Interaction of x rays with matter is an important topic to understand

1. Gamma- and X-ray
Interaction with
Matter
BAEN-625 Advances in Food Engineering

2. Photon Interactions

3. y When a radiation beam passes through material,
energy is lost from the incident beam
y Some energy is imparted to the medium and
some of it leaves the volume
Energy absorbed
l
tr
ab E
E
E Δ
−
Δ
=
Δ
Energy transferred
from the beam
Energy lost

4. y Exponential law
Absorption process
L
en
e
I
I μ
−
= 0
Io = initial intensity of the beam before absorption
I = final intensity of beam
μen = absorption coefficient of the material [1/cm]
L = thickness [cm]
Intensity = photon energy fluence rate [ MeV/s]

5. y Attenuation coefficient, μ
Absorption process
μen for cm2/electron
μ/ρ for cm2/g (mass coefficient)
μa for cm2/atom (atomic coefficient)
μ for cm-1
e
A
e
a
A
Z
N
Z μ
ρ
μ
μ
μ ⎟
⎠
⎞
⎜
⎝
⎛
=
= ;
Avogadro’s number

6. Interactions type of interest
y Three modes of interaction
(depending on the photon
energy)
y Photoelectric effect, PE
y Compton effect, CE
y Pair production, PP
y Photons transfer their
energy to electrons
y Electrons then impart
energy to matter in many
Coulomb-force interactions
along theirs tracks

7. Photon interaction
y Depends on
y Photon energy
y Atomic number Z of the
absorbing medium
y PE dominant at lower
photon energies
y CE at medium
energies
y PP at higher energies
hv
E =
γ
Two kinds of interactions are
equally probable
CE dominance is very broad
for low Z values

8. Compton Effect
y Two aspects
y Kinematics – relates the energies and angles of particles
when Compton event occurs
y Cross section – predicts the probability that a Compton
interaction will occur
y Assumed that the electron struck by the incoming
photon is initially unbound and stationary

9. Compton effect
y Only part of the incident energy is absorbed to
eject an electron (Compton electron)
y During interaction:
y The photon disappears
y A secondary photon is created with reduced energy –
propagating in a changed direction

10. Compton Effect

11. Kinematic of Compton effect
y A photon of energy Eγ
incident from the left
strikes an electron,
scattering it in an angle
θ with KE T
y The scattered photon
departs at angle φ on
the opposite side
y Energy and
momentum are
conserved

12. Kinematics
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
−
=
−
+
=
2
tan
1
cot
'
)
cos
1
)(
/
(
1
'
2
2
φ
θ
φ
c
m
hv
hv
hv
T
c
m
hv
hv
hv
o
o
Rest energy of electron = 0.511 MeV
T, hv and hv’ [MeV]

13. Kinematics
y Max electron energy
resulting from a head-
on Compton collision
(θ=0o) by a photon of
energy hv occurs when
φ =180o
T, hv and hv’ [MeV]
MeV
hv
hv
T
hv
hv
T
c
m
hv
hv
hv
o
511
.
0
2
)
(
2
'
)
cos
1
)(
/
(
1
'
2
max
2
+
=
−
=
−
+
=
φ

14. Kinematics of hv, hv’ and T
'
)
cos
1
)(
/
(
1
' 2
hv
hv
T
c
m
hv
hv
hv
o
−
=
−
+
=
φ

15. Electron and photon scattering
angles
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
2
tan
1
cot 2
φ
θ
c
m
hv
o

16. Interactions Cross Section for
Compton Effect
y Thomson scattering
y Klein-Nishina cross sections (K-N)

17. Total Thomson Scattering Cross
Section
y Can be thought of as an effective target area
y The probability of a Thomson-scattering event
occurring when a single photon passes through a
layer containing 1 electron/cm2
y Fraction of a large number of incident photons that
scatter in passing through the same layer, i.e.,
approximately 665 events for 1027 photons
electron
cm
e /
10
65
.
6 2
23
0
−
×
=
σ

18. Klein-Nishina cross sections
y Thomson’s cross section
y Independent of hv
y value is too large for hv > 0.01 MeV
y K-N differential cross section for low energies
)
cos
1
(
2
)
sin
2
(
2
2
2
0
2
2
0
φ
φ
σ
φ
+
=
−
=
Ω
r
r
d
de
solid angle
cm
c
m
e
r
o
13
2
2
2
0 10
818
.
2 −
×
=
=
[cm2 sr-1 per electron]

19. Total K-N cross section per
electron
MeV
c
m
c
m
MeV
hv
r
o
o
e
511
.
0
;
]
[
)
2
1
(
3
1
2
)
2
1
ln(
)
2
1
ln(
2
1
)
1
(
2
1
2
2
2
2
2
2
0
=
=
⎭
⎬
⎫
⎩
⎨
⎧
+
+
−
+
+
⎥
⎦
⎤
⎢
⎣
⎡ +
−
+
+
+
=
α
α
α
α
α
α
α
α
α
α
α
π
σ

