Interactions: Photons and Neutrons - Atomic Physics · Interactions: Photons and Neutrons ... •...

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 1

Michael Ljungberg

Interactions: Photons and Neutrons


Introduction

Ionizing radiation

• Ionizations• Excitations

Direct Ionizing Radiation,

• electrons, protons, alpha-particles, heavy nucleus, charged particles

Indirect ionizing radiation

• Photons and neutrons – uncharged particles that liberate direct ionizing particles in the interaction which in turn give rise to most of the remaining ionizations.


Introduction

Photons and neutrons

• Low probability for interaction but when it occurs it results in a ‘drastic’ energy loss “statistical slowing-down”.

Charged particles

• Undergoes a large number of interactions each with a small energy deposition “continuous slowing-down”

Photons indirectly ionizing

• free secondary e- and large distances between the ionizations

Heavy charged particles is densely ionizating

• Large difference in biological effect and absorbed dose.


Interaction processes

Processes within keV - MeV that is of importance for our applications.

• Photo Absorption

• Compton Scattering

• Coherent Scattering

• Pair Production

• (Triplet Production)

• (Photo Nucleus reaction)


Photoelectric effect

* High atomic number (Z), i.e. Lead

* Low photon energy


The Photoelectric Process

The photon deliver the whole energy to an atomic e- (very low recoil energy)

• hv = Ee – EB

i. hv = incoming energy of the photon

ii. Ee = kinetic energy of the e‐

iii. EB = binding energy of e‐

Results in a vacancy in the shell => characteristic X-ray emission and Auger electrons

Process only with bounded electrons because nucleus receive an certain momentum.


De-excitation mechanisms


Fluroscence yield


Medical applications of Characteristic X-ray

Lead in fingerbone can be measured by irradiating the finger by two 57Co (122 and 136 keV) sources mounted opposite to each other

The lead characteristic X-rays and incoherently scattered photons are detected using a high purity germanium detector


Coherent scatter

•No energy loss for the photon

•Only change in direction


Compton scatter

•Energy loss for the photon

•Change in direction


The Compton Process

The incoming photon interacts with a free electron or with an electron in an outer electron shell. The binding energy, EB is very small compared to hv.

Relation between energy of the scattered photon and the incoming primary photon as function of scattering angle.

hh

c

,

h

, h

cTp,

e-


The Compton Process

Law of Energy: (1)

Law of momentum: cos cos (2)

h h T

h hp

c c

sin sin (3)h

pc

2 2 22 20

2 20

2 2 22 20 0

2 2 22 2 2 20 0 0

2 2 20

2 (4)

2

mc m c pc

mc m c T

m c T m c pc

m c T m c T m c pc

pc T m c T

Relativistic relations


The Compton Process

2 22 2 2

2

22 2 2

2cos cos cos

sin sin

h h h hp

c c c

hp

c

2 22

2

2 2 2

2cos

2 cos (5)

h h h hp

c c c

pc h hv h h

2 2 202pc h h m c h h

Eqn (2) och (3) results in

From eqn (1) T=hv-hv’ which in Eqn (4) results in

Addition:


The Compton Process

2 2 220

2 2 2 220

20

2 2 cos

2 2 2 cos

1 cos

h h m c hv h h h h h

h h h h m c hv h h h h h

m c hv h h h

Put this into eqn (5) yield

20

2

1 cos

Define 1 cos

1 1 cos

o

m ch h h

hh

h h hm c

hh

Division with hv yield


Compton Scattering

The photon energy before and after scattering

The energy loss

The energy loss of the photon (=Compton-e- energy) is for a given hv determined by the scattering angle .

20

1 1 cos

hh

h

m c

(1 cos )

1 (1 cos )eh h E h


Compton Scattering

The angles and depend on each other

The Compton shift

Change in wave-length depend on .

20

sintan

1 1 cos

h

m c

20

Compton wave-length (2.426nm)

1 cosh

m c


Compton Scattering

From the previous relations one see that for a given initial photon energy and knowing one of the following parameters

Then, the others can be determined!

, , eller e

E h


Pair production

e +

e-* High energies (h>1.022 MeV)

* High atomic numberh >1.022 MeV

Annihilation

h= 511 keV

h= 511keV


Pair production

If hv > 2moc2 = 1.022 MeV then it is energetically possible to create an e-e+ pair.

The photon energy, hv, is transferred to rest masses for e+ and e-

and to kinetic energy.

The process occurs close to a charged particle (nucleus) which takes up a certain momentum

Threshold energy is 2moc2 MeV for interaction with a nucleus

Threshold energy is 4moc2 MeV for interaction with a e-. (triplet of e-+ e+ + recoiling e-)

202 e e rhv m c E E E


Photon attenuation

absorbator

d

collimator

0 d

= photon fluence,= attenuation coefficient

d=thickness

“Narrow-beam geometry”


Photon attenuation

Absorb

X

dN

NoN(d)

( )dN N x dx

( )dN

N xdx

( ) xoN x N e


Photon attenuation

Absorb

X

dN

NoN(d)

( )Probability for transmission > 0x

o

N xe

N

Mean-Free Path = mfp = 1/


Attenuation coefficient

Type of processes

• Photo-electric effect• Incoherent Scattering• Coherent Scattering• Pair-production• Triplet-production• Nuclear photoeffect

Total mass-absorption coefficient is the sum of all interaction processes

photo Compton Coherent pair triplet nuclearn


Data on the internet!

http://www.nist.gov/pml/data/xcom/index.cfm


Cross-section as function of energy


Photoelectric effect

* High atomic number (Z), i.e. Lead

* Low photon energy


The Photoelectric Process

Highest likelihood for tight bounded electrons (hv > EB).

