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1
Compton ScatteringThere 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
2
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
3
Thomson Scattering
2242
2
2
2
T
2
2
2
20
3
42
3
2
0
0
22
10655.03
8
3
8
beam incoming thefrom subtractedpower theis this
3
8
emittedpower 3
2
2
1
3
2
sin
sin
power/area 88
cmrmc
e
Smc
eP
m
E
c
ea
c
eP
m
teEa
teEEeF
cBcES
e
4
Rayleigh ScatteringRayleigh scattering is scattering of light
from a harmonically bound electron
You may recall the probability for Rayleigh scattering goes as 1/λ4
Why is the sky blue?
22
02
4
hom
0 atoman in electron an for frequency with SHO Assuming
sonTRayleigh
5
Compton Scattering
Compton scattering is the scattering of light (photons) from free electrons
6
Compton ScatteringCalculationsThe change in wavelength can be found
by applying Energy conservation
Momentum conservation
2/142222 cmcphEhcmh eeee
cos22 22222 ppppppppp
ppp
e
e
7
Compton ScatteringFrom energy conservation
From momentum conservation
Eliminating pe2
hhmc
hh
c
h
c
hp
cpcmhhcmhhcm
ee
eeee
22
2)(
2
222
22422242
cos2
cos2222
2
22222
c
h
c
h
c
h
c
hp
ppppppppp
e
e
cos12 hhhhcme
8
Compton ScatteringContinuing on
And using v=c/λ we arrive at the Compton effect
And h/mc is called the Compton wavelength
)cos1(2
cm
h
e
cos1 cm
h
e
mcm
h
eC
121043.2
9
Compton ScatteringSummarizing and adding a few other
useful results are
2tan1cot
cos11
cos1
2
2
cm
hv
hhT
cmhv
hvh
cm
h
e
e
e
e
10
Compton ScatteringThe differential and total cross sections
are calculated in a straightforward manner using QED Called the Klein-Nishina formula
2
22
222
2
2
21
3121ln
2
1
21ln1
21
121
2
cos11
cos1cos1
cos11
1
2
eCompton
e
r
r
d
d
11
Compton Scattering
On the previous slide
At low energies
At high energies
2hom 3
8esonTCompton r
2
12ln
8
3
3
8 2
eCompton
r
2cm
hv
e
12
Compton ScatteringThus at high energies, the Compton
scattering cross section C goes as
hv
ZCompton ~
13
Compton Scattering
Graphically, d/d
14
Compton ScatteringIn polar form, assume a photon
incident from the left
15
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
16
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
17
Compton Scattering
A
N
h
T
h
T
h
vh
h
vhhv
h
T
CAvCtrC
CscC
CCtrC
scC
trCC
tcoefficienn attenuatio
nsferenergy tra mass for thesimilarly
18
Compton Scattering
Here en=tr
19
Compton ScatteringAnother useful form of the differential
cross section is d/dT, which gives the energy distribution of the electron
20
Compton ScatteringThe maximum electron kinetic energy is
given by
MeVcm
Thv
hv
hvcm
chvmhvThv
cm
hvhvT
e
e
e
e
2555.02
large for and
221
21
and 21
2
2
max
2
2
max
2max
21
Compton Scattering
In cases where the scattered photon leaves a detector without interaction one would observe
22
Compton Scattering
23
Compton Scattering
keVvh
cm
cmhv
hvvh e
e
255|
2/21|
2
2
24
Pair ProductionPair 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
25
Pair Production
26
Pair ProductionEnergy and momentum conservation give
Energy conservation can be re-written
But momentum conservation (x) shows
Thus energy and momentum are not simultaneously conserved
sinsin0 (y) Momentum
coscos (x) Momentum
Energy
pp
ppc
hf
EEhf
42224222 cmcpcmcphf
cpcphf max
27
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 nucleusDoes the photon see nuclear charge Ze or
0 or something in between? The relevant screening parameter is
3/1
2100
ZEE
hvcme
28
Pair Production In the Born approximation (which is not
very accurate for low energy or high Z) one finds
54
1183ln
9
74
137 and 0 screening Complete
54
1092ln
9
74
137 and 1 screening No
3/122
3/12
222
3/122
ZfZrZ
Zcmh
Zfcm
hrZ
Zcmhcm
epair
e
eepair
ee
29
Pair ProductionNotes
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
30
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
15 and 25For
2
022.1
2
MeVTMeVh
hT
T
cme
31
Pair ProductionThe probability for pair production
32
Pair Production2me (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
33
Pair Production
34
Pair ProductionSimilar to the photoelectric effect
and Compton scattering we define the mass attenuation and mass energy transfer coefficients as
paire
trpair
pairAvpair
hv
cmhv
A
N
22
35
Photonuclear InteractionsHere 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
36
Photonuclear InteractionsGiant dipole resonance
37
Photonuclear InteractionsThese 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
38
Photon Interactions
Typical photon cross sections
39
Photon Interactions
Typical photon cross sections
40
Photon InteractionsNotes
Of course different interactions can occur at a given photon energy
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
pairComptonpe
pairComptonpe
Z
Z
HVLeff
693.0
41
Beam Hardening
42
Photon InteractionsLet’s return to our first slide
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 ’ <
xeII 0
43
Photon Interactions
44
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
45
Photon InteractionsThere are three relevant mass
coefficients
n)annhilatio and hlung(bremsstra
nsinteractio radiative lost toenergy electron
secondary of fraction average theis g where
1
tcoefficien absorptionenergy mass
tcoefficiennsfer energy tra mass
tcoefficien absorption mass
g
A
N
tren
en
tr
Av
46
Photon InteractionsTables 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
elements separate of
fractions weight theare where
...
i
BB
AAmixture
f
ff
47
Photon Interactions
=1/(/)
48
Photon InteractionsSometimes easy to loose sight of real
thickness of material involved
49
Photon Interactions
X-ray contrast depends on differing attenuation lengths
50
Photon InteractionsWhat is a cross section?What is the relation of to the cross
section for the physical process?
common more isin
tcoefficien absorptionlinear theis
atoms ofdensity theis where
/1 units has and units has
2
2
g
cm
A
N
N N
cmcm
Av