Zimmar Physics Accelarator Notes-17.pdf - Choice of sector...

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Choice of sector number N = 3 , 4 , 6 , 8 , ... depends on maximum E/A (energy/nucleon). N = 2 is radially unstable. Odd numbers other than 3 are too complicated. N = 3 is the most frequent choice below 200 MeV/ A because of the fast rise of B 3 near r = 0 . Flutter is defined as F ( r ) = B 2 B 2 B 2 = 1 2 i f i ( r ) 2 (3) where f i ( r ) = B i ( r ) /B 0 ( r ) . Sometimes f is called flutter, F (sometimes written F 2 ) called mean square flutter. With flutter, orbit scallops, r ( θ ) = r 1 + i =1 g i cos( + ζ i ) (4) where g i f i / ( i 2 1) . Field spiral ζ N ( r ) is the angular location of maximum value of the main flutter component. Spiral angle α is the angle between radius vector and tangent line to spiral curve, tan α = rdζ/dr . The acceleration system consists of dees (2 gap drift tube) or cavities (1 gap drift tube), now usually located in valleys of magnet (in all val- leys or, if N is even, in alternate valleys). Dees usually operate at 50-200 kV (peak voltage dee to ground). Cavities operate up to 850 kV per gap. The acceleration system often runs on harmonic of beam orbital frequency f rf = hf 0 . Multiple dees are sometimes phased by selecting a natu- ral mode of the resonator structure (0-mode or π - mode) and sometimes by servoed phase shifters (as in 3 dee h = 1 systems at MSU, Texas A&M, Catania, & AGOR). Without flutter, B isoc = 2 πf 0 m 0 q γ ( r ) = 2 πf 0 m q 1 2 πf 0 r c 2 1 / 2 (5) k isoc = r B dB dr = γ 2 1 ν r = γ (6) As scalloping increases, the radial tune ν r rises faster than γ , hitting resonances. The N/ 2 stop- band limits maximum γ (e.g., γ max 1 . 2 200 MeV/ A for N = 3 ). Axial focusing tune ν z [ k + F (1 + 2 tan 2 α )] 1 / 2 . Normally one picks α ( r ) to in- crease with r to give ν z 0 . 2 to avoid reso- nances. If sectors are radial ( α = 0 ), ν z decreases with r and resonances limit usable energy band. Due to large energy gain per turn and rela- tively rapid change of ν values with energy, cy- clotrons frequently pass through essential beta- tron resonances at ν = N/ 3 , N/ 4 , etc. and im- perfection resonances at 1/1, 2/2. Imperfection

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