G4LivermorePolarizedComptonModel.cc

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//
// Authors: G.Depaola & F.Longo
//
// History:
// -------
//
// 05 Apr 2021   J Allison added quantum entanglement of e+ annihilation.
// If the photons have been "tagged" as "quantum-entangled", for example by
// G4eplusAnnihilation for annihilation into 2 photons, they are "analysed"
// here if - and only if - both photons suffer Compton scattering. Theoretical
// predictions from Pryce and Ward, Nature No 4065 (1947) p.435, and Snyder et al,
// Physical Review 73 (1948) p.440. Experimental validation in "Photon quantum 
// entanglement in the MeV regime and its application in PET imaging",
// D. Watts, J. Allison et al., Nature Communications (2021)12:2646
// https://doi.org/10.1038/s41467-021-22907-5.
//
// 02 May 2009   S Incerti as V. Ivanchenko proposed in G4LivermoreComptonModel.cc
//
// Cleanup initialisation and generation of secondaries:
//                  - apply internal high-energy limit only in constructor 
//                  - do not apply low-energy limit (default is 0)
//                  - remove GetMeanFreePath method and table
//                  - added protection against numerical problem in energy sampling 
//                  - use G4ElementSelector
 
#include "G4LivermorePolarizedComptonModel.hh"
#include "G4PhysicalConstants.hh"
#include "G4SystemOfUnits.hh"
#include "G4AutoLock.hh"
#include "G4Electron.hh"
#include "G4ParticleChangeForGamma.hh"
#include "G4LossTableManager.hh"
#include "G4VAtomDeexcitation.hh"
#include "G4AtomicShell.hh"
#include "G4Gamma.hh"
#include "G4ShellData.hh"
#include "G4DopplerProfile.hh"
#include "G4Log.hh"
#include "G4Exp.hh"
#include "G4Pow.hh"
#include "G4LogLogInterpolation.hh"
#include "G4PhysicsModelCatalog.hh"
#include "G4EntanglementAuxInfo.hh"
#include "G4eplusAnnihilationEntanglementClipBoard.hh"
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
using namespace std;
namespace { G4Mutex LivermorePolarizedComptonModelMutex = G4MUTEX_INITIALIZER; }
 
 
G4PhysicsFreeVector* G4LivermorePolarizedComptonModel::data[] = {nullptr};
G4ShellData*       G4LivermorePolarizedComptonModel::shellData = nullptr;
G4DopplerProfile*  G4LivermorePolarizedComptonModel::profileData = nullptr; 
G4CompositeEMDataSet* G4LivermorePolarizedComptonModel::scatterFunctionData = nullptr;
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4LivermorePolarizedComptonModel::G4LivermorePolarizedComptonModel(const G4ParticleDefinition*, const G4String& nam)
  :G4VEmModel(nam),isInitialised(false)
{ 
  verboseLevel= 1;
  // Verbosity scale:
  // 0 = nothing 
  // 1 = warning for energy non-conservation 
  // 2 = details of energy budget
  // 3 = calculation of cross sections, file openings, sampling of atoms
  // 4 = entering in methods
 
  if( verboseLevel>1 )  
    G4cout << "Livermore Polarized Compton is constructed " << G4endl;
        
  //Mark this model as "applicable" for atomic deexcitation
  SetDeexcitationFlag(true);
  
  fParticleChange = nullptr;
  fAtomDeexcitation = nullptr;
  fEntanglementModelID = G4PhysicsModelCatalog::GetModelID("model_GammaGammaEntanglement");
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4LivermorePolarizedComptonModel::~G4LivermorePolarizedComptonModel()
{  
  if(IsMaster()) {
    delete shellData;
    shellData = nullptr;
    delete profileData;
    profileData = nullptr;
    delete scatterFunctionData;
    scatterFunctionData = nullptr;
    for(G4int i=0; i<maxZ; ++i) {
      if(data[i]) { 
    delete data[i];
    data[i] = nullptr;
      }
    }
  }
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4LivermorePolarizedComptonModel::Initialise(const G4ParticleDefinition* particle,
                                       const G4DataVector& cuts)
{
  if (verboseLevel > 1)
    G4cout << "Calling G4LivermorePolarizedComptonModel::Initialise()" << G4endl;
 
  // Initialise element selector 
  if(IsMaster()) {
    // Access to elements 
    const char* path = G4FindDataDir("G4LEDATA");
 
    G4ProductionCutsTable* theCoupleTable = 
      G4ProductionCutsTable::GetProductionCutsTable();
 
