hadron physics ii

Hadron Physics II

KOI, TatsumiSLAC National Accelerator Laboratory

Geant4 v9.4.p01Geant4 Tutorial at ORNL 10-March-

20111

Overview

• Low Energy Neutron Physics– High Precision Neutron Models

• Ion Physics– Inelastic– Electromagnetic Dissociation – Radio Active Decay


20112

Low energy (< 20MeV) neutrons physics

• High Precision Neutron Models (and Cross Section Data Sets)– G4NDL

• ENDF– Elastic– Inelastic– Capture– Fission

• NeutronHPorLEModel(s)• ThermalScatteringModels ( and Cross Section data

Sets)• JENDL High Energy Files ( cross sections < 3GeV)


20113

G4NDL(Geant4 Neutron Data Library)

• The neutron data files for High Precision Neutron models• The data are including both cross sections and final states.• The data are derived evaluations based on the following evaluated

data libraries (in alphabetic order)– Brond-2.1– CENDL2.2– EFF-3– ENDF/B-VI.0, 1, 4– FENDL/E2.0– JEF2.2– JENDL-FF– JENDL-3.1,2– MENDL-2

• The data format is similar ENDF, however it is not equal to.


20114

Evaluated Nuclear Data File-6• “ENDF” is used in two meanings• One is Data Formats and Procedures

– How to write Nuclear Data files– How to use the Nuclear Data files

• The other is name of recommended libraries of USA nuclear data projects. – ENDF/B-VI.8

• 313 isotopes including 5 isomers• 15 elements

– ENDF/B-VII.0• Released on 2006 Dec• almost 400 isotopes • not yet migrated

• After G4NDL3.8 (3.14 is latest) we concentrated translation from ENDF library.– No more evaluation by ourselves.


20115

G4NeutronHPElastic

• The final state of elastic scattering is described by sampling the differential scattering cross-sections– tabulation of the differential cross-section

– a series of legendre polynomials and the legendre coefficients


2011

cos

2

12,cos

2

0ll

n

l

PEal

Ed

d

E

l

Ed

d

d

d,cos

6

G4NeutronHPInelastic• Currently supported final states are (nA ) nγs (discrete and

continuum), np, nd, nt, n 3He, nα, nd2α, nt2α , n2p, n2α, np , n3α, 2nα, 2np, 2nd, 2nα, 2n2α, nX, 3n, 3np, 3nα, 4n, p, pd, pα, 2p d, dα, d2α, dt, t, t2α, 3He, α, 2α, and 3α.

• Secondary distribution probabilities are supported– isotropic emission– discrete two-body kinematics– N-body phase-space distribution– continuum energy-angle distributions

• legendre polynomials and tabulation distribution• Kalbach-Mann systematic A + a → C → B + b, C:compound nucleus

– continuum angle-energy distributions in the laboratory system


20117

G4NeutronHPCapture• The final state of radiative capture is described by either

photon multiplicities, or photon production cross-sections, and the discrete and continuous contributions to the photon energy spectra, along with the angular distributions of the emitted photons.

• For discrete photon emissions– the multiplicities or the cross-sections are given from data

libraries• For continuum contribution

– E neutron kinetic energy, Eγ photon energies

– pi and gi are given from data libraries


2011

EEgEpEEf ii

i

8

G4NeutronHPFission• Currently only Uranium data are available in G4NDL• first chance, second chance, third chance and forth

chance fission are into accounted. • The neutron energy distributions are implemented in

six different possibilities.– tabulated as a normalized function of the incoming and

outgoing neutron energy -– Maxwell spectrum -– a general evaporation spectrum -– evaporation spectrum -– the energy dependent Watt spectrum -– the Madland Nix spectrum


2011

EEf

EEfEEf

EEeEEEf

EEeEEEf

EEbeEEf EaE )(sinh

hl KEgKEgEEf ,,2

1

9

Verification of High Precision Neutron models Channel Cross Sections


2011

20MeV neutron on 157Gd

0.001

0.01

0.1

1

10

Ela

stic

Cap

ture

Inel

astic

Inel

astic

Inel

astic

Inel

astic

Inel

astic

Inel

astic

(n,nγ ) (n,2n) (n,nα ) (n,np) (n,p) (n,α )

