Effect of a large Oth harmonic octupole on stable betatron oscillations Page: 3 of 9
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- 2 - C18! 73-6
asimthal distance, measured along the equilibrium orbit, K(s) is the gradient
forcing function (a =rs t: , 0(s) is the octupole forcing function, given by
0(s) - .(2.2)
gere, the iagnetic field is assued to be expanded in the series
3=-5 + 3x+' "xZ +'15"x3 . . , (2.3)
and p - op , (2.4)
with p the beading radius.
Our first assumption is the stability of the linear betatroa system. Ve
thus transform to betatron coordinates1:
1i -= , (2.5)
# = (2.6)
1 d ' (2.7)
where C is the circumference of the equilibrium orbit. Equation (2.1) is thea
li + v 1i - , p3oi9 , (2.g)
where differentiation is with respect to the angle variable, *.
The smooth approximation consists of replacing p(s) in this equation by
some average value,
_ ... ...r...,.
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Month, M. Effect of a large Oth harmonic octupole on stable betatron oscillations, report, September 20, 1973; Upton, New York. (digital.library.unt.edu/ark:/67531/metadc1022442/m1/3/: accessed September 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.