    G4int numOfCouples = (G4int)theCoupleTable->GetTableSize();
    
    for(G4int i=0; i<numOfCouples; ++i) {
      const G4Material* material = 
        theCoupleTable->GetMaterialCutsCouple(i)->GetMaterial();
      const G4ElementVector* theElementVector = material->GetElementVector();
      std::size_t nelm = material->GetNumberOfElements();
      
      for (std::size_t j=0; j<nelm; ++j) {
        G4int Z = G4lrint((*theElementVector)[j]->GetZ());
        if(Z < 1)        { Z = 1; }
        else if(Z > maxZ){ Z = maxZ; }
 
        if( (!data[Z]) ) { ReadData(Z, path); }
      }
    }
    
    // For Doppler broadening
    if(!shellData) {
      shellData = new G4ShellData(); 
      shellData->SetOccupancyData();
      G4String file = "/doppler/shell-doppler";
      shellData->LoadData(file);
    }
    if(!profileData) { profileData = new G4DopplerProfile(); }
 
    // Scattering Function 
    if(!scatterFunctionData)
      {
    
    G4VDataSetAlgorithm* scatterInterpolation = new G4LogLogInterpolation;
    G4String scatterFile = "comp/ce-sf-";
    scatterFunctionData = new G4CompositeEMDataSet(scatterInterpolation, 1., 1.);
    scatterFunctionData->LoadData(scatterFile);
      }
    
    InitialiseElementSelectors(particle, cuts);
  }
 
  if (verboseLevel > 2) {
    G4cout << "Loaded cross section files" << G4endl;
  }
  
  if( verboseLevel>1 ) { 
    G4cout << "G4LivermoreComptonModel is initialized " << G4endl
       << "Energy range: "
       << LowEnergyLimit() / eV << " eV - "
       << HighEnergyLimit() / GeV << " GeV"
       << G4endl;
  }
  //  
  if(isInitialised) { return; }
  
  fParticleChange = GetParticleChangeForGamma();
  fAtomDeexcitation  = G4LossTableManager::Instance()->AtomDeexcitation();
  isInitialised = true;
}
 
 
void G4LivermorePolarizedComptonModel::InitialiseLocal(const G4ParticleDefinition*,
                          G4VEmModel* masterModel)
{
  SetElementSelectors(masterModel->GetElementSelectors());
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4LivermorePolarizedComptonModel::ReadData(std::size_t Z, const char* path)
{
  if (verboseLevel > 1) 
    {
      G4cout << "G4LivermorePolarizedComptonModel::ReadData()" 
         << G4endl;
    }
  if(data[Z]) { return; }  
  const char* datadir = path;
  if(!datadir) 
    {
      datadir = G4FindDataDir("G4LEDATA");
      if(!datadir) 
    {
      G4Exception("G4LivermorePolarizedComptonModel::ReadData()",
              "em0006",FatalException,
              "Environment variable G4LEDATA not defined");
      return;
    }
    }
  
  data[Z] = new G4PhysicsFreeVector();
    
  std::ostringstream ost;
  ost << datadir << "/livermore/comp/ce-cs-" << Z <<".dat";
  std::ifstream fin(ost.str().c_str());
  
  if( !fin.is_open()) 
    {
      G4ExceptionDescription ed;
      ed << "G4LivermorePolarizedComptonModel data file <" << ost.str().c_str()
     << "> is not opened!" << G4endl;
      G4Exception("G4LivermoreComptonModel::ReadData()",
          "em0003",FatalException,
          ed,"G4LEDATA version should be G4EMLOW8.0 or later");
      return;
    } else {
    if(verboseLevel > 3) {
      G4cout << "File " << ost.str() 
         << " is opened by G4LivermorePolarizedComptonModel" << G4endl;
    }   
    data[Z]->Retrieve(fin, true);
    data[Z]->ScaleVector(MeV, MeV*barn);
  }   
  fin.close();
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4double G4LivermorePolarizedComptonModel::ComputeCrossSectionPerAtom(
                                       const G4ParticleDefinition*,
                                             G4double GammaEnergy,
                                             G4double Z, G4double,
                                             G4double, G4double)
{
  if (verboseLevel > 3)
    G4cout << "Calling ComputeCrossSectionPerAtom() of G4LivermorePolarizedComptonModel" << G4endl;
 
  G4double cs = 0.0; 
  
  if (GammaEnergy < LowEnergyLimit()) 
    return 0.0;
 
  G4int intZ = G4lrint(Z);
  if(intZ < 1 || intZ > maxZ) { return cs; } 
  
  G4PhysicsFreeVector* pv = data[intZ];
  
  // if element was not initialised
  // do initialisation safely for MT mode
  if(!pv) 
    {
      InitialiseForElement(0, intZ);
      pv = data[intZ];
      if(!pv) { return cs; }
    }
  
  G4int n = G4int(pv->GetVectorLength() - 1);   
  G4double e1 = pv->Energy(0);
  G4double e2 = pv->Energy(n);
  
  if(GammaEnergy <= e1)      { cs = GammaEnergy/(e1*e1)*pv->Value(e1); }
  else if(GammaEnergy <= e2) { cs = pv->Value(GammaEnergy)/GammaEnergy; }
  else if(GammaEnergy > e2)  { cs = pv->Value(e2)/GammaEnergy; }
  
  return cs;
 
