photoelectric effect models in geant4 and geantv€¦ · photoelectric e ect models in geant4 and...

PhotoElectric effect models in Geant4 and GeantV

Marilena Bandieramonte

CERN

[email protected]

June 8, 2017

Marilena Bandieramonte (ep-sft) PhotoElectric effect review June 8, 2017 1 / 46

Overview

1 PhotoElectric effect - IntroductionIntroduction

2 PhotoElectric effect - Literature reviewPhysics OverviewPhotoelectric cross section libraries

3 PhotoElectric effect in Geant4Standard electromagnetic packageLow energy package

4 Developing a new Photoelectric modelPossible actions and open pointsLivermore model analysisNew parameterisation with epics2014 dataBinding energies dataConclusions

5 Appendix


Introduction

In the photo-electric absorption process a photon is absorbed by an atom and an electronis emitted with an energy:

Ephotoelectron = Eγ − Bshell (Zi ) (1)

The atom, left in an excited state with a vacancy in the ionized shell, decays to its groundstate through a cascade of radiative and non-radiative transitions with the emission ofcharacteristic x-rays and Auger and Coster-Kronig electrons.


Energy range

Photoelectric effect dominates at low energies.In the case of Carbon (C) it dominates for energies Eγ < 20KeV .


Photoelectric process in a nutshell

5 main ingredients:

1 Macroscopic p.e. cross-section

2 Total p.e. atomic cross-section

3 Partial cross-section - sub-shell

4 Angular distribution

5 Atomic relaxation


Total and partial cross sections

The total photoelectric cross section σpe exhibits a characteristicsawtooth behaviour corresponding to absorption edges.The K shell electrons are the the most tightly bound, and themost important contribution to the atomic photoelectric crosssection when there is enough energy to release them.


Physics overview - limitations

Current general purpose Monte Carlo codes for particle transportconsider single photon interactions with isolated atoms in theirground state.

So they neglect:

interactions with ions and excited statesmultiple ionizationssolid state effectsnear-edge absorption structures produced by molecular or crystallineordering.

Absorption spectroscopy techniques like:

NEXAFS (Near Edge X-ray Absorption Fine Structure)EXAFS (Extended X-ray Absorption Fine Structure)XANES (X-ray Absorption Near Edge Structure)

cannot be simulated properly.




So they neglect:

interactions with ions and excited states

multiple ionizationssolid state effectsnear-edge absorption structures produced by molecular or crystallineordering.







So they neglect:

interactions with ions and excited statesmultiple ionizations

solid state effectsnear-edge absorption structures produced by molecular or crystallineordering.







So they neglect:

interactions with ions and excited statesmultiple ionizationssolid state effects

near-edge absorption structures produced by molecular or crystallineordering.







So they neglect:

interactions with ions and excited statesmultiple ionizationssolid state effectsnear-edge absorption structures produced by molecular or crystallineordering.






The elementary theory for p.e. has been reviewed by Pratt

The interaction between the active electron and the electromagneticfield is treated in first-order perturbation theory[Francesc Salvat , 2009]

It is usually assumed that the incident photon beam isunpolarized, and that the spin of the photoelectron is not observed.

With these assumptions, the partial DCS for photoionization of theactive shell is a function of only the direction k̂b of thephotoelectron.

dσph,a

dk̂b

= (2π)2e2~c1

Ekb

Eb + mec2

c2~2

∑′| (Mkα)ba |2 (2)


Total and partial cross sections [J H Hubbell , 2006]

Early theoretical calculations of photoionization cross sections werelimited to the K shell (Pratt (1960) , Pratt et al. - 1964) - showingasymptotic behaviourScofield’s (1973) non-relativistic calculations in a HS framework: allsub-shells, elements from Z = 1 to 101, energy range [1keV - 1.5MeV ].Provided renormalization factors to convert his cross section results tovalues expected from a relativistic Dirac-Hartree-Fock (DHF) computationChantler (1995 - 2000) in a self-consistent relativistic DHF framework.Z = 1 to 92, the lower-bound energy varies between 1eV and 10eV , andthe upper-bound energy varies between 0.4MeV and 1.0MeV .Salvat and Sabatucci (2015) formulation of p.e. within theone-active-electron approximation, with a DHFS self-consistent potential(Dirac-Hartree-Fock-Slater) - used in the latest 2015 version of Penelope.



