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Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 1
Michael Ljungberg
Interactions: Photons and Neutrons
Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 2
Introduction
Ionizing radiation
• Ionizations• Excitations
Direct Ionizing Radiation,
• electrons, protons, alpha-particles, heavy nucleus, charged particles
Indirect ionizing radiation
• Photons and neutrons – uncharged particles that liberate direct ionizing particles in the interaction which in turn give rise to most of the remaining ionizations.
Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 3
Introduction
Photons and neutrons
• Low probability for interaction but when it occurs it results in a ‘drastic’ energy loss “statistical slowing-down”.
Charged particles
• Undergoes a large number of interactions each with a small energy deposition “continuous slowing-down”
Photons indirectly ionizing
• free secondary e- and large distances between the ionizations
Heavy charged particles is densely ionizating
• Large difference in biological effect and absorbed dose.
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Interaction processes
Processes within keV - MeV that is of importance for our applications.
• Photo Absorption
• Compton Scattering
• Coherent Scattering
• Pair Production
• (Triplet Production)
• (Photo Nucleus reaction)
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Photoelectric effect
* High atomic number (Z), i.e. Lead
* Low photon energy
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The Photoelectric Process
The photon deliver the whole energy to an atomic e- (very low recoil energy)
• hv = Ee – EB
i. hv = incoming energy of the photon
ii. Ee = kinetic energy of the e‐
iii. EB = binding energy of e‐
Results in a vacancy in the shell => characteristic X-ray emission and Auger electrons
Process only with bounded electrons because nucleus receive an certain momentum.
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De-excitation mechanisms
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Fluroscence yield
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Medical applications of Characteristic X-ray
Lead in fingerbone can be measured by irradiating the finger by two 57Co (122 and 136 keV) sources mounted opposite to each other
The lead characteristic X-rays and incoherently scattered photons are detected using a high purity germanium detector
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Coherent scatter
•No energy loss for the photon
•Only change in direction
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Compton scatter
•Energy loss for the photon
•Change in direction
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The Compton Process
The incoming photon interacts with a free electron or with an electron in an outer electron shell. The binding energy, EB is very small compared to hv.
Relation between energy of the scattered photon and the incoming primary photon as function of scattering angle.
hh
c
,
h
, h
cTp,
e-
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The Compton Process
Law of Energy: (1)
Law of momentum: cos cos (2)
h h T
h hp
c c
sin sin (3)h
pc
2 2 22 20
2 20
2 2 22 20 0
2 2 22 2 2 20 0 0
2 2 20
2 (4)
2
mc m c pc
mc m c T
m c T m c pc
m c T m c T m c pc
pc T m c T
Relativistic relations
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The Compton Process
2 22 2 2
2
22 2 2
2cos cos cos
sin sin
h h h hp
c c c
hp
c
2 22
2
2 2 2
2cos
2 cos (5)
h h h hp
c c c
pc h hv h h
2 2 202pc h h m c h h
Eqn (2) och (3) results in
From eqn (1) T=hv-hv’ which in Eqn (4) results in
Addition:
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The Compton Process
2 2 220
2 2 2 220
20
2 2 cos
2 2 2 cos
1 cos
h h m c hv h h h h h
h h h h m c hv h h h h h
m c hv h h h
Put this into eqn (5) yield
20
2
1 cos
Define 1 cos
1 1 cos
o
m ch h h
hh
h h hm c
hh
Division with hv yield
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Compton Scattering
The photon energy before and after scattering
The energy loss
The energy loss of the photon (=Compton-e- energy) is for a given hv determined by the scattering angle .
20
1 1 cos
hh
h
m c
(1 cos )
1 (1 cos )eh h E h
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Compton Scattering
The angles and depend on each other
The Compton shift
Change in wave-length depend on .
20
sintan
1 1 cos
h
m c
20
Compton wave-length (2.426nm)
1 cosh
m c
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Compton Scattering
From the previous relations one see that for a given initial photon energy and knowing one of the following parameters
Then, the others can be determined!
, , eller e
E h
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Pair production
e +
e-* High energies (h>1.022 MeV)
* High atomic numberh >1.022 MeV
Annihilation
h= 511 keV
h= 511keV
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Pair production
If hv > 2moc2 = 1.022 MeV then it is energetically possible to create an e-e+ pair.
The photon energy, hv, is transferred to rest masses for e+ and e-
and to kinetic energy.
The process occurs close to a charged particle (nucleus) which takes up a certain momentum
Threshold energy is 2moc2 MeV for interaction with a nucleus
Threshold energy is 4moc2 MeV for interaction with a e-. (triplet of e-+ e+ + recoiling e-)
202 e e rhv m c E E E
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Photon attenuation
absorbator
d
collimator
0 d
= photon fluence,= attenuation coefficient
d=thickness
“Narrow-beam geometry”
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Photon attenuation
Absorb
X
dN
NoN(d)
( )dN N x dx
( )dN
N xdx
( ) xoN x N e
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Photon attenuation
Absorb
X
dN
NoN(d)
( )Probability for transmission > 0x
o
N xe
N
Mean-Free Path = mfp = 1/
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Attenuation coefficient
Type of processes
• Photo-electric effect• Incoherent Scattering• Coherent Scattering• Pair-production• Triplet-production• Nuclear photoeffect
Total mass-absorption coefficient is the sum of all interaction processes
photo Compton Coherent pair triplet nuclearn
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Data on the internet!
http://www.nist.gov/pml/data/xcom/index.cfm
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Cross-section as function of energy
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Photoelectric effect
* High atomic number (Z), i.e. Lead
* Low photon energy
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The Photoelectric Process
Highest likelihood for tight bounded electrons (hv > EB).
