1 Centre d’Etudes Nucléaires de Bordeaux - Gradignan 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 13 th -19 th , 2005
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1Centre d’Etudes Nucléaires de Bordeaux - Gradignan
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
2Centre d’Etudes Nucléaires de Bordeaux - Gradignan
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 νproton 0 kexc, k Ω ν
σ (Z a ) ( - E )σ ( )
J
tt
t
No experimental data, but semi-empirical relations with electron excitation cross sections
0 is a constant (0 = 1E-20 m²)Z = 10 number of electrons in the crossed mediumEk 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)
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
• function of t
5 excitation levels
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Ionisation by Protons (DXS)
j
j
jj dW
dσ G
dE
dσ
E is the transfered energyt is the proton kinetic energyRy = 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)
Wj = E - Ij is the secondary electron kinetic energyw = Wj/Bj
Nj is the number of electrons on shell jS = 4πα0²Nj(Ry/Bj)² T = (me/mp) t : kinetic energy of an electron traveling at the same speed as the proton² = T/Bj
wc = 4²-2-Ry/(4Bj)α related to the size of the target molecule
Parameters from vapor data
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
]ν/) w- (w αexp 1[ w)(1
)ν(F w )ν(F
B
S
dw
dσ
c3
21
j
j
ν)(H )ν(L
ν)(H )ν(L )ν(F
)ν(H )ν(L )ν(F
22
222
111
)4(D1
-D1
11
1
νE 1
ν C )ν(L
21
2
21
1 /νB ν
)νln(1 A )ν(H
2D22 ν C )ν(L
ν
B
ν
A )ν(H
42
22
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
• 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
Rudd model
LE term
HE term
5 ionisation shells (K included)
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Ionisation by Protons (TXS)
1
ionilow high
1 1σ ( )
σ σ
F
Ry
T C α π4 σ
D
20low
where T is the kinetic of an electron with the same speed as the proton
B
T
Ry 1ln A
T
Ry α π4 σ 2
0high
σioni
A 2.98
B 4.42
C 1.48
D 0.75
F (4.80)
• function of t
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if W > 100 eV
max
1-
W
W cos
where Wmax = 4Telec and Telec is the kinetic energy of an electron with the same speed as the proton
π0,
φ uniformly shot within [0, 2π]
φφ -
Secondary electrons after ionisation
• if W ≤ 100 eV, θ’ is uniformly shot within
Angles
Energy
E is the transfered energy of an incident electron with kinetic energy T
W = E - Ij is the secondary electron kinetic energy
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Y(X)10σ ( ) 10t
10X log ( )tt in eV
0 0Y(X) a X b a0 , b0 low energy line
c0 , d0 intermediate power
a1 , b1 high energy line
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
Proton charge transfert (TXS)
• function of t• plenty of experimental data• dominant at low energy
0d0 0 0 0Y(X) a X b - c (X - x )
1 1Y(X) a X +b
for X<x0
for X<x1
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1
01low high
1 1σ ( )
σ σ
F
Ry
T C α π4 σ
D
20low
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.