These theoretical calculations provide the basis for the tabulated datalibraries [M. C. Han, 2016]

Various empirical formulations of photoionization cross sections are reportedin the literature


Photoelectron angular distribution

Fischer’s non-relativistic theory addresses the calculation ofdifferential cross sections in the low energy region.

The first relativistic treatment of the photoelectric effect was givenby Sauter who calculated the K-shell cross section in the Bornapproximation; it concerns the lowest order in Zα/β

Gavrila and Nagel extended Sauter’s results to the next order inZα/β.

Further calculations by Gavrila are available for the L shell.


Double Differential Photoelectric cross section correct tofirst order αZ

dσκ

dω(θ,φ) =

4

m2α6Z 5 β3(1− β2)3

[1− (1− β2)12 ]5

{F(1− παZ

β+ παZG)

}(3)

where

F =sin2 θ cos2 φ

(1− β cos θ)4− 1− (1− β2)

12

2(1− β2)

sin2 θ cos2 φ

(1− β cos θ)3+

[1− (1− β2)12 ]2

4(1− β2)32

sin2 θ

(1− β cos θ)3(4)

and

G =[1− (1− β2)

12 ]

12

272β2(1− β cos θ)

52

[4β2

(1− β2)12

sin2 θ cos2 φ

1− β cos θ+

4β

(1− β2)cos θcos2φ− 4

1− (1− β)12

1− β2(5)

(1 + cos2 φ)− β2 1− (1− β2)12

1− β2

sin2 θ

1− β cos θ+ 4β2 1− (1− β2)

12

(1− β2)32

− 4β

[1− (1− β2)

12

]2(1− β2)

32

cos θ]

+1− (1− β2)

12

4β2(1− β cos θ)2[

β

1− β2− 2

1− β2cos θ cos2 φ+

1− (1− β2)12

(1− β2)32

cos θ − β 1− (1− β2)12

(1− β2)32

]


Simplified differential cross section correct to first order αZ

This DCS describes the ionization of the ground state (1s1/2) ofhydrogenic ions, and is obtained in the plane-wave Born approximation:

dσ

d cos θ' sin2 θ

(1− β cos θ)4

{1 +

1

2γ(γ − 1)(γ − 2)(1− β cos θ)

}(6)

where β and γ are the photoelectron Lorentz factors.


Photoelectric cross section libraries



EPDL - Evaluated Photon data Library. It combines two sets of datato compute the total cross-section.

Data calculated using Scofield’s sub-shell cross-sections from the edgeenergy up to 1 MeVHubbell’s total photoelectric cross- sections from 1keV to 100GeV.

PHOTX and XCOM. Both include

Scofield’s 1973 unrenormalized cross sections up to 1.5 MeV.At higher energies a semi-empirical formula connects Scofield’s values at1.5 MeV to the asymptotic high energy limit calculated by Pratt

Ebel. Parameterization based on fitting fifth order polynomials in thelogarithm of the photon energy to Scofield’s 1973 cross section dataBiggs and Lighthill. Analytical approximation using varioussemi-empirical and theoretical sources.

For more details look at:indico.cern.ch/event/629729/contributions/2569301/attachments/1449139/2233786/PhotoElectricEffect MB.pdf







Ebel. Parameterization based on fitting fifth order polynomials in thelogarithm of the photon energy to Scofield’s 1973 cross section data

Biggs and Lighthill. Analytical approximation using varioussemi-empirical and theoretical sources.








Ebel. Parameterization based on fitting fifth order polynomials in thelogarithm of the photon energy to Scofield’s 1973 cross section dataBiggs and Lighthill. Analytical approximation using varioussemi-empirical and theoretical sources.



