Comments on RF for JHF synchrotrons

 The choice of low harmonic number in the Booster machine (h=4) has some advantages: However, some voltage is is also needed to accelerate the beam: (1/2) I_b V Sin(Phi_b)=I_dc V Sin(Phi_s). It would be easier to develop the accelerating voltage at higher frequency, where it would be possible to have higher Quality factor.

JHF Booster

The ring is filled by multi-turn injection over some 400 turns by a linac. One could hope that during this time, with slow build up of charge, that some kind of quasi-equilibrium is built up with a self-consistent adjustment of the beam phase-space and the cavity voltage components at multiples of the RF. This beam is then accelerated. Calculations by the JHF team. suggest that there is is no coupled bunch or within-bunch instability (due to the known sources of impedance). If this is true, one might consider trying to accelerate without the compensation of periodic transients. Of course, what is at stake here, is the worry of particle losses in such a high intensity machine -- and it is for this reason that previous studies were cautious and desired a pure sinusoidal gap voltage. The stakes are high! hands-on versus remote handling for maintenance, etc.
There is a similar scaling law for the periodic voltage transients and for the coherent frequency shift: dV/V=df_s/f_s = I_b Z/V. The periodic transients are very strong, and although the frequency shift is reduced by partial cancellation between sidebands; it still seems that growth rates should be substantial. To elaborate: though the quality factor is low, the revolution harmonics are quite widely spaced (4 of them in the bandwidth). Hence, although the impedance asymmetry is weak around the resonance, it is hard to believe that the cancellation between upper and lower sidebands of the coupled bunch revolution harmonics is in all cases sufficient to prevent a significant instability. I believe, that it may be necessary to tune the Booster cavity, to avoid CB instability throughout the whole acceleration cycle and recommend that this be studied again. If the studies reconfirm stability, then of course I agree that "not having to tune" is an advantage of these novel broadband cavities.
The possibility to accelerate without compensation of the periodic transients, is a fascinating one: a new territory. I would suggest, from the beginning, to include the possibility of a RF feedback at multiples of the RF as a safety measure, but also to study the new territory with simulations. Unfortunately, there is a difficulty with the simulation of multi-turn injection. In order to achieve reasonable representation of the beam (to get good statistics) one needs, say, 1000 particles. If you take 1000 particles for the first injected turn, then after 400 turns, you will have 400,000 particles! Hence, this calculation is quite time consuming. However, I recommend, you should consider to do this at least once. I do not recommend to track 4*10^5 particles through the whole Booster acceleration cycle. Instead, one should adopt some other procedure.
For example:
  1. use random number generator to pick particle that should be thrown away (so randomly discard 360,000 particles) to leave 40,000.
  2. a more sophisticated approach would be to Fourier analyse the voltage and beam components, and find a self-consistent ensemble that produces the same current from the known confining potential.
On the subject of periodic transient compensation, I suggest to use just enough feedback so as to make the Booster impedance look the same as the Main-Ring impedance at transfer; with luck, this could substantially reduce the power requirement.

JHF Main Ring

Thoughts on transients In principle, there are 2 types of transient: (a) injection and (b) periodic. In the case of injection transients, there are in principle 3 times scales involved:
  1. transient response of the cavity,
  2. transient response of the beam,
  3. transient response of the tuner.
The adoption of low Q cavities implies that the voltage follows modulations of the input current very quickly, and that there is no ringing. These same cavities do not have to be tuned. Consequently, one can argue that there is no injection transient in the JHF main Ring, but rather that a periodic transient is established almost instantly -- and the beam responds to this. 4 batches each of 4 bunches are transferred from the Booster to fill 16 of the 17 main ring buckets. Consequently, the beam Fourier components will change 4 times: starting with a case that the revolution harmonic is stronger than the beam RF component -- and finishing with strong RF component and revolution harmonics 17 times smaller. In this case, it may be wondered if it is necessary to compensate the beam-induced-voltage components at the revolution harmonics. I would argue that this is not necessary. The cavity voltage follows the beam current pulse very quickly, and the situation is something like this:

                             ---------------------------
              gap         |      beam batch    |          gap
-------------------|                           |----------------------------
/\   /\   /\   /\   /\  -/\  /\   /\   /\   /\   /\   /\  /\   /\   /\   /\   /\   /\   /\
  \/   \/   \/   \/   \/   \/   \/   \/   \/   \/   \/  |/   \/   \/   \/   \/   \/   \/
Clearly, there is a voltage component at the revolution frequency -- but it is of no consequence whatever to the beam. If one now considers some compensation which is limited to narrow bands around the RF multiples, then the absence of revolution harmonics in the compensation implies that the compensation will distort the RF voltage in the gaps -- but this is of no consequence for the beam! I have a slight preference for the feed-forward compensation in the main ring -- and this is made easier by the small frequency swing in that ring.
Though the stakes are high (in terms of longitudinal particle losses) I suggest to study Daniel Boussard's concept of no compensation of periodic transients -- instead, try and match the beam at transfer between Booster and Main Ring. If the impedances are not well matched between rings, then compensate the revolution harmonics just enough so as to obtain a match -- and this will reduce the power requirement. The wideband cavities and small frequency swing mean that no tuning is required, and I consider this to produce a strong conceptual simplification of the Main Ring RF design (and of the transient regime) -- which is a definite advantage.

Heretical View

I would like to present a heretical view. During the workshop, we heard much about the advantages of low Q -- but I want to argue that low R/Q is a better figure of merit from the beam view point. RF Requirements should be considered from 2 views: (i) beam, (ii) power. From beam view, "figure of merit" is R/Q. R=shunt resistance, Q=quality factor Perturbation due to beam is dV/V=I_b^2/(QI_g^2). The Robinson threshold scales as Ib/Ig. Ig=V/R, so prefer large gap voltage and small R. Hence prefer small R/Q. Hence prefer few cavities and few gaps. However, power = V^2/R and the beam preference leads to very large power requirement. For TRIUMF KAON, the cavities were designed to achieve R/Q of 35 ohm. The present JHF scenario has R/Q approx 250 ohm. With a high enough Q, one could hope to avoid all periodic transients and all coupled bunch instabilities driven by the cavity fundamental -- a loaded Q of 100 would be adequate. The disadvantage, though, is high Q parasitics, the need to tune the fundamental, etc.

General conclusion

 The JHF team are pioneering a new and exciting territory with the development of these compact, low frequency RF cavities. The very low quality factor and the absence of cavity transient response make this a completely new regime of operation -- with exciting possibilities. I encourage the JHF team to continue with R&D studies in this area. If, despite the initial promise, problems become apparent with the new approach, then the team need not worry as some kind of "conventional solution" will be found.

General comment on simulations: time-domain versus mixed time/frequency simulations.

The Robinson power limited instability is largely due to the geometric cross-coupling, and I believe will be reproduced by a mixed time/frequency simulation that relies on an impedance to find an approximation to "the cavity response". The low current Robinson instability, however, will most probably not be reproduced. Using impedance to find transient response becomes "correct" in the limit of very very short cavity time constant; but this is also the limit of a very wideband cavity. For such a cavity, there is no difference between the impedance at upper and lower synchrotron sidebands, and so no instability.

Possible model for resonator

R= shunt resistance
C= capacitance
L/(1+s*T) = inductance that rolls off to constant at high frequency
Impedance Z(s) = 1
---------------------
1    (1+sT)
-  +  ---- + sC
R    sL
=          s/C
------------------------
s^2 +s(1/RC +T/LC) +1/LC
Hence the effect of the time constant T, is to reduce the quality factor.