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4.3 Interaction of Photons
NPRE 441, Principles of Radiation Protection, Spring 2020
Chapter 4: Interaction of Radiation with Matter
Interactions of Photons with Matter
Reading Material:
F Chapter 5 in <<Introduction to Health Physics>>, Third edition, by Cember.
F Chapter 8 in <<Atoms, Radiation, and Radiation Protection>>, by James E Turner.
F Chapter 2 in <<Radiation Detection and Measurements>>, Third Edition, by G. F. Knoll.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Classification of Photon Interactions
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
NPRE 441, Principles of Radiation Protection, Spring 2020
Chapter 4: Interaction of Radiation with Matter – Interaction of Heavy Charged Particles
Coherent or Raleigh Scattering
•www.forbrf.lth.se/Rayleigh_scattering_2009__lecture_2_.pdf
Rayleigh scattering: Dparticle ≤ λ
NPRE 441, Principles of Radiation Protection, Spring 2020
Chapter 4: Interaction of Radiation with Matter – Interaction of Heavy Charged Particles
Coherent or Raleigh Scattering
Rayleigh scattering
F Rayleigh scattering results from the interaction between theincident photons and the target atoms as a whole.
F There is no appreciable energy loss by the photon to the atom.
F The scattering angle is very small.
Photoelectric Effect
F Photoelectric interaction is with the atom in a whole and can not take placewith free electrons.
photon theof frenquency theisconstant sPlanck' theis
vh
EhvE be -=-
F Photoelectric effect creates a vacancy in one of the electron shells, whichleaves the atom at an excited state.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
In photoelectric process, an incident photon transfer its energy to an orbitalelectron, causing it to be ejected from the atom.
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect – Absorption Edges
F Requires sufficient photon energy forP.E. interaction.
F Interaction probability decreasesdramatically with increasing energy.
F P.E. interaction is significant only forlow energy photons, when thephoton energy is close to the bindingenergies of the target atoms.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect Cross Section
Probability of photoelectric absorption per atom is
ïïî
ïïí
ì
µenergyhigh
hvZ
energylowhvZ
5.3
5
5.3
4
)(
)(s
F The interaction cross section for photoelectric effect depends strongly on Z.
Page 49, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
F Photoelectric effect is favored at lower photon energies. It is the majorinteraction process for photons at low hundred keV energy range.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Relaxation Process after Photoelectric Effect
Generation of characteristic X-rays Generation of Auger electrons
Competing
Processes
F The excited atoms will de-excite through one of the following processes:
F Auger electron emission dominates in low-Z elements. Characteristic X-rayemission dominates in higher-Z elements.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Auger Electrons
•NPRE 441, Principles of Radiation Protection
•Chapter 3: Radioactivity
149
Compton ScatteringF In Compton scattering, the incident gamma ray photon is deflected by an orbital
electron in the absorbing material.
F Part of the energy carried by the incident photon is transferred to the targetelectron in the atom, causing it to be ejected from the atom.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Basic Kinematics in Compton Scattering
The energy transfer in Compton scattering may be derived as the following:
F Assuming that the electron binding energy is small compared with the energy ofthe incident photon – elastic scattering.
F Write out the conservation of energy and momentum:
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Conservation of energy
Conservation of momentum
NPRE 441, Principles of Radiation Protection, Spring 2020
Energy Transfer in Compton Scattering
and the photon transfers part of its energy to the electron (assumed to be at restbefore the collision), which is known as the recoil electron. Its energy is simply
,)cos1(1 2
0
q-+=¢
cmhvhvvh
))cos(1(1 20
q-+-=¢-=
cmhv
hvhvvhhvErecoil
If we assume that the electron is free and at rest, the scattered gamma ray has anenergy
Initial photon energy, v: photon frequency
Scattering anglemass of electron
Reading: Page 51, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
In the simplified elastic scattering case, there is an one-to-one relationship betweenscattering angle and energy loss!!
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
assuming the bindingenergy of the electron isnegligible.
