mode - CERN Accelerator School
mode - CERN Accelerator School
mode - CERN Accelerator School
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Longitudinal g dynamics y<br />
→ Ions are accelerated around a<br />
(linac=negative) synchronous phase.<br />
→ Particles around the synchronous one<br />
perform oscillations in the longitudinal<br />
phase space.<br />
→ FFrequency of f small ll oscillations: ill ti<br />
ω<br />
2<br />
l<br />
= ω<br />
2<br />
0<br />
( −ϕ<br />
)<br />
qE0T<br />
sin<br />
2<br />
2π<br />
mc βγ<br />
→ Tends to zero for relativistic particles γ>>1.<br />
→ Note phase damping of oscillations:<br />
Δϕ<br />
=<br />
const<br />
3/<br />
4<br />
( β γ )<br />
3<br />
λ<br />
3/<br />
4<br />
ΔW = const × ( β γ )<br />
At relativistic velocities phase oscillations stop, the<br />
beam is compressed in phase around the initial phase. phase<br />
The crest of the wave can be used for acceleration.<br />
35
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