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Low Energy Electromagnetic Physics. Maria Grazia Pia INFN Genova [email protected] on behalf of the Low Energy Electromagnetic Working Group Geant4 Workshop Helsinki, 30-31 October 2003. http://www.ge.infn.it/geant4/training/. Part 1 Overview Software process OOAD Physics

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Low energy electromagnetic physics

Low Energy Electromagnetic Physics

Maria Grazia Pia

INFN Genova

[email protected]

on behalf of the Low Energy Electromagnetic Working Group

Geant4 Workshop

Helsinki, 30-31 October 2003

http://www.ge.infn.it/geant4/training/


Plan of the tutorial

Part 1

Overview

Software process

OOAD

Physics

Electrons and photons

Hadrons and ions

Atomic relaxation

Polarisation

Part 2

How to use LowE processes

Examples

Some experimental applications

Outlook

Plan of the tutorial


What is
What is

  • A package in the Geant4 electromagnetic package

    • geant4/source/processes/electromagnetic/lowenergy/

  • A set of processes extending the coverage of electromagnetic interactions in Geant4 down to “low” energy

    • 250 eV (in principle even below this limit)/100 eVfor electrons and photons

    • down to the approximately the ionisation potential of the interacting material for hadrons and ions

  • A set of processes based on detailed models

    • shell structure of the atom

    • precise angular distributions

  • Complementary to the “standard” electromagnetic package

    • will learn more on domains of application in the second lecture


Overview of physics

Compton scattering

Rayleigh scattering

Photoelectric effect

Pair production

Bremsstrahlung

Ionisation

Polarised Compton

+ atomic relaxation

fluorescence

Auger effect

following processes leaving a vacancy in an atom

In progress

More precise angular distributions (Rayleigh, photoelectric, Bremsstrahlung etc.)

Polarised g conversion, photoelectric

Development plan

Driven by user requirements

Schedule compatible with available resources

Overview of physics

  • in two “flavours” of models:

  • based on theLivermore Library

  • à laPenelope


Overview of physics1

Compton scattering

Rayleigh scattering

Photoelectric effect

Pair production

Bremsstrahlung

Ionisation

Polarised Compton

+ atomic relaxation

fluorescence

Auger effect

following photoelectric effect and ionisation

In progress

Polarised g conversion, photoelectric

More precise angular distributions (Rayleigh, photoelectric, Bremsstrahlung etc.)

Foreseen

New models, based on different physics approaches

Processes for positrons

Development plan

Driven by user requirements

Schedule compatible with available resources

Overview of physics


Software process
Software Process

  • A rigorous approach to software engineering

    • in support of a better quality of the software

    • especially relevant in the physics domain of Geant4-LowE EM

    • several mission-critical applications (space, medical…)

  • Public URD

  • Full traceability through UR/OOD/implementation/test

  • Testing suite and testing process

  • Public documentation of procedures

  • Defect analysis and prevention

  • etc.…

Spiral approach

A life-cycle model that is both iterative and incremental

Collaboration-wide Geant4 software process, tailored to the WG projects

current

status

Huge effort invested into SPI


User requirements
User requirements

Various methodologies adopted to capture URs

User Requirements

  • Elicitation through interviews and surveys

    • useful to ensure that UR are complete and there is wide agreement

  • Joint workshops with user groups

  • Use cases

  • Analysis of existing Monte Carlo codes

  • Study of past and current experiments

  • Direct requests from users to WG coordinators

Posted on the WG web site


LowE e/g processes

based on Livermore Library


Photons and electrons
Photons and electrons

different approach w.r.t. Geant4 standard e.m. package

  • Based on evaluated data libraries from LLNL:

    • EADL (Evaluated Atomic Data Library)

    • EEDL (Evaluated Electrons Data Library)

    • EPDL97 (Evaluated Photons Data Library)

  • especially formatted for Geant4 distribution(courtesy of D. Cullen, LLNL)

  • Validity range: 250 eV - 100 GeV

    • The processes can be used down to 100 eV, with degraded accuracy

    • In principle the validity range of the data libraries extends down to ~10 eV

  • Elements Z=1 to Z=100

    • Atomic relaxation: Z > 5 (transition data available in EADL)


Data Management

Cross sections, final state

Intelligent data: know how to handle themselves through algorithm objects

e.g.: interpolation algorithms encapsulated in objects

(to let them vary and be interchangeable)

