Analytic Solutions for Compton Scattering in the High
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Transcript Analytic Solutions for Compton Scattering in the High
Analytic Solutions for Compton
Scattering in the High Energy Regime
To d d H o d g e s
A r i zo n a S ta t e U n i v e rs i t y
O l d D o m i n i o n U n i v e rs i t y 2 0 1 4 R E U Pa r t i c i p a nt
M e n t o rs : D r. Wa l l y M e l n i tc h o u k , D r. B a l š a Te r z i ć , & D r. G e o ff r e y Kra ff t
Overview
•Definitions
• Compton scattering and Thomson scattering
•Background
• Thomson scattering with relativistic electrons (Thomson source)
• Applications
• Limitations of Thomson sources
• Current corrections
•Differential cross section for Compton scattering
• Differential cross section in different reference frames
• Truncated series representation
• Comparison of expressions at fixed scattering angles
•Accommodation of a polarized photon beam
•Work in progress
• Contribution of multi-photon emitting processes
Compton Scattering
•Compton scattering
• Scattering of real photons from electrons
xˆ
•Thomson scattering in electron rest frame
• Low-energy limit (ω << m)
• Recoil of electron negligible
• Differential cross section is a function of scattering angle only
d
2
2 1 cos 2
d cos
m
• α = Fine structure constant for QED
• m = Mass of electron
zˆ
e
e
Thomson Scattering with Relativistic Electrons
•Advantages of Thomson sources
• Range of scattered photon energies is small (small bandwidth)
• Scattered photons are at greater energies than incident photons
•Applications of Thomson source photons
• Probes for nuclear physics (E > 1MeV)
• Medicine
• Higher resolution scanning
• Detection of nuclear materials
Limitations of Thomson Sources
•As total incident photon intensity increases, bandwidth of scattered photons increases
• Krafft 2004, PRL 92, 204802
•Solution to bandwidth problem
• Frequency modulation of the laser pulse
• Terzić, Deitrick, Hofler & Krafft 2014, PRL 112, 074801
• Relies on cross section
• Currently, limited to individual photon energies within the Thomson limit
•Desire to maintain low bandwidth at high intensities with photons outside of Thomson limit
• Generalization of cross section to higher energies is needed
Compton Scattering Cross Section
•General differential cross section needed
• Derive with Quantum Electrodynamics (QED)
• Begin with one photon emitting processes
k
k
•Differential cross section in electron rest frame
d
2
2
d cos
m
ω = Photon energy
E = Electron energy
ϴ = Scattering angle
2
2
sin
p
p
k
p = Electron 4-vector
k = Photon 4-vector
Primed ( ʹ ) = Final State
p
k
p
Compton Scattering Cross Section
•Differential cross section in “lab” frame
• Electron beam and photon beam are collinear
•Initial electron four-vector
p m, 0, 0, 0 p E , 0, 0, p z
ω = Photon energy
E = Electron energy
pz = Electron momentum (ẑ)
ϴ = Scattering angle
Primed ( ʹ ) = Final state
•Differential cross section in lab frame
d
2 E p z cos
m 2 2 pz pz E m 2
2
sin
2
pz
d cos
E p z cos
E
p
E
p
cos
z
z
1 E
E
Where E cos p z
Compton Scattering Expansion
•Maclaurin series expansion in powers of ω (incident photon energy)
• Electron rest frame
2
d
2
1 cos 1 cos 2
2 1 3 1 cos
2
2 1 cos 2
1 cos
d cos
m
m
m2
2
• Lab frame
2
2 T sin 2
d
2 2 T sin 2
2 1 3 2 T sin
21 cos
1
cos
d cos
E
2
3
Where :
E pz
E p z cos
T m
2
2 p z m 2
2
2
Differential Cross Section
Differential Cross Section
Differential Cross Section
Polarized Photon Beam
•For unpolarized scattering
• Average over initial electron and photon polarizations
• Sum over final electron and photon polarizations
• Klein-Nishina formula
k 2 p k 2 p
k 2 p k 2 p
1
e4
2
4 g g Tr p m 2 p k
p m
4 spins
2
p
k
2
p
k
2
p
k
• For polarized incident and scattered photons
• Do not average over initial photons polarizations
• Do not sum over scattered photon polarizations
k
k
1
e4
k
k
2
Tr
p
m
p
m
2 p k 2 p k
2 e spins
8m 2 2 p k 2 p k
k Photon 4 vector
Polarization 4 vector
Polarized Photon Beam
•Evaluate trace of polarized expression and impose conditions
k 0
p0
p p k k
k 0
•Final polarized squared amplitude (Note: Averaged and summed electron spins)
1
e4
2
2 e spins
4m 2
k k k k
p k
p 2 k k p k k k
2
4
2
4
p k p k
p k
p k p k
1
1
1
2
m 2
2
k
k
2
k
k
k
k
2 p k p k p k 2
Multi-Photon Processes
•At photon energies outside Thomson limit, the contribution of multi-photon emitting processes
may be significant
k
k
p
p
k
k
p
p
k
k
p
p
k
p
k
p
...
Summary
•Completed work
• Derivation of differential cross section in electron rest frame
• Derivation of differential cross section in “lab” frame
• Expansion of differential cross section in both frames with corrections
• In powers of ω
• Calculation of squared amplitude without incident or scattered photon polarizations
•In progress
• Summation over scattered photon polarizations
• Contribution of multi-photon emitting processes
Acknowledgements
•Special Thanks
• Dr. Wally Melnitchouk
• Dr. Balša Terzić
• Dr. Geoffrey Krafft
•Funding
•
•
•
•
Old Dominion University
National Science Foundation
Thomas Jefferson National Accelerator Facility
U.S. Department of Energy