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5/19/16v1.0
Simulate radiation in high-gradient vacuum RF cavitiesKatsuya Yonehara and Alvin Tollestrup, APC, FermilabAbstractRadiation from high-gradient vacuum RF cavities is induced by the dynamics of fast electrons in the cavity. When electrons are incident into a cavity conductor wall they lose energy via elastic and inelastic scatterings with the conductor material. Fast electron sometimes produces secondary electrons and gamma rays in the inelastic scattering. If the kinetic energy is high enough electron can escape from the cavity enclosure. We are particularly interested in such an escaped electron that can be measured by using a Polaroid film. Radiation generated by fast electrons in the cavity is evaluated by using G4Beamline.
IntroductionWe have studied high-gradient vacuum RF cavities in multi-Tesla magnets for the ionization cooling application. Various kinds of RF cavities were operated with two RF frequencies, 201 and 805 MHz at Lab-G and MTA (Mucool Test Area). All experimental results show that cavities tend an electric breakdown at lower RF gradient in stronger magnetic field without any exemptions (shown some results in Figure 1).[footnoteRef:1] This suggests that electrons in the cavity play a key role to degrade the maximum available RF gradient.
Figure 1: Past magnetic field dependence measured in various vacuum RF cavities.
Various radiation detectors are used to investigate the electron dynamics in high gradient vacuum RF cavities. A plastic scintillator is widely used in this test. A NaI crystal is used for spectroscopy of radiation though it is not functioned in a high radiation environment. An ionization chamber is used for a background radiation monitor. By using these detectors, we found that radiation intensity is exponentially increased with the peak RF gradient. It is interpreted as the Fowler-Nordheim model. Electron are pulled out from the cavity surface by an enhanced local electric field at a micro protrusion, accelerated by the RF field, and incident into the cavity wall. This electron is called the field emission electron. These electrons produce secondary electrons and gamma rays via the elastic and inelastic scattering with the wall material. Some electron can penetrate through the wall if its kinetic energy is high enough, so called the escaped electron. Besides, Alvin suggested that the reflected electrons from the cavity wall should also contribute radiation. The energy spectrum of the reflected electrons is widely distributed since it contains not only a secondary electron but also the backscattered electrons from the wall. Indeed, a low energy secondary electron gains more energy from the RF field than the field emission electron and becomes the escaped electron.
We evaluate radiation that is generated by fast electron processing in the conductor wall material by using G4Beamline that is Geant4 based particle tracking code[footnoteRef:2]. The simulation provides the yield and the 6D phase space of the escaped electrons, the reflected electrons, and the forward and the backward scattering gamma rays with and without magnetic fields. We modeled an old pillbox cavity, called the Lab-G cavity, a new pillbox cavity, called the modular cavity. Especially, we predict the radiation signal in the modular cavity in simulation. The result will provide us the systematic error of the radiation analysis with the modular cavity.Evaluate Lab-G cavityGeometryThe first cavity is the Lab-G cavity. The Polaroid film measurement has been done with this cavity. The observed image on the film is used to validate the present simulation result. Figure 2 shows a schematic drawing of the Lab-G cavity that is modeled in simulation. The cavity has a 0.25 mm – thick Beryllium RF window at two cavity ends at z = ±40 mm. It also has a 0.2 mm – thick Titanium vacuum window outside the RF window at z = ±60 mm. There are virtual detectors at several locations that are pretended as a Polaroid film. Two planer virtual detectors locate z = ±90 mm to measure the escaped electrons. This is the actual location where the film was placed in the Lab-G cavity test. One planer detector locates z = -10 mm to measure the reflected particle from the cavity wall. One cylindrical detector with 160 mm radius is covered to measure particles that hit the cylinder wall.
The initial electron beam is located at the origin and emitted to direction. The initial electron energy is monochromatic. The initial momentum is varied from 0.5 to 2.3 MeV/c with 0.2 MeV/c steps. The magnetic field is 2.2 Tesla that is the same as the experiment. 105 test particles are used except for p = 0.62 MeV/c, at which 106 particles are used for high statistics. For simplicity, there is no RF field in the cavity. There is no material except for the window. The virtual detector is made of vacuum. In reality, there is atmospheric air between the vacuum window and the detector. We assume that the air does not change the simulation result.