PhotoElectric effect in Geant4


Geant4 photoelectric package

The basic set of gamma models in EM physics packages [J Allison , 2016][V. Ivanchenko, 2011] includes:

Models developed for HEP applications: standard electromagneticpackage

G4PEEffectFluoModel : V. Grichine, M. Mairie and V. Ivanchenko

Models based on the Livermore evaluated data Library

G4LivermorePhotoElectricModel : A Ivanchenko and V. IvanchenkoG4LivermorePolarizedPhotoElectricModel : Sebastien Incerti, A.Forti,M.G.Pia, A. Mantero and V. Ivanchenko

C++ implementation of the Penelope 2001 models

G4PenelopePhotoElectricModel : L. Pandola

If photon energy is below the lowest available energy, the cross section iscomputed for this lowest energy, to ensure the gamma is absorbed byphotoabsorption at any energy.

This is done for transport reasons in HEPAs a result, all media is non-transparent to low-energy gamma rays


Standard electromagnetic package: G4PEEffectFluoModel

The atomic cross sections. Biggs et al. parameterization from SANDIAtables, with separate fit of coefficients, is used :

σ(Z ,Eγ) =a(Z ,Eγ)

Eγ+

b(Z ,Eγ)

E 2γ

+c(Z ,Eγ)

E 3γ

+d(Z ,Eγ)

E 4γ

(7)

The sub-shell it’s chosen in a deterministic way, not sampled:

the first inner shell having a binding energy Bshell < Eγ

Shell atomic energies are taken from G4AtomicShells data

The photoelectron angle is calculated according to the Sauter-Gavriladistribution for K shell, which is correct only to zero order in αZ


Low Energy Package: G4LivermorePhotoElectricModel

The total photoelectric and single shell cross-sections aretabulated from threshold to 600keV. Above 600keV EPDL97 crosssections are parameterised as following:

σ(E ) =a1

E+

a2

E 2+

a3

E 3+

a4

E 4+

a5

E 5 (8)

The accuracy of such parameterisation is better than 1%.

The sub-shell is sampled according to the relative cross-sections ofall sub-shells.

Two angular generators:

G4SauterGavrilaAngularDistribution: same as in the standard model -default optionG4PhotoElectricAngularGeneratorPolarized : double differential crosssection generator derived from Gavrila’s calculations, which can alsohandle polarized photons.


Low Energy Package: G4PenelopePhotoElectricModel

The total photoelectric cross section at a given photon energy E iscalculated from the data EPDL89 [D.E.Cullen , 1989].

The sub-shell is selected according to the relative cross sections of sub-shells,determined at the energy E by interpolation of the data. Only K-L-M shellsare taken into account.The direction of the electron is sampled according to the Sauterdistribution. Introducing the variable ν = 1− cosθe , the angular distributioncan be expressed as:

p(ν) = (2− ν)[1

A + ν+

1

2βγ(γ − 1)(γ − 2)]

ν

(A + ν)3 (9)

where

γ = 1 +Ee

mec2, A =

1

β− 1 (10)

where Ee is the electron energy, me its rest mass and β its velocity in units ofthe speed of light c .



The total photoelectric cross section at a given photon energy E iscalculated from the data EPDL89 [D.E.Cullen , 1989].The sub-shell is selected according to the relative cross sections of sub-shells,determined at the energy E by interpolation of the data. Only K-L-M shellsare taken into account.

The direction of the electron is sampled according to the Sauterdistribution. Introducing the variable ν = 1− cosθe , the angular distributioncan be expressed as:

p(ν) = (2− ν)[1

A + ν+

1

2βγ(γ − 1)(γ − 2)]

ν

(A + ν)3 (9)

where

γ = 1 +Ee

mec2, A =

1

β− 1 (10)




The total photoelectric cross section at a given photon energy E iscalculated from the data EPDL89 [D.E.Cullen , 1989].The sub-shell is selected according to the relative cross sections of sub-shells,determined at the energy E by interpolation of the data. Only K-L-M shellsare taken into account.The direction of the electron is sampled according to the Sauterdistribution. Introducing the variable ν = 1− cosθe , the angular distributioncan be expressed as:

p(ν) = (2− ν)[1

A + ν+

1

2βγ(γ − 1)(γ − 2)]