Energy Transfer in Compton Scattering
F The electron recoil angle is confined to the forward direction (0 ≤ j ≤ 90°.
F The scattering angle of the photon can take any value between 0 and 180°.
The scattering angles of the photon and the recoil electron is related by
Reading: Page 51, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Energy Transfer in Compton ScatteringChapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
F The maximum energy carried by the recoil electron is obtained by setting q to180 °,
F The maximum energy transfer is exemplified by the Compton edge in measuredgamma ray energy spectra.
Figure from Atoms, Radiation, andRadiation Protection, James ETurner, p180.
nnhmc
hE 2max 22
+=
NPRE 441, Principles of Radiation Protection, Spring 2020
Energy Transfer in Compton ScatteringChapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
))cos(1(1 20
q-+-=¢-=
cmhv
hvhvvhhvErecoil
Energy Transfer in Compton ScatteringChapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Energy Transfer in Compton Scattering
Figure from Page 321, Radiation Detection and Measurements, Third Edition, G. F. Knoll.
Small angle scattering:
Energy carried by the scattered gamma ray depends strongly on scattering angle
Large angle scattering:
Energy carried by the scattered gamma ray depends only weakly on scattering angle
Initial photon energy
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Derivation of the Relationship Between Scattering Angle and Energy Loss
The relation between energy the scattering angle and energy transfer are derivedbased on the conservation of energy and momentum:
evhehv pppp ¢¢ +=+ !!!!
evhehv EEEE ¢¢ +=+
Are those terms truly zero?
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Compton Scattering with Non-stationary Electrons – Doppler Broadening
F It is so far assumed that (a) the electron is free and stationary and (b) theincident photon is unpolarized.
)())cos(1(1 2
0
vh
cmhv
hvvh ¢±-+
=¢ sq
The one-to-one relationship between scattering angle and energy loss holds onlywhen incident photon energy is far greater than the bonding energy of theelectron…
F When an incident photon is reflected by a non-stationary electron, for examplean bond electron, an extra uncertainty is added to the energy of the scatteredphoton. This extra uncertainty is called Doppler broadening.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Compton Scattering with Non-stationary Electrons – Doppler Broadening
The Doppler broadening is stronger in Cu than in C because of the Cu electrons have greaterbonding energy.
With Dopplerbroadening
Without Doppler broadening
Without Doppler broadening
With Doppler broadening
Comparison of the energy spectra for the photons scattered by C and Cu samples. Ehv=40keV,q=90 degrees
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Angular Distribution of the Scattered Gamma Rays
The differential scattering cross section of per electron is given by the Klein-Nishina formula:
( )
) m 10 (2.818 radiuselectron classic theis and
,)]cos1(1)[cos1(
cos112
cos1)cos1(1
1)(
15-2
0
20
20
2
22222
´==
÷÷ø
öççè
æ
-++-
+÷÷ø
öççè
æ +÷÷ø
öççè
æ-+
=W
cmekr
cmhv
where
rdd
e
e
a
qaqqaq
qaqs
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Angular Distribution of the Scattered Gamma RaysThe differential scattering cross section per electron – the probability of a photonscattered into a unit solid angle around the scattering angle q, when the incidentphoton is passing normally through a thin layer of scattering material that containsone electron per unit area.
𝑑𝜎𝑑Ω
(𝜃) = 𝑟!"1
1 + 𝛼(1 − cos 𝜃)
" 1 + cos" 𝜃2
1 +𝛼" 1 − cos 𝜃 "
(1 + cos" 𝜃)[1 + 𝛼(1 − cos 𝜃)]𝑚"𝑠𝑟#$
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
d𝛺 = 2𝜋 sin 𝜃 d𝜃
Angular Distribution of the Scattered Gamma Rays
Incident photons with higher energiestend to scatter with smaller angles(forward scattering).
Incident photons with lower energy (afew hundred keV) have greater chanceof undergoing large angle scattering(back scattering).