20. K-N Compton effect cross
section
y Is independent of the atomic number Z
y So, the K-N cross section per atom of any Z is:
0
Z
e ∝
σ
]
/
[cm2
atom
Z e
a σ
σ ⋅
=

21. K-N Compton mass attenuation
coefficient
material
of
gram
per
electrons
of
number
]
[g/cm
density
material
of
mole
per
grams
of
number
A
element
an
of
atom
per
electron
of
number
element
any
of
weight
atomic
-
gram
a
in
atoms
of
number
the
constant
s
Avogadro'
10
0022
.
6
]
/
[cm
3
1
23
2
=
=
=
=
=
×
=
=
−
A
Z
N
Z
mole
N
g
A
Z
N
A
A
e
A
ρ
σ
ρ
σ

22. K-N energy transfer cross
section for the Compton effect
)
2
1
ln(
2
1
2
1
1
)
2
1
(
3
4
)
2
1
(
)
1
2
2
)(
1
(
)
2
1
(
3
1
)
2
1
(
)
1
(
2
2
'
sin
'
'
(
'
2
3
3
3
2
2
2
2
2
2
2
0
2
2
2
0
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
π
σ
φ
σ
σ
φ
φ
+
⎟
⎠
⎞
⎜
⎝
⎛
+
−
+
−
+
−
⎥
⎦
⎤
⎢
⎣
⎡
+
−
−
+
−
+
+
−
+
+
=
⎟
⎠
⎞
⎜
⎝
⎛ −
⎟
⎠
⎞
⎜
⎝
⎛
−
+
⎟
⎠
⎞
⎜
⎝
⎛
=
⋅
Ω
=
Ω
r
hv
hv
hv
hv
hv
hv
hv
hv
hv
r
hv
T
d
d
d
d
tr
e
e
tr
e
cm2/e]
[cm2/sr e]
This cross section, multiplied by the unit thickness 1 e/cm2, represents
the fraction of the energy fluence in a photon bean that is diverted to the
recoil electron

23. K-N Compton effect cross
section
photons)
scattered
by the
carried
energy
for the
section
cross
N
-
(K
s
e
tr
e
e σ
σ
σ =
−

24. Average energy of the Compton
recoil electrons
y The average fraction of the incident photon’s
energy given to electron:
y The average energy of the Compton recoil
electrons generated by photons of energy hv:
σ
σ
e
tr
e
hv
T
=
σ
σ
e
tr
e
hv
T ⋅
=

25. Mean fractions
photon
scattered
by
retained
energy
fraction
mean
'
electron
recoil
given to
energy
fraction
mean
=
=
hv
hv
hv
T

26. K-N Compton mass-energy
transfer coefficient
]
/
[cm2
g
A
Z
N
e
A
tr
σ
ρ
σ
=

27. Photoelectric effect
y Most important interaction of low-energy photons
with matter
y Cross-sections for photoelectric effect increase
strongly, specially for high-Z media
y Photoelectric effect totally predominates over the
Compton effect at low photon energies

28. The Photoelectric effect
y A photon is absorbed
completely with the
ejection of an
electron
b
E
hv
T −
=
Energy of a
photon
in the beam
Binding
energy of
an electron
in an atom
KE of the
ejected
electron

29. Photoelectric effect

30. Kinematics of Photoelectric
Effect
y A photon cannot give
up all of its energy in
colliding with a free
electron (see case of
CE)
y For PE effect to take
place the electron to
be ejected must be
bound in a molecule or
atom

31. Kinematics of Photoelectric
Effect
y The PE cannot take
place unless hv>Eb for
that electron
y The smaller hv is, the
more likely is the
occurrence of PE
y Ta = KE given to the
recoiling atom = 0
a
b T
E
hv
T −
−
=

32. Interactions Cross Section for
Photoelectric Effect
y More difficult to derive than for CE
y There is no single equation
y Published tables give results

33. Photoelectric interaction cross
sections
y Interaction cross section per atom, integrated over
all angles of photoelectron emission
y k = constant
y n ~ 4 at hv = 0.1 MeV (4.6 at 3 MeV)
y m ~ 3 at hv = 0.1 MeV (1.0 at 5 MeV)
y For hv < 0.1 MeV
]
/
[cm
)
(
2
atom
hv
Z
k m
n
a ≅
τ
]
/
[
]
/
[cm
)
(
2
3
2
3
4
g
cm
hv
Z
atom
hv
Z
a ⎟
⎠
⎞
⎜
⎝
⎛
∝
≅
ρ
τ
τ

34. Energy-transfer cross section for
the PE
⎥
⎦
⎤
⎢
⎣
⎡ −
−
−
=
−
=
hv
v
h
Y
P
P
v
h
Y
P
hv
hv
E
hv
hv
T
L
L
L
K
K
K
K
tr
b
)
1
(
ρ
τ
ρ
τ

35. Mass-attenuation coefficient for
Carbon

36. Mass-attenuation coefficient for
Lead

37. Pair production
y For photoelectric and
Compton effects the
interaction of photon is with
electrons of atom
y Pair production involves
interaction of photons with
the nucleus of the atom
y The photon disappears
and a positron and an
electron appear
y ENERGY IS CONVERTED
TO MASS!!

