Geant4 DNA Physics processes
overview and current status
Y. Perrot, S. Incerti
Centre d'Etudes Nucléaires de Bordeaux - Gradignan
IN2P3 / CNRS
Université Bordeaux 1
33175 Gradignan
France
Z. Francis, G. Montarou
Laboratoire de Physique Corpusculaire
IN2P3 / CNRS
Université Blaise Pascal
63177 Aubière
France
R. Capra, M.G. Pia
INFN Sezione di Genova
Geant4 DNA meeting
Genova - July 13th-19th, 2005
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Aim
• Extend Geant4 to simulate electron, proton and alpha electromagnetic interactions
in liquid water down to ~7.5 eV
• electrons : elastic scattering, excitation, ionization
• p, H : excitation (p), ionization (p & H), charge transfer (p), stripping (H)
• He++, He+, He : excitation, ionization, charge transfer
• validation
:
two independent computations performed by LPC Clermont & CENBG from
litterature
• References used for the models :
- Dingfelder, Inokuti, Paretzke et al. (2000 for protons, 2005 for He)
- Emfietzoglou et al. (2002 for electrons)
- Friedland et al. (PARTRAC)
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Protons and Hydrogen
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List of processes
Processes p and H
excitation : p + H2O → p + H2O*
ionisation : p + H2O → p + e- + H2O+
charge transfer : p + H2O → H* + H2O+
stripping : H + H2O → p + e- + H2O*
ionisation : H + H2O → H + e- + H2O+
excitation neglected for H
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Excitation by Protons (TXS)
Ω
ν
σ
(Z
a
)
(
t
E
)
proton
0
k
σexc,
k (t ) 
J Ω  ν  t 
• function of t
No experimental data, but semi-empirical relations with electron excitation cross sections
s0 is a constant (s0 = 1E-20 m²)
Z = 10 number of electrons in the crossed medium
Ek excitation energy.
a and  represent the energy superior limit so that this relation is in agreement with First Born Approximation (> 500 keV)
 and J for low energy (FBA not valid)
5 excitation levels
Excitations
Ek (eV)
a (eV)
J (eV)
Ω
ν
A B1
8.17
876
19820
0.85
1
B A1
10.13
2084
23490
0.88
1
Ryd A+B
11.31
1373
27770
0.88
1
Ryd C+D
12.91
692
30830
0.78
1
Diffuse bands
14.50
900
33080
0.78
1
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Ionisation by Protons (DXS)
• function of E and t, for E>Ij
• Nice agreement on TXS by Simpson integration
• analytical formula also available for ionisation TCS
• reproduces ICRU stopping powers
dσ
dσ j
 G j
dE
dW j
j
F1 ( ν)  w F2 ( ν )
dσ j S

3
dw B j (1  w) [1  exp α (w - w c ) / ν]
Rudd model
5 ionisation shells (K included)
E is the transfered energy
t is the proton kinetic energy
Ry = 13.606 eV (1 Ry -> eV)
Ij ionisation energy of shell j (liquid)
Bj is the binding energy of shell j (vapour)
Gj partitioning factor to adjust the shell contributions to the FBA calculations
(Gj is 1 for K shell)
Shell j
Ij (eV)
Bj (eV)
Nj
Gj
1a1
539.00
539.70
2
1.00
2a1
32.30
32.20
2
0.52
1b2
16.05
18.55
2
1.11
3a1
13.39
14.73
2
1.11
1b1
10.79
12.61
2
0.99
W j = E - Ij is the secondary electron kinetic energy
w = W j/Bj
Nj is the number of electrons on shell j
S = 4πα0²Nj(Ry/Bj)²
T = (me/mp) t : kinetic energy of an electron traveling at the same speed as the proton
² = T/Bj
wc = 4²-2-Ry/(4Bj)
C1 ν -D1
α related to the size of the target molecule
L1 ( ν ) 
Parameters from vapor data
LE term
HE term
F1 ( ν )  L1 ( ν )  H 1 ( ν )
F2 ( ν ) 
L 2 ( ν ) H 2 ( ν)
L 2 ( ν )  H 2 ( ν)
1  E1 ν (D1  4)
A ln(1  ν 2 )
H1 ( ν)  1 2
ν  B1 /ν 2
L 2 ( ν)  C 2 ν D 2
Parameter
Valence
K-shell
A1
1.02
1.25
B1
82.0
0.50
C1
0.45
1.00
D1
-0.80
1.00
E1
0.38
3.00
A2
1.07
1.10
B2
14.6
1.30
C2
0.60
1.00
D2
0.04
0.00
α
0.64
0.66
A
B
H 2 ( ν)  22  42
ν
ν
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Ionisation by Protons (TXS)
 1
1
σioni ( )  