It should note that the simulation has the cut-off energy 1 keV for all particles. The particle that is lower than 1 keV is killed as the ranged-out particle. Indeed, the physics below 100 eV is not reliable in Geant4. We plan to evaluate for the low energy physics process with a different simulation code. This effort is not covered in this document so far.
Figure 2: Schematic drawing of the Lab G cavity.
Yield of escaped and reflected electrons and gamma ray
Figure 3 shows the simulated fraction of electrons and gamma rays at the detector with the magnet turned on. Blue, orange, and green points are taken at z = -10, -90, and 90 mm, respectively. Red points are taken at the cylinder wall. First, the lowest initial electron momentum to be the escaped electrons is around 0.6 MeV/c. Such an electron will be produced at the peak RF gradient 12 MV/m in the Lab-G cavity. The transmission efficiency of electrons drastically increases with the initial momentum and reaches the saturation around 0.9 MeV/c. On the other hand, the yield of forward scattering gamma rays that penetrate through windows is at most 5 % in the interesting initial electron momentum range.
Second, the fraction of the reflected electrons is flat in the wide momentum range, and reaches a peak, slightly above 10 % at 1.3 MeV/c. Especially, in the low momentum region, the fraction of the reflected electron is higher than the fraction of the escaped electron. These fractions are equivalent, ~10 % at ~0.9 MeV/c. Such an electron is generated at the RF gradient ~17 MV/m. A hypothesis is made based on the result. When the cavity is excited at 17 MV/m, 10 % of the field emission electrons are accelerated and escaped from the cavity in a half RF cycle. Other 10 % of the electrons are bounced at the wall and regain energy from the RF field in other half RF cycle. The transmission efficiency is almost 100 % when the momentum of the re-accelerated electrons reaches well beyond 0.9 MeV/c. As a result, the electrons become the escaped electrons and its fraction is 10 % at the detector. Detail analysis based on the hypothesis is given in later section.
Third, the solid angle of gamma ray is toward to forward with the high momentum electron (shown green points in Figure 3) because the gamma ray is boosted by an incident electron.
Figure 3: The fraction of electrons (left) and gamma rays (right) in the Lab-G cavity with the magnet on. Green and blue points are the escaped and reflected particles, respectively. Orange points are also the escaped electrons in backward direction. Red points are the particle hitting on the cylinder wall.
Phase space of electron and gamma ray
Figure 4 shows the phase space of escaped and reflected electrons as a function of initial electron momentum. The scale and the bin size of histogram for the transverse distribution and the total kinetic energy are fixed except for the initial momenta 2.1 and 2.3 MeV/c.
Electron
Transverse (escape)
Ttotal (escape)
Transverse (reflected)
Ttotal (reflected)
x-px (escape)
x-px (reflected)
Mm
MeV
mm
MeV
mm-MeV/c
mm-MeV/c
Initial p = 0.5 MeV/c
0.7 MeV/c
0.9 MeV/c
1.1 MeV/c
1.3 MeV/c
1.5 MeV/c
1.7 MeV/c
1.9 MeV/c
2.1 MeV/c
2.3 MeV/c
Figure 4: Simulated transverse and kinetic energy distributions for the escaped and reflected electrons in the Lab-G cavity with the magnet on. The last two 2D histograms show the x-px phase space for the escaped and reflected electrons.
Figure 5 shows the standard deviation of transverse distribution and the momentum of the escaped (blue points) and the reflected electrons (orange points) as a function of the initial electron momentum, respectively. First, the transverse distribution of the escaped and the reflected electrons are very similar. On the other hand, the transverse distribution of the escaped electrons is deviated from the reflected one in the high momentum region. The multiple scattering model is agreed with the distribution of the escaped electrons at the momentum above 1.7 MeV/c. At the momentum range below 1.7 MeV/c, the electron momentum in the matter is reduced too much to apply the multiple scattering model. Indeed, some electron does not have enough energy to penetrate through the windows. Only electrons that have a small scattering angle can pass through the windows. That interprets why the beam size is small at the low momentum region. The RMS transverse beam size was increased at 2.3 MeV/c. No further investigation has been made yet.
