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PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: • photoelectric effect • Compton scattering (including Thomson and Rayleigh scattering) • pair production • photodissociation of the nucleus (only at very high γ ray energies) • we will ignore

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Page 1: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

PHYS 352

Photon Interactions

Photon Interactions in Matter

three interactions to consider:

• photoelectric effect

• Compton scattering (including Thomson and Rayleigh scattering)

• pair production

• photodissociation of the nucleus (only at very high γ ray energies)

• we will ignore

Page 2: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Characteristics of Photon Radiation

• X-rays and γ rays are penetrating because the cross sections for the above processes are much smaller than cross sections for charged particles undergoing inelastic electron collisions

• photons are not degraded in energy as they pass through matter

• the processes either absorb the photon or scatter them out of the beam

• thus, the photon that passes straight through has not interacted

except for: multiple Compton scattersputs photons back into the beam, and they have lower energy

Photon Beam Attenuation

• exponential attenuation of beam intensity:

• where μ is the absorption coefficient, related to the total cross section that includes the cross section for all the processes (including scattering)

I(x) = I0 exp(−µx); µ = Nσ tot

Page 3: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

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Interaction 1- 29 photon attentn 2009.DOC - 21 -

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Lecture order 10

Photoelectric Effect

• the photon is absorbed by the atom; the atomic electron is ejected

• the kinetic energy of the electron is:

• photon energy > binding energy

• requires atomic electrons (does not occur with free electrons) because the atom must take up the momentum

• absorption cross section depends on the material and on which shell electrons are participating

• K-absorption edge, L-absorption edges are clearly visible

Photoelectric Effect Cross Section

• complicated calculation because it involves wavefunctions for many atomic electrons; some simplifying assumptions can be made

• for photon energy above the K-absorption edge and

• where α≅1/137 is the fine structure constant, re is the classical electron radius = 2.817 × 10–13 cm

• cross section falls rapidly with energy (other processes become more important around one to a few MeV

• scales approximately as Z5 or Z4

• higher Z desirable in detectors (and shielding)

• more photoelectric effect interactions

• larger “photopeak”

hν mec2

σ photo = 4α4 2Z 5 8πre

2

3mec

2

hν⎛⎝⎜

⎞⎠⎟

7 /2

Page 4: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Photoelectric Cross Section Over Broad Range

• at high E, cross section goes as 1/(hν); at lower E, σ goes as 1/(hν)3.5

• the dependence on Z varies somewhat with energy too (that’s why Z5 or Z4)

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Interaction 1- 29 photon attentn 2009.DOC - 27 -

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Lecture order 16

X-ray Fluorescence Yield

• as already described in this course, the atom with the K-shell vacancy is excited and can give off characteristic X-rays

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Interaction 1- 29 photon attentn 2009.DOC - 24 -

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Lecture order 13

Page 5: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Auger Electrons

• emission of Auger electrons competes with X-rays

• “important” consideration for detectors

• in thin detectors, the X-rays after photoelectric effect might escape

• Auger electrons will certainly add to the detected signal in all detectors

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Interaction 1- 29 photon attentn 2009.DOC - 23 -

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Lecture order 12Compton Scattering

• the photon scatters off of quasi-free electrons

• if the energy of the photon is large compared to the electron’s binding energy, the binding energy can be ignored and the electron can be considered “free”

• the energy of the photon is degraded

• the photon is scattered out of the beam

• incoming photon energy hνi• outgoing photon energy hνf• recoil electron kinetic energy Te

• recoil electron momentum pe

Page 6: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Derivation of Compton Scattering Kinematics

• using 4-momentum, the quantity P2 is Lorentz invariant

P ≡ (E, pc); P2 ≡ E2 − pcipc = E2 − (pc)2 = (mc2 )2

Pi + Pe = Pr + Pf ; Pi + Pe − Pf = Pr(Ei + mec

2 − Ef )2 − (pic − p f c)

2 = (Pr )2 = (mec

2 )2

Ei

2 + (mec2 )2 + Ef

2 + 2Eimec2 − 2Efmec

2 − 2EiE f − pi2c2 − pf

2c2 + 2pi ip f c2 = (mec

2 )2

2Eimec2 − 2Efmec

2 − 2EiE f + 2pi p f c2 cosθ = 0

mec2 (hν i − hν f ) = hν ihν f (1− cosθ)

hν f =hν i mec

2

mec2 + hν i (1− cosθ)