Radial distance represents the differential cross section.
Chapter 5: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2007
The higher the energy carried by an incident gamma ray, the more likely that the gamma rayundergoes forward scattering …
Total Compton Collision Cross Section for an Electron
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Compton Collision Cross Section is defined as the total cross section per electronfor Compton scattering. It can be derived by integrating the differential crosssection over the 4 p solid angle.
Since
𝑑Ω = 2𝜋 sin 𝜃 𝑑𝜃,
then the Compton scattering cross section per electron is given by
Note that the Compton scattering crosssection per electron is given in unit of m2.
NPRE 441, Principles of Radiation Protection, Spring 2020
𝜎 = 2𝜋 %!
d𝜎d𝛺
⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚"
Question
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Given the Klein-Nishina formula, how do we derive the energy spectrum of recoilelectrons? In other words, how do we derive the probability of a gamma ray thatCompton scatted and transferred a given energy Erecoil?
The differential scattering cross section per electron – the probability of a photon scattered into a unit solidangle around the scattering angle q, when passing normally through a very thing layer of scatteringmaterial that contains one electron per unit area.
( ) ( )122
22222
)]cos1(1)[cos1(cos11
2cos1
)cos1(11)( -
÷÷ø
öççè
æ
-++-
+÷÷ø
öççè
æ +÷÷ø
öççè
æ-+
=W
srmrdd
e qaqqaq
qaqs
Energy Distribution of Compton Recoil Electrons
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Klein-Nishina formula can be used to calculate the expected energy spectrum ofrecoil electrons as the following:
recoilrecoil dEd
dd
dd
dEd q
qss WW
=
NPRE 441, Principles of Radiation Protection, Spring 2020
recoildEdE s
×Dµ
and the probability that a recoil electron possesses an energy between Erecoil-DE/2and Erecoil+DE/2 is given by
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
The three partial derivative terms on the right-hand side of the equation can bederived from the following relationships,
recoilrecoil dEd
dd
dd
dEd q
qss WW
=
𝐸%!&'() = ℎ𝑣 − ℎ𝑣* = ℎ𝑣 −ℎ𝑣
1 + ℎ𝑣𝑚+𝑐"
1 − cos 𝜃⇒
#$#%!"#$%&
= − &''(
%!"#$%&( ()* $
= − &''(
+,- )*
+, )*-'#(
+./01 2
(()* $
𝑑Ω = 2𝜋 sin 𝜃 𝑑𝜃 ⇒𝑑Ω𝑑𝜃 = 2𝜋 sin 𝜃
Energy Distribution of Compton Recoil Electrons
Klein-Nishina Formula 𝑑𝜎𝑑Ω (𝜃) = 𝑟!"
11 + 𝛼(1 − cos 𝜃)
" 1 + cos" 𝜃2 1 +
𝛼" 1 − cos 𝜃 "
(1 + cos" 𝜃)[1 + 𝛼(1 − cos 𝜃)]
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
𝑑𝜎𝑑𝐸%!&'()
=𝑑𝜎𝑑Ω
𝑑Ω𝑑𝜃
𝑑𝜃𝑑𝐸%!&'()
= 𝑟!"$
$,-($#/01 2)
" $,/01! 2"
1 + -! $#/01 2 !
($,/01! 2)[$,-($#/01 2)]× − 6"&!
7#$%&'(! 189 2
× 2𝜋 sin 𝜃
Therefore the differential cross section becomes
Or the above equitation could be re-written as
𝑑𝜎𝑑𝐸%!&'()
= 𝐹 𝜃 ⋅1
𝐸%!&'()"
𝐹 𝜃 = 𝑟!"#
#$%(#'()* +)
" #$()*( +"
1 + %( #'()* + (
(#$()*( +)[#$%(#'()* +)]× −/'0(
*12 +× 2𝜋 sin 𝜃
where
Energy Distribution of Compton Recoil Electrons
Application of the Klein-Nishina Formula (1)
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Therefore the differential cross section becomes
Remember that
𝑑𝜎𝑑𝐸%!&'()
= 𝐹 𝜃 ⋅1
𝐸%!&'()"
𝐹 𝜃 = 𝑟!"#
#$%(#'()* +)
" #$()*( +"
1 + %( #'()* + (
(#$()*( +)[#$%(#'()* +)]× −/'0(
*12 +× 2𝜋 sin 𝜃 .
where
𝐸%!&'() = ℎ𝑣 −ℎ𝑣
1 + ℎ𝑣𝑚+𝑐"
1 − cos 𝜃.
