6) Benchmarking of the trapping in 1
synchrotron oscillation
The sixth step is to benchmark the trapping of 1 test particle during
1 synchrotron oscillation.
The parameters used are those of the steps
4,
5
In order to visualize the increase of the single particle invariant
we take the tunes: Qx = 4.3504, Qy = 3.2.
Parameter used.
The test particle has the following initial coordinates:
x = 5 mm, px = y = py = 0 and z = 2.5 sigma_z, pz = 0.
Note that the probability of trapping is very sensitive to initial
condition. Therefore it may happen that "scattering" is seen in place
of trapping.
A slight variation of the particle initial coordinates should allow to
find the trapping phenomena as in the picture below.
One synchrotron oscillation takes 15000 turns.
We compare the trapping by plotting the evolution of
the single particle emittance
e_x = bet_x px^2 + 2 a_x x px + gamma_x x^2
of the test particle
(normalized with the initial value) vs. number of turns.
Note that the maximum amplitude of the center island of the island
is roughly ~30 mm (see step 5),
and a particle with initial coordinates x = 5 mm px = 0
has maximum amplitude of 8.125 mm.
Therefore one expects a maximum emittance growth of
e_x/e_x0 ~ (30/8.125)^2 = 13.6.
