Gamma- and X-ray Interaction with

Matter

BAEN-625 Advances in Food Engineering

Photon Interactions

When a radiation beam passes through material, energy is lost from the incident beamSome energy is imparted to the medium and some of it leaves the volume

Energy absorbed

ltrab EEE Δ−Δ=Δ

Energy transferred from the beam

Energy lost

Exponential law

Absorption process

LeneII μ−= 0

Io = initial intensity of the beam before absorptionI = final intensity of beamμen = absorption coefficient of the material [1/cm]L = thickness [cm]Intensity = photon energy fluence rate [ MeV/s]

Attenuation coefficient, μ

Absorption process

μen for cm2/electronμ/ρ for cm2/g (mass coefficient)μa for cm2/atom (atomic coefficient)μ for cm-1

eAea AZNZ μρμμμ ⎟

⎠⎞

⎜⎝⎛== ;

Avogadro’s number

Interactions type of interest

Three modes of interaction (depending on the photon energy)

Photoelectric effect, PECompton effect, CEPair production, PP

Photons transfer their energy to electronsElectrons then impart energy to matter in many Coulomb-force interactions along theirs tracks

Photon interaction

Depends onPhoton energy

Atomic number Z of the absorbing medium

PE dominant at lower photon energiesCE at medium energiesPP at higher energies

hvE =γ

Two kinds of interactions areequally probableCE dominance is very broad for low Z values

Compton Effect

Two aspectsKinematics – relates the energies and angles of particles when Compton event occursCross section – predicts the probability that a Compton interaction will occur

Assumed that the electron struck by the incoming photon is initially unbound and stationary

Compton effect

Only part of the incident energy is absorbed to eject an electron (Compton electron)During interaction:

The photon disappearsA secondary photon is created with reduced energy –propagating in a changed direction

Compton Effect

Kinematic of Compton effect

A photon of energy Eγincident from the left strikes an electron, scattering it in an angle θ with KE TThe scattered photon departs at angle φ on the opposite sideEnergy and momentum are conserved

Kinematics

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−=−+

=

2tan1cot

')cos1)(/(1

'

2

2

φθ

φ

cmhv

hvhvTcmhvhvhv

o

o

Rest energy of electron = 0.511 MeVT, hv and hv’ [MeV]

Kinematics

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]

MeVhvhvT

hvhvTcmhvhvhv

o

511.02)(2

')cos1)(/(1

'

2

max

2

+=

−=−+

=φ

Kinematics of hv, hv’ and T

')cos1)(/(1

' 2

hvhvTcmhvhvhv

o

−=−+

=φ

Electron and photon scattering angles

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2tan1cot 2

φθcm

hv

o

Interactions Cross Section for Compton Effect

Thomson scatteringKlein-Nishina cross sections (K-N)

Total Thomson Scattering Cross Section

Can be thought of as an effective target areaThe probability of a Thomson-scattering event occurring when a single photon passes through a layer containing 1 electron/cm2

Fraction of a large number of incident photons that scatter in passing through the same layer, i.e., approximately 665 events for 1027 photons

electroncme /1065.6 2230

−×=σ

Klein-Nishina cross sections

Thomson’s cross section Independent of hvvalue is too large for hv > 0.01 MeV

K-N differential cross section for low energies

)cos1(2

)sin2(2

22

022

0 φφσ

φ

+=−=Ω

rrdd e

solid angle

cmcm

er

o

132

22

0 10818.2 −×==

[cm2 sr-1 per electron]

Total K-N cross section per electron

MeVcmcm

MeVhv

r

oo

e

511.0;][)21(

312

)21ln()21ln(21

)1(212

22

222

0

==

⎭⎬⎫

⎩⎨⎧

++

−+

+⎥⎦⎤

⎢⎣⎡ +

−+

++=

α

αα

αα

αα

αα

ααπσ

K-N Compton effect cross section

Is independent of the atomic number Z

So, the K-N cross section per atom of any Z is:

0Ze ∝σ

]/[cm 2 atomZ ea σσ ⋅=

K-N Compton mass attenuation coefficient

material of gramper electrons ofnumber

][g/cmdensity material of moleper grams ofnumber A

elementan of atomper electron ofnumber elementany

of weight atomic-gram ain atoms ofnumber theconstant sAvogadro' 100022.6

]/[cm

3

123

2

=

=

==

=×=

=

−

AZN

Z

moleN

gAZN

A

A

eA

ρ

σρσ

K-N energy transfer cross section for the Compton effect

)21ln(2

1211

)21(34

)21()122)(1(

)21(31

)21()1(22

'sin''

('2

333

2

22

2

222

0

222

0

αααα

αα

α

ααααα

αα

αααπσ

φσσ

φφ

+⎟⎠⎞

⎜⎝⎛ +−

+−

+−

⎥⎦

⎤⎢⎣

⎡+

−−+−

++

−++

=

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛=⋅

Ω=

Ω

r

hvhvhv

hvhv

hvhv

hvhvr

hvT

dd

dd

tre

etre

cm2/e]

[cm2/sr e]

This cross section, multiplied by the unit thickness 1 e/cm2, representsthe fraction of the energy fluence in a photon bean that is diverted to the recoil electron

K-N Compton effect cross section

photons) scattered by the carriedenergy for thesection cross N-(K setree σσσ =−

Average energy of the Compton recoil electrons

The average fraction of the incident photon’s energy given to electron:

The average energy of the Compton recoil electrons generated by photons of energy hv:

