phys586-lec08-photons2

Report
Compton Scattering
 There are three related processes

Thomson scattering (classical)
 Photon-electron

Compton scattering (QED)
 Photon-electron

Rayleigh scattering (coherent)
 Photon-atom
 Thomson and Rayleigh scattering are elastic-
only the direction of the photon changes, not
its energy

Plus Thomson and Rayleigh scattering are only
important at low energies where the photoelectric
effect dominates
1
Thomson Scattering
 In Thomson scattering an electromagnetic
(EM) wave of frequency f is incident on an
electron

What happens to the electron?
 Thus the electron will emit EM waves of the
same frequency and in phase with the
incident wave
 The electron absorbs energy from the EM
wave and scatters it in a different direction
 In particular, the wavelength of the scattered
wave is the same as that of the incident wave
2
Thomson Scattering
S 
 2
E c

 2
B c
8
8

F  eE  eE0 sin t
power/area
eE0 sin t
a
m
2 e2 2
1 2 e4 E02
power emitted
P
a 
3
3
2
3c
23c m
8  e2 
 2  S
P
3  mc 
thisis thepower subtract edfrom theincomingbeam
8
T 
3
2
 e  8 2
 2  
re  0.655 1024 cm2
3
 mc 
2
3
Rayleigh Scattering
 Rayleigh scattering is scattering of light from a
harmonically bound electron
AssumingSHO with frequency0 for an electronin an atom
4


 Rayleigh   T hom son  2
  2
0




2 


 You may recall the probability for Rayleigh
scattering goes as 1/λ4

Why is the sky blue?
4
Compton Scattering
Compton scattering is the scattering of
light (photons) from free electrons
5
Compton Scattering
 Calculations
 The change in wavelength can be found by
applying

Energy conservation

h  me c  h   Ee  h   p c  m c
2

2 2
e

2 4 1/ 2
e
Momentum conservation
  
p  p  pe
 
p  p  p  2 p  p  p 2  p2  2 p  p cos
2
e
2
2
6
Compton Scattering
 From energy conservation
me2 c 4  (h  h ) 2  2me c 2 h  h   me2 c 4  pe2 c 2
 h   h   2hh 
2
pe  
 2me h  h 
 
 
2
c
 c   c 
2
2
 From momentum conservation
 
2

p  p  p  2 p  p  p 2  p2  2 p  p cos
2
e
2
h h 
 h   h  
p 
cos
 
 2
c c
 c   c 
2
2
2
e
 Eliminating pe2
mec2 h  h   hh 1  cos 
7
Compton Scattering
 Continuing on
  
h

(1  cos )
2
  me c
 And using v=c/λ we arrive at the Compton
effect
h
1 cos 
   
mec
 And h/mc is called the Compton wavelength
h
C 
 2.43 1012 m
mec
8
Compton Scattering
 Summarizing and adding a few other useful
results are
h
1  cos 
   
me c
hv
h  
 hv 
1  cos 
1  
2 
 me c 
Te  h  h 

hv    
 t an 
cot  1 
2 
 me c   2 
9
Compton Scattering
 The differential and total cross sections are
calculated in a straightforward manner using
QED
Called the Klein-Nishina formula

2
2

d r
1
 1  cos  
2
1  cos  


2 
d 2 1   1  cos  
1   1  cos  
2
e
 Compton
1    21    1
 
 ln1  2   
 2 
1  2




2 
 2re 

 1 ln1  2   1  3

2
 2

1  2 
10
Compton Scattering
 On the previous slide
 At low energies
 Compton   T hom son
hv

2
mec
8 2

re
3
 At high energies
 Compton
8r 3 
1

  ln 2  
3 8 
2
2
e
11
Compton Scattering
 Thus at high energies, the Compton scattering
cross section C goes as
 Compton
Z
~
hv
12
Compton Scattering
Graphically, d/d
13
Compton Scattering
In polar form, assume a photon incident
from the left
14
Compton Scattering
At high energies, say > 10 MeV, most of
the photons are scattered in the
forward direction
Because of the high forward
momentum of the incident photons,
most of the electrons will also be
scattered in the forward direction
15
Compton Scattering
Concerning kerma and absorbed dose,
we are particularly interested in the
scattered electron because it is ionizing
We can split the Compton cross section
into two parts: one giving the fraction
of energy transferred to the electron
and the other the fraction of energy
contained in the scattered photon
16
Compton Scattering
C   
tr
C
sc
C
T
hv  hv
  C
 C
h
h
hv 
sc
C  C
h
similarlyfor t hemass energy t ransfer
at t enuat ion coefficient
tr
C

T C
T N Av C


 h  h A
tr
C
17
Compton Scattering
Here en=tr
18
Compton Scattering
 Another useful form of the differential cross
section is d/dT, which gives the energy
distribution of the electron
19
Compton Scattering
 The maximum electron kinetic energy is given
by
 2 
hv
 and  
Tmax  hv
2
me c
 1  2 

hvme c 2
2 
 
hv  Tmax  hv1 
2
1

2

m
c
 2hv


e
and for hv large
me c 2
hv  Tmax 
 0.2555MeV
2
20
Compton Scattering
In cases where the scattered photon
leaves a detector without interaction
one would observe
21
Compton Scattering
22
Compton Scattering
me c 2
hv
hv |  

2
1  2hv / me c
2
hv |   255keV
23
Pair Production
Pair production is the dominant photon
interaction at high energies (> 10 MeV)
In order to create a pair, the photon
must have > 2me = 1.022 MeV
In order to conserve energy and
momentum, pair production must take
place in the Coulomb field of a nucleus
or electron