ν

(A + ν)3 (9)

where

γ = 1 +Ee

mec2, A =

1

β− 1 (10)



Validation of Photoelectric effect models


Validation of photoelectric model


Validation of the Geant4 electromagnetic photoncross-sections for elements and compounds

[G.A.P. Cirrone, 2010]

Validation against NIST, SANDIA and EPDL97 data libraries15 elements, spanning the range of atomic number Z from 1 to 82, havebeen selected for validation and three compounds, namely Air, Water andNaIThe deviation of the Geant4 Standard model with respect to NIST andEPDL97 is within 10%.The two Geant4 LowEnergy models exhibit a very close agreement withthese libraries, as it is expected, because both NIST and EPDL97 use thesame data source below 1 MeV.The agreement of the Livermore and Penelope photoelectric models withthe SANDIA database is within 10% and in the energy range[1KeV − 1MeV ] is lower.


Validation of cross sections for Monte Carlo simulation ofthe photoelectric effect

[M. C. Han, 2016]

The validation process identifies Scofield’s 1973 non-relativisticcalculations, tabulated in EPDL97, as the one best reproducingexperimental measurements of total cross sections.

In a few test cases Ebel’s parameterisation [1KeV-300KeV], producesmore accurate results close to absorption edges.

G4Sandia parameterization differs from Biggs and Lighthill’s and isreported to significantly reduce the accuracy of total cross sectionsbelow 1KeV.


Modified Biggs-Lighthill parameterisation

The data sample was limited to test cases where the two cross sectioncalculations produce different values at the same energy of an experimentalmeasurement.

These test cases concern only noble gases, oxygen and hydrogen, andconcerns data below 1 keV, since the coefficients of the twoparameterisations differ only in the low energy range.


Ebel’s parameterisation

This parameterisation is applicable to energies between [1keV − 300keV ].

The efficiencies show similar compatibility with experiment for Ebel’s and EPDLcross sections, which is confirmed by the statistical comparison of the twocategories.


Ebel’s parameterisation - K shell cross sections

Ebel’s parameterisation appears more efficient than EPDL tabulations atreproducing experimental K-shell cross sections.

The failures of EPDL-based cross sections at reproducing experimental datamostly concern test cases close to absorption edges


New G4/GV photoelectric models development


Possible actions and open points

Revert to original Sandia coefficients: not an option. Sandiaparameterization is not statistically superior to EPDL. Furthermore subshellscross sections are missing and the selection is currently done in a deterministicway.

Use EPDL/livermore in standard physics: livermore model is more complex(but also more accurate) than the current standard. It can be improved:

Accuracy of cross-sections with new available data (livermore/epics2014)CPU perfomanceAngular generator sampling algorithm

Other possibilities under investigation:

Move to Ebel’s parametrization: K- shell cross sections based on Ebel’sparameterisation produce more accurate results than EPDL in some test casesclose to absorption edges for energy range [1KeV- 300 KeV]Salvat and Sabatucci new formulation (Penelope 2015 version):uses the sametheory as in Scofield (1973) of the EPDL tables, but more accurate numericalalgorithms.







Move to Ebel’s parametrization: K- shell cross sections based on Ebel’sparameterisation produce more accurate results than EPDL in some test casesclose to absorption edges for energy range [1KeV- 300 KeV]

Salvat and Sabatucci new formulation (Penelope 2015 version):uses the sametheory as in Scofield (1973) of the EPDL tables, but more accurate numericalalgorithms.







Move to Ebel’s parametrization: K- shell cross sections based on Ebel’sparameterisation produce more accurate results than EPDL in some test casesclose to absorption edges for energy range [1KeV- 300 KeV]Salvat and Sabatucci new formulation (Penelope 2015 version):uses the sametheory as in Scofield (1973) of the EPDL tables, but more accurate numericalalgorithms.