Figure 5: The RMS transverse position (left) and the transverse momentum (right) of the escaped (blue) and the reflected (orange) electrons in the Lab-G cavity with the magnet on. Large fluctuation is due to poor statistics.
We treated the gamma ray distribution as same as the electron. Figure 6 shows the phase space of the escaped and the reflected gamma rays, respectively. Not like the electron transverse distribution, the gamma ray is distributed very widely. Since the gamma ray is not confined by the field the x-px has a phase evolution.
Gamma ray
Transverse (escape)
Ttotal (escape)
Transverse (reflected)
Ttotal (reflected)
x-px (escape)
x-px (reflected)
Mm
MeV
mm
MeV
mm-MeV/c
mm-MeV/c
Initial p = 0.5 MeV/c
0.7 MeV/c
0.9 MeV/c
1.1 MeV/c
1.3 MeV/c
1.5 MeV/c
1.7 MeV/c
1.9 MeV/c
2.1 MeV/c
2.3 MeV/c
Figure 6: Simulated transverse and kinetic energy distributions for the forward and the backward scattering gamma rays in the Lab-G cavity with the magnet on. The last two 2D histograms show the x-px phase space for these gamma rays.
Figure 7 shows the standard deviation of the transverse distribution and the momentum distribution of the escaped and the reflected gamma rays, respectively. Because the gamma ray is not confined by the magnetic field the transverse distribution is two orders of magnitude larger than that of electrons. The gamma ray has narrower distribution at higher momentum due to the Lorentz boost.
Figure 7: The RMS transverse position (left) and the transverse momentum (right) of the forward (blue) and the backward (orange) scattering gamma rays in the Lab-G cavity with the magnet on. Large fluctuation is due to poor statistics.
Magnetic field effect
We carried out other simulation in the Lab-G cavity without a magnetic field. The phase space of radiation is drastically changed. Figure 8 shows the fraction of electrons and gamma rays at the detector, respectively. The result shows that the fraction of electrons at the cylinder wall is non-zero with no magnetic field. This suggests that the electron is widely spread in the cavity via scattering. Interestingly, the critical momentum is 0.5 MeV/c without the magnetic field. It is lower than that with the field is turned on. No additional investigation has been made yet.
Figure 8: The fraction of electrons (left) and gamma rays (right) in the Lab-G cavity with the magnet off. Green and blue points are the escaped and the reflected particles, respectively. Orange points are also the escaped electrons in backward direction. Red points are the particle hitting on the side wall.
Figure 9 shows the RMS of transverse and momentum of the escaped and the reflected electrons, respectively. The beam size is two orders of magnitude larger than the beam size taken with the magnetic field. The phase space of gamma rays is not changed by the magnetic field. Thus, we omit the plot. The simulation result supports the experimental result that we do not find a clear spot on the Polaroid film when the cavity operates without a magnetic field.
Figure 9: The RMS transverse position (left) and the transverse momentum (right) of the escaped (blue) and the reflected (orange) electrons in the Lab-G cavity with the magnet off. Large fluctuation is due to poor statistics.
Establish hypothesis to explain the experimental result
Figure 8 is the note what Al Moretti made in 2003 for the Lab-G cavity experiment. The top part shows the cavity geometry. The first picture (top left) was taken at 2/14/03 when the cavity was excited at 17.5 MV/m in a magnetic field 1.3 Tesla. Two weeks later (2/28/03), they took the second picture when the cavity was excited 17.5 MV/m at 2.17 Tesla. Then they reduced the RF power down to 13.6 MV/m and taken the third picture at 2.17 Tesla. They ramped down the RF power 12 MV/m at 2.17 Tesla.
First, the spot pattern in the last three shots seems to be identical. This suggests that the field emitter stays the same location. Some emitter seems to fade out at the low RF gradient.
Alvin made a single particle tracking code written in Mathematica. It estimates the maximum achievable momentum of the surface emission electron in a pillbox cavity with a micro protrusion (Pfe). It computes the maximum achievable momentum for the secondary electron (Psec). Table 1 shows the summary of result.