B
T
Ry 1ln A
T
Ry α π4 σ 2
0high
(50)
Hydrogen stripping (TXS)
σ01
A 2.835
B 0.310
C 2.100
D 0.760
F -
• function of t• two contributions
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hydrogen proton
dσ dσ g( )
dE dEt
1log( ) - 4.2
g( ) 0.8 1 exp 0.90.5
tt
Ionisation by Hydrogen (DCS)
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 energyat low energ, g(t) > 1at high energy, g(t) <1 to take into account the screening effect by the Hydrogen electron
• function of E and t• integration by Simpson
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He, He+, He2+
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Processes He
ionisation : W + He → W+ + He + e-
excitation : W + He → W * + He
charge transfer σ01 : W + He → W + He+ + e-
charge transfer σ02 : W + He → W + He++ + e- + e-
Processes He+
ionisation : W + He+ → W+ + He+ + e-
excitation : W + He+ → W * + He+
charge transfer σ12 : W + He+ → W + He++ + e-
charge transfer σ10 : W + He+ → W+ + He
Processes He++
ionisation : W + He++ → W+ + He++ + e-
excitation : W + He++ → W *+ He++
charge transfer σ21 : W + He++ → W+ + He+
charge transfer σ20 : W + He++ → W++ + He
List of processes
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)(v E d
σ d (E) Z )(v
E d
σ di
proton2effi
ion
2p2s1s0
2p2s1s
2
S(R) 0.25 S(R) 0.25 S(R) 0.50 S(R) : He
S(R) 0.15 S(R) 0.15 S(R) 0.70 S(R) : He
0 S(R) : He
Zeff = Z - S(R)
elec eff2 t QR
E n
Qeff = 2.0 for 1s electron, Qeff = 1.7 pour 2 electrons on 1s, Qeff = 1.15 for an electron on 2s or 2p
Takes into account the screening by the projectile’s electrons
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
21sS (R) 1 - exp(-2R) (1 2R 2R )
2 42sS (R) 1 - exp(-2R) (1 2R 2R 2R )
2 3 42pS (R) 1 - exp(-2R) (1 2R 2R +(4/3) R (2/3) R )
eelec
He
mt T
m
Excitation & Ionisation for He, He+ and He++ (DCS)
• FBA• from p excitation or ionisation DXS• function of E and t
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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
Charge transfer for He, He+ and He++ (TXS)
• from p charge transfer XS• function of t
Y(X)ijσ ( ) 10t
10X log ( )t
0 0Y(X) a X b 0d
0 0 0 0Y(X) a X b - c (X - x )
1 1Y(X) a X +b
for X<x0
for X<x1
01/( 1)
0 11 0
0 0
da a
x xc d
01 0 1 1 0 0 1 0( ) ( )db a a x b c x x
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electrons
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E : energy transfer (energy loss)T = mv 2 / 2 : electron kinetic energyR = 1 RyN = 0.3343x1023 molecules.cm-3 for liquid H2OB = 537 eV : binding energy of the K-shelln = 2 : electron occupation numberU = 809 eV : average kinetic energy of electron in K-shellContribution not neglected for T above 540 eV (~10% beyond 10 keV)
• function of E and T, E and T > 540 eV• E integrated over [T, (T+540)/2]
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Valence shells excitation and ionisation (DXS)
• Corrections at low energies (exchange and higher-order contributions)
ioniz
Y
• Differential FBA cross section for a single excitation or ionisation
Smearing of four outer shells
• First Born Approximation• non relativisic limit• Dielectric Response Func
1, 3a b
• function of E and T• E integration over [7.5,max(T,0.5*(T+32.2)]
ELFj (E,K)
bajexcj, ]T) / (E[1Y if Ej < T < 500 eV
baexcj, ]T) / (7.5[1Y if 7.5 eV < T ≤ Ej
if cut(j)<T<500 eV
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• Real part of the DRF function (K=0)
• Imaginary 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)
• Dispersion to non-zero momentum transfers (K>0)
fj : ocillator strength
Ej : transition energy
j : damping coefficient
Ep = 21.46 eV plasmon energy
Generalized Oscillator Strength functions
Impulse approximation
Valence shells excitation and ionisation
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Valence shells excitation and ionisation partitioning
shell 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
Excitation
Ionisation
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.
