Gamma ray interaction with matter
A) Primary interactions
1) Coherent scattering (Rayleigh scattering)
2) Incoherent scattering (Compton scattering)
3) Photoelectric effect
4) Production of electron and positron pairs
5) Interactions with small contributions
6) Total attenuation of gamma rays at matter
B) Secondary interactions
γ
1) X-rays
2) Auger electrons
3) Annihilation of positron and electron
4) Bremsstrahlung radiation
eγ
eγ
e-
e+
γ
e-
Coherent scattering
Coherent scattering on bounded electrons (whole atom) (energy is not transfered
only direction of momentum is changed) – in the limit Rayleigh scattering
R 
4
   0   R   T 4
0
8 2
4
4
r0


T
2
2
3
 02   2   0 2
 02   2   0 2




 02
  0   R  T 2
0
   0   R   T
d R 1 2
 r0 (1  cos2 )  F (q, Z )
d 2
F(q,Z) – probability of momentum transfer on Z
electron atom without energy transfer
r0 – classical electron radius (SI units):
e2
α  c
r0 
 2,821015 m 
 2,82fm
2
4π 0 me c
me c 2
Eγ`, ν´
Eγ, ν
Eγ = Eγ´,ν = ν´
Polar graph of cross-section without
inclusion of F(q,Z) influence, classical
limit of Thomson scattering
α = 1/137
ħc = 197 MeVfm
mec2 = 0,511 MeV
High energy → scattering to small angles
Thomson scattering – scattering on free electrons
in classical limit (coherent as well as incoherent)
Polarized:
Unpolarized:
d TP
 r02 sin 2 
d
d T 1 2
 r0 (1  cos 2 )
d 2
Θ
 TP 
8 2
r0   T
3
T 
8 2
r0  6,65  10  29 m 2  0,665 barn
3
Diffraction on crystal lattice
Usage of interference during coherent scattering on layers of crystal lattice
Bragg law:
n·λ = 2d·sin Θ
d – grid spacing
λ – radiation wave length
n – diffraction order
Eγ [keV]
1
10
50
100
500
1000
2000
ν [EHz = 1018 Hz] 0,242 2,42 12,1 24,2
121
242
484
λ [nm]
1,24 0,124 0,025 0,0124 0,0025 0,00124 0,00062
Grid spacing is in the order of 0,1 – 1 nm
Dependency of first diffraction maximum angle on
X-ray and gamma ray energies for two grid spacings
Spectrometers with sizes up to ten meters were built:
Eγ = 1000 keV, d = 0,6 nm, r = 10 m → Θ = 0,059O, x = 10 mm
Eγ = 100 keV
→ Θ = 0,59O , x = 100 mm
Incoherent (Compton scattering)
Θ
Eγ, pγ=Eγ/c
Eγ’, pγ’=Eγ’/c
φ
mec2, pe= 0
Ee , pe 
Ee 2  m2c2
e
c2
We assumed: 1) scattering on free electron (Eγ>>Be)
2) electron is in the rest
We obtain relations between energies and angles
of scattering and reflection from the energy and
momentum conservation laws
Scattered photon energy:
Where parameter:  
E
me c 2
Reflected electron energy:
Reflection angle:
E 
E
1   1  cos
  1800  E 
Ee  E  E 
cot   1     tan

2
E
Relation between scattered photon
energy Eγ and scattering angle Θ
1 2 
E  1  cos
1   1  cos
  1800  Ee 
2    E
1 2 
Diferential cross-section is described by
Klein-Nishin equation (on free electrons):
d C 1 2  E
 r0 Z
E
d
2
 




2
 E E



 sin 2  
 E E


 

We introduce energy of scattered photon:
d C 1 2 
1

 r0 Z 
2
d 2

 1   1  cos

 2 1  cos2  
2
1  cos  
1   1  cos 

inclusion of influence of electron binding at atom →
multiplying by function S(q,Z) – probability of momentum
q transfer to electron during ionization or excitation
Polar graph of cross-section without
inclusion of S(q,Z) influence. In the
limit E → 0 we obtain graph
for coherent scattering
Total cross section (can be obtained by integration):
1    21    1
 1
1  3 
 ln1  2  
ln1  2  