Livermore vs Standard - execution time

Model level test, #particles = 107, #repetitions = 10


Livermore vs Standard percent variation

Max percent variation: all energies vs E >= 1MeV


Livermore vs Standard - Angular distribution ON

The standard angular distribution has a threshold at ∼ 26MeVGreater differences are observable at low energies and for heavierelements


New Penelope-like angular distribution

Achievements:

Increased threshold from 26MeV to 100MeVSpeedup between 4% and 10% in most cases


Unit test and benchmark of improved PELivermore model

The improved Livermore model has been included in 10.3ref04 and tested:

Results for 10.3ref04 are stable compared to 10.3ref03

No visible degradation of any result

Validation results available athttps://geant4-tools.web.cern.ch/geant4-tools/emtesting/

Version 10.3ref04 with Livermore model for Rayleigh and Photo-effectused as default in Em physics list, to check CPU performance:

Improved quality of photon cross sections that may slightly affectshower shape

1% slowdown for Higgs sample

2% slowdown for e- showers

− > Optimization on photoelectric model might recover CPUperformance.


Livermore vs Standard - Angular distribution OFF

Switching off the angular distribution we clearly see a dependency on theenergy and on the Z of the material.This is dependent on a 600KeV threshold between tabulated andparameterised cross sections (EPDL97)


New parameterisation - livermore/epics2014 data

New cross sections data recently introduced in Geant4: livermore/epics2014 (S. Incerti)New fit: adding one more parameter and performing two separate fits we can reduce thethreshold from 600KeV to 5KeV.

σ(E ) =a1

E+

a2

E 2+

a3

E 3+

a4

E 4+

a5

E 5+

a6

E 6 (11)

Eγ >= 5KeV : Two new fits in two different energy ranges

Eγ ∈ [5KeV − 50KeV ]Eγ > 50KeV

Eγ < 5KeV :

Keep the parameterisation (under evaluation): high-energy physics simulationsTabulated cross sections: low-energy physics simulations


Two separate fits - new data

Achievements:

Threshold moved from 600KeV to 5KeV

Speedup has to be measuredMarilena Bandieramonte (ep-sft) PhotoElectric effect review June 8, 2017 38 / 46

Two separate fits - new data

In some cases we are able to cover all the spectrum withparameterization

Especially for inner shells


New parameterisation: example


Two separate fits - new data: special cases

Outer shells present a non-monotonic behaviour, significantly at verylow energies


Two separate fits - new data: special cases


Binding energies data analysis

At least four different sources of binding energies data in Geant4:

G4AtomicShells: used in the standard physics

G4AtomicShellsEADL: recently introduced by L. G. Sarmiento

livermore/phot/pe-: livermore parametrisation

fluor/binding.dat: used for de-excitation model - same as theparametrisation ones

EADL vs livermore/phot/pe-MeV

G4AtomicShells vs G4AtomicShells_EADLMeV


Binding energies data analysis

G4AtomicShellsEADL and livermore/phot/pe-For Z < 60 differences are within 35eV .From Z = 60 (mostly on k-shells) we have differences from around 50eV up to630eV for Z = 98

G4AtomicShells and G4AtomicShellsEADLTotal differentNumShell: 38Total n. energies that differ more than 10 eV: 19 (72 eV is one of the biggest)G4AtomicShellsEADL has always one more shell than G4AtomicShells - and it’sthe right number of shells (ionisation potential taken mostly from theoreticalcalculations)

Decoupling of models from binding energies would be desirable. But not easyto achieve:

Data should not disagreeAt least the deexcitation model has to have the same number of shellsThe source of data of EADL class need to be clarified - detailed study need tobe doneTo guarantee the consistency the decision (today) is to include binding energiesinside the model.


Conclusions and plans

Review of G4 photoelectric models has been doneUpdated and improved Livermore model:

Photoelectron angular distribution sampling has been improved.Measurements report improvements between 4% and 10%.

The use of Alias sampling in GeantV for the rejection part will eventuallyproduce greater gains.

