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V/pC/m2 ‘kick’ along bunch: ∆y′( z ) = 8000 ∞ qb W⊥ ( z '− z ) ρ ( z ′) y ( s; z ′)dz ′ E ( z ) z′=∫ z 6000 4000 2000 Note: y(s; z) describes a free betatron oscillation along linac (FODO) lattice (as a function of s) tail 0 -1000 head -500 0 500 1000 z/σz CAS Zeuthen 2 particle model Effect of coherent betatron oscillation - head resonantly drives the tail tail head eom (Hill’s equation): y1′′ + k β2 y1 = 0 solution: y1 ( s ) = a β ( s ) sin (ϕ ( s ) + ϕ 0 ) tail eom: head y2′′ + k 2 y2 = y1 W '⊥ q 2σ z 2 Ebeam resonantly driven oscillator CAS Zeuthen 33 BNS Damping If both macroparticles have an initial offset y0 then particle 1 undergoes a sinusoidal oscillation, y1=y0cos(kβs).

What happens to particle 2? W '⊥ qσ z y2 = y0 cos ( k β s ) + s sin ( k β s ) 2k β Ebeam Qualitatively: an additional oscillation out-of-phase with the betatron term which grows monotonically with s. How do we beat it? Higher beam energy, stronger focusing, lower charge, shorter bunches, or a damping technique recommended by Balakin, Novokhatski, and Smirnov (BNS Damping) curtesy: P. Tenenbaum (SLAC) CAS Zeuthen BNS Damping Imagine that the two macroparticles have different betatron frequencies, represented by different focusing constants kβ1 and kβ2 The second particle now acts like an undamped oscillator driven off its resonant frequency by the wakefield of the first.

Orbit not quite Dispersion Free, but very close µm CAS Zeuthen 44 DFS practicalities • Need to align linac in sections (bins), generally overlapping. • Changing energy by 20% – quad scaling: only measures dispersive kicks from quads. Other sources ignored (not measured) – Changing energy upstream of section using RF better, but beware of RF steering (see initial launch) – dealing with energy mismatched beam may cause problems in practise (apertures) • Initial launch conditions still a problem – coherent β-oscillation looks like dispersion to algorithm.