Choice of harmonic number with the consideration of ion beam formation Jiquan Guo 9/17/2015

Ion beam formation cycles: Multi-turn strip-injection of up polarized ion from linac to booster (non-polarized for heavy ions) Capture beam into a bucket (~0.7× booster ring circumference) Ramp energy (to 7.9GeV proton equivalent) and perform DC cooling in the booster Compress the bunch length to ~0.7/Nh0× of the collider ring circumference Bucket-to-bucket transfer beam into collider ring Repeat step 1-5 for (Nh0/2-1) times Switch to down polarized ions and repeat step 1-5 for another (Nh0/2-1) times Ramp collider ring to collision energy, perform bunch splitting to Nh If needed, manipulate the beam to create extra empty buckets in the gap, as required by beam synchronization. (w/o barrier bucket, bunch splitting needs to be done in steps, and the final harmonic number Nh must be multiple of Nh0, but beam sync requires varying Nh by increment of 1, so sometimes Nh is impossible to be multiple of Nh0) BB cooler 14 up polarized long bunches 2 empty buckets for gap 14 down polarized long bunches Desired Nh0 depends on linac energy and target collider ring current (0.5A in baseline) Nh0=36-40 for ~42MeV/u Pb Nh0=30-32 for ~130MeV proton or ~55MeV/u Pb Nh0=16-18 for ~210MeV proton or ~100MeV/u Pb

Possible choices of Nh Nh 3600 3584 2266 2256 16 30 40 28 32 Circumference (m) 2266 2256 Nh0 16 30 40 28 32 Occupied long bunches 14 36 26 Gap length (ns, ×2) 472 252 378 470 269 235 Linac Energy (MeV/u) Pb 100 55 42 57 50 Proton 215 125 130 115 Occupied short bunches 3150 3360 3240 3136 3328 Split scheme ×3×3×5×5 ×5×4×3×2 ×5×3×3×2 ×7×25 ×27 ×7×24 Plenty of options to reduce bunch reprate Binary split, might be the easiest Factor of 7 split, might be beneficial for e-ring injection with reduced reprate ~110m longer than 2154m, just enough for the bypass lines