σ
 low σ high
  T D

  F 
σ low  4 π α  C 
  Ry 



2
0



1
 Ry 
σ high  4 π α 02 

 T 
• function of t


 Ry 
 A ln 1 
  B 
T 



where T is the kinetic of an electron with the same speed as the proton
σioni
A
2.98
B
4.42
C
1.48
D
0.75
F
(4.80)
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Secondary electrons after ionisation
Energy
E is the transfered energy of an incident electron with kinetic energy T
W = E - Ij is the secondary electron kinetic energy
Angles
 if W > 100 eV
  cos -1
W
Wmax
where W max = 4Telec and Telec is the kinetic energy of an electron with the same speed as the proton
• if W ≤ 100 eV, θ’ is uniformly shot within
  0, π
 uniformly shot within [0, 2π]
  -
• proton scattering neglected (nuclear scattering < 1 keV ?)
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Proton charge transfert (TXS)
X  log10 (t )
t in eV
σ10 (t )  10Y(X)
• function of t
• plenty of experimental data
• dominant at low energy
Y(X)  a 0 X  b0
Y(X)  a 0 X  b0 - c0 (X - x 0 )d0
Y(X)  a1X +b1
a0 , b0 low energy line
c0 , d0 intermediate power
a1 , b1 high energy line
for X<x0
for X<x1
Parameters calculated from vapor data and in order that stopping powers match recommendations for liquid water
Parameters
a0
-0.180
b0
-18.22
c0
0.215
d0
3.550
a1
-3.600
b1
-1.997
x0
3.450
x1
5.251
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Hydrogen stripping (TXS)
 1
1 
σ 01 ( )  


σ
σ
high 
 low
  T D

  F 
σ low  4 π α  C 
  Ry 



2
0
1
 Ry 
σ high  4 π α 02 

 T 
• function of t
• two contributions


 Ry 
 A ln 1 
  B 
T 



where T is the kinetic of an electron with the same speed as the proton
Parameters adjusted to reproduce Dagnac & Toburen data, as well as stopping powers.
σ01
(50)
A
2.835
B
0.310
C
2.100
D
0.760
F
-
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Ionisation by Hydrogen (DCS)
 dσ 
 dσ 
 g(t )  
 
