Transverse collective instability excited by a nonuniform $nu$-shift in intense beams. Page: 3 of 18
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Observations at the CERN ISR have demonstrated the presence of a current
limiting transverse collective instability, termed the "brickwall" effect.1
The classical transverse resistive wall instability does not in the first
instance seem adequate to explain the phenomenon. It has, in fact, been shown
that the threshold for the instability is related to the parameter I/A, where
I is the circulating current and A is a measure of the width of the v-value
(betatron tune) distribution. Now, since in the ISR, current accumulation is
performed with I/A kept constant, or in any case below the design threshold
for the transverse instability, it is necessary to seek another physical factor
to explain the ob:nervntion.
The suggestion has been made that ipace-charge effects can cause a change
in the beam distribution of v-values1'4 and that this distortion of the
v-distribution will produce a current limiting instability threshold, even
though the parameter I/A remains constant.1,3,5 (A is assumed to be a nominal
value, independent of space-charge considerations.) A qualitative explanation
is given, as follows: The low current v-value is a linear funcion of momen-
tum, p or radius, R. As the current is increased the function is altered into
a curved, approximately parabolic shape. At some current value, the curve
v(R) will become flat, v' - 0, at one edge of the beam. Thus the v-value
density at this edge of the beam tends to infinity and, the idea is, the beam
becomes locally unstable. This is a very appealing explanation in view of the
observation that beam loss does indeed occur at the edge of a stacked coasting
However, this explanation, attractive as it is, is not entirely satisfying
in that one has not reconciled the effect with a "collective growth of the beam."
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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/m1/3/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.