38. Pair production
y It can only occur in a
Coulomb force field,
usually near the field of
an atomic nucleus
y It can also take place,
with lower probability,
in the field of an atomic
electron
y A min. photon energy
2m0c2=1.022 MeV is
required

39. Pair production in the nuclear
Coulomb force field
)
(
2
022
.
1
022
.
1
2
2
0
2
0
radians
T
c
m
MeV
hv
T
T
T
MeV
T
T
c
m
hv
≅
−
=
+
+
=
+
+
=
+
−
+
−
θ

40. Atomic differential cross section
( )
electron
cm
c
m
e
r
atom
cm
dT
c
m
hv
P
Z
d a
/
10
80
.
5
137
1
137
)
/
(
2
2
28
2
0
2
2
0
0
2
2
0
2
0
−
+
×
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
−
=
σ
σ
κ
P = Figure 7.18

41. Total nuclear pair-production
cross section/atom
( )
( )
P
Z
c
m
hv
T
Pd
Z
c
m
hv
PdT
Z
d
atom
cm
dT
c
m
hv
P
Z
d
c
m
hv
a
T
a
a
2
0
1
0
2
0
2
0
)
2
(
0
2
0
2
0
2
2
0
2
0
2
2
)
/
(
2
2
0
σ
σ
σ
κ
κ
σ
κ
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
−
=
=
−
=
∫
∫
∫
+
− +
+
+
P = Figure 7.18

42. Mass attenuation coefficient for
nuclear PP
hydrogen)
for
(
05
.
0
45
.
0
constant
)
/
( 2
except
A
Z
g
cm
A
NA
a
±
=
≈
= κ
ρ
κ

43. Total Coefficients for attenuation, energy
transfer, and energy absorption
y Mass attenuation coefficient
y Mass energy-transfer coefficient
y Mass energy-absorption coefficient
y Coefficient for compounds and mixtures
y Tables of photon interaction coefficients

44. Mass attenuation coefficient
y The total mass attenuation coefficient for gamma-
ray interactions
]
/
[ 2
g
cm
ρ
κ
ρ
τ
ρ
σ
ρ
μ
+
+
=

45. Mass energy-transfer coefficient
y The total mass energy-transfer coefficient for
gamma-ray interactions
⎥
⎦
⎤
⎢
⎣
⎡ −
+
⎥
⎦
⎤
⎢
⎣
⎡ −
+
⎥
⎦
⎤
⎢
⎣
⎡
=
+
+
=
hv
c
m
hv
hv
v
h
Y
p
hv
hv
T K
k
K
tr
tr
tr
tr
2
0
2
ρ
κ
ρ
τ
ρ
σ
ρ
κ
ρ
τ
ρ
σ
ρ
μ

46. Mass energy-absorption
coefficient
y The total mass energy-absorption coefficient for
gamma-ray interactions
y g = average fraction of secondary-electron energy
that is lost in radiative interactions
y For low Z and hv, g~0
y For increasing Z and hv, g increases gradually
)
1
( g
tr
en
−
=
ρ
μ
ρ
μ

47. Coefficient for Compounds and
Mixtures
y For compounds or mixtures of elements the Bragg
rule applies
y fA , fB …= are the weight fractions of separate
elements (A,B,…)
..
...
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
B
B
tr
A
A
tr
mix
tr
B
B
A
A
mix
f
f
f
f
ρ
μ
ρ
μ
ρ
μ
ρ
μ
ρ
μ
ρ
μ

48. Coefficient for Compounds and
Mixtures
y Same rule also applies to the mass energy-
absorption coefficient
y gA , gB …= are radiation yield fractions for
elements (A,B,…)
( ) ..
)
1
(
1
..
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≅
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
B
B
B
tr
A
A
A
tr
B
B
en
A
A
en
mix
en
f
g
f
g
f
f
ρ
μ
ρ
μ
ρ
μ
ρ
μ
ρ
μ

49. Coefficient for Compounds and
Mixtures
y For water, for example
Atom Z A H2O B F=B*A μ/ρ (@1MeV)
H 1 1 2
2*0.0556=
0.1111
1*0.1111=0.
1111 1.26E-01
O 8 16 1
1*0.0556=
0.0556
16*0.0556=
0.8889 6.37E-02
MW 18 FH*1.26e‐1+FO*6.37e‐2
Weig Fra/MW 1/18=0.0556 1 7.07E‐02
...
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
B
B
A
A
mix
f
f
ρ
μ
ρ
μ
ρ
μ

50. Tables of photon interaction
coefficients
y Appendix D.1, D.2, D.3 and D.4