Composite pattern to treat different physical entities (e.g. whole atom and atom with shell structure) transparently


Calculation of cross sections
Calculation of cross sections

Interpolation from the data libraries:

E1 and E2 are the lower and higher energy

for which data (s1 and s2) are available

Mean free path for a process, at energy E:

ni = atomic density of the ith element contributing to the material composition



Compton scattering
Compton scattering

  • Energy distribution of the scattered photon according to the Klein-Nishina formula, multiplied by scattering functions F(q) from EPDL97 data library

  • The effect of scattering function becomes significant at low energies

    • suppresses forward scattering

  • Angular distribution of the scattered photon and the recoil electron also based on EPDL97

Klein-Nishina cross section:


Rayleigh scattering
Rayleigh scattering

  • Angular distribution: F(E,q)=[1+cos2(q)]F2(q)

    • where F(q) is the energy-dependent form factor obtained from EPDL97

  • Improved angular distribution to be available in next Geant4 release, December 2002


Photoelectric effect
Photoelectric effect

  • Cross section

    • Integrated cross section (over the shells) from EPDL + interpolation

    • Shell from which the electron is emitted selected according to the detailed cross sections of the EPDL library

  • Final state generation

    • Direction of emitted electron = direction of incident photon

  • Deexcitation via the atomic relaxation sub-process

    • Initial vacancy + following chain of vacancies created


G conversion
g conversion

  • The secondary e- and e+ energies are sampled using Bethe-Heitler cross sections with Coulomb correction

  • e- and e+ assumed to have symmetric angular distribution

  • Energy and polar angle sampled w.r.t. the incoming photon using Tsai differential cross section

  • Azimuthal angle generated isotropically

  • Choice of which particle in the pair is e- or e+ is made randomly


Photons mass attenuation coefficient

LowE

Fe

NIST-XCOM

G4 Standard

standard

G4 LowE

Photons: mass attenuation coefficient

Tests by IST - Natl. Inst. for Cancer Research, Genova (F. Foppiano et al.)

LowE accuracy ~ 1%

2N-L=13.1 – =20 - p=0.87

Comparison against NIST data

LowE accuracy ~ 1%

2N-S=23.2 – =15 - p=0.08


Photons evidence of shell effects
Photons, evidence of shell effects

Photon transmission, 1 mm Pb

Photon transmission, 1 mm Al


Polarisation

x

x

f

hn

A

hn0

10 MeV

100 keV

q

1 MeV

a

small 

z

O

small 

small 

C

large 

large 

large 

y

Cross section:

Polarisation

Scattered Photon Polarization

250 eV -100 GeV

 Polar angle

 Azimuthal angle

 Polarization vector

Low Energy

Polarised Compton

More details: talk on Geant4 Low Energy Electromagnetic Physics

Other polarised processes under development


Polarisation1
Polarisation

theory

500 million events

simulation

Polarisation of a non-polarised photon beam, simulation and theory

Ratio between intensity with perpendicular and parallel polarisation vector w.r.t. scattering plane, linearly polarised photons


Electron bremsstrahlung
Electron Bremsstrahlung

  • Parameterisation of EEDL data

    • 16 parameters for each atom

    • At high energy the parameterisation reproduces the Bethe-Heitler formula

    • Precision is ~ 1.5 %

  • Plans

    • Systematic verification over Z and energy


Electron ionisation
Electron ionisation

  • Parameterisation based on 5 parameters for each shell

  • Precision of parameterisation is better then 5% for 50 % of shells, less accurate for the remaining shells

  • Work in progress to improve the parameterisation and the performance


Electron ionisation1
Electron ionisation

  • New parameterisations of EEDL data library recently released

    • precision is now better than 5 % for ~ 50% of the shells, poorer for the 50% left

  • Plans

    • Systematic verification over shell, Z and energy

    • Need Test & Analysis Project for automated verification (all shells, 99 elements!)


Electrons range

NIST-ESTAR

G4 Standard

G4 LowE

Electrons: range

Range in various simple and composite materials

Compared to NIST database

Al


Electrons de dx
Electrons: dE/dx

Ionisation energy loss in various materials

Compared to Sandia database

More systematic verification planned

Also Fe, Ur


Electrons transmitted
Electrons, transmitted

20 keV electrons, 0.32 and 1.04 mm Al


The problem of validation finding reliable data
The problem of validation: finding reliable data

Note: Geant4 validation is not always easy

experimental data often exhibit large differences!