0
5
10
15
0 5 10 15 20 25 30 35 40 45 50
Energy Transfer (eV)
dS
igm
a(j)
/dE
(1/
µm
/eV
)
excitations
1b1
3a1
1b2
2a1
Total
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22
el
) cos-(T)2(1
(T)
) cos-(T)2(1
1 R(T))T(
dΩ
dσ
Elastic scattering DCS and TCS
2
4
20 T 4
e 1)(Z Z
4
1R(T)
5
0n
nnTγ exp (T) for 0.35 eV ≤ T ≤ 10 eV
4
0n
n6n Tγ exp (T) for 10 eV < T ≤ 100 eV
2
0n
n11n Tγ (T) for 100 eV < T ≤ 200 eV
4
0n
nnT exp (T)
4
0n
nnT exp (T)
2el
cosθ2s(T)1
R(T)(T)
dΩ
dσ
)2mc / T ( T
mcZ1.7x10(T)s s(T)
2
22/3-5
c
1s(T) T)s(Z,
R(T) dθ θsin
dθ
dσ2)T(σ
π
0
function of T
Rutherford term
Below 200 eV : Brenner-Zaider Above 200 eV : Rutherford « screened »
ln(T) 0.0825-1.64 (T)sc
• function of T• valid over whole enrgy range
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Secondary electrons after ionisation
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.
if W > 100 eV 1 )2mc / (T T) W / -(1
TW / sin
22
if W ≤ 100 eV, θ shot uniformly within
4
π0,
π0,2 φ φ shot uniformly within
22
2mc W / 1
T W / - 1 sin
if W > 200 eV
if 50 ≤ W ≤ 200 eV : π π90% , and 10% 0, π
4 2
if W < 50 eV, θ’ shot uniformly within 0, π
'φ φ
Angles
Energy
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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) A AElectron inelastic on valence T TElectron inelastic on Oxygen K shell A T
Proton excitation T (>100keV*) AProton ionisation A T or AProton charge transfer - AHydrogen ionisation A THydrogen stripping - A
Helium excitation T (>100keV*) AHelium ionisation A THelium charge transfer - 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)
H ionisation + stripping
104 105 106 107 eV10310210
p excitation
p ionisation
He excitation + ionisation + charge transfer
e- ionisation+ excitation + elastic scattering
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Final states kinematics
Excitation (5 shells)
W + e → W* + eW + p → W* + pW + H → W* + HW + → W* + W + + → W* + +
W + ++ → W* + ++
• Outgoing direction same as incoming• E out = E in – E excitation for e, p, H,
Ionisation (5 shells + K shell)
W + e → W+ + e + eW + p → W+ + p + eW + H → W+ + H + eW + → W+ + + eW + + → W+ + + + eW + ++ → W+ + ++ + e
• Outgoing electron : analytical (energy, angle)• Outgoing p, H, : energy + momentum conservation
Charge changing and stripping
W + ++ → W+ + + 21 E+ = E++ - 1/2me(p++/m++)2 + C C = B+-Bw
W + ++ → W++ + 20E = E++ - 2x1/2me(p++/m++)2 + C C = B*-B*w
W + + → W + ++ + e 12 E++ = E+ - D D = B+
W + + → W+ + 10E = E+ - 1/2me(p+/m+)2 + C C = B-Bw
W + → W + + + e 01 E+ = E - D D = B
W + → W + ++ + e + e 02 E++ = E - D D = B*
W + p → W+ + H 10 EH = Ep – 1/2me(pp/mp)2 + C C = BH-Bw
W + H → W + p + e 01 Ep = EH - D 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
10 20 30 40 50
0.5
1
1.5
2
2.5
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10 20 30 40 50EeV0.2
0.4
0.6
0.8
1
Im1epsilon
Energy Loss Function (ELF) without dispersion
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Energy Loss Function (ELF) with dispersion
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Bethe surface : ELF in two dimensions
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SP and MFP
10 50 100 500 1000 5000100000
5
10
15
20
25
50 100 500 1000 5000 10000
1
1.5
2
3
5
7
10
15
• Born-corrections included
• no corrections
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Definitions (liquid H2O molecule)
• Collision Stopping Power = average energy loss per unit path length
• Inelastic Mean Free Path = distance between successive energy loss events
• Valence and core (K shell) processes
dE : energy loss
d/ dE : prob. per unit path length that an electronof kinetic energy T will experience an energy loss between E and E+dE
T = mv 2 / 2 : electron kinetic energy
Emin = 0, Emax = T / 2
Justified by large difference in binding energy between valence and core shells
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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.
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.