3 Non-linear Dynamics in Accelerators
93
sin(π/6)) = 0.5, while in the right column we get additional numbers sorted
according to the array index.
(0,0) 0.50000000E+00
(0,0) 0.50000000E+00
(1,0) 0.86602540E+00
(0,1) 0.00000000E+00
(2,0) -0.25000000E+00
(0,2) 0.00000000E+00
(1,1) 0.00000000E+00
(3,0) -0.14433756E+00
(0,3) 0.00000000E+00
(2,1) 0.00000000E+00
(1,2) 0.00000000E+00
The inspection shows that these numbers are the coefficients of the Taylor expansion
of sin(x) around x = π/6:
sin(
π
6
+x) = sin(
π
6
)+cos(
π
6
))x
1
−
1
2
sin(
π
6
))x
2
−
1
6
cos(
π
6
))x
3
(3.165)
We have indeed obtained the derivatives of our “algorithm” through the tracking
code.
Some examples related to the analysis of accelerator physics lattices.
In example 1 a lattice with 8 FODO cells is constructed and the quadrupole is
implemented as a thin lens “kick” in the center of the element. Note that the example
is implemented in the horizontal and the longitudinal planes. For the second example
an octupole kick is added to demonstrate the correct computation of the non-linear
effect, i.e. the detuning with amplitude.
The procedure is:
1. Track through the lattice and get Taylor coefficients
2. Produce a map from the coefficients
3. Perform a Normal Form Analysis on the map
program ex1
use my_own_da
use my_analysis
type(my_taylor) z(3)
type(normalform) NORMAL
type(my_map) M,id
real(dp) L,DL,k1,k3,fix(3)
do j = 1,8
z(1) = z(1)+DL/2*z(2)
z(2) = z(2)-kf*DL*z(1)/(1 + z(3))
z(1) = z(1)+DL/2*z(2)
z(1) =z(1)+LC*z(2)
z(1)
= z(1)+DL/2*z(2)
z(2)
= z(2)-kd*DL*z(1)/(1 + z(3))