80% of all interactions by K-electrons
defined for a scatter angle and a solid angle d
d
hv
e-
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The atomic photo-effect cross section
Cross-section per atom for photo absorption
Bethe (Quantum mechanical model)
Probability for photo process increase rapidly with increasing Z and decreasing energy
,a
e
27 5 7 / 2 2, 4 10 ( , ) ( 0.3) ( ) mae s Z h Z h
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Coherent scatter
•Low atomic number (tissue, water)
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Thompson cross-section
Starting point for the classical description of coherent x-ray scattering is the Lorentz equations
• Relates the force on an unbound electron to the amplitude of an electromagnetic wave with which it interacts.
• The electromagnetic wave train causes the electron to oscillate• The electron thus giving rise to its own radiation, with a frequency identical to that of
the incoming wave.
Such considerations lead to the Thomson form of the differential (with angle) scatter cross section of a free electron
22(Thompson) (1 cos )
2ord
d
r0 is the classical electron radius, θ is the angle between the incident photon momentum and the scattered photon momentum.
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Coherent Cross-section
Thomson cross-section per electron
Cross-section per atom Independent of energy – not realistic
Formfactor include energy by the parameter x
Total cross-section
22 2(atom) (1 cos ) ( , )
2ord
F x Zd
22(electron) (1 cos )
2ord
d
2 22 2
0 0
(1 cos ) ( , )2or F x Z d d
22(atom) (1 cos )
2ord
Zd
sin( / 2)x
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Coherent Scattering
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Compton scatter
•Low atomic number (tissue, water)
Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 35
Compton scatter
hh
c
,
h
, h
cTp,
e-
1 1 cos
hh
sintan
1 1 cos
Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 36
The Klein-Nishina Cross-section
Quantum mechanical extension to the Thomson Cross- section
As energy decreases α (=hv/moc2) tends towards zero and regain Thompson differential cross-section
)cos1)(cos1(1
)cos1(1
)cos1(1
1
)cos1(2
2
222
22
KN
KNoTh
F
where
Fr
d
d
d
d
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222
, sin2
e oe
r h h hd d
h h h
The Klein-Nishina Cross-section
Differential cross-section per electron for a Compton process where the photon is scattered an angle θ in the solid angle element dΩ is given by:
, ,a e
e eZ
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Direction distribution for photons
Distribution for low energies are symmetrical around 90o
Increase in the forward direction for higher energies
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Direction distribution for photo electrons
Low energies means electron emissions symmetrical around 90 degrees.
Photons incoming with higher energies results in photo electrons that are forward directed.
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Cross-section per atom for Klein-Nishina
Cross-section for a Compton process is equal for all electrons and independent of Z (assuming that the binding energy are neglectable)
Additional corrections necessary when the photon energy is in the order of the binding energies of the atomic electron.
, ,a e
e eZ
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Pair production
e +
e-* High energies (h>1.022 MeV)
* High atomic numberh >1.022 MeV
Annihilation
h= 511 keV
h= 511keV
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Bethe cross-section for Pair Production
Valid in the range hv< 15 MeV.
For higher energies the process can occur at larger distances from the nucleus
Increase slowly with photon energy
22
2,
1 28 218ln(2 )
137 4 9 27a
eeo o
eZ
m c
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Bethe cross-section for pair production
22 1/ 3
2,
1 28 218ln(183 2 )
137 4 9 27a
eeo o
eZ
m c
For very high energies:
No energy dependence - no
Same Z dependence for all energies
2
,
a
eeZ
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Energy dependence
7/2h
If photo absorption dominates!
Decrease slowly with increasing energy of compton scattering dominates.
Increase slowly with hv if pair productiondominates.
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Z dependence
5
2
Z
Z
Z
If photo absorption dominates
Independent of Z if comptonscattering dominates
If pair production dominates
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Dominating cross section
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Differential cross-sections - Hydrogen
Low-Z material
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Differential cross-sections – Iodine
Z=52
K-Xrays at 30 keV
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Differential cross-sections - Lead
Z=82
K-Xrays at 80 keV
L,M edges visible
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Neutrons
Neutrons interacts by
• Elastic scattering by direct collision with a nucleus
• Inelastic scattering which means that the neutron is absorbed by the nucleus that will become in an excitated state and thereby emit a neutron in a different direction.
• Absorption where the nucleus is excitated. The excess of energy is emitted as gamma and/or particles
For each type above a cross-section can be defined. The total is the sum of
Compare with photons
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Cross-sections