=hν i

1+ hν i

mec2 (1− cosθ)

Te = hν i − hν f

Pr

Pe

Pi

Pf

Aside: Feynman Diagrams

• Compton scattering, photoelectric effect, Bremsstrahlung cross sections are all related

Compton scattering

time

space

photoelectric effect

bremsstrahlung and photoelectric effect must occur in the vicinity of the electric field of the nucleus

Page 7: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Compton Scattering Cross Section

• Klein-Nishina formula (calculated to 1st order in quantum electrodynamics)

• integrate over θ to get total Compton scattering cross section

• expressed as differential cross section versus recoil electron kinetic energy

• gives the spectrum of Compton recoil electrons seen in a detector

dσdΩ

= FKNre2

2(1+ cos2θ)

FKN =1

1+ A(1− cosθ)⎛⎝⎜

⎞⎠⎟

2

1+ A2 (1− cosθ)2

[1+ A(1− cosθ)](1+ cos2θ)⎧⎨⎩

⎫⎬⎭

A =hν i

mec2

ratio of hνf / hνi

dσdTe

=πre

2

A2mec2 2 + s2

A2 (1− s)2+

s1− s

s − 2A

⎛⎝⎜

⎞⎠⎟

⎣⎢

⎦⎥

s = Tehν i

Compton edge

Total Compton Cross Section

• correction for binding of electrons appears at low energies

• here, photoelectric effect dominates, so it almost doesn’t matter if the free electron calculation overestimates

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Interaction 2009 30-52.DOC - 36 -

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Lecture order 25

on Pb

Page 8: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Features of Compton Scattering

• the atomic cross section (versus the electronic one): multiply by Z electrons in the atom

• # of atoms per gram = NA/A

• # of electrons per gram = Z NA/A

• but Z/A ~ 1/2 for most materials

• thus, the Compton cross section per gram is roughly independent of the material

• the probability for Compton scattering goes as the number of grams in the path or equivalently proportional to density

• energy dependence goes approximately as 1/hν (at higher energies)

Classical Limit – Thomson Scattering

• J.J. Thomson calculated the classical scattering cross section for photons by free electrons; classical limit of Klein-Nishina formula is:

hν i mec2; A→ 0; FKN → 1

dσdΩ

=re2

2(1+ cos2θ); σ =

8π3re2 = 0.665 barns

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Interaction 2009 30-52.DOC - 35 -

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Lecture order 24

A

AA

A

A

Page 9: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

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Lecture order 4

Rayleigh Scattering

• is classical Thomson that takes place coherently for all electrons in the atom

• atom is polarized by the electric field; oscillating polarization emits EM radiation

• Z electrons participate; the effect is coherent so goes as Z2 (amplitude of the fields add and not just incoherent sum of the power)

• Rayleigh has coherence (and potentially resonance) on top of Thomson; and materials have atoms and not plasma (free electrons)

• hence the low energy behaviour of photons in matter becomes Rayleigh scattering and not Thomson

27. Passage of particles through matter 19

Photon Energy

1 Mb

1 kb

1 b

10 mb10 eV 1 keV 1 MeV 1 GeV 100 GeV

(b) Lead (Z = 82)- experimental !tot

!p.e.

"e

Cros

s sec

tion

(bar

ns/a

tom

)Cr

oss s

ectio

n (b

arns

/ato

m)

10 mb

1 b

1 kb

1 Mb(a) Carbon (Z = 6)

!Rayleigh

!g.d.r.

!Compton

!Compton

!Rayleigh

"nuc

"nuc

"e

!p.e.

- experimental !tot

Figure 27.14: Photon total cross sections as a function of energy in carbon andlead, showing the contributions of di!erent processes:

!p.e. = Atomic photoelectric e!ect (electron ejection, photon absorption)

!Rayleigh = Rayleigh (coherent) scattering–atom neither ionized nor excited

!Compton = Incoherent scattering (Compton scattering o! an electron)

"nuc = Pair production, nuclear field

"e = Pair production, electron field

!g.d.r. = Photonuclear interactions, most notably the Giant Dipole Reso-nance [46]. In these interactions, the target nucleus is broken up.