 dE hydrogen
 dE proton

 log(t ) - 4.2  
g(t )  0.8 1  exp 
 
0.5



• function of E and t
• integration by Simpson
1
 0.9
Differ from proton cross sections because of :
• screening effect of the H electron
• contribution of the stripping to the electron
spectrum
• interaction of H electron with water electrons
• Obtained from proton spectrum taking into
account Bolorizadeh and Rudd data,
as well as ICRU recommandations for liquid water.
t incident particle energy
at low energ, g(t) > 1
at high energy, g(t) <1 to take into account
the screening effect by the Hydrogen electron
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He,
+
He ,
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2+
He
12
List of processes
Processes He
ionisation : W + He → W+ + He + eexcitation : W + He → W * + He
charge transfer σ01 : W + He → W + He+ + echarge transfer σ02 : W + He → W + He++ + e- + eProcesses He+
ionisation : W + He+ → W+ + He+ + eexcitation : W + He+ → W * + He+
charge transfer σ12 : W + He+ → W + He++ + echarge transfer σ10 : W + He+ → W+ + He
Processes He++
ionisation : W + He++ → W+ + He++ + eexcitation : W + He++ → W *+ He++
charge transfer σ21 : W + He++ → W+ + He+
charge transfer σ20 : W + He++ → W++ + He
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Excitation & Ionisation for He, He+ and He++ (DCS)
d σ proton
d σ ion
2
(v i )  Z eff
(E)
(v i )
dE
dE
• FBA
• from p excitation or ionisation DXS
• function of E and t
Zeff = Z - S(R)
Takes into account the screening by the projectile’s electrons
R 
2 t elec Qeff
E
n
telec 
me
T
mHe
We have :
• Zeff : ion effective charge
• S(R) : screening at distance R from nucleus
• telec : kinetic energy of an electron with the same speed as the incident particle
• E : transfered energy
• Qeff : Slater effective charge for an electron on shell n for the considered ion
Qeff = 2.0 for 1s electron, Qeff = 1.7 pour 2 electrons on 1s, Qeff = 1.15 for an electron on 2s or 2p
S (R)1s  1 - exp(-2R) (1  2R  2R 2 )
S (R)2s  1 - exp(-2R) (1  2R  2R 2  2R 4 )
S (R)2p  1 - exp(-2R) (1  2R  2R 2 +(4/3) R 3  (2/3) R 4 )
He 2 : S(R)  0
He  : S(R)  0.70 S(R) 1s  0.15 S(R) 2s  0.15 S(R) 2p
He 0 : S(R)  0.50 S(R) 1s  0.25 S(R) 2s  0.25 S(R) 2p
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Charge transfer for He, He+ and He++ (TXS)
σ01
σ02
σ12
σ21
σ20
σ10
a0
2.25
2.25
2.25
0.95
0.95
0.65
b0
-30.93
-32.61
-32.10
-23.00
-23.73
-21.81
a1
-0.75
-0.75
-0.75
-2.75
-2.75
-2.75
c0
0.590
0.435
0.600
0.215
0.250
0.232
d0
2.35
2.70
2.40
2.95
3.55
2.95
x0
4.29
4.45
4.60
3.50
3.72
3.53
• from p charge transfer XS
• function of t
σij (t )  10Y(X)
X  log10 (t )
Y(X)  a 0 X  b0
Y(X)  a 0 X  b0 - c0 (X - x 0 )d0
Y(X)  a1X +b1
for X<x0
for X<x1
1/( d0 1)
a a 
x1   0 1 
 c0 d 0 
 x0
b1  (a0  a1 ) x1  b0  c0 ( x1  x0 ) d0
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electrons
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Oxygen K-shell ionisation (DXS)
• Binary Encounter Approximation (BEA)
• function of E and T, E and T > 540 eV
• E integrated over [T, (T+540)/2]
E : energy transfer (energy loss)
T = mv 2 / 2 : electron kinetic energy
R = 1 Ry
N = 0.3343x1023 molecules.cm-3 for liquid H2O
B = 537 eV : binding energy of the K-shell
n = 2 : electron occupation number
U = 809 eV : average kinetic energy of electron in K-shell
Contribution not neglected for T above 540 eV (~10% beyond 10 keV)
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Valence shells excitation and ionisation (DXS)
• function of E and T
• E integration over [7.5,max(T,0.5*(T+32.2)]
• Differential FBA cross section for a single excitation or ionisation
• First Born Approximation
• non relativisic limit
• Dielectric Response Func
ELFj (E,K)
Smearing of four outer shells
• Corrections at low energies (exchange and higher-order contributions)
Yj,exc  [1  (E j / T) a ]b
if Ej < T < 500 eV
Yj,exc  [1  (7.5 / T) a ]b
if 7.5 eV < T ≤ Ej
a  1, b  3
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Y ioniz 
if cut(j)<T<500 eV
18
Valence shells excitation and ionisation
• Real part of the DRF function (K=0)
• Dielectric formalism accounts for condensed-phase effects
• Superposition of Drude functions : optical model of the liquid
• Sum rule constraints
• only if E>cut(j)
fj : ocillator strength
Ej : transition energy
gj : damping coefficient
Ep = 21.46 eV plasmon energy
• Imaginary part of the DRF function (K=0)
• Dispersion to non-zero momentum transfers (K>0)
Generalized Oscillator Strength functions
Impulse approximation
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Valence shells excitation and ionisation partitioning
The energy loss function is cut just below the shell binding energy
and redistributed over the lower shells, to prevent the contribution to the
cross section below the binding energy :
• if E>=13 eV and E<17 eV, shell 8 is redistributed on shells 6 and 7
• if E>=10 eV and E<13 eV, shells 7-8 are redistributed on shell 6
• if E>=7.5 eV and E<10 eV, shells 6-7-8 are redistributed on shells 1 & 2
E is the transfered energy.
Differential IMFP for an incident electron energy T = 1 keV
15
Ionisation
Cut (eV)
1
7.5
2
7.5
3
7.5
4
7.5
5
7.5
6
10
7
13
8
17
9
32.2
dSigma(j)/dE (1/µm/eV)
Excitation
shell
excitations
1b1
3a1
1b2
2a1
Total
10
5
0
0
5
10
15
20
25
30
35
40
45
50
Energy Transfer (eV)
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Elastic scattering DCS and TCS
dσ el
(T)  R(T)
dΩ