Cro

ssS

ectio

n [b

arn]

G4ENDF

Geant4 results are derived from thin target calculations

10

Verification of High Precision Neutron models

Energy Spectrum of Secondary Particles

Gd154 (n,2n) channel

0.0E+005.0E- 081.0E- 071.5E- 072.0E- 072.5E- 073.0E- 073.5E- 074.0E- 074.5E- 07

0 2E+06 4E+06 6E+06 8E+06 1E+07 1E+07

secondary neutron energy [eV]

ENDFG4 result


201111

G4NeutornHPorLEModels• Many elements remained without data for High

Precision models.• Those models make up for such data deficit.• If the High Precision data are not available for a

reaction, then Low Energy Parameterization Models will handle the reaction.

• Those can be used for not only for models (final state generator) but also for cross sections.

• Elastic, Inelastic, Capture and Fission models are prepared.


201112

Thermal neutron scattering from chemically bound atoms

• At thermal neutron energies, atomic translational motion as well as vibration and rotation of the chemically bound atoms affect the neutron scattering cross section and the energy and angular distribution of secondary neutrons.

• The energy loss or gain of incident neutrons can be different from interactions with nuclei in unbound atoms.

• Only individual Maxwellian motion of the target nucleus (Free Gas Model) was taken into account the default NeutronHP models.


201113

Thermal neutron scattering files from the evaluated nuclear data files

ENDF/B-VI, Release2• These files constitute a thermal sub-library • Use the File 7 format of ENDF/B-VI• Divides the thermal scattering into different

parts: – Coherent and incoherent elastic; no energy change– Inelastic; loss or gain in the outgoing neutron energy

• The files and NJOY are required to prepare the scattering law S(α,β) and related quantities.

Geant4 v9.4.p01

; ,2

,:section cross Scattering SE

E

kTEE b

kT

EE

AkT

EEEE

:nsferenergy tra ,

2: transfermomentum

14Geant4 Tutorial at ORNL 10-March-

2011

Cross section and Secondary Neutron Distributions using S(α, β) model

Angular distributionneutron scattering on Hydrogen within polyethylene 300 K

0.00

0.01

0.02

0.03

0.04

-1 -0.5 0 0.5 1

μ = cos(θ ）

rela

tive

rati

o

0.1eV FreeGas0.1eV S(α , β ) 1eV FreeGas1eV S(α , β )


2011

Energy Spectrum ofNeutron Scattering on Hydrogen within polyethylene at 300 K

0.0

4.0

8.0

12.0

16.0

1.0E- 05 1.0E- 03 1.0E- 01 1.0E+01

energy [eV]

rela

tive

rat

io [

/eV

/bin

]

0.1eV FreeGas0.1eV S(α , β ) 1eV FreeGas1eV S(α , β )

15

Neutron scattering on Hydrogen within polyethylene at 300K

1.0E- 01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E- 05 1.0E- 02 1.0E+01 1.0E+04 1.0E+07

energy [eV]

cros

s se

ctin

[ba

rn]

Free Gas ModelInelasticIncoherent elastic

Japanese Evaluated Nuclear Data Library (JENDL)

High Energy Files 2007• JENDL are been making by the Nuclear Data

Evaluation Center of Japan Atomic Energy Agency with the aid of Japanese Nuclear Data Committee

• High Energy Files 2007– Neutron- and proton-induced reaction data up to

3 GeV for 107 nuclides.– We introduced its cross sections except for Am


201116


2011

Neutron Cross Section on Iron

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11

Neutron Energy [eV]

Cro

ss S

ectio

n [b

arn] J ENDL Fe56 tot

J ENDL Fe56 elaJ ENDL Fe56 inelaQGSP_BERT_HP totQGSP_BERT_HP elaQGSP_BERT_HP inela

Difference of Cross Sectionσ J ENDL_HE - σ QGSP_BERT_HP

- 5.0E- 01

- 4.0E- 01

- 3.0E- 01

- 2.0E- 01

- 1.0E- 01

0.0E+00

1.0E- 01

2.0E- 01

3.0E- 01

4.0E- 01

5.0E- 01

1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11

Primary Neutron Energy [eV]

delta

Cro

ss S

ectio

n [b

arn]

