Interaction theory –Electrons
Lesson FYSKJM4710
Eirik Malinen
Source: F. H. Attix: Introduction to radiological physics and radiation dosimetry
(ISBN 0-471-01146-0)
Excitation / ionization
•
Incoming charged particle interacts with atom /
molecule:
Excitation
Ionisation
•
An ion pair is created
Elastic collision 1
•
Interaction between two particles where kinetic
energy is preserved:
m2, v2
m2
m1, v
•
m1, v1
χ
θ
Classical mechanics:
1
1
1
m1v 2 = m1v12 + m 2 v 22
2
2
2
m1v = m1v1 cos θ + m 2 v 2 cos χ
T0 =
0 = m1v1 sin θ − m 2 v 2 sin χ
Elastic collision 2
2m1v cos χ
⇒ v2 =
m1 + m 2
tan θ =
•
,
4m1m 2 cos 2 χ
v1 = v 1 −
(m1 + m 2 ) 2
sin 2χ
m1
− cos 2χ
m2
Equations give, among others, maximum energy
transferred:
E max =
1
m1m 2
2
m 2 v 2,max
=4
T0
2
(m1 + m 2 ) 2
Elastic collision 3
a) m1>>m2
0 ≤ χ ≤ π/2
m
0 ≤ θ ≤ tan −1 ( 2 sin 2χ)
m1
m
E max = 4 2 T0
m1
b) m1=m2
c) m1<<m2
0 ≤ χ ≤ π/2
0 ≤ χ ≤ π/2
0 ≤ θ ≤ π/2
0≤θ≤π
E max = T0
E max = 4
m1
T0
m2
•
Proton-electron collision:
•
θmax = 0.03º , Emax = 0.2 %
Electron-electron (or e.g. neutron-neutron) coll.:
θmax = 90º , Emax = 100 %
Elastic collision – cross section
•
Rutherford showed that the cross section is:
dσ
1
∝
4
dΩ sin (θ / 2)
•
→ small scattering angles are most probable
With respect to energy:
dσ
1
∝ 2
dE E
•
→ small energy transfers most probable
Stopping power
•
S=dT/dx; expected energy loss per unit lenght
dx
T0
nV targets per volume
T0 - dT
dT = En V dxσ = n V dx
E max
∫
E min
E
max
dσ
dσ
⎛ NA Z ⎞
EdE = ρ ⎜
EdE
⎟ dx ∫
dE
A
dE
⎝
⎠ Emin
E
⎛ dT ⎞ S ⎛ N A Z ⎞ max dσ
EdE
⎟ ∫
⎜
⎟= =⎜
dx
A
dE
ρ
ρ
⎝
⎠
⎝
⎠
E min
Impact parameter
•
•
•
Charged particles: Coulomb interactions
Most impprtant: interactions with electrons
Impact parameter b:
a: classical atomic radius
Soft collisions 1
•
•
•
•
•
•
b >> a : incoming particle passes atom at long
distance
Weak forces, small energy transfers to the atom
Inelastic collisions: Predominantly excitations, some
ionizations
Energy transfer range from ”Emin” to ”H”
Hans Bethe. Quantum mechanical considerations
In the following, theory for heavy charged particles
Soft collisions 2
Sc,soft
ρ
•
•
•
•
•
•
•
⎛ dTsoft ⎞
N A Z 2πr02 m e c 2 z 2 ⎡ 2m e c 2β2 H
2⎤
ln
=⎜
=
−
β
⎢ 2
⎥
⎟
2
A
β2
⎝ ρdx ⎠c
⎣ I (1 − β )
⎦
r0: classical electron radius = e2/4πε0mec2
I: mean excitation potential
β = v/c
z: charge of incoming particle
ρ: density of medium
NAZ/A : numbers of electron per gram
H: maksimum energy transferred by soft collisions
Soft collisions 2
•
Quantum mechanics (atomic structure) is reflected in
the mean excitation potential
Hard collisions 1
•
•
•
•
b << a : charged pass particle pass ’through’ atom
Large (but few) energy transfers
Energy transfers from H to Emax
May be considered as an elastic collision between
free particles (binding energy is negligible)
Sc,hard
ρ
⎛ dThard ⎞
N A Z 2πr02 m e c 2 z 2 ⎡ E max
2⎤
=⎜
−
β
ln
⎟ =
⎢ H
⎥
β2
A
⎣
⎦
⎝ ρdx ⎠c
Collision stopping power
•
For inelastic collisions, the total cross section is thus:
Sc Sc,soft Sc,hard
=
+
ρ
ρ
ρ
⎛ N Z ⎞⎛ z ⎞
= 4πr02 m e c 2 ⎜ A ⎟ ⎜ ⎟
⎝ A ⎠⎝ β ⎠
•
2
⎡ ⎛ 2m e c 2β2 ⎞ 2 ⎤
⎢ ln ⎜
⎟ −β ⎥
2
(1
)I
−
β
⎠
⎣ ⎝
⎦
Important: Increases with z2, decreases with v2, not
dependent on particle mass
Sc/ρ, different substances
Singly charged heavy particles
•
I and electron density (ZNA/A) give differences
Sc/ρ, electrons and positrons
•
Electron-electron scattering is more complicated;
scattering between two identical particles
•
Sc, hard/ρ (el-el) is described by the Möller cross
section
•
•
•
Sc, hard/ρ (pos-el) is described by the Bhabha c.s.
