1
B. Azadegan, S. A. Mahdipour, W. WagnerB. Azadegan, S. A. Mahdipour, W. Wagner
26.09.2013,26.09.2013, Sevan Sevan
Simulation of positron energy spectra generated by channeling radiation of GeV electrons in a tungsten
single crystal
Outline
1.Introduction
2.Theory of channeling radiation in thick crystal
3.Numerical results
4.Solution of Fokker-Plank equation
5.Positron production in a hybrid schame
6.Summary
2
1. Introduction
Channelingradiation
1
)1(
2)(
220
2
E
EX
fiE 0
CR properties
quasi-monochromatic directed
intense
tunable
- Lorentz factor
- observation angle
3
2. Theory of planar channeling radiation
ingx
nnevxV )(
Planar Continuum potential:
2
2))(
4(
4
14
1
.)(0
220 )/(
2 ngb
ii
rgi
j
gM
cn
i
jj eaeeaeaV
v
C 11 0 S i 11 0 G e 11 0 W 11 0
1 .0 0 .5 0 .0 0 .5 1 .00
2 0
4 0
6 0
8 0
1 0 0
In te rp lan ar p o s itio n Å
Pote
ntia
leV
rgi
gg
m
m
mevyxV
.),(
Axial Continuum potential:
2)2224
(4
14
1
.0
220 )/(
2 mgjuib
ii
j
jrgi
cmg eaeaea
Vv
Planar continuum potential of (110) plane of C, Si, Ge and W single crystal
Axial continuum potential of <100> axis of W single crystal
4
Classical model MeVEe 100
x
xVFtxm
)()(Planar : Planar :
2. Theory of planar channeling radiation (classical)
0
),(
),(
zm
y
yxVym
x
yxVxm
Axial : Axial :
2
20
).(2
22
).1(
))((
4dt
n
nne
c
e
dd
Ed rkti
Angular energy distribution: Angular energy distribution:
2~
2
124
2
)2
1(1
x
Tc
e
zd
dEnn
nn
Total radiated energyTotal radiated energy in thick crystal: in thick crystal:
T tin exx
T
πnω
n
T0
~~
2 ;
2~ ;4
5
3. Numerical results
0 5 . 1 0 1 5 1 . 1 0 1 4 1 .5 1 0 1 4 2 . 1 0 1 4 2 .5 1 0 1 4 3 . 1 0 1 40
5
1 0
1 5
t s
xA°
0 1 0 0 2 0 0 3 0 0 4 0 0
0
1 0
2 0
3 0
4 0
5 0
P hoton energy M eVdd
(b)
Θ0=0.25 mrad
0 5 . 1 0 1 5 1 . 1 0 1 4 1 .5 1 0 1 4 2 . 1 0 1 4 2 .5 1 0 1 4 3 . 1 0 1 4
0 .2
0 .1
0 .0
0 .1
0 .2
t s
xA°
0 1 0 0 2 0 0 3 0 0 4 0 00
5 0
1 0 0
1 5 0
2 0 0
P hoton energy M eV
dd (a)
Θ0=0.0
Planar: Planar:
Axial: Axial:
x 0 0 .3 1 9 Åy 0 0 .2 1 3 Å
0 .3 0 .2 0 .1 0 .0 0 .1 0 .2 0 .3
0 .3
0 .2
0 .1
0 .0
0 .1
0 .2
0 .3
x A°
yA°
0 50 100 150 200 250 300
0
1000
2000
3000
4000
5000
P h o to n e n e rg y M e V
dEddz
cm
1
Trajectories and CR spectra forone incident point with a) zero b) 0.25 mrad incidence angles for 2 GeV electron channeled
along the (110) plane of a tungsten single crystal.
Trajectory (rosette motion) and CR spectra for one incident point
for 1 GeV electron channeled along the <100> axis of W
single crystal.
6
4. Solution of Fokker-plank equation
Fokker-Plank equation:Fokker-Plank equation:
Boundary conditions:Boundary conditions:
Drift coefficient:Drift coefficient:
Diffusion coefficient:Diffusion coefficient:
EEss=13.6 MeV=13.6 MeV
Time parameter: Time parameter:
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For the numerical solution of the Fokker-Plank equation, a uniform distribution of the For the numerical solution of the Fokker-Plank equation, a uniform distribution of the electron across the transverse electron across the transverse xx coordinate, and a Gaussian scattering distribution tilted coordinate, and a Gaussian scattering distribution tilted by an angle by an angle θθ00, and with standard deviation , and with standard deviation yy for the angular divergence were assumed. for the angular divergence were assumed.
C
S iG e
W
a
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0
2
4
6
8
1 0
1 2
E eVCT
m W
G e
S i
C
b
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0
125
1 02 05 0
1 0 02 0 0
E e V
D1eVm
W
G e S i
C c0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0
1
1 0
1 0 0
1 0 0 0
E e V
D2eV2
m C
S iG e
W
d
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 .0 0
0 .0 2
0 .0 4
0 .0 6
0 .0 8
0 .1 0
E eV
F 0eV1
(a)(a) Time parameter Time parameter c.Tc.T(b) Drift coefficient D1,(c) Diffusion coefficient D2
(d) Initial probability density distribution F0 at z=0
calculated with standard deviation y=100 µrad and θ0=0 all as function of transverse
energy for 2 GeV electrons.