A new fit (in two steps) on updated cross-sections data has been performed

More accuracyMore efficiency: threshold from 600KeV to 5KeV

We may produce an alternative standard model:

More accurate parameterizationThat take into account shells samplingFast

Open issue: have consistency in the atomic binding energies data

PhotoelectricDeexcitation modelOther modelsWe are not ready to drop G4AtomicShells data



Review of G4 photoelectric models has been done

Updated and improved Livermore model:











Review of G4 photoelectric models has been doneUpdated and improved Livermore model:










Thanks for your attention.


Appendix


Experimental data: Total cross sections


References

D.E.Cullen et al

Tables and graphs of photon-interaction cross sections from 10 eV to 100 GeV derived from theLLNL evaluated photon data library (EPDL)

Report UCRL-50400 (Lawrence Livermore National Laboratory

J Allison et al., by the Geant4 Collaboration

Recent Developments in Geant4

Nucl. Instrum. Meth. A 835 186-225, 2016

V. Ivanchenko et al.

Recent Improvements in Geant4 Electromagnetic Physics Models and Interfaces

Progress in NUCLEAR SCIENCE and TECHNOLOGY, 2011 Vol. 2, pp.898-903,

J H Hubbell

Review and history of photon cross section calculations

Phys. Med. Biol. 51 (2006) R245-R262 .

Francesc Salvat and Jose M Fernandez-Varea

Overview of physical interaction models for photon and electron transport used in Monte Carlocodes

Metrologia 46 (2009) S112-S138 2010.

Vladimir Ivanchenko et al. (2010)

Recent Improvements in Geant4 Electromagnetic Physics Models and Interfaces

Joint International Conference on Supercomputing in Nuclear Applications and Monte Carlo 2010618(1-3), 315 - 322.

G.A.P. Cirrone and G. Cuttone and F. Di Rosa and L. Pandola and F. Romano and Q. Zhang(2010)

Validation of the Geant4 electromagnetic photon cross-sections for elements and compounds

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,Detectors and Associated Equipment 618(1-3), 315 - 322.

M. C. Han and H. S. Kim and M. G. Pia and T. Basaglia and M. Batic and G. Hoff and C. H. Kimand P. Saracco (2016)

Validation of Cross Sections for Monte Carlo Simulation of the Photoelectric Effect

IEEE Transactions on Nuclear Science 63(2), 1117-1146.


The End


Backup slides


Standard electromagnetic package - Geant4-10.2 - Finalstate sampling

Differential Photoeffelectric cross section correct to first order αZ

dσ

d cos θ' sin2 θ

(1− β cos θ)4

{1 +

1

2γ(γ − 1)(γ − 2)(1− β cos θ)

}(12)

where β and γ are the photoelectron Lorentz factors. cosθ is sampled from the probability density function:

f (cos θ) =1− β2

2β

1

(1− β cos θ)2(13)

so

cos θ =(1− 2r) + β

(1− 2r)β + 1(14)

The rejection function is :

g(cos θ) =1− cos2θ

(1− β cos θ)2[1 + b(1− β cos θ)] (15)

with b = γ(γ − 1)(γ − 1)/2It can be shown that g(cosθ) is positive for each cosθ ∈ [−1,+1], and can be majored by :

gsup =

{γ2[1 + b(1− β)] if γ ∈]1, 2]

γ2[1 + b(1 + β)] if γ > 2(16)

The efficiency of this method is ∼ 50% if γ < 2,∼ 25% if ∈ [2, 3].


Standard electromagnetic package - Geant 10.2 -Macroscopic and atomic cross-sections

Once the photoelectric process is selected, the model must choosewhich atom is involved. In compound materials the i th element ischosen randomly according to the probability:

Prob(Zi ,Eγ) =niσi (Zi ,Eγ)∑

i [niσi (Eγ)](17)

where ni is the number of atom per volume of the i th elementcomposing the material.

For each process the total cross section at a given energy E isobtained with a log-log interpolation of cross section data σ1 and σ2

available in the data libraries from the closest energies.


