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 






 21  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
p0
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