4. Solution of Fokker-plank equation
8
4. Solution of Fokker-plank equation
probability density for 2 GeV electrons channeled along (110) plane probability density for 2 GeV electrons channeled along (110) plane
9
4. Solution of Fokker-plank equation
C
S i
G eW
0 5 0 1 0 0 1 5 0 2 0 00 .0
0 .2
0 .4
0 .6
0 .8
z m
f chz
.
Electron dechanneling function: Electron dechanneling function:
Dechanneling length: Dechanneling length:
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4. Solution of Fokker-plank equation
L d ,W 2 .8 m EG eV
L d ,S i 1 9 .0 1 m EG eV
L d ,G e 7 .3 9 m EG eV
L d ,C 39.28 m EG eV
0 1 2 3 4 50
5 0
1 0 0
1 5 0
2 0 0
E e G eV
Ldm
Dechanneling lengths for (110)-planar channeling of electrons as function of the electron energy
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4. Solution of Fokker-plank equation
Radiation intensity in thick crystals: Radiation intensity in thick crystals:
3 8 .9 8 m
7 7 .9 6 m
1 1 6 .9 4 m
1 5 5 .9 3 m
1 9 4 .9 1 m
0 5 0 1 0 0 1 5 0 2 0 00
2
4
6
8
1 0
1 2
1 4
P hoton energy M eV
Inte
nsit
y
2 .8 2 m
5 .6 4 m8 .4 5 m
1 1 .2 7 m
1 4 .0 9 m
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 00
1
2
3
4
5
P hoton energy M eVIn
tens
ity
Thickness dependence of radiation spectrum for 2 GeV electrons channeled along Thickness dependence of radiation spectrum for 2 GeV electrons channeled along (110) plane of a) diamond b) Tungsten (110) plane of a) diamond b) Tungsten
(a) C(a) C (b) W(b) W
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4. Solution of Fokker-plank equation
0 100 200 300 400
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
Dependence of radiation spectrum on incidence angle of electrons for 2 Dependence of radiation spectrum on incidence angle of electrons for 2 GeV electrons channeled along (110) plane of GeGeV electrons channeled along (110) plane of Ge
Inte
nsity
Inte
nsity
Photon energy (MeV)Photon energy (MeV)
Red Red θθ=0.0=0.0Blue Blue θθ=45.35 µrad=45.35 µrad Yellow Yellow θθ=90.69 µrad=90.69 µradOrange Orange θθ=136.04 µrad=136.04 µrad Pink Pink θθ=181.39 µrad =181.39 µrad Green Green θθ=272.0 µrad =272.0 µrad
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5. Positron production in a hybrid schame
Crystalline radiator
e-
Crystal
e+
e-
e-
Amorphous convertor
e+
e-
γ
e+
e-
(a) (b)
Schemes of non-conventional positron sources. a)One single crystal. b)Crystalline target combined with an amorphous convertor.
Positron spectra are simulated by means of Positron spectra are simulated by means of GEANT4GEANT4 Monte Carlo code taking the Monte Carlo code taking the CR/CB spectra as input dataCR/CB spectra as input data
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5. Positron production in a hybrid schame
Num
ber
of p
ositr
ons
Num
ber
of p
ositr
ons
Comparison of positron energy distributions between C, Si, Ge, W radiator Comparison of positron energy distributions between C, Si, Ge, W radiator crystalscrystals
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5. Positron production in a hybrid schame
Num
ber
of p
ositr
ons
Num
ber
of p
ositr
ons
Dependence of positron energy distribution on thickness of radiator crystal Dependence of positron energy distribution on thickness of radiator crystal for 2 GeV electrons channeled along (110) plane of W radiator crystalfor 2 GeV electrons channeled along (110) plane of W radiator crystal
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5. Positron production in a hybrid schame
radrad
Num
ber
of p
ositr
ons
Num
ber
of p
ositr
ons
Dependence of positron energy distributions on incidence angle of electrons Dependence of positron energy distributions on incidence angle of electrons for 2 GeV electrons channeled along (110) plane of Gefor 2 GeV electrons channeled along (110) plane of Ge
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Planar CR emitted by electrons channeled in thick crystals has been investigated Planar CR emitted by electrons channeled in thick crystals has been investigated theoretically on the base of the solution of Fokker-Plank equation.theoretically on the base of the solution of Fokker-Plank equation.
Dependence of CR spectrum on the incidence angle of electron has been investigated. Dependence of CR spectrum on the incidence angle of electron has been investigated.
Dependence of positron energy distribution on the thickness of radiator crystal and the Dependence of positron energy distribution on the thickness of radiator crystal and the incidence angle of electron has been investigated in a hybrid positron production scheme.incidence angle of electron has been investigated in a hybrid positron production scheme.
Positron energy distributions of C, Si, Ge and W radiator crystals have been compared.Positron energy distributions of C, Si, Ge and W radiator crystals have been compared.
W radiator crystal with small channeling length produce more positron in comparison with W radiator crystal with small channeling length produce more positron in comparison with C, Si and Ge.C, Si and Ge.
Comparison of positron energy distribution in planar and axial CR needs the solution of Comparison of positron energy distribution in planar and axial CR needs the solution of Fokker-Plank equation for axial CR in two dimentions.Fokker-Plank equation for axial CR in two dimentions.
6. Summary6. Summary
