1

Photon InteractionsWhen a photon beam enters matter, it

undergoes an interaction at random and is removed from the beam

2

2

/00

cm / g length nattenuatio mass theis /

1

/cmt coefficien nattenuatio mass theis

cm path free meanor length nattenuatio theis

1/cm coefficent nattenuatiolinear theis

g

eIeII xx

2

Photon Interactions

3

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

Gives an indirect measure of the photon energies of a beam (under the conditions of a narrow-beam geometry)

In shielding calculations, you will see TVL used a lot

693.0

2

1lnln

0

HVLx

I

I

4

Beam Hardening

5

Photon Interactions

6

Photon InteractionsWhat is a cross section?What is the relation of to the cross

section for the physical process?

common more is in ,mentioned as

tcoefficien nattenuatiolinear theis

atoms ofdensity theis where

/1 units has and units has

2

2

g

cm

A

N

N N

cmcm

Av

7

Cross Section

Consider scattering from a hard sphere

What would you expect the cross section to be?

b

θ

αR

αα

8

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

9

Cross SectionOne can find the scattering rate by

g

cmcmgcm

A

NN

cmcmsNsN

NIR

AvT

T

32

22

0

//

///

10

Cross SectionFor students working at collider

accelerators

pb160 bemight TeV 7at section cross typicalA

fb 1deliver toexpects LHC the2011 In

luminosity integrated of pb 35 delivered LHC the2010 In

/ in luminosity integrated theis

// in luminosity theis

1-

1-

2

2

tt

cmLdt

LdtN

scmL

LR

11

Photon Interactions

In increasing order of energy the relevant photon interaction processes are Photoelectric effect Rayleigh scattering Compton scattering Photonuclear absorption Pair production

12

Photon InteractionsRelative importance of the photoelectric

effect, Compton scattering, and pair production versus energy and atomic number Z

13

Photoelectric EffectAn approximate expression for the

photoelectric effect cross section is

What’s important is that the photoelectric effect is important

For high Z materials At low energies (say < 0.1 MeV)

2252

0

2/725

04

1065.63

8

24

cmr

hv

cmZ

e

epe

14

Photoelectric EffectMore detailed calculations show

3.5and 5 just take llwe'

1 - 3.5 from varies

5-4 from varies

~

mn

m

nhv

Zm

n

pe

15

Photon Interactions

Typical photon cross sections

16

Photoelectric EffectThe energy of the (photo)electron is

Binding energies for some of the heavier elements are shown on the next page

Recall from the Bohr model, the binding energies go as

be EEE

HeVE

eVn

Z

n

eZmE e

n

for 6.13

6.131

24

1

2

2

2220

42

17

Photoelectric Effect

18

Photoelectric EffectThe 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

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 InteractionsSometimes easy to loose sight of real

thickness of material involved

Photon Interactions

X-ray contrast depends on differing attenuation lengths

23

Photoelectric EffectRelated to kerma (Kinetic Energy

Released in Mass Absorption) and absorbed dose is the fraction of energy transferred to the photoelectron

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

hv

Ehv

hv

T b

24

Photoelectric EffectThus a better approximation of the

energy transferred to the photoelectron is

We can then define e.g.

rays-x K by theaway carriedenergy mean theis

shell K thein occurring nsinteractio PEof fraction theis

vacantis shell Kray when-x an ofy probabilit theis

K

K

K

KKK

hv

P

Y

hvYPhvT

peKKKtrpe

peKKKtr

pe

hv

hvYPhv

hv

hvYPhv

25

Photoelectric EffectFluorescence yield Y for K shell

26

Cross SectiondΩ=dA/r2=sinθdθdφ

27

Cross Section

28

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

29

Cross SectionFrom the figure on slide 7 we see

This is the relation between b and θ for hard sphere scattering

)/(cos2or

)2/cos( and

)2/cos()2/2/sin(sin then

2 and sin

1 Rb

Rb

Rb

30

Cross Section

We have

And the proportionality constant dσ/dΩ is called the differential cross section

ddd

bdbdd

dd

dd

sin and

where

31

Cross Section

Then we have

And for the hard sphere example

d

dbb

d

d

sin

4sin22

sin2

cos

sin22

sin

2sin

2

22

RRRb

d

d

R

d

db

32

Cross SectionFinally

This is just as we expectThe cross section formalism developed

here is the same for any type of scattering (Coulomb, nuclear, …)

Except in QM, the scattering is not deterministic

22

4Rd

Rd

d

dd

33

Cross Section

We have

And the proportionality constant dσ/dΩ is called the differential cross section

The total cross section σ is just

ddd

bdbdd

dd

dd

sin and

where

dd

dd