Finally, the distribution for energy carried by the recoil electron is given by
01029:;<=>
= 3 4 29:;<=>29:;<=>?
Application of the Klein-Nishina Formula (1)
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Klein-Nishina formula can be used to calculate the expected energy spectrum ofrecoil electrons as the following:
𝑑𝜎𝑑𝐸!"#$%&
=𝑑𝜎𝑑Ω
𝑑Ω𝑑𝜃
𝑑𝜃𝑑𝐸!"#$%&
(𝑚' ⋅ keV())
NPRE 441, Principles of Radiation Protection, Spring 2020
𝑃 𝐸 − ⁄*+' < 𝐸!"#$%& ≤ 𝐸 + ⁄*+
' = Δ𝐸 ⋅ ;⁄,-,+345678
𝜎+345678.+
,
and the probability that a recoil electron possesses an energy between Erecoil-DE/2and Erecoil+DE/2 is given by
where 𝜎 is the Compton scattering cross section per electron and is given by
𝜎 = 2𝜋%!
./
.!⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚" .
Application of the Klein-Nishina Formula (1)
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
From Page 311, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
F The energy distribution of the recoilelectrons derived using the Klein-Nishinaformula is closely related to the energyspectrum measured with “small”detectors (in particular, the so-calledCompton continuum).
Average fraction of energy transfer to the recoil electron through a single Compton Collision
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Of special interest for dosimetry is the average recoil electron energyEavg_recoil, since it is an approximation of the radiation dose by each photonthrough Compton scattering.
The average fraction of energy transfer to the recoil electron through asingle Compton scattering is given by
=012_456789>?
= ∫=456789=456789>?
⋅ #01029:;<=>
𝜎 ⋅ 𝑑𝐸@ABCDE ,
NPRE 441, Principles of Radiation Protection, Spring 2010
where 𝜎 is the Compton scattering cross section per electron and is given by
𝜎 = 2𝜋%!
./
.!⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚" .
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Average fraction of energy transfer to the recoil electron through a single Compton Collision
Linear Attenuation Coefficient through Compton Scattering
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
The differential Compton cross section given by the Klein-Nishina Formula can alsobe related to another important parameter for gamma ray dosimetry – the linearattenuation coefficient.
which is the probability of a photon interacting with the absorber throughCompton scattering while traversing a unit distance. NZ is the electron density ofthe absorber materials (number of electrons per 𝑚:)
𝜎&'()*+ = 𝑁𝑍𝜎 𝑚,- ,
Note that 𝜎 is the Compton scattering cross section per electron and is given by
𝜎 = 2𝜋%!
./
.!⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚" .
Pair Production
Definition:Pair production refers to the creation ofan electron-positron pair by an incidentgamma ray in the vicinity of a nucleus.
CharacteristicsF The minimum energy required is
MeVcmm
cmcmE enucleus
ee 022.1222 2
222 =»+³g
F The process is more probable with a heavy nucleus and incident gamma rayswith higher energies.
F The positrons emitted will soon annihilate with ordinary electrons near by andproduces two 511keV gamma rays.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Photonuclear ReactionF A photon can be absorbed by an atomic nucleus and knock out a nucleon.
This process is called photonuclear reaction. For example,
F The photon must possess enough energy to overcome the nuclear bindingenergy, which is generally several MeV.
