Efficient computation of matched solutions of the KV envelope equations for periodic focusing lattices

Sven H. Chilton(1), Steven M. Lund(2), Edward P. Lee(1)

1 = Lawrence Berkeley National Laboratory
2 = Lawrence Livermore National Laboratory

Note: for the paper with the same title, Steven M. Lund is first author and
Sven H. Chilton is second

The Kapchinskij-Vladimirskij (KV) Equations are coupled ODE's
providing a simple description of the transverse evolution of intense
ion beams. The equations describe the evolution of the beam edge
or RMS radii in response to linear focusing forces of a lattice and
defocusing forces due to space-charge and emittances. The
equations describe low-order evolution of real distributions of
particles when statistical evolution of beam emittances is
negligible; This model is most accurate in systems with ideal
applied field optics, uniform space-charge, and negligible species
contamination.

Of particular interest are the so-called matched solutions, with the
same periodicity as the focusing lattice. The matched beam envelope is
believed to be the most radially compact beam supported by a periodic
linear focusing channel. Typically, matched solutions are calculated
by numerically integrating trial solutions of the KV equations from
assumed initial conditions over one lattice period and searching for
the four initial envelope coordinates and angles that generate the
solution with the periodicity of the lattice. The procedure can be
surprisingly problematic, even for relatively simple focusing
lattices. The basin of attraction for this numerical method is quite
small, particularly for systems with low degrees of symmetry and/or
large focusing strengths. Thus, generating matched solutions may
require a priori knowledge of initial conditions.