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Analytical Approach to Eigen-Emittance Evolution in Storage Rings
This dissertation develops the subject of beam evolution in storage rings with nearly
uncoupled symplectic linear dynamics. Linear coupling and dissipative/diffusive processes
are treated perturbatively. The beam distribution is assumed Gaussian and
a function of the invariants. The development requires two pieces: the global invariants
and the local stochastic processes which change the emittances, or averages
of the invariants. A map based perturbation theory is described, providing explicit
expressions for the invariants near each linear resonance, where small perturbations
can have a large effect. Emittance evolution is determined by the damping and diffusion
coefficients. The discussion is divided into the cases of uniform and non-uniform
stochasticity, synchrotron radiation an example of the former and intrabeam scattering
the latter. For the uniform case, the beam dynamics is captured by a global
diffusion coefficent and damping decrement for each eigen-invariant. Explicit expressions
for these quantities near coupling resonances are given. In many cases, they are
simply related to the uncoupled values. Near a sum resonance, it is found that one
of the damping decrements becomes negative, indicating an anti-damping instability.
The formalism is applied to a number of examples, including synchrobetatron coupling
caused by a crab cavity, a case of current interest where there is concern about
operation near half integer \nu_x. In the non-uniform case, the moment evolution is
computed directly, which is illustrated through the example of intrabeam scattering.
Our approach to intrabeam scattering damping and dicusion has the advantage of not
requiring a loosely-defined Coulomb Logarithm. It is found that in some situations
there is a small difference between our results and the standard approaches such as
Bjorken-Mtingwa, which is illustrated by comparison of the two approaches and with
a measurement of Au evolution in RHIC. Finally, in combining IBS with the global
invariants some general statements about IBS equilibrium can be made. Specifically,
it is emphasized that no such equilibrium is possible in a non-smooth lattice, even
below transition. Near enough to a synchrobetatron coupling resonance, it is found
that even for a smooth ring, no IBS equilibrium occurs.
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