F The threshold, or the minimum photon energy required, for (g,p) reaction isgenerally higher than that for (g,n) reactions. Since the repulsive Coulombbarrier that a proton must overcome to escape from the nucleus.
F Other nuclear reaction s are also possible, such as (g, 2n), (g, np), (g, a) andphoton induced fission reaction.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
MeVvalueQnHhvHMeVvalueQnBehvBe
226.2:,666.1:,
10
11
21
10
84
94
--+Þ+
--+Þ+
Interaction of Photons in Matter
From Page 50, Radiation Detection and
Measurements, Third Edition, G. F. Knoll, John
Wiley & Sons, 1999.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons
NPRE 441, Principles of Radiation Protection, Spring 2020
The Relative Importance of the Three Major Type of X and Gamma Ray Interactions
Page 52, Chapter 2 in Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
p.e.
cros
s sec
tion
= Co
mpt
on c
ross
sect
ion
Pair
prod
uctio
n cr
oss s
ectio
n =
Com
pton
cro
ss se
ctio
n
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Interactions of Photons with Matter (Revisited)
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Interaction of Photons in Matter
From Page 50, Radiation Detection and
Measurements, Third Edition, G. F. Knoll, John
Wiley & Sons, 1999.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons
NPRE 441, Principles of Radiation Protection, Spring 2020
The Relative Importance of the Three Major Type of X and Gamma Ray Interactions
Page 52, Chapter 2 in Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
p.e.
cros
s sec
tion
= Co
mpt
on c
ross
sect
ion
Pair
prod
uctio
n cr
oss s
ectio
n =
Com
pton
cro
ss se
ctio
n
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect
F Photoelectric interaction is with the atom in a whole and can not take placewith free electrons.
photon theof frenquency theisconstant sPlanck' theis
vh
EhvE be -=-
F Photoelectric effect creates a vacancy in one of the electron shells, whichleaves the atom at an excited state.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
In photoelectric process, an incident photon transfer its energy to an orbitalelectron, causing it to be ejected from the atom.
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect – Absorption Edges
F Requires sufficient photon energy forP.E. interaction.
F Interaction probability decreasesdramatically with increasing energy.
F P.E. interaction is significant only forlow energy photons, when thephoton energy is close to the bindingenergies of the target atoms.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Photoelectric Effect Cross Section
Probability of photoelectric absorption per atom is
ïïî
ïïí
ì
µenergyhigh
hvZ
energylowhvZ
5.3
5
5.3
4
)(
)(s
F The interaction cross section for photoelectric effect depends strongly on Z.
Page 49, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
F Photoelectric effect is favored at lower photon energies. It is the majorinteraction process for photons at low hundred keV energy range.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Relaxation Process after Photoelectric Effect
Generation of characteristic X-rays Generation of Auger electrons
Competing
Processes
F The excited atoms will de-excite through one of the following processes:
F Auger electron emission dominates in low-Z elements. Characteristic X-rayemission dominates in higher-Z elements.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Auger Electrons
•NPRE 441, Principles of Radiation Protection
•Chapter 3: Radioactivity
187
Compton Scattereing
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Basic Kinematics in Compton Scattering
The energy transfer in Compton scattering may be derived as the following:
F Assuming that the electron binding energy is small compared with the energy ofthe incident photon – elastic scattering.
F Write out the conservation of energy and momentum:
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Conservation of energy
Conservation of momentum
NPRE 441, Principles of Radiation Protection, Spring 2020
Energy Transfer in Compton Scattering
and the photon transfers part of its energy to the electron (assumed to be at restbefore the collision), which is known as the recoil electron. Its energy is simply
,)cos1(1 2
0
q-+=¢
cmhvhvvh
))cos(1(1 20
q-+-=¢-=
cmhv
hvhvvhhvErecoil
If we assume that the electron is free and at rest, the scattered gamma ray has anenergy
Initial photon energy, v: photon frequency
Scattering anglemass of electron
Reading: Page 51, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
In the simplified elastic scattering case, there is an one-to-one relationship betweenscattering angle and energy loss!!
