The Definition of a1: Circumference, Johnsen, or ESME?

J. MacLachlan, K.Y. Ng, S. Peggs

October, 1990

For reference purposes, this note describes how the three definitions of a1 and the one definition of aP, all of which are common in the literature, are related. The "circumference" definition of a1 comes from expanding the difference in the circumference of the closed orbit, DC = C - C0, as a polynomial in the off momentum parameter, d = (p - p0) / p0,

The definition introduced by Johnsen is very similar

The variation of the transition energy gT with d is directly described by aP, defined through

introduces the ESME definition of a1. Note that equation 3 represents the local derivative at some momentum p. WARNING - some lattice design codes return aP(d), but some return

when they are used with a constant momentum offset. Comparing equations 1 and 2 gives

Performing the differentiation in equation 3 and expanding gives

The reader is NOT implored to adopt one or another of the definitions introduced here, but rather is asked to be careful to specify which definition he or she id using. He or she IS implored not to invent any more definitions for a1.