TotalElasticInelastic

Comparison JEND HE filesto Cross Sections which used inQGSP_BERT_HP physics lists

Comparison carried out at Geant4 v8.0.p01

17

Physics List for NeutronHP• //For example Elastic scattering below 20 MeV • G4HadronElasticProcess* theNeutronElasticProcess = new

G4HadronElasticProcess();• // Cross Section Data set• G4NeutronHPElasticData* theHPElasticData = new

G4NeutronHPElasticData();• theNeutronElasticProcess->AddDataSet( theHPElasticData ); • // Model• G4NeutronHPElastic* theNeutronElasticModel = new

G4NeutronHPElastic();• theNeutronElasticProcess->RegisterMe(theNeutronElasticModel)

• G4ProcessManager* pmanager = G4Neutron::Neutron()-> GetProcessManager();

• pmanager->AddDiscreteProcess( theNeutronElasticProcess );


201118

Physics List for NeutronHPorLE• //For example Elastic scattering below 20 MeV • G4HadronElasticProcess* theNeutronElasticProcess = new

G4HadronElasticProcess();• // Model• G4NeutronHPorLElasticModel* theNeutronElasticModel = new

G4NeutronHPorLElasticModel();• theNeutronElasticProcess->RegisterMe(theNeutronElasticModel)• // Cross Section Data set• theNeutronElasticProcess->AddDataSet( theNeutronElasticModel-

>GiveHPXSectionDataSet() );




201119

Physics List for NeutronHPThermalScattering

• G4HadronElasticProcess* theNeutronElasticProcess = new G4HadronElasticProcess();• // Cross Section Data set• G4NeutronHPElasticData* theHPElasticData = new G4NeutronHPElasticData();• theNeutronElasticProcess->AddDataSet( theHPElasticData );• G4NeutronHPThermalScatteringData* theHPThermalScatteringData = new

G4NeutronHPThermalScatteringData();• theNeutronElasticProcess->AddDataSet( theHPThermalScatteringData ); • // Models• G4NeutronHPElastic* theNeutronElasticModel = new G4NeutronHPElastic();• theNeutronElasticModel->SetMinEnergy ( 4.0*eV );• theNeutronElasticProcess->RegisterMe(theNeutronElasticModel);• G4NeutronHPThermalScattering* theNeutronThermalElasticModel = new

G4NeutronHPThermalScattering();• theNeutronThermalElasticModel->SetMaxEnergy ( 4.0*eV );• theNeutronElasticProcess->RegisterMe(theNeutronThermalElasticModel);

• // Apply Processes to Process Manager of Neutron• G4ProcessManager* pmanager = G4Neutron::Neutron()-> GetProcessManager();• pmanager->AddDiscreteProcess( theNeutronElasticProcess );


201120

Material Definitions for NeutronHPThermalScattering

• // Create Element for Thermal Scattering • G4Element* elTSHW = new G4Element( "TS_H_of_Water" , "H_WATER" , 1.0 ,

1.0079*g/mole );

• G4Element* elTSH = new G4Element( "TS_H_of_Polyethylene" , "H_POLYETHYLENE" , 1.0 , 1.0079*g/mole );

• // Create Materials from the elements• G4Material* matH2O_TS = new G4Material( "Water_TS" , density = 1.0*g/cm3 ,

ncomponents = 2 );• matH2O_TS -> AddElement(elTSHW,natoms=2);• matH2O_TS -> AddElement(elO,natoms=1);

• G4Material* matCH2_TS = new G4Material( "Polyethylene_TS" , density = 0.94*g/cm3 , ncomponents = 2 );

• matCH2_TS -> AddElement(elTSH,natoms=2);• matCH2_TS -> AddElement(elC,natoms=1);


201121

Physics List for JENDL High energy cross sections

• //For example Elastic scattering below 3 GeV • G4HadronElasticProcess* theNeutronElasticProcess = new

G4HadronElasticProcess();• // Cross Section Data set ( HP < 20MeV < JENDL HE)• G4NeutronHPElasticData* theHPElasticData = new G4NeutronHPElasticData(); • theNeutronElasticProcess->AddDataSet( theNeutronElasticModel-

>GiveHPXSectionDataSet() );• theNeutronElasticProcess->AddDataSet( theHPElasticData ); • G4NeutronHPJENDLHEData* theJENDLHEElasticData = new

G4NeutronHPJENDLHEData();• theNeutronElasticProcess->AddDataSet(theJENDLHEElasticData);




201122

Ion PhysicsInelastic Reactions

• Cross Sections• Model

– G4BinaryLightIon– G4WilsonAbrasion– G4QMD


201123

Cross Sections

• Many cross section formulae for NN collisions are included in Geant4– Tripathi, Shen, Kox and Sihver

• These are empirical and parameterized formulae with theoretical insights.

