JLAB-TN-09-028
Damping Requirements for Longitudinal BBU in 12 GeV CEBAF
L. Harwood, G. A. Krafft, and I. Shin
May 11, 2009
Executive summary
A specification and stretch goal have been developed for the damping of cavity modes that could lead to longitudinal BBU in the 12 GeV upgrade of CEBAF.
Baseline specification (for delivery of 85 μA of beam):(R/Q)Q < 6.5x1011
Stretch goal (for delivery of 200 μA of beam):(R/Q)Q < 2.7x1011
Discussion and background
In the optics for the 12 GeV accelerator, it is intended that the arcs, particularly the higher arcs, are run in a mode where there is non-zero isochronicity. Because of this choice, it is possible that multipass BBU in the longitudinal HOMs could cause instability. This document presents numerical calculations that suggest that even if the longitudinal HOMs remainentirely undamped, the instability will not arise.
The phenomenon of longitudinal BBU instability was investigated by Bisognano and Fripp1. Their conclusion was that, under the worst condition where the HOM was fully resonant with the beam frequency,, where is the resonance frequency of the HOM and with the time between bunches, the threshold current for instability must be greater than
whereis the recirculation arc energy, quantifies the time slip for a given energy error, is the recirculation time, and and are the HOM resonance parameters. Using the conventional definition for the matrix element it is easy to show that where is the one-pass path length.
The baseline requirement for the 12 GeV accelerator is to be able to deliver 85 μA of current at 11 GeV to the existing halls. To provide margin in the specification, we will use a larger value for the calculation. The target used for the specification for the transverse HOM damping was 175 μA with the linacs operating at 600MV each; 175 μA will be used for the calculations here as well thus providing a margin of 2x in the final number. No other explicit safety margin will be included; all other parameters will come from credible operating scenarios.
Examining the equation for Ith above, we see that the threshold current: 1) increases was the arc energy, 2) it decreases with the mode frequency, and 3) decreases with M56. To calculate what (R/Q)Q is needed to have a given threshold current, we need to be aware of the potential values for these parameters. In particular, we are interested in the credible values that yield the most stringent conditions.
The optics planned for the 12 GeV accelerator have M56 = 0 in the first 5 arcs. From that point on, M56 is <1m. Thus, using M56=1m would give us the most stringent condition.
For low energy operation, there is much less emittance growth and thus the advantage of a setup with M56≠0 is greatly reduced and an optics with M56=0 everywhere would be chosen. While the intent is to set up the accelerator in the optimal configuration for a given set of experiments, it is conceivable that a setup might inadvertently occur for low energy beams that has M56 ≠0. We do not feel that is credible to consider a case in which the setup would intentionally have M56 ≠0 for arcs 1-5, however. So, the lowest credible arc energy of interest is arc 6 when the linacs are set for 600MV (This is the bottom end of the intended operating range after the Upgrade.) Thus, the energy of interest is 6x600MV = 3600MV.
Modes with frequencies above 3GHz can escape the cavity through the beam pipe. Thus, the highest credible mode frequency is 3GHz.
Rewriting the earlier equation in a suggestive manner, for a given threshold current goal
Inserting 175μA for Ith, we get
Thus we find that(R/Q)Q <6.5x1011 must be achieved in order to avoid longitudinal BBU at 175μA.
It is credible, however, that a setup could occur in which the optics are notset up perfectly and we have M56≠0 in the first arc. If M56=0.1m (sometimes seen at present), then we get
Inserting 175 μA for Ith as before, we get
As this is less stringent than the earlier value, the earlier value will be used for the specification. As with the specification for damping of transverse modes, when delivering beam at 11 GeV in 5 passes, there is an additional margin of 2x, i.e. the threshold current with (R/Q)Q = 6.5x1011 is 350μA.
A stretch goal was identified for the transverse damping. It increased the damping by 2.4x relative to the baseline specification, which yields threshold currents of 400μA for 6GeV in 5 passes and 840μA for 11GeV in 5 passes. Here a similar approach leads to a stretch goal of (R/Q)Q < 2.7x1011.
[1] J. J. Bisognano and M. Fripp, “Requirements for Longitudinal HOM Damping in Superconducting RecirculatingLinacs”, CEBAF-PR-89-018