Backscattering low energies - Au


LowE e/g processes

based on Penelope models


Processes la penelope
Processes à la Penelope

  • The whole physics content of the Penelope Monte Carlo code has been re-engineered into Geant4 (except for multiple scattering)

    • processes for photons: release 5.2, for electrons: release 6.0

  • Physics models by F. Salvat et al.

    • analytical approach

  • Power of the OO technology:

    • extending the software system is easy

    • all processes obey to the same abstract interfaces

    • using new implementations in application code is simple

  • Profit of Geant4 advanced geometry modeling, interactive facilities etc.

    • same physics as original Penelope


LowE hadron/ion processes


Hadrons and ions
Hadrons and ions

  • Variety of models, depending on

    • energy range

    • particle type

    • charge

  • Composition of models across the energy range, with different approaches

    • analytical

    • based on data reviews + parameterisations

  • Specialised models for fluctuations

  • Open to extension and evolution


Hadrons and ions

Physics models handled through abstract classes

Algorithms encapsulated in objects

Interchangeable and transparent access to data sets

Transparency of physics, clearly exposed to users


Positive charged hadrons

Stopping power

Z dependence for various energies

Ziegler and ICRU models

Ziegler and ICRU, Fe

Ziegler and ICRU, Si

Straggling

Nuclear stopping power

Positive charged hadrons

  • Bethe-Bloch model of energy loss, E > 2 MeV

  • 5 parameterisation models, E < 2 MeV

    • based on Ziegler (1977,1985,2000) and ICRU reviews

  • 3 models of energy loss fluctuations

  • Density correction for high energy

  • Shell correction term for intermediate energy

  • Spin dependent term

  • Barkas and Bloch terms

  • Chemical effect for compounds

  • Nuclear stopping power

  • PIXE included(preliminary)


Bragg peak (with hadronic interactions)

The precision of the stopping power simulation for protons in the energy from 1 keV to 10 GeV is of the order of a few per cent


Models for antiprotons

Proton

Proton

G4 Antiproton

Antiproton exp. data

G4 Antiproton

Antiproton exp. data

Antiproton from Arista et. al

Antiproton from Arista et. al

Models for antiprotons

  •  > 0.5 Bethe-Bloch formula

  • 0.01 <  < 0.5 Quantum harmonic oscillator model

  •  < 0.01 Free electron gas model


Positive charged ions

Deuterons

Positive charged ions

  • Scaling:

  • 0.01 < b < 0.05 parameterisations, Bragg peak

    • based on Ziegler and ICRU reviews

  • b < 0.01: Free Electron Gas Model

  • Effective charge model

  • Nuclear stopping power



Fluorescence

X-ray fluorescence spectrum in Iceand basalt

(EIN=6.5 keV)

Counts

Fe lines

GaAs lines

Scattered

photons

Energy (keV)

Fluorescence

Experimental validation:

test beam data, in collaboration with ESA Advanced Concepts Division

Microscopic validation: against reference data

Spectrum from a Mars-simulant rock sample

Anderson-Darling

Ac (95%) =0.752


Auger effect
Auger effect

New implementation, validation in progress

Auger electron emission from various materials

Sn, 3 keV photon beam,

electron lines w.r.t. published experimental results


Contribution from users
Contribution from users

  • Many valuable contributions to the validation of LowE physics from users all over the world

    • excellent relationship with our user community

  • User comparisons with data usually involve the effect of several physics processes of the LowE package

    • sometimes combining LowE + Standard e.m. processes

  • A small sample in the next slides

    • no time to show all!


15x15 cm2

Differences

Differences

10x10 cm2

10x10 cm2

15x15 cm2

Homogeneous Phantom

P. Rodrigues, A. Trindade, L.Peralta, J. Varela, LIP

  • Simulation of photon beams produced by a Siemens Mevatron KD2 clinical linear accelerator

  • Phase-space distributions interface with GEANT4

  • Validation againstexperimental data: depth dose and profile curves

LIP – Lisbon


Dose calculations with 12c
Dose Calculations with 12C

P. Rodrigues, A. Trindade, L.Peralta, J. Varela, LIP

  • Bragg peak localization calculated with GEANT4 (stopping powers from ICRU49 and Ziegler85) and GEANT3 in a water phantom

  • Comparison with GSI data

preliminary


Uranium irradiated by electron beam
Uranium irradiated by electron beam

Jean-Francois Carrier, Louis Archambault, Rene Roy and Luc Beaulieu

Service de radio-oncologie, Hotel-Dieu de Quebec, Quebec, Canada

Departement de physique, Universite Laval, Quebec, Canada

The following results will be published soon. They are part of a general Geant4 low energy validation project.

