Compton Scattering

1

Compton ScatteringThere are three related processes

Thomson scattering (classical) Photon-electron

Compton scattering (QED) Photon-electron

Rayleigh scattering (coherent) Photon-atom

Thomson and Rayleigh scattering are elastic-only the direction of the photon changes, not its energy Plus Thomson and Rayleigh scattering are

only important at low energies where the photoelectric effect dominates

2

Thomson Scattering In Thomson scattering an electromagnetic

(EM) wave of frequency f is incident on an electron What happens to the electron?

Thus the electron will emit EM waves of the same frequency and in phase with the incident wave

The electron absorbs energy from the EM wave and scatters it in a different direction

In particular, the wavelength of the scattered wave is the same as that of the incident wave

3

Thomson Scattering

2242

2

2

2

T

2

2

2

20

3

42

3

2

0

0

22

10655.03

8

3

8

beam incoming thefrom subtractedpower theis this

3

8

emittedpower 3

2

2

1

3

2

sin

sin

power/area 88

cmrmc

e

Smc

eP

m

E

c

ea

c

eP

m

teEa

teEEeF

cBcES

e

4

Rayleigh ScatteringRayleigh scattering is scattering of light

from a harmonically bound electron

You may recall the probability for Rayleigh scattering goes as 1/λ4

Why is the sky blue?

22

02

4

hom

0 atoman in electron an for frequency with SHO Assuming

sonTRayleigh

5

Compton Scattering

Compton scattering is the scattering of light (photons) from free electrons

6

Compton ScatteringCalculationsThe change in wavelength can be found

by applying Energy conservation

Momentum conservation

2/142222 cmcphEhcmh eeee

cos22 22222 ppppppppp

ppp

e

e

7

Compton ScatteringFrom energy conservation

From momentum conservation

Eliminating pe2

hhmc

hh

c

h

c

hp

cpcmhhcmhhcm

ee

eeee

22

2)(

2

222

22422242

cos2

cos2222

2

22222

c

h

c

h

c

h

c

hp

ppppppppp

e

e

cos12 hhhhcme

8

Compton ScatteringContinuing on

And using v=c/λ we arrive at the Compton effect

And h/mc is called the Compton wavelength

)cos1(2

cm

h

e

cos1 cm

h

e

mcm

h

eC

121043.2

9

Compton ScatteringSummarizing and adding a few other

useful results are

2tan1cot

cos11

cos1

2

2

cm

hv

hhT

cmhv

hvh

cm

h

e

e

e

e

10

Compton ScatteringThe differential and total cross sections

are calculated in a straightforward manner using QED Called the Klein-Nishina formula

2

22

222

2

2

21

3121ln

2

1

21ln1

21

121

2

cos11

cos1cos1

cos11

1

2

eCompton

e

r

r

d

d

11

Compton Scattering

On the previous slide

At low energies

At high energies

2hom 3

8esonTCompton r

2

12ln

8

3

3

8 2

eCompton

r

2cm

hv

e

12

Compton ScatteringThus at high energies, the Compton

scattering cross section C goes as

hv

ZCompton ~

13

Compton Scattering

Graphically, d/d

14

Compton ScatteringIn polar form, assume a photon

incident from the left

15

Compton Scattering

At high energies, say > 10 MeV, most of the photons are scattered in the forward direction

Because of the high forward momentum of the incident photons, most of the electrons will also be scattered in the forward direction

16

Compton Scattering

Concerning kerma and absorbed dose, we are particularly interested in the scattered electron because it is ionizing

We can split the Compton cross section into two parts: one giving the fraction of energy transferred to the electron and the other the fraction of energy contained in the scattered photon

17

Compton Scattering

A

N

h

T

h

T

h

vh

h

vhhv

h

T

CAvCtrC

CscC

CCtrC

scC

trCC

tcoefficienn attenuatio

nsferenergy tra mass for thesimilarly

18

Compton Scattering

Here en=tr

19

Compton ScatteringAnother useful form of the differential

cross section is d/dT, which gives the energy distribution of the electron

20

Compton ScatteringThe maximum electron kinetic energy is

given by

MeVcm

Thv

hv

hvcm

chvmhvThv

cm

hvhvT

e

e

e

e

2555.02

large for and

221

21

and 21

2

2

max

2

2

max

2max

21

Compton Scattering

In cases where the scattered photon leaves a detector without interaction one would observe

22

Compton Scattering

23

Compton Scattering

keVvh

cm

cmhv

hvvh e

e

255|

2/21|

2

2

24

Pair ProductionPair production is the dominant

photon interaction at high energies (> 10 MeV)

In order to create a pair, the photon must have > 2me = 1.022 MeV

In order to conserve energy and momentum, pair production must take place in the Coulomb field of a nucleus or electron For nuclear field, Ethreshold > 2 x me

