Particle Motion Inside and Near a Linear Half-Integer Stopband Page: 3 of 17
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near an edge of the stopband. It is found that the beta function becomes infinite inversely
as the square root of the distance of the unperturbed tune from the edge of the stopband.
The basic equations used are also valid for small accelerators, where the large accelerator
approximations is not used, and all the results found are valid for small accelerators.
2. Results when the tune is not in the stopband
Before treating the interesting case where the tune is inside the stopband, it will be
helpful to first treat the case where the tune is not inside the stopband. It will then become
clear where the perturbation solution breaks down, when the tune is inside the stopband,
and how the perturbation solution can be repaired.
It will be assumed that in the absence of the perturbing fields, the tune of the particle
is vo and that the motion is stable when vo is close to q/2, where q is an integer.
It is assumed that a perturbing field is present which is given on the median plane by
ABy = -G (s) x (2-1)
G(s) is periodic in s and contains the field harmonics that can excite the stopband around
vo = q/2.
Introducing q defined by
q = x/01/2, (2-2)
where 0 is the beta function of the unperturbed field, the equation of motion can be written
d82 + V207 = .f
f = vo /3/2ABy/Bp (2-3)
f = -v#2G2/Bp
Bp = pc/e, de = ds/vo
Eqs. (2-3) are valid for large accelerators, and a small change is required' to make them
valid for small accelerators (see section 7). However, the final results found below are valid
for small accelerators that require the use of the exact linearized equations. This is shown
in section 7.
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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/m1/3/: accessed March 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.