For nuclear field, Ethreshold > 2 x me
For atomic electron field, Ethreshold> 4 x me
24
Pair Production
25
Pair Production
 Energy and momentum conservation give
Energy hf  E  E
hf
Momentum(x)
 p cos   p cos 
c
Momentum(y) 0  p sin    p sin  
 Energy conservation can be re-written
hf 
p2 c 2  m 2 c 4 
p2 c 2  m 2c 4
 But momentum conservation (x) shows
hfmax  pc  pc
 Thus energy and momentum are not
simultaneously conserved
26
Pair Production
The processes of pair production and
bremsstrahlung are related (crossed
processes)

Thus we’d expect the cross section to
depend on the screening of atomic electrons
surrounding the nucleus
 Does the photon see nuclear charge Ze or 0 or
something in between?

The relevant screening parameter is
2
100me c hv

1/ 3
E E Z
27
Pair Production
 In the Born approximation (which is not very
accurate for low energy or high Z) one finds
No screening  1 and me c 2  h  137me c 2 Z 1/ 3
 7  2h
 109
 pair  4 Z r   ln
 f Z  

2
 54 
 9  me c
Completescreening  0  and h  137me c 2 Z 1/ 3
2
 pair
2
e


7
 1
1 / 3
 4 Z r  ln 183Z
 f Z  
9
 54
2
2
e
28
Pair Production
Notes




pair ~ Z2
Above some photon energy (say > 1 GeV),
pair becomes a constant
In order to account for pair production
from the Coulomb field of atomic electrons,
Z2 is replaced by Z(Z+1) approximately
since the cross section is smaller by a
factor of Z
Usually we don’t distinguish between the
source of the field
29
Pair Production
Notes

In the case of the nuclear field and for large
photon energies, the mean scattering angle
of the electron and positron is
me c 2
 
T
h  1.022
T
2
For h  5MeV  T  2 MeV and  15
30
Pair Production
The probability for pair production
31
Pair Production
 2me (1.022 MeV) of the photon’s energy goes
into creating the electron and positron
 The electron will typically be absorbed in a
detector
 The positron will typically annihilate with an
electron producing two annihilation photons
of energy me (0.511 MeV) each
 If these photons are not absorbed in the
detector than the pair production energy
spectrum will look like
32
Pair Production
33
Pair Production
 Similar to the photoelectric effect and
Compton scattering we define the
mass attenuation and mass energy
transfer coefficients as
 pair N Av

 pair

A
 trpair  hv  2me c 2   pair

 

hv

 
34
Photonuclear Interactions
Here a nucleus is excited by the
absorption of a photon, subsequently
emitting a neutron or proton
Most important when the energy of the
photon is approximately the binding
energy of nucleons (5-15 MeV)


Called giant nuclear dipole resonance
Still a small fraction compared to pair
production however
35
Photonuclear Interactions
 Giant dipole resonance
36
Photonuclear Interactions
These interactions would be observed
with higher energy x-ray machines


A 25 MV x-ray beam will contain neutron
contamination from photonuclear
interactions
Small effect compared to the photon beam
itself
Also important in designing shielding
since ~MeV neutrons are difficult to
contain
37
Photon Interactions
Typical
photon
cross
sections
38
Photon Interactions
Typical
photon
cross
sections
39
Photon Interactions
 Notes

Of course different interactions can occur at a given
photon energy     Z
pe
Compton   pair
  pe ZCompton  pair








A polyenergetic beam such as an x-ray beam is not
attenuated exponentially
 Lower energy x-rays have higher attenuation coefficients
than higher energy x-rays
 Thus the attenuation coefficient changes as the beam
proceeds through material
 An effective attenuation length eff can be estimated as
eff
0.693

HVL
40
Beam Hardening
41
Photon Interactions
 Let’s return to our first slide
I  I0e
 x
 As we’ve seen in the different photon
interactions


Secondary charged particles are produced
Photons can lose energy through Compton
 We define

Narrow beam geometry and attenuation
 Only primaries strike the detector or are recorded

Broad beam geometry and attenuation
 All or some of the secondary or scattered photons strike
the detector or are recorded
 Effective attenuation coefficient
’ < 
42
Photon Interactions
43
Photon Interactions
In ideal broad beam geometry all
surviving primary, secondary, and
scattered photons (from primaries
aimed at the detector) is recorded

In this case ’ = en
44
Photon Interactions
 There are three relevant mass coefficients
 N Av

mass absorptioncoefficient

A
 tr
mass energy transfer coefficient

 en
mass energy absorptioncoefficient

 en  tr
1  g 



where g is theaveragefractionof secondary
electronenergy lost toradiativeinteractions
(bremsstrahlung and annhilation)
45
Photon Interactions
 Tables of photon cross sections, mass
attenuation, and mass-energy absorption
coefficients can be found in numerous places



http://physics.nist.gov/PhysRefData/contents.html
NIST also gives material constants and composition
Useful since




fA 
f B  ...
 mixture  A
B
where f i are the weight fractions
of separateelements
46
Photon Interactions
=1/(/)
47
Photon Interactions
 Sometimes easy to loose sight of real thickness
of material involved
48
Photon Interactions
X-ray contrast depends on differing
attenuation lengths
49
Photon Interactions
 What is a cross section?
 What is the relation of  to the cross section 
for the physical process?
 has units cm2 and  has units1 / cm
  N where N is thedensityof atoms
N Av 

 is thelinear absorptioncoefficient
A
 cm2
in
is morecommon

g
50

similar documents