80% of all interactions by K-electrons

defined for a scatter angle and a solid angle d

d

hv

e-


The atomic photo-effect cross section

Cross-section per atom for photo absorption

Bethe (Quantum mechanical model)

Probability for photo process increase rapidly with increasing Z and decreasing energy

,a

e

27 5 7 / 2 2, 4 10 ( , ) ( 0.3) ( ) mae s Z h Z h


Coherent scatter

•Low atomic number (tissue, water)


Thompson cross-section

Starting point for the classical description of coherent x-ray scattering is the Lorentz equations

• Relates the force on an unbound electron to the amplitude of an electromagnetic wave with which it interacts.

• The electromagnetic wave train causes the electron to oscillate• The electron thus giving rise to its own radiation, with a frequency identical to that of

the incoming wave.

Such considerations lead to the Thomson form of the differential (with angle) scatter cross section of a free electron

22(Thompson) (1 cos )

2ord

d

r0 is the classical electron radius, θ is the angle between the incident photon momentum and the scattered photon momentum.


Coherent Cross-section

Thomson cross-section per electron

Cross-section per atom Independent of energy – not realistic

Formfactor include energy by the parameter x

Total cross-section

22 2(atom) (1 cos ) ( , )

2ord

F x Zd

22(electron) (1 cos )

2ord

d

2 22 2

0 0

(1 cos ) ( , )2or F x Z d d

22(atom) (1 cos )

2ord

Zd

sin( / 2)x


Coherent Scattering


Compton scatter

•Low atomic number (tissue, water)


Compton scatter

hh

c

,

h

, h

cTp,

e-

1 1 cos

hh

sintan

1 1 cos


The Klein-Nishina Cross-section

Quantum mechanical extension to the Thomson Cross- section

As energy decreases α (=hv/moc2) tends towards zero and regain Thompson differential cross-section

)cos1)(cos1(1

)cos1(1

)cos1(1

1

)cos1(2

2

222

22

KN

KNoTh

F

where

Fr

d

d

d

d


222

, sin2

e oe

r h h hd d

h h h

The Klein-Nishina Cross-section

Differential cross-section per electron for a Compton process where the photon is scattered an angle θ in the solid angle element dΩ is given by:

, ,a e

e eZ


Direction distribution for photons

Distribution for low energies are symmetrical around 90o

Increase in the forward direction for higher energies


Direction distribution for photo electrons

Low energies means electron emissions symmetrical around 90 degrees.

Photons incoming with higher energies results in photo electrons that are forward directed.


Cross-section per atom for Klein-Nishina

Cross-section for a Compton process is equal for all electrons and independent of Z (assuming that the binding energy are neglectable)

Additional corrections necessary when the photon energy is in the order of the binding energies of the atomic electron.

, ,a e

e eZ


Pair production

e +

e-* High energies (h>1.022 MeV)

* High atomic numberh >1.022 MeV

Annihilation

h= 511 keV

h= 511keV


Bethe cross-section for Pair Production

Valid in the range hv< 15 MeV.

For higher energies the process can occur at larger distances from the nucleus

Increase slowly with photon energy

22

2,

1 28 218ln(2 )

137 4 9 27a

eeo o

eZ

m c


Bethe cross-section for pair production

22 1/ 3

2,

1 28 218ln(183 2 )

137 4 9 27a

eeo o

eZ

m c

For very high energies:

No energy dependence - no

Same Z dependence for all energies

2

,

a

eeZ


Energy dependence

7/2h

If photo absorption dominates!

Decrease slowly with increasing energy of compton scattering dominates.

Increase slowly with hv if pair productiondominates.


Z dependence

5

2

Z

Z

Z

If photo absorption dominates

Independent of Z if comptonscattering dominates

If pair production dominates


Dominating cross section


Differential cross-sections - Hydrogen

Low-Z material


Differential cross-sections – Iodine

Z=52

K-Xrays at 30 keV


Differential cross-sections - Lead

Z=82

K-Xrays at 80 keV

L,M edges visible


Neutrons

Neutrons interacts by

• Elastic scattering by direct collision with a nucleus

• Inelastic scattering which means that the neutron is absorbed by the nucleus that will become in an excitated state and thereby emit a neutron in a different direction.

• Absorption where the nucleus is excitated. The excess of energy is emitted as gamma and/or particles

For each type above a cross-section can be defined. The total is the sum of

Compare with photons


Cross-sections

Interactions: Photons and Neutrons - Atomic Physics · Interactions: Photons and Neutrons ... •...

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