Data from [47]; parameters for !g.d.r. from [48]. Curves for these and otherelements, compounds, and mixtures may be obtained fromhttp://physics.nist.gov/PhysRefData. The photon total cross section isapproximately flat for at least two decades beyond the energy range shown. Originalfigures courtesy J.H. Hubbell (NIST).

August 29, 2007 11:19

Rayleigh Cross Section in Comparison

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Interaction 1- 29 photon attentn 2009.DOC - 19 -

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Lecture order 8

dσdΩ

=re2

2(1+ cos2θ){F(θ,hν i ,Z )}

2

• Rayleigh form factor multiplies Thomson differential cross section

Page 10: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Rayleigh and Thomson are Elastic Scattering

• photon does not lose energy; atom is not ionized or excited

• nucleus/atom absorbs momentum recoil and takes a negligible amount of energy from the photon

• hence, the processes are not relevant for dose calculations (e.g. in medical physics)

• on the other hand, they do scatter photons out of the beam; add to the overall attenuation

• ~5% or less of the X-rays (i.e. X-ray energies, in a typical diagnostic beam) suffer Rayleigh scattering

• photoelectric effect dominates at low energies also (see previous slide)

Pair Production

• photon must have energy greater than 2 mec2 or 1.022 MeV then a photon can transform into an electron and positron

• must occur in the vicinity of the nuclear Coulomb field also

• otherwise cannot simultaneously conserve energy and momentum if a photon turned into an electron and positron in free space

• the nucleus is able to absorb the momentum of the photon

• can follow same approach as for bremsstrahlung

• swap (outgoing e+ and incoming γ) for (incoming e– and outgoing γ)

• screening of the nucleus can be important

• e.g. total cross section for pair production (at high energies)

incoming photon

x

outgoing e–

outgoing e+

nucleus

σ pair = 4Z2αre

2{79 [ln(183Z1/3 ) − f (Z )]−1 / 54}

Page 11: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Pair Production Cross Section

• scales as Z2

• energy dependence: yes

27. Passage of particles through matter 19

Photon Energy

1 Mb

1 kb

1 b

10 mb10 eV 1 keV 1 MeV 1 GeV 100 GeV

(b) Lead (Z = 82)- experimental !tot

!p.e.

"e

Cros

s sec

tion

(bar

ns/a

tom

)Cr

oss s

ectio

n (b

arns

/ato

m)

10 mb

1 b

1 kb

1 Mb(a) Carbon (Z = 6)

!Rayleigh

!g.d.r.

!Compton

!Compton

!Rayleigh

"nuc

"nuc

"e

!p.e.

- experimental !tot

Figure 27.14: Photon total cross sections as a function of energy in carbon andlead, showing the contributions of di!erent processes:

!p.e. = Atomic photoelectric e!ect (electron ejection, photon absorption)

!Rayleigh = Rayleigh (coherent) scattering–atom neither ionized nor excited

!Compton = Incoherent scattering (Compton scattering o! an electron)

"nuc = Pair production, nuclear field

"e = Pair production, electron field

!g.d.r. = Photonuclear interactions, most notably the Giant Dipole Reso-nance [46]. In these interactions, the target nucleus is broken up.

Data from [47]; parameters for !g.d.r. from [48]. Curves for these and otherelements, compounds, and mixtures may be obtained fromhttp://physics.nist.gov/PhysRefData. The photon total cross section isapproximately flat for at least two decades beyond the energy range shown. Originalfigures courtesy J.H. Hubbell (NIST).

August 29, 2007 11:19

Electron-Photon Showers

• at very high energies, pair production dominates and we can define a mean free path length for pair production to occur Lpair:

• and it turns out to be about equal to Lrad

• Lrad is the mean free path length for bremsstrahlung

• not surprising since bremsstrahlung and pair production are Feynman diagrams are closely related to each other

• in high energy particle physics, electrons (and positrons) and gammas produce electron-photon showers and the radiation length Lrad characterizes the distance over which the showers develop

µ = Nσ pair =1Lpair

Lpair 97Lrad

X0 ≡ Lrad

Page 12: PHYS 352 Photon Interactions - Engineering Physicsphys352/lect17.pdf · 2010-03-30 · PHYS 352 Photon Interactions Photon Interactions in Matter three interactions to consider: •photoelectric

Simulation 5 GeV e– on Lead-Glass

• from S. Menke’s web page

MPI Munich