Erf (MV/m)
17.5
13.6
12
Pfe (MeV/c)
0.97
0.74
0.62
Psec (MeV/c)
1.62
1.37
1.22
Table 1: Estimated maximum momentum for the field emission electrons and the secondary electrons in the Lab-G cavity.
Figure 10: Snapshot of Al’s experimental note for the Lab-G cavity test.
Figure 11 shows a schematic of the hypothesis to explain the observed image on the Polaroid film.
Figure 11: Schematic of the possible electron dynamics in a high gradient vacuum RF cavity.
Erf = 17.5 MV/m
The maximum achievable electron momentum is 0.97 MeV/c at this RF gradient. The fraction of escaped electrons is 30 % while the fraction of reflected one is 13 %. The RMS radius of the escaped electrons is 0.7 mm. On the other hand, the reflected electrons gain the energy via the RF field and reaches to 1.62 MeV/c. The transmission efficiency of such an electron is almost 100 %. However, the RMS size of the reflected electrons is 0.8 mm. Therefore, the overall RMS radius of the escaped secondary electron is 1.5 + 0.8 = 2.3 mm. The beam size becomes three times larger than the size of the field emission electrons. The hypothesis also predicts that the number of small spots should be three times larger than the number of large spots. If the brightness of the image on the film corresponds to the energy of incident electrons, the large spot is brighter than the small spot.
Erf = 13.6 MV/m
The maximum available momentum of escaped electrons and the accelerated secondary electrons are 0.74 and 1.37 MeV/c, respectively. Since the momentum for the field emission electron is so low that the fraction of reflected electrons is three orders of magnitude larger than the fraction of the escaped electrons. Thus, the number of small spots should be smaller than the number of large spots. The observed image shows that the number of large spots seems to be more than the number of small spots although it is not like three orders of magnitude difference.
Erf = 12 MV/m
The last case, when the peak RF gradient is Erf = 12 MV/m the maximum available momenta of the first and the second escaped electrons are 0.62 and 1.22 MeV/c, respectively. The former momentum is slightly above the boundary momentum to penetrate through two windows. Therefore, the observed spot will represent the reflected electrons which has the latter momentum. We claimed that the observed spot at 12 MV/m is the same location of the large spots taken at 13.6 MV/m. It suggests that the spot at 12 MV/m will be the reflected electrons.
Counterpart model
Of course, we can have other hypothesis that is simpler than the present one. There is no multiple electron production process. The multiple scattering of a single beamlet that is produced from a single micro protrusion makes the small spot image. While the large spot image is produced from a clustered micro protrusion emitter.
I claimed that the spot image on the film is very circle, which may not be possible to explain by the multiple protrusion model. I am also interested in the brightness of the image. The brightness depends on the integrated energy deposition of the incident particle on the film. It corresponds to the intensity and the kinetic energy of the beamlet. The large spot image is always brighter than the small spot image. The clustered micro protrusion model cannot explain the brightness. We should investigate the surface of the wall. The scale of the protrusion can be a few micro-meter or smaller.
Evaluate Modular Cavity
The modular cavity is designed to systematic study of the electric breakdown in the cavity. In order to investigate the material property, the RF window is replaceable. We have Cu and Be RF windows. Since the RF window seals a vacuum condition as well as a RF field there is no vacuum window in the modular cavity. The configuration significantly changes the characteristic of radiation.
Geometry
Figure 12 shows the geometry of the modular cavity in simulation. It has a 10 mm - thick Be RF-vacuum window on both ends at z = ±50 mm. The accelerating gap in the modular cavity is longer than the Lab-G cavity. The window has a counterbored structure. The thickness of thin window part is 1 mm. A virtual detector is located several places. A planer virtual detector is located at z = -60, -50, -10, 50, and 60 mm while there is a cylindrical detector. The detectors at z = -50 and 50 mm are in the counterbored window to minimize the phase space evolution in a free space. The applied magnetic field for the modular cavity is 3.0 Tesla.