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
assuming the bindingenergy of the electron isnegligible.
Angular Distribution of the Scattered Gamma RaysThe differential scattering cross section per electron – the probability of a photonscattered into a unit solid angle around the scattering angle q, when the incidentphoton is passing normally through a thin layer of scattering material that containsone electron per unit area.
𝑑𝜎𝑑Ω
(𝜃) = 𝑟!"1
1 + 𝛼(1 − cos 𝜃)
" 1 + cos" 𝜃2
1 +𝛼" 1 − cos 𝜃 "
(1 + cos" 𝜃)[1 + 𝛼(1 − cos 𝜃)]𝑚"𝑠𝑟#$
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
d𝛺 = 2𝜋 sin 𝜃 d𝜃
Angular Distribution of the Scattered Gamma Rays
Incident photons with higher energiestend to scatter with smaller angles(forward scattering).
Incident photons with lower energy (afew hundred keV) have greater chanceof undergoing large angle scattering(back scattering).
Radial distance represents the differential cross section.
Chapter 5: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2007
The higher the energy carried by an incident gamma ray, the more likely that the gamma rayundergoes forward scattering …
Energy Distribution of Compton Recoil Electrons
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Klein-Nishina formula can be used to calculate the expected energy spectrum ofrecoil electrons as the following:
recoilrecoil dEd
dd
dd
dEd q
qss WW
=
NPRE 441, Principles of Radiation Protection, Spring 2020
recoildEdE s
×Dµ
and the probability that a recoil electron possesses an energy between Erecoil-DE/2and Erecoil+DE/2 is given by
Application of the Klein-Nishina Formula (1)
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
From Page 311, Radiation Detection and Measurements, Third Edition, G. F. Knoll, John Wiley & Sons, 1999.
F The energy distribution of the recoilelectrons derived using the Klein-Nishinaformula is closely related to the energyspectrum measured with “small”detectors (in particular, the so-calledCompton continuum).
Average fraction of energy transfer to the recoil electron through a single Compton Collision
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
Of special interest for dosimetry is the average recoil electron energyEavg_recoil, since it is an approximation of the radiation dose by each photonthrough Compton scattering.
The average fraction of energy transfer to the recoil electron through asingle Compton scattering is given by
=012_456789>?
= ∫=456789=456789>?
⋅ #01029:;<=>
𝜎 ⋅ 𝑑𝐸@ABCDE ,
NPRE 441, Principles of Radiation Protection, Spring 2010
where 𝜎 is the Compton scattering cross section per electron and is given by
𝜎 = 2𝜋%!
./
.!⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚" .
Linear Attenuation Coefficient through Compton Scattering
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
The differential Compton cross section given by the Klein-Nishina Formula can alsobe related to another important parameter for gamma ray dosimetry – the linearattenuation coefficient.
which is the probability of a photon interacting with the absorber throughCompton scattering while traversing a unit distance. NZ is the electron density ofthe absorber materials (number of electrons per 𝑚:)
𝜎&'()*+ = 𝑁𝑍𝜎 𝑚,- ,
Note that 𝜎 is the Compton scattering cross section per electron and is given by
𝜎 = 2𝜋%!
./
.!⋅ sin 𝜃 ⋅ 𝑑𝜃 𝑚" .
Pair Production
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Pair Production
Definition:Pair production refers to the creation ofan electron-positron pair by an incidentgamma ray in the vicinity of a nucleus.
CharacteristicsF The minimum energy required is
MeVcmm
cmcmE enucleus
ee 022.1222 2
222 =»+³g
F The process is more probable with a heavy nucleus and incident gamma rayswith higher energies.
F The positrons emitted will soon annihilate with ordinary electrons near by andproduces two 511keV gamma rays.
Chapter 4: Interaction of Radiation with Matter – Interaction of Photons with Matter
NPRE 441, Principles of Radiation Protection, Spring 2020
Annihilation photons