mode - CERN Accelerator School

Mechanical errors differences in
frequency q y between cells
to respect the new boundary conditions
the electric field will be a linear
combination of all modes, with weight
1
2 2
f − f
(general case of small perturbation to an
eigenmode system, system
the new solution is a linear combination
of all the individual modes)
The nearest modes have the highest
effect, and when there are many modes
on the dispersion curve (number of
modes = number of cells !) the
difference in E-field between cells can
be extremely high.
Stability of long chains of
coupled resonators
0
mode 0
mode d π/2 /2
mode 2π/3
mode π
ω
E
0 6.67 13.34 20.01 26.68 33.35
0
π/2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250
1.5
1
0.5
0
-0.5 0
-1
-1.5
50 100 150 200 250
1.5
1
0.5
0
-0.5 0 50 100 150 200 250
-1
-1.5
1.5
1
0.5
0
-0.5 0
-1
-1.5
50 100150 200 250
π
k
z
45