For atomic electron field, Ethreshold> 4 x me

25

Pair Production

26

Pair ProductionEnergy and momentum conservation give

Energy conservation can be re-written

But momentum conservation (x) shows

Thus energy and momentum are not simultaneously conserved

sinsin0 (y) Momentum

coscos (x) Momentum

Energy

pp

ppc

hf

EEhf

42224222 cmcpcmcphf

cpcphf max

27

Pair Production

The processes of pair production and bremsstrahlung are related (crossed processes) Thus we’d expect the cross section to

depend on the screening of atomic electrons surrounding the nucleusDoes the photon see nuclear charge Ze or

0 or something in between? The relevant screening parameter is

3/1

2100

ZEE

hvcme

28

Pair Production In the Born approximation (which is not

very accurate for low energy or high Z) one finds

54

1183ln

9

74

137 and 0 screening Complete

54

1092ln

9

74

137 and 1 screening No

3/122

3/12

222

3/122

ZfZrZ

Zcmh

Zfcm

hrZ

Zcmhcm

epair

e

eepair

ee

29

Pair ProductionNotes

pair ~ Z2

Above some photon energy (say > 1 GeV), pair becomes a constant

In order to account for pair production from the Coulomb field of atomic electrons, Z2 is replaced by Z(Z+1) approximately since the cross section is smaller by a factor of Z

Usually we don’t distinguish between the source of the field

30

Pair Production

Notes In the case of the nuclear field and for

large photon energies, the mean scattering angle of the electron and positron is

15 and 25For

2

022.1

2

MeVTMeVh

hT

T

cme

31

Pair ProductionThe probability for pair production

32

Pair Production2me (1.022 MeV) of the photon’s energy

goes into creating the electron and positron

The electron will typically be absorbed in a detector

The positron will typically annihilate with an electron producing two annihilation photons of energy me (0.511 MeV) each

If these photons are not absorbed in the detector than the pair production energy spectrum will look like

33

Pair Production

34

Pair ProductionSimilar to the photoelectric effect

and Compton scattering we define the mass attenuation and mass energy transfer coefficients as

paire

trpair

pairAvpair

hv

cmhv

A

N

22

35

Photonuclear InteractionsHere a nucleus is excited by the

absorption of a photon, subsequently emitting a neutron or proton

Most important when the energy of the photon is approximately the binding energy of nucleons (5-15 MeV) Called giant nuclear dipole resonance Still a small fraction compared to pair

production however

36

Photonuclear InteractionsGiant dipole resonance

37

Photonuclear InteractionsThese interactions would be

observed with higher energy x-ray machines A 25 MV x-ray beam will contain

neutron contamination from photonuclear interactions

Small effect compared to the photon beam itself

Also important in designing shielding since ~MeV neutrons are difficult to contain

38

Photon Interactions

Typical photon cross sections

39

Photon Interactions

Typical photon cross sections

40

Photon InteractionsNotes

Of course different interactions can occur at a given photon energy

A polyenergetic beam such as an x-ray beam is not attenuated exponentially Lower energy x-rays have higher attenuation

coefficients than higher energy x-rays Thus the attenuation coefficient changes as the

beam proceeds through material An effective attenuation length eff can be estimated

as

pairComptonpe

pairComptonpe

Z

Z

HVLeff

693.0

41

Beam Hardening

42

Photon InteractionsLet’s return to our first slide

As we’ve seen in the different photon interactions Secondary charged particles are produced Photons can lose energy through Compton

We define Narrow beam geometry and attenuation

Only primaries strike the detector or are recorded Broad beam geometry and attenuation

All or some of the secondary or scattered photons strike the detector or are recorded

Effective attenuation coefficient ’ <

xeII 0

43

Photon Interactions

44

Photon Interactions

In ideal broad beam geometry all surviving primary, secondary, and scattered photons (from primaries aimed at the detector) is recorded In this case ’ = en

45

Photon InteractionsThere are three relevant mass

coefficients

n)annhilatio and hlung(bremsstra

nsinteractio radiative lost toenergy electron

secondary of fraction average theis g where

1

tcoefficien absorptionenergy mass

tcoefficiennsfer energy tra mass

tcoefficien absorption mass

g

A

N

tren

en

tr

Av

46

Photon InteractionsTables of photon cross sections, mass

attenuation, and mass-energy absorption coefficients can be found in numerous places http://physics.nist.gov/PhysRefData/

contents.html NIST also gives material constants and

composition Useful since

elements separate of

fractions weight theare where

...

i

BB

AAmixture

f

ff

47

Photon Interactions

=1/(/)

48

Photon InteractionsSometimes easy to loose sight of real

thickness of material involved

49

Photon Interactions

X-ray contrast depends on differing attenuation lengths

50

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

section for the physical process?

common more isin

tcoefficien absorptionlinear theis

atoms ofdensity theis where

/1 units has and units has

2

2

g

cm

A

N

N N

cmcm

Av