• G4GeneralSpaceNNCrossSection was prepared to assist users in selecting the appropriate cross section formula.


201124

References to NN Cross Section Formulae implemented in Geant4

• Tripathi Formula– NASA Technical Paper TP-3621 (1997)

• Tripathi Light System– NASA Technical Paper TP-209726 (1999)

• Kox Formula– Phys. Rev. C 35 1678 (1987)

• Shen Formula– Nuclear Physics. A 49 1130 (1989)

• Sihver Formula– Phys. Rev. C 47 1225 (1993)


201125

Inelastic Cross SectionC12 on C12


201126

Binary Cascade ~Model Principals~

• In Binary Cascade, each participating nucleon is seen as a Gaussian wave packet, (like QMD)

• Total wave function of the nucleus is assumed to be direct product of these. (no anti-symmetrization)

• This wave form have same structure as the classical Hamilton equations and can be solved numerically.

• The Hamiltonian is calculated using simple time independent optical potential. (unlike QMD)


201127

xtip

tqxLLtpqx ii

ii 2

43

2exp2,,,

Binary Cascade ~nuclear model ~

• 3 dimensional model of the nucleus is constructed from A and Z.

• Nucleon distribution follows– A>16 Woods-Saxon model– Light nuclei harmonic-oscillator shell model

• Nucleon momenta are sampled from 0 to Fermi momentum and sum of these momenta is set to 0.

• time-invariant scalar optical potential is used.


201128

Binary Cascade ~ G4BinaryLightIonReaction ~

• Two nuclei are prepared according to this model (previous page).

• The lighter nucleus is selected to be projectile.• Nucleons in the projectile are entered with

position and momenta into the initial collision state.

• Until first collision of each nucleon, its Fermi motion is neglected in tracking.

• Fermi motion and the nuclear field are taken into account in collision probabilities and final states of the collisions.


201129

Validation resultsNeutrons from 400MeV/n Ne20 on

Carbon


201130

Neutron YieldFe 400 MeV/n beams


2011

Copper Thick Target Lead Thick Target

T. Kurosawa et al., Phys. Rev. C62 pp. 04461501 (2000)

31

Fragment Production

Si 453 MeV/ n on Al

1

10

100

1000

Al Mg Na Ne F O N C

Particle Species

Cro

ss S

ectio

n [m

b]

DATAG4

Si 490 MeV/ n on Cu

1

10

100

1000

Al Mg Na Ne F O N C

Particle Species

Cro

ss S

ectio

n [m

b]

DATAG4


2011

F. Flesch et al., J, RM, 34 237 2001

32

G4WilsonAbrasionModel & G4WilsonAblationModel

• G4WilsonAbrasionModel is a simplified macroscopic model for nuclear-nuclear interactions based largely on geometric arguments

• The speed of the simulation is found to be faster than models such as G4BinaryCascade, but at the cost of accuracy.

• A nuclear ablation has been developed to provide a better approximation for the final nuclear fragment from an abrasion interaction.

• Performing an ablation process to simulate the de-excitation of the nuclear pre-fragments, nuclear de-excitation models within Geant4 (default).

