mode - CERN Accelerator School
mode - CERN Accelerator School
mode - CERN Accelerator School
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Transverse equilibrium in ion<br />
and electron linacs<br />
The equilibrium between external focusing force and internal defocusing forces<br />
defines the frequency of beam oscillations.<br />
Oscillations are characterized in terms of phase advance per focusing period σ t<br />
or phase advance per unit length k t.<br />
Ph Ph. advance d = Ext. E t quad d focusing f i - RF ddefocusing f i - space charge h – IInstabilities t biliti<br />
k<br />
2<br />
t<br />
2<br />
2<br />
( −ϕ<br />
) 3q<br />
I λ(<br />
1−<br />
)<br />
⎛ σ t ⎞ ⎛ qGl<br />
⎞ π q E0T<br />
sin<br />
f<br />
= ⎜ ⎟ = ⎜ ⎟ −<br />
−<br />
− ...<br />
2 3 3<br />
3 3 2 3<br />
⎝ βλ ⎜<br />
⎠ 2 β ⎟<br />
2 3 3<br />
3 3 2 3<br />
⎝ Nβλ<br />
⎠ ⎝ mc βγ ⎠ mc λλ<br />
ββ<br />
γ 8πε<br />
0 r0<br />
mc ββ<br />
γ<br />
Approximate expression valid for:<br />
F0D0 lattice, smooth focusing approximation, space charge of a uniform 3D ellipsoidal bunch.<br />
Electron Linac:<br />
Ph. advance = Ext. focusing + RF defocusing + space charge + Instabilities<br />
For γ>>1 γ (electron ( linac): ) RF defocusing g and space p charge g disappear, pp phase p advance ￫0.<br />
External focusing is required only to control the emittance and to stabilize the<br />
beam against instabilities (as wakefields and beam breakup).<br />
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