The initial electron beam is located at the origin and emitted to direction. The initial momentum is varied from 0.5 to 2.9 MeV/c with 0.2 MeV/c steps. The dynamic range is wider than the Lab-G cavity since the modular cavity may operate higher RF gradient, up to 50 MV/m, than the Lab-G cavity. For simplicity, there is no RF field in the cavity. There is no material except for the window.
Figure 12: Schematic drawing of the modular cavity.
Yield of primary and secondary of electrons and gamma ray
Figure 13 shows the fraction of electrons and gamma rays at the detector. Since the RF window is thicker than the Lab-G cavity the lowest momentum to penetrate through the window is around 0.75 MeV/c that is higher than the Lab-G cavity. On the other hand, the fraction of reflected electrons is one order of magnitude smaller than the RF windows in the Lab-G cavity. Besides, the yield of gamma rays in the cavity is also significantly lower than the Lab-G cavity. Radiation in the modular cavity with Be window must be much cleaner than the Lab-G cavity.
Figure 13: The fraction of electrons (left) and gamma rays (right) in the modular cavity with the magnet on. Green and blue points are the escaped and reflected particles, respectively. Orange points are also the escaped electrons in backward direction. Red points are the particle hitting on the cylinder wall.
Phase space of electron and gamma ray
Figure 14 shows the standard deviation of the transverse distribution and the total momentum for the escaped electrons as a function of the initial electron momentum, respectively. The response functions of phase space of electrons and gamma rays are quite similar as the Lab-G cavity. However, the beam size is 3-5 times smaller than that in the Lab G cavity. The maximum RMS beam is 0.3 mm in the modular cavity.
Figure 14: The RMS transverse position (left) and the transverse momentum (right) of the escaped (blue) and the reflected (orange) electrons in the modular cavity with the magnet on. Large fluctuation is due to poor statistics.
Figure 15 shows the standard deviation of the transverse and the total momentum of the escaped and the reflected gamma rays as a function of the initial electron momentum, respectively. Again, the yield of gamma rays in the modular cavity is smaller than the Lab-G cavity. Further study needs to understand why the forward scattering gamma rays have such a small transverse distribution.
Figure 15: The RMS transverse position (left) and the transverse momentum (right) of the forward (blue) and the backward (orange) scattering gamma rays in the modular cavity with the magnet on. The plot on the bottom left is the enlarged RMS transverse size of the forward scattering gamma rays. Large fluctuation is due to poor statistics.
Magnetic field effect
Figure 16 shows the fraction of electrons and gamma rays at the detector when the magnet is turned off. Some electrons hit on the cylinder wall although its fraction is extremely low.
Figure 16: The fraction of electrons (left) and gamma rays (right). Green and blue points are the escaped and reflected particles, respectively in the modular cavity with the magnet off. Orange points are also the escaped electrons in backward direction. Red points are the particle hitting on the cylinder wall.
The RMS transverse beam distribution and the RMS transverse momentum are shown in Figure 17. Interestingly, the beam spread in the modular cavity is quite small. It is because the detector is located next to the thin RF window without a gap. Possible thickness of a Polaroid film is 0.1 mm. In this case the phase space is evolved and the beam size is enlarged about 20 % in the film. The RMS distribution of the gamma rays are quite similar as the phase space with a magnetic field. Thus, we omit the plot.
Figure 17: The RMS transverse position (left) and the transverse momentum (right) of the escaped (blue) and the reflected (orange) electrons in the modular cavity with the magnet off. The plot on the bottom left is the enlarged RMS transverse size of the escaped electrons. Large fluctuation is due to poor statistics.
Possible radiation measurement for modular cavity
We found that radiation in the modular cavity will be low. Although we do not know the yield of low energy gamma rays yet the gamma ray background in the modular cavity may be ignorable. The fraction of the reflected electrons is also very small in the modular cavity. Therefore, if our hypothesis is correct we may have a large spot image on the film but the number of large spots should be smaller than the number of small spots in the modular cavity.
If the position sense radiation detector can locate next to the thin Be window, the size of the beamlet is not affected by the external magnetic field at all. This gives us an opportunity to observe the small spot image on the film even the beamlet is not confined the external magnetic field.