• G4WilsonAblationModel also prepared and uses the same approach for selecting the final-state nucleus as NUCFRG2 (NASA TP 3533)


201133

Abrasion & Ablation


2011

Ablation process

Abrasion process

target nucleus

projectile

34

Validation of G4WilsonAbrasion model

12C-C 1050 MeV/nuc

0.1

1.0

10.0

100.0

C11 C10 B11 B10 Be10 Be9 Be7 Li8 Li7 Li6 He6

Fragment

cro

ss-s

ecti

on

[m

b]

Abrasion + ablation

Experiment

NUCFRG2


201135

G4QMD• Binary Light Ion Cascade is an Ion extension of Binary Cascade• However, in the model

– Neglects participant-participant scattering– Uses simple time independent optical potential– Does not provide ground state nucleus which can be used in molecular

dynamics

• The solution for overcoming these limitation and enabling the simulation of real HZE reactions is QMD (Quantum Molecular Dynamics)

• QMD is quantum extension of classical molecular-dynamics model. – Each nucleon is seen as a Gaussian wave packet– Propagation with scattering term which takes into account Pauli principal


201136

G4QMD• G4QMD create ground state nucleus based on JQMD, which can be

used in MD• Potential field and field parameters of G4QMD is also based on

JQMD with Lorentz scalar modifications– “Development of Jaeri QMD Code” Niita et al, JAERI-Data/Code 99-

042• Self generating potential field is used in G4QMD• G4QMD uses scattering and decay library of Geant4

– Following 25 resonances are taken into account– Δ from 1232 up to 1950– N from 1400 up to 2250

• G4QMD includes Participant-Participant Scattering• All major limitations of Binary cascade for Nucleus-Nucleus

calculations are cleared in G4QMD


201137

Ar40 560MeV/n on Lead Secondary neutron spectra


2011

+ Data+ G4BinaryCascade+ G4QMD

38

“Energy deposition in intermediate-energy nucleon–nucleus collisions,” Kwiatkowski et al., Phys. Rev. Lett.,

vol. 50, no. 21, pp. 1648–1651, 1983

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 5 10 15 20 25 30 35 40 45

cro

ss

se

cti

on

[mb

/Me

V/s

r]

frgament energy [MeV]

180MeV Proton on AlFragment A=7

exp 20°

exp 40°

exp 70°

G4BC 20°

G4BC 40°

G4BC 70°

G4QMD 20°

G4QMD 40°

G4QMD 70°


201139

Ion Physics EelectorMagnetic Dissociation

• Electromagnetic dissociation is liberation of nucleons or nuclear fragments as a result of electromagnetic field by exchange of virtual photons, rather than the strong nuclear force

• It is important for relativistic nuclear-nuclear interaction, especially where the proton number of the nucleus is large

• G4EMDissociation model and cross section are an implementation of the NUCFRG2 (NASA TP 3533) physics and treats this electromagnetic dissociation (ED).


201140

Validation of G4EMDissociaton Model

Projectile

Energy[GeV/nuc]

Product from ED

G4EMDissociation

[mbarn]

Experiment[mbarn]

Mg-24 3.7 Na-23 + p 124 2 154 31

Si-28 3.7 Al-27 + p 107 1 186 56

14.5 Al-27 + p 216 2 165 24†128 33‡

O-16 200 N-15 + p 331 2 293 39†342 22*


2011

M A Jilany, Nucl Phys, A705, 477-493, 2002.

Target Emulsion nuclei: Ag 61.7%, Br 34.2%, CNO 4.0% and H 0.1%Target Emulsion nuclei: Ag 61.7%, Br 34.2%, CNO 4.0% and H 0.1%

41

Physics List for Binary Light Ion• G4HadronInelasticProcess* theIPGenericIon = new

G4HadronInelasticProcess("IonInelastic", G4GenericIon::GenericIon() );• // Cross Section Data Set• G4TripathiCrossSection * TripathiCrossSection= new G4TripathiCrossSection;• G4IonsShenCrossSection * aShen = new G4IonsShenCrossSection;• theIPGenericIon->AddDataSet(aShen);• theIPGenericIon->AddDataSet(TripathiCrossSection);• // Model• G4BinaryLightIonReaction * theGenIonBC= new G4BinaryLightIonReaction;• theIPGenericIon->RegisterMe(theGenIonBC);• //Apply Processes to Process Manager of Neutron• G4ProcessManager* pmanager = G4GenericIon:: GenericIon()->

GetProcessManager();• pmanager->AddDiscreteProcess( theIPGenericIon );


201142

Physics List for WilsonAbrasion• G4HadronInelasticProcess* theIPGenericIon = new G4HadronInelasticProcess("IonInelastic",