1
 (T)


2
2
(1  2 (T) - cos  ) 
 (1  2g (T) - cos  )
R(T) 
Below 200 eV : Brenner-Zaider

5

g (T)  exp  γ n T n 
1
4   0 2
Z (Z  1) e 4
4 T2
Rutherford term
Above 200 eV : Rutherford « screened »
dσ el
R(T)
(T) 
dΩ
1  2s(T)  cosθ2
for 0.35 eV ≤ T ≤ 10 eV
 n 0

4


g (T)  exp  γ n 6 T n  for 10 eV < T ≤ 100 eV
 n 0

2
n
g (T)   γ n 11T
for 100 eV < T ≤ 200 eV
s(T)  s c (T)
n 0

4
function of T

 (T)  exp   n T n 
1.7x10 -5 Z 2/3mc 2
T ( T / mc 2  2)
s c (T)  1.64 - 0.0825 ln(T)
 n 0

4


 (T)  exp   n T n 
 n 0

dσ
 R(T)
sin θ dθ 
dθ
s(Z, T) s(T)  1
0
π
σ(T)  2 
• function of T
• valid over whole enrgy range
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Secondary electrons after ionisation
Energy
E is the transfered energy of an incident electron with kinetic energy T
The incident electron energy becomes T-E
The secondary electron energy is W = E - Bj where Bj is the binding energy of the ejected electron.
Angles
 if W > 100 eV sin 2 
W/T
(1 - W / T) (T / 2mc 2 )  1
 if W ≤ 100 eV, θ shot uniformly within
  shot uniformly within
 if W > 200 eV
 0,2 π
sin 2  

