issues related to crossing angles Frank Zimmermann

Super-KEKB crab cavity scheme 2 crab cavities / beam / IP

voltage of crab cavities tolerance on IP offset jitter translates into tolerance on left-right crab-cavity phase and crab-main-rf phase

R12 & R22(R11) from MAD nominal LHC optics |R 12,34 |~30-45 m |R 22,44 |~1 (from crab cavity to IP)

voltage required for Super-LHC

crossing angle0.3 mrad1 mrad8 mrad 800 MHz2.1 MV7.0 MV56 MV 400 MHz4.2 MV13.9 MV111 MV 200 MHz8.4 MV27.9 MV223 MV crab cavity voltage for different  c ’s & rf frequencies

tolerance on R22 z-dependent additional crossing angle corresponding Piwinski angle should be small not a problem [for  c =1 mrad,  x =12 mm, R 12 =30 m,  z =7.55 cm]

KEKB crab cavity Squashed cell operating in TM2-1-0 (x-y-z) Coaxial coupler is used as a beam pipe Designed for B-factories (1 〜 2A) Courtesy K. Akai ~1.5 m K. Ohmi, HHH-2004 ~1.5 MHz

longitudinal space required for crab cavities scales roughly linearly with crab voltage; desired crab voltage depends on rf frequency); achievable peak field also depends on rf frequency; 2 MV ~ 1.5 m, 20 MV ~ 15 m frequency must be compatible with bunch spacing; wavelength must be large compared with bunch length; 1.2 GHz probably too high; 400 MHz reasonable; 800 MHz perhaps ok longitudinal space & crab frequency

noise amplitude noise introduces small crossing angle; e.g., 1% jitter translates into 1%  c /2 crossing angle – not critical phase noise causes beam-beam offset; → tight tolerance on left-right crab phase and on crab-main-rf phase differences

emittance growth from turn-by-turn random offsets  x requiring less than 10%/hr emittance growth  x rms <12 nm ~  *  <0.012 o at  c =1 mrad & 500 MHz  <0.04 o at  c =0.3 mrad & 500 MHz SuperLHC:   x,y =0.25 m, n IP =2,  HO =0.005,  =7500,  =3.75  m p emittance growth due to random offsets

diffusion rate from strong-strong simulation with BBSS  x 2 =  x0 2 +Dt t: turn D~1.4x  x[  m] 2  z= K. Ohmi, HHH-2004

tolerance from Ohmi san’s strong-strong simulation For  x=1.6  m (  =5 degree) and  =100, D~1.4x  x[  m] 2, where  x 2 =  x0 2 +Dt, t: turn. Tolerance is  x=0.016  m,  = 0.05 degree for  =100, and  x=  m, degree for  =1, for luminosity life time ~ 1 day K. Ohmi, HHH-2004 for 300  rad crossing angle and 400 MHz slightly worse than my “pessimistic estimate”!?

analytic theory of b-b diffusion (T. Sen et al., PRL77, 1051 (1996) M.P.Zorzano et al., EPAC2000) Diffusion rate due to offset noise. (round beam) K. Ohmi, HHH-2004

comparison with the simulation D(a=1)= =1.5x m 2 /turn D(sim)=(  -  0 2 ) 2 /  2 = m 2 /turn Need to check K. Ohmi, HHH-2004 analytical diffusion rate from Sen-Ellison-Zorzano model even much larger!! } 3 orders of magnitude discrepancy!

impedance of crab cavities transverse impedance is an issue due to large beta function rise time due to 1 crab cavity = rise time from ~10 normal rf cavities with the same voltage

dispersion correction if large crossing angles are realized by placing single D1 dipoles first, and the triplet between D1 and 2, the dispersion correction could be an issue to be studied

minimum crossing angle from LR b-b “Irwin scaling” coefficient from simulation note: there is a threshold - a few LR encounters may have no effect! (2nd PRST-AB paper with Yannis Papaphilippou) minimum crossing angle with wire compensator need dynamic aperture of 5-6  and wire compensation not efficient within 2  from the beam center