Transverse coherence due to a time varying dipole deflection. Page: 3 of 22
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Here, we propose to give a detailed account of the coherent effect on a
beam of particles of a time-dependent dipole perturbation. The case of rf
knockout will be analyzed as an example. We will derive the beam response,
as given in Ref. 2, and show clearly the proper integration contour in the
v-plane that is to be taken. First we will treat the case with no space
charge or image forces. Then in a later section the influence of these forces
on the beam response is included. It will be shown that a conceptually
simple experiment can be performed to determine completely the density distri-
bution of v-values in the beam. The presence of self-forces and image forces
introduces some difficulty in the measuring procedure and we will discuss
these. Another example considered is the case of a short-pulse perturbation.
For the case of no v-spread we reproduce the results given in Ref. 3. We
further obtain the beam response when a spread in v-values is present.
2. Equation for the Bear Response
A. Motion of a particle:
We consider a system of variables describing the motion of a single
particle, as follows. The position, x, and its time derivative, x, are
variables representing the transverse motion of a particle. We denote by v
the betatron wave number which is assumed to be independent of x and A. The
azimuth, A, defines the actual geometric longitudinal position of the particle,
i.e., has the range 0 s 9 < 2T7.
The variables 9, x, A and v are time dependent, and we consider a system
whose time evolution is defined by
3. K. Hubner, CERN Report ISR-TH/69-17 (1969).
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Month, M. Transverse coherence due to a time varying dipole deflection., report, January 1, 1973; Upton, New York. (digital.library.unt.edu/ark:/67531/metadc1028178/m1/3/: accessed October 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.