August, Friday the 31st
Determination of the cavity matrix
Emmanuel Branlard
August, Friday the 31st
Presentation of the cavity
Matrix formalism
The pill box approximation
Numerical matrix determination
Plots and final comparisons
August, Friday the 31st
Presentation of the cavity
●
Copper cavity
●
3.9 Ghz
●
5 cells
●
TM-110 mode
●
Pi mode
August, Friday the 31st
Usual notations
August, Friday the 31st
Ez
●
pi/5
The pi mode
Ez
1.5
1
1
0.5
0.5
z
0
0
Ez
●
3pi/5
1.5
50
100
150
200
0
250
-0.5
-1
-1
-1.5
-1.5
Ez
1.5
1
0.5
0.5
z
0
0
50
100
150
200
-1
-1
-1.5
-1.5
5pi/5
50
100
1.5
1
0.5
z
0
0
-0.5
-1
-1.5
50
150
200
100
150
●
4pi/5
250
z
0
250
-0.5
●
100
0
-0.5
Ez
50
1.5
1
2pi/5
z
0
-0.5
●
200
250
150
200
250
August, Friday the 31st
The aim of the cavity
●
Exchange transverse and longitudinal emittance
August, Friday the 31st
Matrix formalism
●
For a simple drift
August, Friday the 31st
Problem definition
-> Finding the cavity matrix A
August, Friday the 31st
The pill box approximation
with the work of Don Edwards
●
The fields
●
Equation of motion :
August, Friday the 31st
●
Doing the same for the reference particle, after
integration we have :
August, Friday the 31st
Matrix in the pill box approximation
August, Friday the 31st
Numerical matrix determination
●
The method
●
The input vectors
August, Friday the 31st
The RF phase
●
●
Ez field
For a particle at speed of light : z(t)=ct
August, Friday the 31st
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
6.00E-02
4.00E-02
2.00E-02
Ez given by HFSS
0.00E+00
0
50
100
150
200
tyscle
arb
Ezin
-2.00E-02
-4.00E-02
-6.00E-02
Z in mm
250
300
350
RF time oscillator sine function
with arbitrary scale
August, Friday the 31st
The scaling factor
August, Friday the 31st
August, Friday the 31st
Plots and final comparisons
x trajectory with z for different phases for a particle
starting at x=0mm p=13MeV
1.80E+00
1.60E+00
100
1.40E+00
102
1.20E+00
104
m mm
xin
X in
1.00E+00
106
8.00E-01
108
6.00E-01
110
4.00E-01
112
2.00E-01
0.00E+00
0.00E+00
-2.00E-01
5.00E-02
1.00E-01
1.50E-01
2.00E-01
z in m
2.50E-01
3.00E-01
3.50E-01
August, Friday the 31st
X deflection
X of the probe particles with z
2.50E+00
2.00E+00
1.50E+00
X in
m mm
xin
1.00E+00
dp/p offset
z offset
y' offset
5.00E-01
y offset
x' offset
0.00E+00
0.00E+00
x offset
5.00E+01
1.00E+02
1.50E+02
2.00E+02
-5.00E-01
-1.00E+00
-1.50E+00
z in mm
2.50E+02
3.00E+02
3.50E+02
Ref Particle
August, Friday the 31st
The final matrix
August, Friday the 31st
Y deflection
3.50E+00
Y of the probe particles with z
3.00E+00
2.50E+00
2.00E+00
dp/p offset
z offset
y' offset
1.50E+00
m
yin
y offset
x' offset
1.00E+00
x offset
Ref particle
5.00E-01
0.00E+00
0.00E+00
-5.00E-01
5.00E+01
1.00E+02
1.50E+02
2.00E+02
z in mm
2.50E+02
3.00E+02
3.50E+02
August, Friday the 31st
8.00E+05
Ez seen by the probe particles with z
6.00E+05
4.00E+05
dp/p offset
2.00E+05
z offset
y' offset
/m
V
Ezin
y offset
0.00E+00
0.00E+00
x' offset
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
x offset
Ref Particle
-2.00E+05
-4.00E+05
-6.00E+05
z in mm
August, Friday the 31st
X of the probe particles with z
4.00E+00
3.50E+00
3.00E+00
2.50E+00
dp/p offset
z offset
2.00E+00
m
xin
y' offset
y offset
1.50E+00
x' offset
x offset
1.00E+00
Ref Particle
5.00E-01
0.00E+00
0.00E+00
-5.00E-01
5.00E+01
1.00E+02
1.50E+02
2.00E+02
z in mm
2.50E+02
3.00E+02
3.50E+02
August, Friday the 31st
2.00E+05
Ez seen by the probe particles with z
0.00E+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
-2.00E+05
-4.00E+05
dp/p offset
z offset
y' offset
-6.00E+05
y offset
/m
V
Ezin
x' offset
-8.00E+05
x offset
Ref Particle
-1.00E+06
-1.20E+06
-1.40E+06
z in mm
August, Friday the 31st
Ez with z for different phases for a particle
starting at x=0mm p=13MeV
1.00E+05
0.00E+00
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
-1.00E+05
2.50E-01
3.00E-01
100
102
-2.00E+05
xisTtle
A
104
106
-3.00E+05
108
110
-4.00E+05
-5.00E+05
112
Axis Title
3.50E-01