength Particle Tracking Figure 35: Average SEY and average impact energy computed at different mesh densities and initial electron number. Although the largest number of mesh cells eliminates the unphysical peak, we have not found any convergence on the xseyy and xe i y curve with increasing number of cells, as recommended and found by other expert designers. 34

35 Fig. 35 shows the average SEY and the average impact energy depending on the number of cells. We found no convergence although we used many more cells than those suggested in the literature. This can be attributed to the fact that the geometry in question is very different from the simplest ones simulated by other designers. Inside the slim crab cavity, due to the curved geometry, many particles are accelerated by strong electric fields at energies above a few MeV. In this manner the high energy particles making the average impact energy not trustworthy. A.2 Choise of running parameters Table 5 shows the parameters chosen after the optimization process of the simulator. The results presented in Section 7 were obtained using these values, the parameters that are not in this table were left to the solver default values. Table 5: Parameters used in simulations MWS parameter Value Eigenmode meshing - LPW 100 (11.1M cells) Tracking mesching - LPW 150 (35.3M cells) Electron source area OOM side iris Number of initial electrons 900 Emission model Fixed emission Emission energy - [ev] 1 Number of secondary per hit 10 Maximum generation 20 B The cavity geometry The cavity geometry was constructed for the HFSS and MWS simulator codes following these steps: 1. Draw a simple pillbox having a radius of mm to resonate at 800 MHz; 2. Round off the two extreme edges and increase the radius up to 243 mm to compensate the change in resonant frequency due to the corners rounding perturbation; 3. Squash the cavity horizontally and stretch it vertically; 4. Add the two beam pipes; 5. Round off the edges of the irises; 6. Add the two coaxial coupler; Table 6 shows all geometrical parameters (in millimeters) needed to design the cavity. All the source files used in the development of this crab cavity data are conserved for possible further reaching studies or mechanical design work at CERN s EDMS system under the folder AB-Department / ABRF Group / Sections / BR - Beams & RF / CRAB cavity design. In Fig. 36 the three main sections of the structure with the geometric dimensions highlighted are displayed. 35

36 Figure 36: Three principal cross sections of the cavity 3D model with the parameters name highlighted. The parameters values are reported in Table 6. 36