Non-Dipolarity of Channeling Radiation at GeV Beam Energies B. - - PowerPoint PPT Presentation

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• B. Azadegan, W. Wagner

26.09.2016, del Garda Italy

Non-Dipolarity of Channeling Radiation at GeV Beam Energies

Outline

1. Introduction 2. Continuum potential 3. Theory of planar channeling radiation (Quantum) 4. Theory of planar channeling radiation (classical) dipole approximation 5. Non-dipole approximation 6. Comparison of non-dipole with dipole approximation 7. Summary

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• 1. Introduction

Planar channeling: one-dimensional problem Axial channeling: two-dimensional problem

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• 2. Continuum potential

Planar channeling

ingx n ne

v x V

∑

= ) (

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − −

∑ ∑

− =

2 2

) )( 4 ( 4 1 4 1 . ) ( 2 2

) / ( 2

ng b i i r g i j g M c n

i j j

e a e e a e a V v

π

π

Axial channeling

⊥

∑

=

r g i g g

m m me

v y x V

• .

) , (

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ > < + − = −

∑ ∑

− =

2 ) 2 2 2 4 ( 4 1 4 1 . 2 2

) / ( 2

m g j u i b i i j j r g i c m g

e a e a e a V v

• π

π

The planar continuum potentials of diamond for electrons The <100> axial continuum potential

• f germanium for electrons

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• 3. Theory of planar channeling radiation (Quantum)

Quantum mechanical model

MeV Ee 100 <

) ( ) ( ) ( ) ( 2

2 2 2

x E x x V dx x d me ψ ψ ψ γ = + −

• Wave functions and eigenvalues

) (x

i

ψ

i

E

25 . ) ( 2 / ) ( ) ( ) ( ) ( 2 ) (

2 tot 2 tot 2 2 2 2

∫

+ − × − = Ω →

z i i f f i c CR

Γ E E Γ z dzP x dx d x E E c dE d f i N d

γ γ γ

ψ ψ γ π αλ

• )

( 2

2 f i

E E E − = γ

0.6 0.4 0.2

0.2 0.4 0.6 Interplanar position HÅL

10 20 30 40

l a i t n e t

• P

H V

e L Diamond

H110L plane

Ee=14.6 MeV n=0 n=1 n=2 n=3

5 10 15 20 25 30 0.02 0.04 0.06 0.08 0.1 0.12

Photon energy (keV) Yield (photons/e sr keV)

Computer Physics Communications 184 (2013) 1064-1069

A Mathematica package for calculation of planar channeling radiation spectra of relativistic electrons channeled in a diamond-structure single crystal (quantum approach)

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Classical model

MeV Ee 100 > x x V F t x m ∂ ∂ − = = ) ( ) (

• γ

Planar :

• 4. Theory of planar channeling radiation (classical)

dipole approximation

2 2 ) . ( 2 2 2

) . 1 ( ) ) (( 4 dt n n n e c e d d E d

r k t i

• β

β β π ω

τ ω

− × − × = Ω

∫

−

Angular-energy distribution:

[ ]

2 ~ 2 1 2 4 2

) 2 1 ( 1

ω

η η η ω x T c e z d dE

n n n n

• ⋅

+ − − Θ = Δ

∑

∞ =

Total radiated energy in thin crystal:

∫

= = =

T t i n

e x x T πn ω n T

~ ~ 2

; 2 ~ ; 4

ω ω

πγ ω η

• )

(t r c

• =

β

c n k /

• ω

=

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• 4. Dipol approximation

Trajectories, velocities and CR spectra for two different incidence points of 2 GeV electrons to (110) plane of a Si crystal. Radiation spectrum of 2 GeV electrons channeled along (110) plane of Si in dipol approximation. Simulation of planar channeling-radiation spectra of relativistic electrons and positrons channeled in a diamond-structure or tungsten single crystal (classical approach) Paper: Nucl. Instrum. Methods B 342 (2015) 144 Program: Classical Planar Channeling Radiation Package, http://profs.hsu.ac.ir/azadegan/

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• 5. Non-dipole approximation

At relativistic energies the longitudinal velocity component is coupled with the transverse component through conservation law for the longitudinal momentum component longitudinal component:

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• 5. Non-dipole approximation

2 ) . ( 2 2 2 2

4 dt e n c e d d E d

r k t i

∫

−

× = Ω

τ ω

β π ω ω

• ( )

( )

( )

( )⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − = Ω

∑

∞ =

ϑ β ω ω δ ϑ β π ω ω cos 1 cos 1 2

1 2 2 2 2 z n n z n

I c e d d E d

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• 5. Non-dipole approximation

( ) ( )

( )

( )

( )

( ) ( ) ( )

∑ ∑∫

∞ = ∞ = −

= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − =

1 2 2 1 1 1 2 2 2 2

2 cos cos 1 cos 1 2

n m n n z n z n

I c e d I c e d E d θ π ω ϑ ϑ β ω ω δ ϑ β ϑ π ω ω

The frequency spectrum is obtained by integration over all emission angles ϑ and φ. Integration over angle φ can be taken easily. Due to δ-function under the integral over angle ϑ, the spectrum is restricted by two limits ωmin=ωn/(1+β) and ωmax=ωn/(1-β), J0(x) is the zero order Bessel function and integration over time must be done numerically.

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.

• 5. Non-dipole approximation

Longitudinal component Transversal component 2 GeV electron (110) plane of Si

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• 5. Comparison of non-dipole with dipol approximation

Ee=200 MeV

non-dipole dipole Total radiation spectra for 200 MeV electrons Si (110) plane Radiation spectra for different incidence points

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• 5. Comparison of non-dipole with dipol approximation

Ee=800 MeV

non-dipole dipole Total radiation spectra for 800 MeV electrons Si (110) plane Radiation spectra for different incidence points

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• 6. Comparison of non-dipole with dipol approximation

Ee=2 GeV

Radiation spectra for different incidence points non-dipole dipole Total radiation spectra for 2 GeV electrons Si (110) plane

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• 6. Comparison of non-dipole with dipol approximation

Ee=5 GeV

Total radiation spectra for 5 GeV electrons Si (110) plane non-dipole dipole Radiation spectra for different incidence points

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Ø

We treated planar as well as axial channeling radiation at different energies and developed several software codes (Mathematica) appropriate for quantum as well as for classical

• calculations. Users can download the codes from the internet.

Ø

We investigated the influence of non-dipolarity of channeling radiation. This effect can not be neglected at beam energies larger than about 1 GeV.

Ø

This effect is also important for the simulation of positron production by means of channeling radiation.

• 7. Summary

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Thank you