Transverse collective instability excited by a nonuniform $nu$-shift in intense beams. Page: 4 of 18
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- 3 - AADD- 73-2
In other words, one must remember that the instability is, after all, excited
by the dipole moment of the entire beam.
In order to make the connection between the qualitative explanation given
above and the dispersion relation theory of the collective instability,2 we
propose a simple model that takes account of the nonuniform space charge
v-shift. What is suggested here is that the space-charge effect be included
by altering the distribution of v-values in a manner consistent with the per-
turbed v versus R curve. With the perturbed v-distribution function, one can
then solve the proper dispersion relation, finding the stability threshold and
growth rate when unstable conditions apply.
This model has the basic property that the current limitation, i.e. the
onset of the transverse instability, is a function only of the space-charge
distortion. Thus, quantitative knowledge of this latter effect as a function
of the charge distribution in the beam will allow the prediction of the current
limitation. Here we will assume that the space-charge effect is a parabolic
distortion and we will show how the instability threshold changes as a function
of a single distortion parameter. We will also show how the model predicts
initial beam losses at the beam edge when the instability threshold is crossed.
2. Effect of Space Charae v-Shift
v-Value Density Distribution
According to our model, the actual density distribution, that is, the one
to be used in the dispersion relation, can be found from an unperturbed density
distribution, an unperturbed v value and a function describing the nonuniform
space charge v-shift. We work in terms of normalized v-values, x, defined by
v - v
X d c (2.1)
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Month, M. & Jellett, K. Transverse collective instability excited by a nonuniform $nu$-shift in intense beams., report, January 1, 1973; Upton, New York. (https://digital.library.unt.edu/ark:/67531/metadc1032327/m1/4/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.