Collective Effects and Impedances in the RLHC(s)

We estimate coupling impedances and collective instability thresholds for the high- and low-field options for the 50 TeV on 50 TeV Really Large Hadron Colliders discussed at the 1996 Snowmass workshop. Because of the large circumference of these machines, transverse instabilities tend to be dominant. The challenges for the high-field machine are similar in kind and magnitude to those faced by the LHC, and are expected to be solved at the LHC. The low-field machine faces new challenges. Because of its warm, small radius aluminum beampipe and large circumference, there is a large broadband resistive wall impedance which must be controlled to avoid the transverse mode coupling instability. A new set of low-field RLHC parameters was produced by the RLHC working group to avoid this instability. Both the high- and low-field machines require transverse feedback to stabilize the beam against a multibunch resistive wall instability. The resistive wall risetime is less than one turn for the low-field machine, which will require a feedback system based on yet-to-be-developed technology. I