2 
1  2 2 
 2
   1  2 
 C  2  r02 Z 
Eγ >
mec2
→ζ>1:
C ~
Z

~
Z
E
Scattering of high energy electron
and low energy photon –
inverse Compton scattering (see exercise)
Distribution of energy transferred
to electrons
Photoelectric effect
γ
Can pass only on bounded electron
e-
Total photon energy is transfered
Electron energy:
Ee = Eγ - Be
Accurate calculation of photoeffect process (solution of Dirac equation) is very
sophisticated:
If it is enough energy (Eγ > BeK binding energy on K-shell) the photoeffect will pass
almost only on these electrons
7
1 2
4
5
 F  4   2Z  T  
Cross-section (for Eγ << mec2):
 
where
and so

e2
4 0c

1
137
σF = ~ Z5·Eγ-3,5
is fine structure constant
near to K-shell
More accurate equation for σF near to K-shell see Leo
Dependency of K-shell electron binding energy on
proton number Z of atom (Si – 1,8389 keV,
Ge – 11.1031 keV, Pb – 88.0045 keV)
σF = ~ Z4,5·Eγ-3
Production of electron and positron pairs
Transformation of photon to electron and positron pair.
e+
Energy and momentum conservation laws → only
γ
In nucleus field (mostly) Eγ > 2mec2 = 1022 keV
eventually electrons Eγ > 4mec2 = 2044 keV (je 1-2Z smaller)
e-
Description is equivalent to description of bremsstrahlung radiation (necessity to include
screening influence: predominance of pair production near nucleus – without screening):
There is valid for cross-section in special cases:
Without screening: 2me c  E 
2
Complete screening:
E 
me c 2 Z 1 3

me c 2 Z 1 3

7
ln 2  f (Z )  109
54 
9
 P  4    Z 2 r02 
 P  4    Z 2 r02  ln183Z 1/ 3   f (Z ) 
7
9
1
54
where f(Z) is coulomb correction of order α2
2
Production in the electron field     Zr0  3   42
PT
2235
(„triplet“ production)
Dependency of σP on Z and Eγ is:
(for „lower energies“ range)
σP ~ Z2ln(2Eγ)
Eγ>>mec2 electrons and positrons are peaked forward Θ ≈ 1/ζ
Cross-section dependency
On photon energy
Interaction with small contribution
Nuclear Rayleigh scattering
Nuclear Thomson scattering – substitutions e →Ze , me → Mj
Z 2e2
Z2

 1,5  1018 m
and then r0  r j 
2
A
4 0 M J c
Total cross-sections:
 TJ
8 2 Z 4
Z4
36
2

r j  2  12,6  10 m  2  0,126  barn
3
A
A
Nuclear resonance scattering ( for example giant dipole resonance)
Photonuclear reactions – resonance processes with small probability
Photonuclear reactions in order of mbarn up to barn in narrow energy range
interaction with electrons in order of barns up to 105 barns in broad energy range
Photon interaction with coulomb field of nucleus (Delbrück scattering) –
we can look on it as on virtual pair production and following annihilation
Secondary processes
X-ray
Fluorescent efficiency (coefficient):

NX
NA  NX
NX – X-ray photons NA – Auger electrons
Auger electrons
Released energy is transferred during electron transition at atomic cloud on other electron
Annihilation of positron and electron
Positrons are stopped by ionization losses and they annihilate in the rest → 2 quanta
of 511 keV (they are not fully in the rest → energy smearing of annihilation quanta)
Bremsstrahlung radiation during movement of electrons and positrons
Passage of electrons and positrons:
1) Ionization losses
2) Bremsstrahlung radiation
elektron záření gama
proton
Charged particle moves in nuclear field with acceleration → it emits photons
Total absorption of gamma rays at matters
Photon can loss big part (even all) its energy in one
interaction → beam weakens, it has not fixed range
dI = -μIdx
Equation for decreasing
of photon number:
I
 ex
I0
μ – total absorption coefficient – inverse value of
mean free path of photon at material
Review of main processes
σ = σF + σC + σP
Total cross-sections:
Multiply by number of atoms per volume unit N:
  N 
N a 
A
where Na – Avogadro constant, A – atomic mass,
ρ – material density
For compound or mixture Bragg rule is valid:



 w1 1  w2 2  ...

1
2
Total cross-section