Fig 1. Depth-dose curve for a semi-infinite uranium slab irradiated by a 0.5 MeV broad parallel electron beam

1Chibani O and Li X A, Med. Phys. 29 (5), May 2002


Ions

Independent validation at Univ. of Linz (H. Paul et al.)

Geant4-LowE reproduces the right side of the distribution precisely, but about 10-20% discrepancy is observed at lower energies


The future
The future…

  • In progress

    • More precise angular distributions (Rayleigh, photoelectric, Bremsstrahlung etc.)

  • Foreseen

    • Penelope processes for electrons (December 2003 release)

    • Processes for positrons (Penelope, December 2003 release)

    • Performance optimisation (later time scale)

    • Polarised g conversion, photoelectric

  • Development plan

    • Driven by user requirements

    • Schedule compatible with available resources


Low energy em physics implementation

Compton scattering

Rayleigh scattering

Photoelectric effect

Pair production

Bremsstrahlung

Ionisation

Polarised Compton

+ atomic relaxation

fluorescence

Auger effect

following photoelectric effect and ionisation

The following code is required in your PhysicsList.cc

[All code has been lifted from the relevant advanced examples]

Low Energy Em Physics Implementation:


Brachytherapy
brachytherapy

  • Low energy electromagnetic processes for precise calculation of dose distribution

    // gamma

    #include "G4LowEnergyRayleigh.hh"

    #include "G4LowEnergyPhotoElectric.hh"

    #include "G4LowEnergyCompton.hh"

    #include "G4LowEnergyGammaConversion.hh"

    // e-

    #include "G4LowEnergyIonisation.hh"

    #include "G4LowEnergyBremsstrahlung.hh"

    // e+

    #include "G4eIonisation.hh"

    #include "G4eBremsstrahlung.hh"

    #include "G4eplusAnnihilation.hh"


Brachytherapy implementation
Brachytherapy Implementation

void BrachyPhysicsList::ConstructEM()

{

theParticleIterator->reset();

while( (*theParticleIterator)() ){

G4ParticleDefinition* particle = theParticleIterator->value();

G4ProcessManager* pmanager = particle->GetProcessManager();

G4String particleName = particle->GetParticleName();

//processes

lowePhot = new G4LowEnergyPhotoElectric("LowEnPhotoElec");

loweIon = new G4LowEnergyIonisation("LowEnergyIoni");

loweBrem = new G4LowEnergyBremsstrahlung("LowEnBrem");

if (particleName == "gamma") {

//gamma

pmanager->AddDiscreteProcess(new G4LowEnergyRayleigh);

pmanager->AddDiscreteProcess(lowePhot);

pmanager->AddDiscreteProcess(new G4LowEnergyCompton);

pmanager->AddDiscreteProcess(new G4LowEnergyGammaConversion);

} else if (particleName == "e-") {

//electron

pmanager->AddProcess(new G4MultipleScattering, -1, 1,1);

pmanager->AddProcess(loweIon, -1, 2,2);

pmanager->AddProcess(loweBrem, -1,-1,3);

} else if (particleName == "e+") {

//positron

pmanager->AddProcess(new G4MultipleScattering, -1, 1,1);

pmanager->AddProcess(new G4eIonisation, -1, 2,2);

pmanager->AddProcess(new G4eBremsstrahlung, -1,-1,3);

pmanager->AddProcess(new G4eplusAnnihilation, 0,-1,4);

}

}

}


X ray fluorescence

sample

detector

beam

Fe lines

GaAs lines

Scattered

photons

X-ray fluorescence

  • Physics: Low Energy processes, atomic relaxation

    #include "G4LowEnergyCompton.hh"

    #include "G4LowEnergyGammaConversion.hh"

    #include "G4LowEnergyPhotoElectric.hh"

    #include "G4LowEnergyRayleigh.hh"

