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Linac Basics Homework – Chris Adolphsen
Problem 1: Cavity Fill Time and External Q
The stored energy in a cavity (W) can be expressed as
W = Vc2 / s, where s is the ‘elastance’ =  R/Qo and
Vc is the integrated field along the cavity witnessed by a speed-of-light particle that
travels on the crest of the rf waveform
R is the cavity ‘shunt’ impedance, defined such that the power loss on the cavity
walls is Vc2 /R
 = 2  times the cavity frequency (= 1.3 GHz for the ILC)
Qo is the internal (or ‘intrinsic’) quality factor of the cavity (from the expression for
W, the power loss on the cavity walls can also be expressed as  W / Qo).
Note that the ratio R/Qo depends only on the cavity geometry, but that R and Qo
individually depend on the cavity wall impedance. The beauty of superconducting
cavities is that the wall losses are very low, for example, Qo ~ 1e10 for the TESLA
cavities, which means the energy decay time in a closed cavity, Qo/, is around 1 sec !
For the ILC linac cavities, the external quality factor, Qext, is adjusted to optimally
convert the rf input power into beam power. For the ILC, Qext ~ 1e6, which is much
smaller than Qo, so we can ignore the power loss in the cavity walls when considering
this energy transfer (which we do in the rest of this problem). In this case, the power
discharged from the cavity after the input rf and beam are shut off is  W / Qext.
During the filling of the cavity, before the beam arrives, the energy transfer to the cavity
is governed by the equation
W/t = Vi2 / Z – Vr2 / Z
where Vi is the field strength of the input rf, Vr is the field strength of reflected rf from
the cavity and Z is the effective impedance of the cavity input power coupler, which is
located at one end of the cavity. Vi and Vr are normalized so the following continuity
equation holds: Vc = Vi +Vr. Assuming the input power is a square pulse, compute the
time after the start of the pulse when the reflected power goes to zero (express this in
terms of Qext and note Qext and Z are simply related).
For the ILC, the timing of the rf pulse is adjusted so the beam arrives at the cavity at the
time derived above. After this time, one wants all of the input power to be transferred to
the beam (with zero reflected power and constant stored cavity energy). In this case, the
input power needs to be I Vc, the power absorbed by the beam, where I is the beam
current (~ 10 mA) and Vc is the cavity field at the injection time. This 100% transfer
efficiency only occurs for a particular value of Qext, which is adjusted by changing the
depth at which the center conductor in the input coupler extends into the cavity. Derive
Qext in this case in terms of Vc, I, Qo and R (hint, equate the power absorbed by the
beam to the change in the rate of stored energy in the cavity if the beam were not
present).
Problem 2: Linac Cost
As a simplified model for a superconducting linac, consider a sting of identical cavities
each with integrated field strength Vc. Assume that the linac cost is the sum of three cost
contributions:
1) Cost to cryogenically cool each cavity = A Vc2
2) Cost of each cavity (independent of Vc) = B
3) Cost of everything else (independent of Vc) = C
where A, B and C are constants.
Derive an expression for Vc that yields the minimum linac cost for a fixed final beam
energy.