ILC Crab Cavity Wakefield Analysis

ILC Crab Cavity Wakefield Analysis Yesterday, Peter McIntosh discussed the overall requirements for the ILC crab cavities, the system-level design, and the team that is working on it. Here, I will discuss some specifics about the beam-cavity interaction Leo ( ) Bellantoni Fermi National Accelerator Lab 1

Long range wakes RF cavities by design support eigenmodes that interact strongly with beam bunches In SC cavities, very small intrinsic damping means the energy left in the cavity by a passing bunch is potentially a problem for many following bunches For accelerating cavities, the azimuthal symmetry is broken by the φ of the incoming bunch, taken to be the same for the entire train of bunches; in crab cavities the azimuthal symmetry is broken by the cavity shape This long-range wake is amenable to frequency-domain analysis; this is a well-known technique. For case of azimuthal non-symmetry, see (with errata 19 Sep 2007) http://hom e.fnal.gov/~bellanto/scrf/13-cells/ Wakefields/TM-2356.pdf 2

Long range wakes [ ( )] The wakefield W r ( r 1, r 2,s)= 1 + dz E r ( r 2,z,t)+ c?z B r r 2,z,t q 1 t =(s+ z)/c for a single mode n of azimuthal order m when witness charge enters at same location as the exciting charge) has longitudinal component W // Parameter for beam-cavity coupling ( ) ( r,s)= R n Q ω n cos ω ns e s cτ n c cosine - exponential function of s, the interbunch distance r 2m cos 2 [ m( φ φ n )] Dependence on beam position For the transverse wake, there is a similar expression with the cosine replaced by the sine 3

Long range wakes Then the sum of the wakefields over all the proceeding bunches in a train of length N includes a factor that looks like N 1 j=1 ( ) cos ω n s N j sin c e ( ) cτ n s N j N 1 j=1 exp( jd), D C This simple harmonic series is amenable to analytic manipulation G. Burt, R.M.Jones and A.Dexter, IEEE Transactions on Nuclear Science, 54 (October 2007) 0018-9499 L.Bellantoni, H.Edwards and R.Wanzenberg, FNAL TM-2404/DESY M 08-01 (March 2008) ILC Crab specific Encyclopedia for TM 010 type cavities 4

Long range dipoles Our primary concern has been dipole modes 0.284σ σ/4 displacement will cost 2% luminosity Bunch width over throw arm = 4.0 x10-8 in x, 2.4 x10-9 in y For dipoles, on-resonance means phase difference between bunches and HOM is small but not zero 8 6 Fr y is worse! Fi Long bunchtrain limit N valid 4 Q EXT = ω t cqr R Q N 2 sinh 1 CAVITY 0.568σE b 2 0-180 -150-120 -90-60 -30 0 30 60 90 120 150 180-2 bunch/hom phase (degrees) -4 5

Long range monopoles Monopole modes could be a problem because of: Peak, in-train power limit of H/L/SOM coupler Long-term average power limit (in cable, actually) Peak fields in-train could provoke quenching ILCTA optics want <200MeV dispersion S.Tariq, T.Khabibouline From 3rd Harmonic cavity work, we think we should expect ~1kW peak power handling through the L/HOM feedthroughs That makes the peak coupler power during the bunch train the limiting factor for most modes 6

Q EXT design goals 2.83GHz TM 010 1.6 x10 4 3.90GHz 3.90GHz 6.02GHz 7.15GHz ~7.1GHz 8.05GHz 8.74GHz 9.31GHz 9.98GHz 10.86GHz Operating mode SOM TM 020 m = 0 m = 1 with lots of pipe energy m = 1 with low phase velocity m = 0 E MAX limited @12MV/m m = 0 m = 1 m = 0 4.4 x10 5 5.7 x10 4 4.7 x10 5 1.3 x10 5 1.1 x10 6 1.7 x10 6 5.3 x10 6 6.1 x10 5 4.8 x10 6 5.4 x10 5 These numbers susceptible to change as we learn more about coupler power handling, R/Q for actual manufactured units, quench points in higher order modes etc. 12.11GHz m = 0 2.1 x10 5 12.84GHz 13.28GHz m = 1 m = 0 8.0 x10 6 2.9 x10 5 Values from MAFIA, ~20k points end-cell compensation issues 7