Macroscopic cross section and mean free path

For compound materials (and also for mixtures) the molecular crosssection σph(E ) is evaluated by means of the additivity approximation.

It consists in the sum of the atomic cross sections of all the elementsin the molecule:

σ(Eγ) =∑

i

niσ(Zi ,Eγ) (18)

where ni is the number of atoms per volume of the i th element of thematerial.The mean free path, λ, for a photon to interact via the photoelectriceffect is given by:

λ(Eγ) =(∑

i

niσ(Zi ,Eγ))−1

(19)



For compound materials (and also for mixtures) the molecular crosssection σph(E ) is evaluated by means of the additivity approximation.It consists in the sum of the atomic cross sections of all the elementsin the molecule:

σ(Eγ) =∑

i

niσ(Zi ,Eγ) (18)

where ni is the number of atoms per volume of the i th element of thematerial.

The mean free path, λ, for a photon to interact via the photoelectriceffect is given by:

λ(Eγ) =(∑

i

niσ(Zi ,Eγ))−1

(19)



For compound materials (and also for mixtures) the molecular crosssection σph(E ) is evaluated by means of the additivity approximation.It consists in the sum of the atomic cross sections of all the elementsin the molecule:

σ(Eγ) =∑

i

niσ(Zi ,Eγ) (18)

where ni is the number of atoms per volume of the i th element of thematerial.The mean free path, λ, for a photon to interact via the photoelectriceffect is given by:

λ(Eγ) =(∑

i

niσ(Zi ,Eγ))−1

(19)


PhotoElectric SG angular distribution

0

0.02

0.04

0.06

0.08

0.1

0.12

-12 -10 -8 -6 -4 -2 0 2

’photoelectric_G4_angular_Pb.ascii’ u 2:3’photoelectric_GV_angular_NIST_MAT_Pb.ascii’ u 2:3


Total Photoeffect cross section correct to first order αZ

σκ =3

2φ0α

4Z 5 β3(1− β2)[1− (1− β2)

12

]5

[M

(1− παZ

β

)+ παZN

](20)

where

M =4

3+

1− 3(1− β2)12 + 2(1− β2)

β2(1− β2)12

[1 +

1− β2

2βln

1− β1 + β

](21)

and

N =1

β3{− 4

15

1

(1− β2)12

}+34

15− 63

15

(1− β2

) 12 +

25

15

(1− β2

)+

8

15

(1− β2

) 32 + (1− β2)

12

[1− 3(1− β2)

12 + 2(1− β2)

] 1

2βln

1− β1 + β

(22)


Standard electromagnetic package - Geant 10.2 - Totalcross-section

For each process the total cross section at a given energy E is obtained witha log-log interpolation.

log(σ(E )) = log(σ1)log(E2)− log(E )

log(E2)− log(E1)+ log(σ2)

log(E )− log(E1)

log(E2)− log(E1)(23)

where E1 and E2 are the closest lower and higher energy for which crosssection data σ1 and σ2 are available in the data libraries respectively.


Notes

According to G4PEFluoModel for energies Eγ > 25.55MeV it isassumed that the emitted photoelectron has the same direction as theincident gamma. This means that for Eγ > 25.55MeV cosθ = 1always.

N.B.: Need to test/check and correct the number of bin and thenumber of tested energies. (with nBins=200 there aren’timprovements with respect to nBins=151 which is working betterthan nBin=99

Da 0.5MeV in su abbiamo problemi nel riprodurre la distribuzioneangolare. Update: Improved, but still need to test.


Photoelectric effect in Geant4 - low energy package

Two implementations of the p.e. effect. The Livermore and Penelopecross-sections are tabulated according to EPDL97 and EPDL89[D.E.Cullen , 1989], respectively.

G4Livermore model provides two options of computing the angulardistribution of the emitted photoelectron

Based on Gavrila’s distribution of the polar angle for the K shell andthe L1 sub-shell - default as the standard modelBased on a double differential cross section derived from Gavrila’scalculations, which can also handle polarized photons.

G4Penelope model is reengineered from the 2001 Penelope code.


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