Particle Motion Inside and Near a Linear Half-Integer Stopband Page: 2 of 17
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Particle Motion Inside and Near a
Linear Half-Integer Stopband
July 20, 1995
This paper studies the motion of a particle whose tune is inside and
near a linear half-integer stopband. Results are found for the tune and beta
functions in the stable region close to an edge of the stopband. Results are
found for the growth rates and for the exponentially growing particle motion
inside the stopband. It is shown that the eigenvalues and the eigenfunctions
of the transfer matrix are real inside the stopband. All the results found are
also valid for small accelerators where the large accelerator approximation
is not used.
Inside a linear-half integer stopband the particle motion can be unstable and grow
exponentially. A result is found for the growth parameter g which is the imaginary part of
the tune. It is shown that the real part of the tune inside the stopband is constant at the
value q/2, where v = q/2 is the center of the stopband, q being some integer. The proof
given does not depend on perturbation theory, and the result follows from the symplectic
properties. The eigenvalues and eigenfunctions of the transfer matrix are shown to be real
inside the stopband. In the stable region near an edge of the stopband, the tune varies
rapidly and the beta function becomes infinite as the unperturbed tune approaches the
edge of the stopband. Results are found for the tune and beta function in the stable region
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Parzen, G. Particle Motion Inside and Near a Linear Half-Integer Stopband, report, July 1, 1995; United States. (digital.library.unt.edu/ark:/67531/metadc872467/m1/2/: accessed November 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.