Sc, soft /ρ was given by Bethe, as for heavy particles
Characteristics similar to that for heavy charged
particles
Shell correction
•
•
•
•
•
Derivation of Sc assumes v >> vatomic electrons
When v ~ vatomic electrons, no ionizations
Most important for K-shell electrons
Shell correction C/Z takes this into accout, and thus
reduces Sc /ρ
C/Z depends on particle energy and medium
Density correction
•
Charged particles polarizes medium which is being
traversed
→
ur
E part
→
ur
→
→
ur
ur
ur→
ur→
E eff = E part + E pol , E eff < E part
→
ur
+
+
+
+
+
-
-
-
-
-
E pol
Charged (+z) particle
-
-
+
+
+
+
•
•
+
•
Weaker interactions with remote atoms due to
reduction in electromagnetic field strenght
Polarization increases with energy and density
Most important for electrons and positrons
Density correction
•
•
Density correction δ reduces Sc /ρ for liquids and
solids
Sc/ρ (water vapor) > Sc/ρ (water)
Dashed line:
Sc/ρ without δ
Linear Energy Transfer 1
•
•
LETΔ is denoted the restricted stopping power
dT/dx: mean energy loss per unit lenght – but how
much is deposited ’locally’?
Electron released by ionization
Track from charged particle
Electrons leaving ’local volume’ → energy transfer > Δ
•
•
Sc: energy transfers from Emin to Emax
How much energy is deposited within the range of
an electron given energy Δ?
Linear Energy Transfer 2
•
Energy loss (soft + hard) per unit lenght for
Emin < E < Δ:
Δ
⎛ N A Z ⎞ dσ
⎛ dT ⎞
=
ρ
LΔ = ⎜
EdE
⎟
⎜
⎟ ∫
⎝ dx ⎠ Δ
⎝ A ⎠ Emin dE
⎛ N Z ⎞⎛ z ⎞
= ρ2πr02 m e c 2 ⎜ A ⎟ ⎜ ⎟
⎝ A ⎠⎝ β ⎠
•
•
•
2
⎡ ⎛ 2m e c 2β2 Δ ⎞
⎤
2
−
β
ln
2
⎢ ⎜
⎥
⎟
2
−
β
(1
)I
⎝
⎠
⎣
⎦
For Δ=Emax, we have L∞=Sc ; unrestricted LET
LETΔ is often given in [keV/μm]
30 MeV protons in water: LET100 eV / L∞ = 0.53
Brehmsstrahlung 1
•
Photon may be emitted from charged particle
accelerated in the field from an electron or nucleus
e.g. electron
e.g. nucleus
•
Larmor’s formula (classical electromagnetism) for
radiated effect from accelerated charged particle:
(ze) 2 a 2
P=
6πε 0 c3
Brehmsstrahlung 2
•
For particle accelerated in nuclear field:
zZe 2
F = ma =
4πε 0 r 2
•
⎛Z⎞
⇒P∝⎜ ⎟
⎝m⎠
⇒
zZe 2
a=
4πε 0 mr 2
2
Comparison of protons and electrons:
2
⎛ me ⎞
1
=⎜
≈
⎟
Pe ⎜⎝ m p ⎟⎠ 18362
Pp
•
Brehmsstrahlung not important for heavy charged
particles
Brehmsstrahlung 3
•
•
•
Energy loss by brehmsstrahlung is called radiative
loss
Maksimum energy loss is the totale kinetic energy T
Radiative loss per unit lenght: radiative stopping
power:
2
⎛ S ⎞ ⎛ dT ⎞
2 NA Z
(T + m e c 2 )Br (T, Z)
⎜ ⎟ =⎜
⎟ ≈ αr0
A
⎝ ρ ⎠ r ⎝ ρdx ⎠ r
•
•
Br (T, Z) weakly dependent on T and Z
Brehmsstrahlung increases with energy and atomic
number
Total stopping power, electrons
•
Total mass stopping power:
⎛ dT ⎞
⎛ dT ⎞ ⎛ dT ⎞
=
⎜
⎟
⎜
⎟ +⎜
⎟
dx
dx
ρ
ρ
⎝
⎠ tot ⎝
⎠c ⎝ ρdx ⎠ r
Radiation yield
