1 Centre d’Etudes Nucléaires de Bordeaux - Gradignan G eant4 DNA Physics processes overview and current status Y. Perrot, S. Incerti Centre d'Etudes Nucléaires

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


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)


Protons and Hydrogen


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


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


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)


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


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

• proton scattering neglected (nuclear scattering < 1 keV ?)


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


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


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


He, He+, He2+


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


)(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


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


electrons


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)

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]


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


• 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


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


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


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


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


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


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


Thank you for your attention


Dielectric Response Function at the optical limit

10 20 30 40 50

0.5

1

1.5

2

2.5


10 20 30 40 50EeV0.2

0.4

0.6

0.8

1

Im1epsilon

Energy Loss Function (ELF) without dispersion


Energy Loss Function (ELF) with dispersion


Bethe surface : ELF in two dimensions


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


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


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.

Orders of magnitude

Documents

1 Centre d’Etudes Nucléaires de Bordeaux - Gradignan G eant4 DNA Physics processes overview and current status Y. Perrot, S. Incerti Centre d'Etudes Nucléaires