π
  0, 
 4
1- W / T
1  W / 2mc 2
 if 50 ≤ W ≤ 200 eV : 90%     ,  and 10%   
 4 2 
π π
 if W < 50 eV, θ’ shot uniformly within
0, π 
0, π
 '   
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Status : where are we now ?
We have all C codes available for the following processes :
Process
DiffXS
TotalXS
Electron elastic (Brenner and Rutherford)
Electron inelastic on valence
Electron inelastic on Oxygen K shell
A
T
A
A
T
T
Proton excitation
Proton ionisation
Proton charge transfer
Hydrogen ionisation
Hydrogen stripping
T (>100keV*)
A
A
-
A
T or A
A
T
A
Helium excitation
Helium ionisation
Helium charge transfer
T (>100keV*)
A
-
A
T
A
All analytical formulas (A) can produce tables (T)…
* Tables for proton excitation > 100 keV from Dingfelder’s code
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Energy ranges (usual)
e- ionisation+ excitation + elastic scattering
p ionisation
p excitation
H ionisation + stripping
He excitation + ionisation + charge transfer
10
102
103
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104
105
106
107
eV
24
Final states kinematics
Excitation (5 shells)
Ionisation (5 shells + K shell)
W + e → W* + e
W + p → W* + p
W + H → W* + H
W + a → W* + a
W + a+ → W* + a+
W + a++ → W* + a++
W + e → W+ + e + e
W + p → W+ + p + e
W + H → W+ + H + e
W + a → W+ + a + e
W + a+ → W + + a+ + e
W + a++ → W + + a++ + e
• Outgoing direction same as incoming
• E out = E in – E excitation for e, p, H, a
• Outgoing electron : analytical (energy, angle)
• Outgoing p, H, a : energy + momentum conservation
Charge changing and stripping
W + a++ → W + + a+
W + a++ → W ++ + a
s21
s20
Ea+ = Ea++ - 1/2me(pa++/ma++)2 + C
Ea = Ea++ - 2x1/2me(pa++/ma++)2 + C
C = Ba+-Bw
C = B*a-B*w
W + a+ → W + a++ + e
W + a+ → W + + a
s12
s10
Ea++ = Ea+ - D
Ea = Ea+ - 1/2me(pa+/ma+)2 + C
D = Ba+
C = Ba-Bw
W + a → W + a+ + e
W + a → W + a++ + e + e
s01
s02
Ea+ = Ea - D
Ea++ = Ea - D
D = Ba
D = B*a
W + p → W+ + H
W+H→W+p+e
s10
s01
EH = Ep – 1/2me(pp/mp)2 + C
Ep = E H - D
C = BH-Bw
D = BH
• Outgoing direction same as incoming
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Thank you for your attention
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Dielectric Response Function at the optical limit
2.5
2
1.5
1
0.5
10
20
30
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40
50
27
Energy Loss Function (ELF) without dispersion
Im
1 epsilon
1
0.8
0.6
0.4
0.2
10
20
30
40
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50
E
eV
28
Energy Loss Function (ELF) with dispersion
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Bethe surface : ELF in two dimensions
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SP and MFP
25
20
15
10
5
0
10
50 100
500 1000
5000 10000
15
• Born-corrections included
10
• no corrections
7
5
3
2
1.5
1
50 100
500 1000
5000 10000
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31
Definitions (liquid H2O molecule)
• Collision Stopping Power = average energy loss per unit path length
dE : energy loss
dS / dE : prob. per unit path length that an electron
of kinetic energy T will experience an energy loss
between E and E+dE
T = mv 2 / 2 : electron kinetic energy
• Inelastic Mean Free Path = distance between successive energy loss events
Emin = 0, Emax = T / 2
• Valence and core (K shell) processes
Justified by large difference in binding energy between valence and core shells
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Orders of magnitude
Partial ionization cross section for
each subshell of a water molecule as a
function of impact energy for (full
curves) electrons and (broken curves)
protons. The 1a1 curve for electrons is
multiplied by 100.
Centre d’Etudes Nucléaires de Bordeaux - Gradignan
For electrons, elastic collisions are increasingly
the most probable interaction event below
about 2 keV, while ionization takes over above
that energy. For both protons and electrons (T
> 100 eV) ionizations account for 75% of
inelastic collisions, the remaining 25% being
excitation events. For electron impact and as
threshold energies are approached excitations
become increasingly important and eventually
dominate the inelastic scattering probability.
33