Y(T) =
(dT / ρdx) r
S TZ
= r ≈
n
(dT / ρdx)c + (dT / ρdx) r S
n=750 MeV
0.8
Y(T)
0.6
0.4
0.2
vann
Water
Tungsten
wolfram
0.0
0
20
40
60
80
100
T [MeV]
Sc/ρ [MeV/cm2 g]
Sc/ρ, protons and electrons
Electrons, total
Electrons, collisions
Electrons, radiative
Protons, collision
T [MeV]
Cerenkov effect
•
•
High energy electrons (v > c/n) polarizes medium
(e.g. water) and blueish light (+ UV) is emitted
Low energy loss
Other interactions
•
•
•
•
Nuclear interactions: Inelastic process where
charged particle (e.g. proton) excites nucleus →
– Scattering of charged particle
– Emission of neutron, photon, or α-particle (42 He )
Not important below ~10 MeV (protons)
Positron annihilation: Positron interacts with
electron → a pair of photons with energy ≥ 2 x 0.511
MeV is created. Photons are emitted in opposite
directions.
Probability decreases as ~ 1/v
Range 1
•
•
•
•
The range ℜ of a charged particle in matter is the
(expectation value) of it’s total pathlenght p
The projected range <t> er is the (expectation
value) of the largest depth tf a charged particle can
reach along it’s incident direction
Electrons:
<t> < ℜ
Heavy charged particles:
<t> ≈ ℜ
CSDA-range
•
•
The range may be approximated by ℜCSDA
(continuous slowing down approximation)
Energy loss per unit lenght dT/dx – gives implicitly a
measure of the range:
T0 − ΔT = T0 −
T0
Δx =
Δx
⇒ ℜCSDA
T0
dT
Δx
dx
n
n
dx
⎛ dx ⎞
ΔT , ⇒ ℜ = ∑ Δ x i = ∑ ⎜
⎟ ΔT
dT
i =1
i =1 ⎝ dT ⎠i
−1
⎛ dT ⎞
= ∫⎜
⎟ dT
dx ⎠
0 ⎝
Range 3
•
The range is often given multiplied by the density:
ℜCSDA
•
•
T0
−1
⎛ dT ⎞
= ∫⎜
⎟ dT
ρ
dx
⎠
0 ⎝
Unit thus becomes [cm] [g/cm3] = [g/cm2]
Range of charged particle depends on:
– Charge and kinetic energy
– Density, electron density and mean excitation
potential of absorber
Range 4
Multiple scattering and straggling
•
In a beam of charged particles, one has:
– Variations in energy deposition (straggling)
– Variations in angular scattering
→ The beam, where all particles originally had the same
velocity, will be smeared out as the particles
traverses matter
→
v1
→
v2
→
→
v
→
v4
v3
Number of electrons
Straggling
Primary beam
Beam at shallow depths
Beam at large depths
Energy [MeV]
Projected range
Energy deposition, protons
187 MeV
Energy deposition, electrons
Monte Carlo simulations
•
•
Monte Carlo simulations of the track of an electron
(0.5 keV) and an α-particle (4 MeV) in water
Note:
e- is most scattered
α has the highest dT/dx
eExcitation
2 nm
Ionisation
α
Hadron therapy
•
Heavy charged particles may be used for radiation
therapy – conforms better to the target than photons
or electrons
Web pages
•
For stopping powers:
http://physics.nist.gov/PhysRefData/Star/Text/contents.html
•
For attenuation coefficients:
http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html