Collective Instabilities in RHIC Page: 48 of 55
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longitudinal stability calculations [40].
Consider injection, which appears to be the most worrisome part of the cycle.
The quantity of interest is the growth rate amax of the most unstable mode. For
a normalized 95% emittance of 157rpm, the incoherent space charge tune shifts
for protons and gold satisfy Awj/w, ^ 20 and Awj/w, ~ 13, respectively. Since
the frequency shift of the first rigid mode is comparable in magnitude to the
space charge frequency shift, one requires max ? 20 and Amax ? 13 for protons
and gold, respectively. This is a lot of modes. Calculations for such large values
of max are very time consuming in the current computational scenario. In an
attempt to understand the parameter dependence, calculations are performed
for a range of values of Awi/w, < 10 and max < 15.
At the time of writing, this promising calculational technique is not ready
for firm predictions. However, development of the technique continues.
6.3 Landau damping in RHIC
The two main sources of Landau damping in RHIC are nonlinearity of the rf
restoring force, and nonlinearity of the lattice. While the former leads to an am-
plitude dependent synchrotron tune Q,(A6), the latter leads to an amplitude
dependent betatron tune Q(Ap). The size of the synchrotron tune spread de-
pends on the relative size of the bunch and bucket areas. Gold ion bunches that
diffuse through intrabeam scattering during storage eventually fill the rf bucket
(and spill out of it), with a consequently large synchrotron tune spread. A
significant betatron tune spread is present even at injection, when it is mainly
driven by the chromaticity sextupoles, to second order in their strength. At
storage it is dominated by IR quadrupole triplet nonlinearities. If necessary,
octupole circuits in RHIC can be powered to enhance the betatron tune spread.
Longitudinal instabilities may be stabilized by the synchrotron tune spread.
For a longitudinal density of the form
P(O) a (02 - 02)2 (6.8)
dipole longitudinal instabilities are damped if the tune shift AQo, calculated
using a pure quadratic rf potential, satisfies
AQo _ Q,(0) - Q,( )/4 (6.9)
where Q, (r) is the synchrotron tune as a function of amplitude in the octupole
approximation [41]. For Gaussian bunches the stability criterion becomes [42]
AQo < Q,(0) - Q,(o-)/2 (6.10)
For nearly full buckets the octupole approximation breaks down and a more
careful analysis is needed. Quadrupole and higher order mode longitudinal
oscillations are usually stable if the dipole modes are stable.45
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Blaskiewicz, M.; Deng, D. P.; MacKay, W. W.; Mane, V.; Peggs, S.; Ratti, A. et al. Collective Instabilities in RHIC, report, September 20, 1994; United States. (https://digital.library.unt.edu/ark:/67531/metadc867889/m1/48/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.