The Hopf's limiting cycle -- A method to measure the electron cooling force Page: 2 of 12
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1 ELECTRON COOLING FORCE
For a Gaussian distribution, the electron cooling force resembles the derivative of a Gaussian
with its extrema determined by the rms velocity spread of the electron beam. However, the
velocity spread is in general much smaller in the longitudinal than the transverse directions.
The cooling force may exhibit two peaks instead. Understanding the cooling force is crucial
in understanding the electron cooling mechanism. In this paper, we try to understand the
appearance and disappearance of the Hopf's limit cycle, from which the electron cooling
force can be deduced. A solvable example is given. The analytic solution is compared with
simulations. Part of this paper was first written in 1995 to explain experimental results [1]
gathered at the Cooler Ring of the Indiana University Cyclotron Facility (IUCF). Some
additions are included later.
2 FIXED POINT
The equations of motion of a proton to be cooled in the longitudinal phase space with electron
cooling are
=n+1 = 8n - h s (sin #n - sin #5) - f (n - 6e) , (2.1)
=n+ - n 27ThlJ bj+1 . 2.2)
where r < 0 here for the Cooler Ring at IUCF because the machine is below transition.
The cooling force is f( - te) which is so defined that it is antisymmetric with respect
to its argument. Here b is the fractional momentum offset of the particle with respect to
the synchronous particle which has a synchronous rf phase of #i, and 61 is the fractional
momentum deviation of a proton traveling at the same mean velocity as the electrons. To
make the two equations of motion more symmetric, let us redefine the momentum offset as
6 - h . (2.3)
v-s
Then (2.2) becomes
n+1= 8n + 2ffVs(sin n - sin #58) - _ (_ n - te) , (2.4)
v-s(2.5)
n+1 =n - 2Usn+1.
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Ng, King-Yuen. The Hopf's limiting cycle -- A method to measure the electron cooling force, report, September 30, 2002; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc741927/m1/2/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.