Observations on Betatron Collimation and the Effect of Tune Splitting

We find the conditions for optimal collimator location by the method of “Lagrange undetermined multipliers”. We give a ‘global’ explanation for the advantage of tune splitting on betatron collimation in terms of the envelope of all possible optimal collimator positions. We make analytic calculations of the maximum amplitude of surviving particles for elliptic and jaw-type collimators at a variety of optimal locations. All statements and results reported in this brief resume are substantiated and derived in a lengthy 1997 TRIUMF internal report[6] along with further elaboration of the properties of the A, C and J functions.