    // e+

    #include "G4MultipleScattering.hh"

    #include "G4eIonisation.hh"

    #include "G4eBremsstrahlung.hh"

    #include "G4eplusAnnihilation.hh"

    #include "G4LowEnergyIonisation.hh"

    #include "G4LowEnergyBremsstrahlung.hh"

    #include "G4hLowEnergyIonisation.hh"


Process registration x rayfluo
Process Registration X-rayFluo

else if (particleName == "proton") {

//proton

pmanager->AddProcess(new G4MultipleScattering,-1,1,1);

pmanager->AddProcess(new G4hLowEnergyIonisation,-1, 2,2);

}

else if ( particleName == "alpha" )

{

pmanager->AddProcess(new G4MultipleScattering,-1,1,1);

G4hLowEnergyIonisation* iIon = new G4hLowEnergyIonisation() ;

pmanager->AddProcess(iIon,-1,2,2);

}

}

}

void XrayFluoPhysicsList::ConstructEM()

{

theParticleIterator->reset();

while( (*theParticleIterator)() ){

G4ParticleDefinition* particle = theParticleIterator->value();

G4ProcessManager* pmanager = particle->GetProcessManager();

G4String particleName = particle->GetParticleName();

if (particleName == "gamma") {

// gamma

pmanager->AddDiscreteProcess(new G4LowEnergyCompton);

LePeprocess = new G4LowEnergyPhotoElectric();

//LePeprocess->ActivateAuger(false);

//LePeprocess->SetCutForLowEnSecPhotons(10000 * keV);

//LePeprocess->SetCutForLowEnSecElectrons(10000 * keV);

pmanager->AddDiscreteProcess(LePeprocess);

pmanager->AddDiscreteProcess(new G4LowEnergyRayleigh);

}

else if (particleName == "e-") {

//electron

pmanager->AddProcess(new G4MultipleScattering,-1, 1,1);

LeIoprocess = new G4LowEnergyIonisation();

//LeIoprocess->ActivateAuger(false);

//LeIoprocess->SetCutForLowEnSecPhotons(10000 keV);

//LeIoprocess->SetCutForLowEnSecElectrons(10000 keV);

pmanager->AddProcess(LeIoprocess, -1, 2, 2);

LeBrprocess = new G4LowEnergyBremsstrahlung();

pmanager->AddProcess(LeBrprocess, -1, -1, 3);

} else if (particleName == "e+") {

//positron

pmanager->AddProcess(new G4MultipleScattering,-1, 1,1);

pmanager->AddProcess(new G4eIonisation, -1, 2,2);

pmanager->AddProcess(new G4eBremsstrahlung, -1,-1,3);

pmanager->AddProcess(new G4eplusAnnihilation, 0,-1,4);

}


Underground physics

mirror

LXe

GXe

source

PMT

Underground physics

// Electromagnetic Processes // all charged particles

// gamma

#include "G4LowEnergyRayleigh.hh"

#include "G4LowEnergyPhotoElectric.hh"

#include "G4LowEnergyCompton.hh"

#include "G4LowEnergyGammaConversion.hh"

// e-

#include "G4LowEnergyIonisation.hh"

#include "G4LowEnergyBremsstrahlung.hh"

// e+

#include "G4eIonisation.hh"

#include "G4eBremsstrahlung.hh"

#include "G4eplusAnnihilation.hh"

// alpha and GenericIon and deuterons, triton, He3:

#include "G4hLowEnergyIonisation.hh"

#include "G4EnergyLossTables.hh"

//muon:

#include "G4MuIonisation.hh"

#include "G4MuBremsstrahlung.hh"

#include "G4MuPairProduction.hh"

#include "G4MuonMinusCaptureAtRest.hh"


Process creation dmx

void DMXPhysicsList::ConstructEM() {

// processes

G4MultipleScattering* aMultipleScattering = new G4MultipleScattering();

G4LowEnergyPhotoElectric* lowePhot = new G4LowEnergyPhotoElectric();

G4LowEnergyIonisation* loweIon = new G4LowEnergyIonisation();

G4LowEnergyBremsstrahlung* loweBrem = new G4LowEnergyBremsstrahlung();

