Non-Dipolarity of Channeling Radiation at GeV Beam Energies B. Azadegan, W. Wagner 26.09.2016, del Garda Italy 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 1
1. Introduction Planar channeling: one-dimensional problem Axial channeling: two-dimensional problem 2
2. Continuum potential Planar channeling ∑ = ingx V ( x ) v n e n ⎛ − 1 b ⎞ π 2 4 2 i ⎜ ( )( ng ) ⎟ ∑ ∑ − − M ( g ) i g . r = − 2 2 2 ⎝ 4 π ⎠ v a ( e / a ) e e a e 4 j j n 0 0 i V = j i 1 c Axial channeling The planar continuum potentials of diamond for electrons � � ∑ = i g . r V ( x , y ) v m e ⊥ m � g � g m ⎛ ⎞ 1 b � 2 2 i ⎜ ⎟ − + < > ( 2 u ) g π m 2 4 ⎜ ⎟ j − 2 i g . r 4 2 2 ∑ j ∑ π = − ⎝ ⎠ 4 v a ( e / a ) e a e � g 0 0 i m V = j i 1 c The <100> axial continuum potential of germanium for electrons 3
3. Theory of planar channeling radiation (Quantum) < E e 100 MeV Quantum mechanical model ψ 2 2 � d ( x ) ψ − + ψ = ψ ( x ) E V ( x ) ( x ) E ( x ) Wave functions and eigenvalues i i γ 2 2 m e dx = γ − 2 E 2 ( E E ) 0 i f 2 → αλ 2 2 d N ( i f ) d Γ / 2 z ∫ = γ − ψ ψ × 2 CR c 2 ( E E ) ( x ) ( x ) dzP ( z ) tot Ω π i f f i i − + 2 2 d dE � c dx ( E E ) 0 . 25 Γ 0 γ γ γ 0 tot 0.12 Yield (photons/e sr keV) 0 n = 3 0.1 � 10 0.08 e L H V n = 2 l � 20 n = 1 a 0.06 i t H 110 L plane n e Diamond t o 0.04 P � 30 n = 0 E e = 14.6 MeV 0.02 � 40 position H Å L � 0.6 � 0.4 � 0.2 0 0.2 0.4 0.6 5 10 15 20 25 30 Interplanar Photon energy (keV) A Mathematica package for calculation of planar channeling radiation spectra of relativistic electrons channeled in a diamond-structure single crystal (quantum approach) Computer Physics Communications 184 (2013) 1064-1069 4
4. Theory of planar channeling radiation (classical) dipole approximation > E e 100 MeV Classical model ∂ V ( x ) Planar : γ = = − � � m x ( t ) F ∂ x � � � � 2 � × − β × β 2 2 d E e n (( n ) ) Angular-energy distribution: � τ � ∫ = ω − i ( t k . r ) � e dt � ω Ω π 2 − β d d 4 c 2 ( 1 . n ) 0 � � � � � β = c r ( t ) = ω k n / c 2 ∞ dE e 1 [ ] ∑ 2 = Θ − η η − η + ⋅ Total radiated energy 2 � 1 ( ) x ~ ω n n n ω Δ 4 2 d z c T 2 in thin crystal: = n 1 ω T 2 π n ~ ~ T ω i t η = = = � ∫ � ; ω ; x x e ~ ω n 2 πγ 0 T 4 n 5
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/ 6
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: 7
5. Non-dipole approximation � ω 2 2 2 � 2 d E e � τ � ∫ = × β ω − i ( t k . r ) n e dt ω Ω π 2 d d 4 c 0 ⎛ ⎞ ω ω 2 2 2 ∞ d E e I ∑ = δ ⎜ ω − ⎟ n n ( ) ( ) ⎟ ⎜ ( ) ( ) ω Ω π 2 − β ϑ − β ϑ d d 2 c 1 cos 1 cos ⎝ ⎠ = n 1 z z 8
5. Non-dipole approximation The frequency spectrum is obtained by integration over all emission angles ϑ and φ . Integration over angle φ can be taken easily. J 0 (x) is the zero order Bessel function and integration over time must be done numerically. Due to δ -function under the integral over angle ϑ , the spectrum is restricted by two limits ω min = ω n /(1+ β ) and ω max = ω n /(1- β ), ( ) ⎛ ⎞ ω ϑ ω 2 2 2 ∞ d E e I ( ( ) ) ∑∫ 1 ⎜ ⎟ = δ ω − ϑ n n d cos ( ) ( ) ⎜ ⎟ ( ) ( ) ω π − β ϑ − β ϑ 2 d 2 c 1 cos 1 cos − 1 ⎝ ⎠ = n 1 z z ω 2 ∞ e ( ) ∑ = θ I n m π 2 2 c = n 1 9
5. Non-dipole approximation . 2 GeV electron (110) plane of Si Transversal component Longitudinal component 10
5. Comparison of non-dipole with dipol approximation E e =200 MeV Radiation spectra for different incidence points Total radiation spectra non-dipole for 200 MeV electrons dipole Si (110) plane 11
5. Comparison of non-dipole with dipol approximation Ee=800 MeV Radiation spectra for different incidence points Total radiation spectra non-dipole for 800 MeV electrons dipole Si (110) plane 12
6. Comparison of non-dipole with dipol approximation Ee=2 GeV Radiation spectra for different incidence points Total radiation spectra non-dipole for 2 GeV electrons dipole Si (110) plane 13
6. Comparison of non-dipole with dipol approximation Ee=5 GeV Radiation spectra for different incidence points Total radiation spectra non-dipole for 5 GeV electrons dipole Si (110) plane 14
7. Summary 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. 15
Thank you 16
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