Photon Interactions
 When a photon beam enters matter, it
undergoes an interaction at random and is
removed from the beam
I  I 0e  x  I 0e  x / 
 is thelinear attenuat ion coefficent1/cm
 is theattenuat ion length or mean free pat hcm

is themass attenuat ion coefficient cm2 / g 

1
2
is themass attenuat ion length g / cm 
 /  
1
Photon Interactions
2
Photon Interactions
 Notes



 is the average distance a photon travels before
interacting
 is also the distance where the intensity drops by a
factor of 1/e = 37%
For medical applications, HVL is frequently used
 Half Value Layer
 Thickness needed to reduce the intensity by ½
I 
0.693
1


ln   ln    x  HVL 

2
 I0 
 Gives an indirect measure of the photon energies

of a beam (under the conditions of a narrowbeam geometry)
In shielding calculations, you will see TVL used a 3lot
Beam Hardening
4
Photon Interactions
5
Photon Interactions
 What is a cross section?
 What is the relation of  to the cross section s
for the physical process?
s has units cm and  has units1 / cm
  Ns where N is thedensity of atoms
N Av 

s is thelinear attenuation coefficient
2
A
 cm2
as mentioned, in
is morecommon

g
6
Cross Section
Consider scattering from a hard sphere
What would you expect the cross
section to be?
α
θ
α
b
α
R
7
Cross Section
The units of cross section are barns


1 barn (b) = 10-28m2 = 10-24cm2
The units are area. One can think of the
cross section as the effective target area
for collisions. We sometimes take σ=πr2
8
Cross Section
 One can find the scattering rate by
R  I 0 NTs
N / s  N / s / cm cm
N Av 
NT 
A
3

g / cm cm
2
/ cm 
g
2
2

9
Cross Section
 For students working at collider accelerators
R  Ls
L is t heluminosit yin / cm2 / s
N  s  Ldt
2
Ldt
is
t
he
int
egrat
ed
luminosit
y
in
/
cm

In 2010 t he LHC delivered 35 pb-1 of int egrat edluminosit y
In 2011t he LHC expect st o deliver 1 fb-1
A t ypicalcross sect ionat 7 T eV might be s tt   160pb
10
Photon Interactions
In increasing order of energy the
relevant photon interaction processes
are





Photoelectric effect
Rayleigh scattering
Compton scattering
Photonuclear absorption
Pair production
11
Photon Interactions
 Relative importance of the photoelectric effect,
Compton scattering, and pair production
versus energy and atomic number Z
12
Photoelectric Effect
 An approximate expression for the
photoelectric effect cross section is
7/2
 me c 

s pe  4 2  0 Z 
 hv 
8re2
 25
2
0 
 6.65 10 cm
3
2
4
5
 What’s important is that the photoelectric
effect is important


For high Z materials
At low energies (say < 0.1 MeV)
13
Photoelectric Effect
 More detailed calculations show
n
Z
s pe ~ m
hv
n variesfrom 4 - 5
m variesfrom 3.5 - 1
we' ll just taken  5 and m  3.5
14
Photon Interactions
Typical
photon
cross
sections
15
Photoelectric Effect
 The energy of the (photo)electron is
Ee  E  Eb
 Binding energies for some of the heavier
elements are shown on the next page
 Recall from the Bohr model, the binding
energies go as
me Z 2e4 1
13.6Z 2
En  

eV
2
2
2
2
n
40  2 n
E1  13.6eV for H
16
Photoelectric Effect
17
Photoelectric Effect
The energy spectrum looks like
This is because at these
photon/electron energies the electron is
almost always absorbed in a short
distance

As are any x-rays emitted from the ionized
atom
18
Photoelectric Effect and X-rays
 PE proportionality to Z5 makes diagnostic x-
ray imaging possible
 Photon attenuation in
 Air – negligible
 Bone – significant (Ca)
 Soft tissue (muscle e.g.) – similar to water
 Fat – less than water
 Lungs – weak (density)
 Organs (soft tissue) can be differentiated by
the use of barium (abdomen) and iodine
(urography, angiography)
19
Photoelectric Effect and X-rays
Typical diagnostic x-ray spectrum

1 anode, 2 window, 3 additional filters
20
Photon Interactions
 Sometimes easy to loose sight of real thickness
of material involved
Photon Interactions
X-ray contrast depends on differing
attenuation lengths
Photoelectric Effect
 Related to kerma (Kinetic Energy Released in
Mass Absorption) and absorbed dose is the
fraction of energy transferred to the
photoelectron
T hv  Eb

hv
hv
 As we learned in a previous lecture, removal of
an inner atomic electron is followed by x-ray
fluorescence and/or the ejection of Auger
electrons

The latter will contribute to kerma and absorbed
dose
23
Photoelectric Effect
 Thus a better approximation of the energy
transferred to the photoelectron is
T  hv  PKYK  hvK
YK is theprobability of an x - ray when K shell is vacant
PK is thefractionof PE interactions occurringin the K shell
hvK is themean energy carried away by theK x - rays
 We can then define e.g.
s
tr
pe
 hv  PKYK hvK 

s pe
hv


 trpe  hv  PKYK hvK   pe


 
hv
 
24
Photoelectric Effect
 Fluorescence yield Y for K shell
25
Cross Section
 dΩ=dA/r2=sinθdθdφ
26
Cross Section
27
Cross Section
If a particle arrives with an impact
parameter between b and b+db, it will
emerge with a scattering angle between
θ and θ+dθ
If a particle arrives within an area of dσ,
it will emerge into a solid angle dΩ
28
Cross Section
 From the figure on slide 7 we see
b  R sin  and 2    
then sin   sin( / 2   / 2)  cos( / 2)
and b  R cos( / 2)
or   2 cos1 (b / R )
 This is the relation between b and θ for hard
sphere scattering
29
Cross Section
We have
ds
ds 
d
d
where ds  bdbd
and d  sin dd
And the proportionality constant dσ/dΩ
is called the differential cross section
30
Cross Section
Then we have
ds
b db

d sin  d
And for the hard sphere example
db
R

  sin
d
2
2

2


ds Rb sin 2 R cos 2 sin 2 R 2



d
2 sin 
2 sin 
4
31
Cross Section
 Finally
ds
R
s   ds   d   d  R 2
d
4
2
 This is just as we expect
 The cross section formalism developed here is
the same for any type of scattering (Coulomb,
nuclear, …)

Except in QM, the scattering is not deterministic
32
Cross Section
We have
ds
ds 
d
d
where ds  bdbd
and d  sin dd
And the proportionality constant dσ/dΩ is
called the differential cross section
The total cross section σ is just
ds
s   ds   d
d
33