// fluorescence: specific cuts for flourescence

// from photons, electrons and bremsstrahlung photons

G4double fluorcut = 250*eV;

lowePhot->SetCutForLowEnSecPhotons(fluorcut);

loweIon ->SetCutForLowEnSecPhotons(fluorcut);

loweBrem->SetCutForLowEnSecPhotons(fluorcut);

Process Creation - DMX


Hadron ionisation choosing a model
Hadron Ionisation – Choosing a model

G4hLowEnergyIonisation* ahadronLowEIon = new G4hLowEnergyIonisation();

ahadronLowEIon->SetNuclearStoppingPowerModel("ICRU_R49") ;

ahadronLowEIon->SetNuclearStoppingOn() ;

// Switch off the Barkas and Bloch corrections

ahadronLowEIon->SetBarkasOff();

// Switch off hadron-induced fluorescence (for now)

ahadronLowEIon->SetFluorescence(false);


Attaching processes to particles
Attaching Processes to Particles:

theParticleIterator->reset();

while( (*theParticleIterator)() ){

G4ParticleDefinition* particle = theParticleIterator->value();

G4ProcessManager* pmanager = particle->GetProcessManager();

G4String particleName = particle->GetParticleName();

G4String particleType = particle->GetParticleType();

G4double particleCharge = particle->GetPDGCharge();

// gamma

if (particleName == "gamma")

{

pmanager->AddDiscreteProcess(new G4LowEnergyRayleigh());

pmanager->AddDiscreteProcess(lowePhot);

pmanager->AddDiscreteProcess(new G4LowEnergyCompton());

pmanager->AddDiscreteProcess(new G4LowEnergyGammaConversion());

}


Attaching processes to particles1
Attaching Processes to Particles:

{

// electron

else if (particleName == "e-")

{

// process ordering: AddProcess(name, at rest, along step, post step)

// -1 = not implemented, then ordering

pmanager->AddProcess(aMultipleScattering, -1, 1, 1);

pmanager->AddProcess(loweIon, -1, 2, 2);

pmanager->AddProcess(loweBrem, -1,-1, 3);

}

// positron

else if (particleName == "e+")

{

pmanager->AddProcess(aMultipleScattering, -1, 1, 1);

pmanager->AddProcess(new G4eIonisation(), -1, 2, 2);

pmanager->AddProcess(new G4eBremsstrahlung(), -1,-1, 3);

pmanager->AddProcess(new G4eplusAnnihilation(), 0,-1, 4);

}


Attaching processes to particles2
Attaching Processes to Particles:

// muons

else if( particleName == "mu+" ||

particleName == "mu-" )

{

pmanager->AddProcess(aMultipleScattering, -1, 1, 1);

pmanager->AddProcess(new G4MuIonisation(), -1, 2, 2);

pmanager->AddProcess(new G4MuBremsstrahlung(), -1,-1, 3);

pmanager->AddProcess(new G4MuPairProduction(), -1,-1, 4);

if( particleName == "mu-" )

pmanager->AddProcess(new G4MuonMinusCaptureAtRest(), 0,-1,-1);

}

// charged hadrons

else if (particleName == "proton" ||

particleName == "alpha" ||

particleName == "deuteron" ||

particleName == "triton" ||

particleName == "He3" ||

particleName == "GenericIon" ||

(particleType == "nucleus" && particleCharge != 0))


Attaching processes to particles3
Attaching Processes to Particles:

{

// OBJECT may be dynamically created as either a GenericIon or nucleus

pmanager->AddProcess(aMultipleScattering, -1, 1, 1);

pmanager->AddProcess(ahadronLowEIon, -1, 2, 2);

}

// all other charged particles except geantino

else if ((!particle->IsShortLived()) &&

(particleCharge != 0.0) &&

(particleName != "chargedgeantino"))

{

pmanager->AddProcess(aMultipleScattering, -1, 1, 1);

pmanager->AddProcess(ahadronLowEIon, -1, 2, 2);

}

}

}


Setting energy cuts

//special for low energy physics

G4double lowlimit=250*eV;

G4Gamma ::SetEnergyRange(lowlimit,100*GeV);

G4Electron::SetEnergyRange(lowlimit,100*GeV);

G4Positron::SetEnergyRange(lowlimit,100*GeV);

Setting Energy Cuts:


To learn more
To learn more

  • Geant4 Physics Reference Manual

  • Application Developer Guide

  • User Forum: Electromagnetic

    http://www.ge.infn.it/geant4/lowE


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