Short range wakes Short-range (single bunch) longitudinal wakes are O [ 10 MeV ] << E BEAM Transverse wakes occur if the bunch is off-axis; they increase as a power of the bunch length and can be estimated with either timedomain computation, analytic formulas, or a hybrid of the two. Concern is banana-shape effect: 8

Short range transverse wakes The short (~300mm) ILC bunch length reduces the magnitude of this problem Early studies with MAFIA and ECHO 2-D time domain analysis, and analytic forms suggest that this is not a large source of luminosity loss V/C 1.0E+13 8.0E+12 6.0E+12 4.0E+12 2.0E+12 0.0E+00 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007-2.0E+12 m -4.0E+12-6.0E+12-8.0E+12-1.0E+13 Point transverse wakefield Full transverse wakefield Deflection/Qbunch Bunch shape 1mm off-axis k tot (V/pC) 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 Analytic 2D zoom4 σ (mm) 11

Things to be aware of In this structure, field flatness and hence R/Q values can be very sensitive to mechanical variations within reasonable manufacturing tolerances. We do have f/ (dimension) however, and have allowed for frequency scatter in the eigenmodes. Couplers will of course also be manufactured with some range of variations in Q EXT There is large E Z near but not at cavity center axis. Consequently, Consequently, an off-axis beam train will add or remove a lot of energy to the operating mode and SOM. LLRF and beam steering feedback systems moderate this. Our analysis (conservatively) neglects feedback. Bunch timing jitter not yet studied in a systematic way. 12

Things to be aware of» The canonical analysis assumes that only one mode will be a problem; in reality there might be significant excitation of multiple modes. We summed the wakes over 13 large R/Q modes, varying their frequencies coherently for a single cavity. At f = 0.89 MHz there is an interaction with two modes that are near resonance, producing deflection larger by ~ 2 FNAL-2404/DESY TM 08-01 addresses incoherent sums over modes - this is next step 13

If only that was all there was to it! There are a range of cavity-beam interactions not treated by this canonical analysis» Couplers can break the orthogonality of different polarizations 3.91209 GHz 3.91212 GHz L.Xiao, Z.Li et.al. Beam energy put into horizontal mode can migrate to vertical and not be removed with coupler designed for horizontal mode power Change of mode frequencies in principle solves this - might damp these modes a bit more, too» A coaxial-like coupler with Q EXT ~ 10 4 will protrude far into the beam pipe with beam-coupler hook distances of maybe 7 to 11mm. The peak Biot-Savart field from a bunch will be O [20-30mT] albeit for ~1psec. 14

If only that was all there was to it!» A.Kalinin pointed out to us an even/odd effect from the end groups asymmetry that we need to study a little more If the end groups differ, this will deflect the beam. In the deflection plane, this is the same as a phase offset in the LLRF, and can be ignored unless it s huge. In y direction, will be some beating of the wakefield SOM against the bunch frequency; there will also be a 3.9GHz component due to imperfect roll adjustment coupling to tail of SOM resonance. (Peter s talk re. polarization) These effects all have to fit in the limit of less than ~160V of deflection for each 2-cavity cryostat (5.7nm σ y at IP). 15

Did we forget anything? Modern Computing is a wonderful thing Comprehensive Monte Carlo simulations can (and often do) uncover effects you yourself would never think of... 16

Did we forget anything? PLACET was used to simulate 200 ILC-like bunches and tracks them through the accelerator modelling all cavities and magnets, including x-y coupling n-poles, and applying both long and short range wakefields. Vertical Offset (nm) 2 1.5 1 0.5 0-0.04-0.03-0.02-0.01 0-0.5 0.01 0.02 0.03 0.04 Analytical -1 PLACET -1.5-2 Percentage change in frequency Analytic and PLACET results very similar so far Octopole alignment not yet studied 17

Conclusion Long and short range wakes as well as a number of other beam - cavity interactions have been studied both computationally and semianalytically for the ILC Crab cavity application. Manufacturing deviations from perfect geometry have been incorporated for frequency terms but not for R/Q terms; beam positioning and LLRF feedback has (conservatively) been neglected, and some design constraints are assumptions yet to be backed up with laboratory measurements. Notwithstanding, the results from different methods are quite consistent and we see the analyses as plausible enough to provide design goals. Focus now is really on design of couplers to meet the requirements elucidated from these studies. 18