STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
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M ∗ ≡ M12 cos 2 α + (M14 + M32) cos α sin α + M34 sin 2 α (4.19)<br />
The threshold current is defined as the condition for which the power dissipated<br />
by the cavity is exactly compensated by the power deposited by the beam and occurs<br />
when ˙ U = 0. Applying this condition to Eq. (4.18) yields<br />
qM<br />
Ith<br />
∗<br />
sin(ωTr) +<br />
2ωpb<br />
1<br />
k 2 (Rd/Qo)QL<br />
and the threshold current for instability is given by<br />
2Vb<br />
Ith = −<br />
k(Rd/Qo)QLM ∗ sin(ωTr)<br />
87<br />
= 0 (4.20)<br />
(4.21)<br />
where Vb = pb(c/q) is the beam voltage and M ∗ is defined in Eq. (4.19). Despite<br />
the simplicity of Eq. (4.21), it is important for gaining insights into the parametric<br />
dependence of the threshold current. The formula depends on parameters that<br />
characterize the electron beam, the machine optics and properties of the HOM.<br />
The threshold is directly proportional to the beam energy. That is, at higher<br />
energies, the beam is more rigid and will be deflected less for a given HOM angular<br />
kick. This implies that the front end of the linac, where the injected beam is only<br />
a few MeV, can be particularly susceptible to BBU.<br />
The threshold also depends on the details of the machine lattice. Specifically<br />
it is inversely proportional to the matrix elements that transform an HOM-induced<br />
kick on the first pass to a displacement on the second pass. Large values of these<br />
matrix elements (M12 or M34 with decoupled optics, for example) will lead to large<br />
off-axis displacements in the cavity and, given the correct phase sin(ωTr), the beam<br />
will deposit energy proportional to the displacement into the HOM. In fact, a poorly<br />
designed optics can contribute to a low threshold current just as surely as insuffi-<br />
ciently damped HOMs.