# The Definition of a_{1}:
Circumference, Johnsen, or ESME?

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

### October, 1990

For reference purposes, this note describes how the three definitions of a_{1} and the one definition of a_{P},
all of which are common in the literature, are related. The "circumference"
definition of a_{1} comes from expanding the difference
in the circumference of the closed orbit, DC = C - C_{0},
as a polynomial in the off momentum parameter, d = (p - p_{0})
/ p_{0},

The definition introduced by Johnsen is very similar

The variation of the transition energy g_{T} with d is directly described by a_{P},
defined through

introduces the ESME definition of a_{1}. Note that
equation 3 represents the __local__ derivative at some momentum p. WARNING - some
lattice design codes return a_{P}(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 a_{1}.