What will we learn from the spot image analysis? The location of spot image tells us the location of a protrusion. The spot size would indicate whether our hypothesis is plausible or not. A long-term mystery when we observed in the all seasons cavity test, a pair of damage spots on each end RF window seems to be rotated and/or off positioned, may be solved by this analysis. However, the transverse momentum generated in the multiple scattering is larger than the original transverse momentum induced by the enhanced local electric field.
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-50510152025
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-4 -2 0 2 4 605000
10000
15000
20000
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
500010000150002000025000
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-4 -2 0 2 4 60100200300400500600
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
50
100
150
200
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1
2
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0.20.40.60.81.01.21.4
0
5000
10000
15000
20000
25000
-4-20246
0
100
200
300
400
500
600
0.20.40.60.81.01.21.4
0
50
100
150
200
-505
-2
-1
0
1
2
-6-4-202468
-1.5
-1.0
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10000
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0.5 1.0 1.5 2.00
5000100001500020000250003000035000
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-4 -2 0 2 4 6050100150200250300350
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0.5 1.0 1.5 2.0020406080100120
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0
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10000
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0
5000
10000
15000
20000
25000
30000
35000
-4-20246
0
50
100
150
200
250
300
350
0.51.01.52.0
0
20
40
60
80
100
120
-505
-2
-1
0
1
2
-6-4-202468
-2
-1
0
1
2
-4 -2 0 2 4 6020004000600080001000012000
�
0.5 1.0 1.5 2.00
5000100001500020000250003000035000
�
-4 -2 0 2 4 6050
100
150
200
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0.5 1.0 1.5 2.00
20
40
60
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-5 0 5-2-10
1
2
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-5 0 5-2-10
1
2
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0
2000
4000
6000
8000
10000
12000
0.51.01.52.0
0
5000
10000
15000
20000
25000
30000
35000
-4-20246
0
50
100
150
200
0.51.01.52.0
0
20
40
60
-505
-2
-1
0
1
2
-505
-2
-1
0
1
2
0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
2.0
Initial Momentum [MeV/c]
Electronσ x[m
m]
0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
Initial Momentum [MeV/c]
Electronσ px[
MeV
/c]
-4 -2 0 2 4 601
2
3
4
�
0.2 0.4 0.6 0.8 1.0 1.2 1.40
10
20
30
40
50
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-4 -2 0 2 4 60.00.51.01.52.02.53.0
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0.2 0.4 0.6 0.8 1.0 1.2 1.4010203040506070
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0.05
0.10
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-150-100 -50 0 50 100 150-0.04-0.020.00
0.02
0.04
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0
1
2
3
4
0.20.40.60.81.01.21.4
0
10
20
30
40
50
-4-20246
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.20.40.60.81.01.21.4
0
10
20
30
40
50
60
70
-150-100-50050100
-0.05
0.00
0.05
0.10
0.15
-150-100-50050100150
-0.04
-0.02
0.00
0.02
0.04
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4
6
8
10
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0.2 0.4 0.6 0.8 1.0 1.2 1.4050100150200250
�
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2
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
50
100
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-150-100 -50 0 50 100 150-0.20-0.15-0.10-0.050.000.05
0.10
0.15
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-150-100-50 0 50 100 150-0.10-0.050.00
0.05
0.10
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0
2
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8
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0.20.40.60.81.01.21.4
0
50
100
150
200
250
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0
1
2
3
4
5
0.20.40.60.81.01.21.4
0
50
100
150
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-0.20
-0.15
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-0.05
0.00
0.05
0.10
0.15
0.20
-150-100-50050100150
-0.10
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0.00
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10
15
20
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100
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0.1
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5
10
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0
100
200
300
400
500
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0
1
2
3
4
5
6
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0.20.40.60.81.01.21.4
0
50
100
150
200
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-0.3
-0.2
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0.0
0.1
0.2
0.3
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100200300400500600
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0.2