G4GenericIon::GenericIon() );• // Cross Section Data Set• G4TripathiCrossSection * TripathiCrossSection= new G4TripathiCrossSection;• G4IonsShenCrossSection * aShen = new G4IonsShenCrossSection;• theIPGenericIon->AddDataSet(aShen);• theIPGenericIon->AddDataSet(TripathiCrossSection);• // Model• G4BinaryLightIonReaction * theGenIonBC= new G4BinaryLightIonReaction;• theGenIonBC->SetMinEnergy(0*MeV);• theGenIonBC->SetMaxEnergy(0.07*GeV);• theIPGenericIon->RegisterMe(theGenIonBC);• G4WilsonAbrasionModel* theGenIonAbrasion = new G4WilsonAbrasionModel();• theIPGenericIon->RegisterMe(theGenIonAbrasion);• //Apply Processes to Process Manager of Neutron• G4ProcessManager* pmanager = G4GenericIon:: GenericIon()-> GetProcessManager();• pmanager->AddDiscreteProcess( theIPGenericIon );


201143

Physics List for QMD• G4HadronInelasticProcess* theIPGenericIon = new

G4HadronInelasticProcess("IonInelastic", G4GenericIon::GenericIon() );• // Cross Section Data Set• G4TripathiCrossSection * TripathiCrossSection= new

G4TripathiCrossSection;• G4IonsShenCrossSection * aShen = new G4IonsShenCrossSection;• theIPGenericIon->AddDataSet(aShen);• theIPGenericIon->AddDataSet(TripathiCrossSection);• // Model• G4QMDReaction * theGenIonQMD= new G4QMDReaction;• theIPGenericIon->RegisterMe(theGenIonQMD);• //Apply Processes to Process Manager of Neutron• G4ProcessManager* pmanager = G4GenericIon:: GenericIon()->



201144

Physics List for EMDissociation• G4HadronInelasticProcess* theIPGenericIon = new

G4HadronInelasticProcess("IonInelastic", G4GenericIon::GenericIon() );

• // Cross Section Data Set• G4EMDissociationCrossSection* theEMDCrossSection = new

G4EMDissociationCrossSection; • theIPGenericIon->AddDataSet( theEMDCrossSection );• // Model• G4EMDissociation* theEMDModel = new G4EMDissociation;• theIPGenericIon->RegisterMe(theEMDModel);• //Apply Processes to Process Manager of Neutron• G4ProcessManager* pmanager = G4GenericIon:: GenericIon()->



201145

Ion PhysicsRadio Active Decay

• To simulate the decay of radioactive nuclei • Empirical and data-driven model• α, β+, β- decay electron capture (EC) are implemented • Data (RadioactiveDecay) derived from Evaluated

Nuclear Structure Data File (ENSDF) – nuclear half-lives– nuclear level structure for the parent or daughter nuclide– decay branching ratios– the energy of the decay process.

• If the daughter of a nuclear decay is an excited isomer, its prompt nuclear de-excitation is treated using the G4PhotonEvapolation


201146

Radio Active Decay

• Analog sampling is default• Biasing sampling also implemented

– The decays occur more frequently at certain times – For a given decay mode the branching ratios can

be sampled with equal probability – split parent nuclide into a user-defined number of

nuclides


201147

Physics List for RadioactiveDecay• G4RadioactiveDecay *theRadioactiveDecay = new

G4RadioactiveDecay();

• G4ProcessManager* pmanager =• G4ParticleTable::GetParticleTable()-

>FindParticle("GenericIon")->GetProcessManager();

• pmanager ->AddProcess(theRadioactiveDecay);• pmanager ->SetProcessOrdering(theRadioactiveDecay,

idxPostStep);• pmanager ->SetProcessOrdering(theRadioactiveDecay,

idxAtRest);


201148

Attention : if you want calculate “triton” decay,You must NOT create “triton”(G4Triton) in your physics list.

Summary

• High Precision Neutron models are data driven models and its used evaluated data libraries.

• However the library is not complete because there are no data for several key elements.

• Geant4 has abundant processes for Ion interactions with matter and also without matter.

• Without any extra modules, users may simulate ion transportation in the complex and realistic geometries of Geant4.


201149

hadron physics ii

Documents

p01geant4 tutorial

data formats

data librariesgeant4

nuclear data fileshow

nuclear data filesthe

laboratory system geant4

photon energy spectra

legendre coefficientsgeant4