σσ

e

tre

hvT

=

σσ

e

trehvT ⋅=

Mean fractions

photon scatteredby retainedenergy fraction mean '

electron recoil given toenergy fraction mean

=

=

hvhvhvT

K-N Compton mass-energy transfer coefficient

]/[cm 2 gAZN

eAtr σ

ρσ

=

Photoelectric effect

Most important interaction of low-energy photons with matterCross-sections for photoelectric effect increase strongly, specially for high-Z mediaPhotoelectric effect totally predominates over the Compton effect at low photon energies

The Photoelectric effect

A photon is absorbed completely with the ejection of an electron

bEhvT −=Energy of a

photon

in the beam

Binding energy of

an electron in an atom

KE of the ejectedelectron

Photoelectric effect

Kinematics of Photoelectric Effect

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

Kinematics of Photoelectric Effect

The PE cannot take place unless hv>Eb for that electronThe smaller hv is, the more likely is the occurrence of PETa = KE given to the recoiling atom = 0 ab TEhvT −−=

Interactions Cross Section for Photoelectric Effect

More difficult to derive than for CEThere is no single equationPublished tables give results

Photoelectric interaction cross sections

Interaction cross section per atom, integrated over all angles of photoelectron emission

k = constantn ~ 4 at hv = 0.1 MeV (4.6 at 3 MeV)m ~ 3 at hv = 0.1 MeV (1.0 at 5 MeV)

For hv < 0.1 MeV

]/[cm )(

2 atomhvZk m

n

a ≅τ

]/[]/[cm )(

23

23

4

gcmhvZatom

hvZ

a ⎟⎠⎞

⎜⎝⎛∝≅

ρττ

Energy-transfer cross section for the PE

⎥⎦⎤

⎢⎣⎡ −−−

=

−=

hvvhYPPvhYPhv

hvEhv

hvT

LLLKKKKtr

b

)1(ρτ

ρτ

Mass-attenuation coefficient for Carbon

Mass-attenuation coefficient for Lead

Pair production

For photoelectric and Compton effects the interaction of photon is with electrons of atomPair production involves interaction of photons with the nucleus of the atomThe photon disappears and a positron and an electron appearENERGY IS CONVERTED TO MASS!!

Pair production

It can only occur in a Coulomb force field, usually near the field of an atomic nucleusIt can also take place, with lower probability, in the field of an atomic electronA min. photon energy 2m0c2=1.022 MeV is required

Pair production in the nuclear Coulomb force field

)(

2022.1

022.1

2

20

20

radiansTcm

MeVhvT

TTMeV

TTcmhv

≅

−=

++=

++=+−

+−

θ

Atomic differential cross section

( )

electroncmcm

er

atomcmdTcmhv

PZd a

/1080.5137

1137

)/(2

2282

0

220

0

22

0

20

−

+

×=⎟⎟⎠

⎞⎜⎜⎝

⎛==

−=

σ

σκ

P = Figure 7.18

Total nuclear pair-production cross section/atom

( )

( )

PZcmhv

TPdZ

cmhvPdTZd

atomcmdTcmhv

PZd

cmhv

aT

a

a

20

1

02

0

20

)2(

02

0

20

22

0

20

2

2

)/(2

20

σσ

σκκ

σκ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

−==

−=

∫

∫∫+

− +

+

+

P = Figure 7.18

Mass attenuation coefficient for nuclear PP

hydrogen)for (05.045.0constant

)/( 2

exceptAZ

gcmA

N Aa

±=≈

= κρκ

Total Coefficients for attenuation, energy transfer, and energy absorption

Mass attenuation coefficientMass energy-transfer coefficientMass energy-absorption coefficientCoefficient for compounds and mixturesTables of photon interaction coefficients

Mass attenuation coefficient

The total mass attenuation coefficient for gamma-ray interactions

]/[ 2 gcmρκ

ρτ

ρσ

ρμ

++=

Mass energy-transfer coefficient

The total mass energy-transfer coefficient for gamma-ray interactions

⎥⎦

⎤⎢⎣

⎡ −+⎥⎦

⎤⎢⎣⎡ −

+⎥⎦

⎤⎢⎣

⎡=

++=

hvcmhv

hvvhYphv

hvT KkK

trtrtrtr

202

ρκ

ρτ

ρσ

ρκ

ρτ

ρσ

ρμ

Mass energy-absorption coefficient

The total mass energy-absorption coefficient for gamma-ray interactions

g = average fraction of secondary-electron energy that is lost in radiative interactions

For low Z and hv, g~0For increasing Z and hv, g increases gradually

)1( gtren −=ρμ

ρμ

Coefficient for Compounds and Mixtures

For compounds or mixtures of elements the Bragg rule applies

fA , fB …= are the weight fractions of separate elements (A,B,…)

..

...

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

BB

trA

A

tr

mix

tr

BB

AAmix

ff

ff

ρμ

ρμ

ρμ

ρμ

ρμ

ρμ

Coefficient for Compounds and Mixtures

Same rule also applies to the mass energy-absorption coefficient

gA , gB …= are radiation yield fractions for elements (A,B,…)

( ) ..)1(1

..

+−⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛≅

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛≅⎟⎟

⎠

⎞⎜⎜⎝

⎛

BBB

trAA

A

tr

BB

enA

A

en

mix

en

fgfg

ff

ρμ

ρμ

ρμ

ρμ

ρμ

Coefficient for Compounds and Mixtures

For water, for example

Atom Z A H2O B F=B*A μ/ρ (@1MeV)

H 1 1 22*0.0556=

0.11111*0.1111=0.

1111 1.26E-01

O 8 16 11*0.0556=

0.055616*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

BA

Amix

ffρμ

ρμ

ρμ

Tables of photon interaction coefficients

Appendix D.1, D.2, D.3 and D.4