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-150-100 -50 0 50 100 150-0.4-0.20.0
0.2
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5
10
15
20
25
30
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0.20.40.60.81.01.21.4
0
100
200
300
400
500
600
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0
1
2
3
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0.20.40.60.81.01.21.4
0
50
100
150
200
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-0.6
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-0.2
0.0
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0.2
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10
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0.20.40.60.81.01.21.4
0
100
200
300
400
500
600
700
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0
1
2
3
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0.20.40.60.81.01.21.4
0
50
100
150
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-0.6
-0.4
-0.2
0.0
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20
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-150-100 -50 0 50 100 150-0.4-0.20.0
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10
20
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0
100
200
300
400
500
600
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0.0
0.5
1.0
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2.5
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0.20.40.60.81.01.21.4
0
20
40
60
80
100
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-0.5
0.0
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10
20
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0
100
200
300
400
500
600
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0
1
2
3
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0.20.40.60.81.01.21.4
0
10
20
30
40
50
60
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-1.0
-0.5
0.0
0.5
1.0
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0.0
0.2
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30
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50
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
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10
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0.5
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20
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0.20.40.60.81.01.21.4
0
100
200
300
400
500
600
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0.0
0.5
1.0
1.5
2.0
0.20.40.60.81.01.21.4
0
10
20
30
40
50
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-1.0
-0.5
0.0
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0
100
200
300
400
500
600
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0.0
0.5
1.0
1.5
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0.20.40.60.81.01.21.4
0
5
10
15
20
25
30
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-1.0
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0.0
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
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0.2 0.4 0.6 0.8 1.0 1.2 1.40
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10
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0.5
1.0
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0.2
0.4
0.6
0.8
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0
10
20
30
40
50
60
0.20.40.60.81.01.21.4
0
100
200
300
400
500
600
-4-20246
0.0
0.2
0.4
0.6
0.8
1.0
0.20.40.60.81.01.21.4
0
5
10
15
-150-100-50050100150
-1.0
-0.5
0.0
0.5
1.0
-150-100-50050100150
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0.5 1.0 1.5 2.00
10
20
30
40
50
60
70
Initial Momentum [MeV/c]
Gammarayσ x[m
m]
0.5 1.0 1.5 2.00.00
0.05
0.10
0.15
Initial Momentum [MeV/c]
Gammarayσ px[
MeV
/c]
0.5 1.0 1.5 2.0
10-510-40.001
0.010
0.100
1
Initial Momentum [MeV/c]
Probabilityfunction
0.5 1.0 1.5 2.05.×10-51.×10-45.×10-40.0010.0050.010
Initial Momentum [MeV/c]
Probabilityfunction
0.5 1.0 1.5 2.00
20
40
60
80
Initial Momentum [MeV/c]
Electronσ x[m
m]
0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
Initial Momentum [MeV/c]
Electronσ px[
MeV
/c]
0.5 1.0 1.5 2.0 2.5 3.0
10-40.001
0.010
0.100
1
Initial Momentum [MeV/c]
Probabilityfunction
0.5 1.0 1.5 2.0 2.5 3.0
10-510-40.001
0.010
Initial Momentum [MeV/c]
Probabilityfunction
0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
1.0
1.2
Initial Momentum [MeV/c]
Electronσ x[m
m]
0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
1.0
1.2
Initial Momentum [MeV/c]
Electronσ px[
MeV
/c]
0.5 1.0 1.5 2.0 2.50
20
40
60
80
Initial Momentum [MeV/c]
Gammarayσ x[m
m]
0.5 1.0 1.5 2.0 2.50.00
0.05
0.10
0.15
Initial Momentum [MeV/c]
Gammarayσ px[
MeV
/c]
0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Initial Momentum [MeV/c]
Electronσ x[m
m]
1.0 1.5 2.0 2.5
10-510-40.001
0.010
0.100
1
Initial Momentum [MeV/c]
Probabilityfunction
1.0 1.5 2.0 2.5
10-510-40.001
0.010
Initial Momentum [MeV/c]
Probabilityfunction
1.0 1.5 2.0 2.50
10
20
30
40
50
60
Initial Momentum [MeV/c]
Electronσ x[m
m]
1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
1.0
Initial Momentum [MeV/c]
Electronσ px[
MeV
/c]
1.0 1.5 2.0 2.50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Initial Momentum [MeV/c]
Electronσ x[m
m]