Determination of the cavity matrix Determination of the cavity matrix
Emmanuel Branlard
Emmanuel Branlard
Presentation of the cavity
The pill box approximation Matrix formalism
Numerical matrix determination
Plots and final comparisons
Presentation of the cavity Presentation of the cavity
● Copper cavity
● 3.9 Ghz
● 5 cells
● TM-110 mode
● Pi mode
Usual notations
Usual notations
The pi mode The pi mode
● pi/5
-1.5 -1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250
-1.5 -1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250
-1.5 -1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250
-1.5 -1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250
-1.5 -1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250
● 3pi/5
● 2pi/5
● 4pi/5
● 5pi/5
z Ez
Ez
Ez
Ez
Ez
z z
z
z
The aim of the cavity The aim of the cavity
● Exchange transverse and longitudinal emittance
Matrix formalism Matrix formalism
● For a simple drift
Problem definition Problem definition
-> Finding the cavity matrix A
-> Finding the cavity matrix A
The pill box approximation The pill box approximation
with the work of Don Edwards with the work of Don Edwards
● The fields
● Equation of motion :
● Doing the same for the reference particle, after
integration we have :
Matrix in the pill box approximation
Matrix in the pill box approximation
Numerical matrix determination Numerical matrix determination
● The method
● The input vectors
The RF phase The RF phase
● Ez field
● For a particle at speed of light : z(t)=ct
-6.00E-02 -4.00E-02 -2.00E-02 0.00E+00 2.00E-02 4.00E-02 6.00E-02
0 50 100 150 200 250 300 350
Ez in arbitrary scale
Z in mm
The Ez field seen by a "particle" travelling on the axis x=1mm, split in a product of two functions :
a sine function and a spatial function
Ez given by HFSS
RF time oscillator sine function
with arbitrary scale
The scaling factor
The scaling factor
Plots and final comparisons Plots and final comparisons
-2.00E-01 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 1.60E+00 1.80E+00
0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 3.00E-01 3.50E-01
x in mm
z in m
x trajectory with z for different phases for a particle starting at x=0mm p=13MeV
100
102
104
106
108
110
112
X in mm
X deflection X deflection
-1.50E+00 -1.00E+00 -5.00E-01 0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00
0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02
x in mm
z in mm
X of the probe particles with z
dp/p offset z offset y' offset y offset x' offset x offset Ref Particle
X in mm
The final matrix
The final matrix
Y deflection Y deflection
-5.00E-01 0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00 3.00E+00 3.50E+00
0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02
y in mm
z in mm
Y of the probe particles with z
dp/p offset z offset y' offset y offset x' offset x offset Ref particle
-6.00E+05 -4.00E+05 -2.00E+05 0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05
0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02
Ez in V/m
z in mm
Ez seen by the probe particles with z
dp/p offset z offset y' offset y offset x' offset x offset Ref Particle
-5.00E-01 0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00 3.00E+00 3.50E+00 4.00E+00
0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02
x in mm
z in mm
X of the probe particles with z
dp/p offset z offset y' offset y offset x' offset x offset Ref Particle
-1.40E+06 -1.20E+06 -1.00E+06 -8.00E+05 -6.00E+05 -4.00E+05 -2.00E+05 0.00E+00 2.00E+05
0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02
Ez in V/m
z in mm
Ez seen by the probe particles with z
dp/p offset z offset y' offset y offset x' offset x offset Ref Particle
-5.00E+05 -4.00E+05 -3.00E+05 -2.00E+05 -1.00E+05 0.00E+00 1.00E+05
0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 3.00E-01 3.50E-01
Axis Title
Axis Title
Ez with z for different phases for a particle starting at x=0mm p=13MeV
100 102 104 106 108 110 112
