CERN LHC Project Report 121 (1997),
presented at the IEEE Particle Accelerator Conference (session 7V-30), Vancouver, 1997
Stability Diagrams for Landau Damping
J.S. Berg and F. Ruggiero
May 1997
Abstract:
Coherent modes which are present when there is no incoherent tune
spread may be absent when such a spread exists. Such modes are
``Landau damped.'' There is instead an incoherent spectrum, a
continuum of an infinite number of frequencies, which will decohere
(filament), thus not leading to collective instabilities. A
stability diagram indicates when Landau damping will be effective.
It divides the effective impedance plane, or equivalently the plane
of coherent frequency in the absence of tune spread, into regions.
The region which contains $+i\infty$ corresponds to instability.
Thus, one can substitute a simpler computation (finding discrete
eigenvalues) for a more complex computation (solving an eigenvalue
system with both a discrete and a continuous eigenvalue spectrum).
We present stability diagrams assuming a linear tune shift with
amplitude, allowing tune spread in two transverse planes or in the
longitudinal plane alone. When there is longitudinal tune spread,
this can not be done exactly, and we describe
approximations which make the computation tractable.
F. Ruggiero
Mon May 26 17:11:35 1997