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4LivermorePolarizedComptonModel::SampleSecondaries(std::vector<G4DynamicParticle*>* fvect,
                          const G4MaterialCutsCouple* couple,
                          const G4DynamicParticle* aDynamicGamma,
                          G4double,
                          G4double)
{
  // The scattered gamma energy is sampled according to Klein - Nishina formula.
  // The random number techniques of Butcher & Messel are used (Nuc Phys 20(1960),15).
  // GEANT4 internal units
  //
  // Note : Effects due to binding of atomic electrons are negliged.
 
  if (verboseLevel > 3)
    G4cout << "Calling SampleSecondaries() of G4LivermorePolarizedComptonModel" << G4endl;
 
  G4double gammaEnergy0 = aDynamicGamma->GetKineticEnergy();
 
  // do nothing below the threshold
  // should never get here because the XS is zero below the limit
  if (gammaEnergy0 < LowEnergyLimit())     
    return ; 
 
  G4ThreeVector gammaPolarization0 = aDynamicGamma->GetPolarization();
 
  // Protection: a polarisation parallel to the
  // direction causes problems;
  // in that case find a random polarization
  G4ThreeVector gammaDirection0 = aDynamicGamma->GetMomentumDirection();
 
  // Make sure that the polarization vector is perpendicular to the
  // gamma direction. If not
  if(!(gammaPolarization0.isOrthogonal(gammaDirection0, 1e-6))||(gammaPolarization0.mag()==0))
    { // only for testing now
      gammaPolarization0 = GetRandomPolarization(gammaDirection0);
    }
  else
    {
      if ( gammaPolarization0.howOrthogonal(gammaDirection0) != 0)
    {
      gammaPolarization0 = GetPerpendicularPolarization(gammaDirection0, gammaPolarization0);
    }
    }
  // End of Protection
 
  G4double E0_m = gammaEnergy0 / electron_mass_c2 ;
 
  // Select randomly one element in the current material
  //G4int Z = crossSectionHandler->SelectRandomAtom(couple,gammaEnergy0);
  const G4ParticleDefinition* particle =  aDynamicGamma->GetDefinition();
  const G4Element* elm = SelectRandomAtom(couple,particle,gammaEnergy0);
  G4int Z = (G4int)elm->GetZ();
 
  // Sample the energy and the polarization of the scattered photon
  G4double epsilon, epsilonSq, onecost, sinThetaSqr, greject ;
 
  G4double epsilon0Local = 1./(1. + 2*E0_m);
  G4double epsilon0Sq = epsilon0Local*epsilon0Local;
  G4double alpha1   = - G4Log(epsilon0Local);
  G4double alpha2 = 0.5*(1.- epsilon0Sq);
 
  G4double wlGamma = h_Planck*c_light/gammaEnergy0;
  G4double gammaEnergy1;
  G4ThreeVector gammaDirection1;
 
  do {
    if ( alpha1/(alpha1+alpha2) > G4UniformRand() )
      {
    epsilon   = G4Exp(-alpha1*G4UniformRand());  
    epsilonSq = epsilon*epsilon; 
      }
    else 
      {
    epsilonSq = epsilon0Sq + (1.- epsilon0Sq)*G4UniformRand();
    epsilon   = std::sqrt(epsilonSq);
      }
 
    onecost = (1.- epsilon)/(epsilon*E0_m);
    sinThetaSqr   = onecost*(2.-onecost);
 
    // Protection
    if (sinThetaSqr > 1.)
      {
    G4cout
      << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
      << "sin(theta)**2 = "
      << sinThetaSqr
      << "; set to 1"
      << G4endl;
    sinThetaSqr = 1.;
      }
    if (sinThetaSqr < 0.)
      {
    G4cout
      << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
      << "sin(theta)**2 = "
      << sinThetaSqr
      << "; set to 0"
      << G4endl;
    sinThetaSqr = 0.;
      }
    // End protection
 
    G4double x =  std::sqrt(onecost/2.) / (wlGamma/cm);;
    G4double scatteringFunction = scatterFunctionData->FindValue(x,Z-1);
    greject = (1. - epsilon*sinThetaSqr/(1.+ epsilonSq))*scatteringFunction;
 
  } while(greject < G4UniformRand()*Z);
 
 
  // ****************************************************
  //        Phi determination
  // ****************************************************
  G4double phi = SetPhi(epsilon,sinThetaSqr);
 
  //
  // scattered gamma angles. ( Z - axis along the parent gamma)
  //
  G4double cosTheta = 1. - onecost;
 
  // Protection
  if (cosTheta > 1.)
    {
      G4cout
    << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
    << "cosTheta = "
    << cosTheta
    << "; set to 1"
    << G4endl;
      cosTheta = 1.;
    }
  if (cosTheta < -1.)
    {
      G4cout 
    << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
    << "cosTheta = " 
    << cosTheta
    << "; set to -1"
    << G4endl;
      cosTheta = -1.;
    }
  // End protection      
 
  G4double sinTheta = std::sqrt (sinThetaSqr);
  
  // Protection
  if (sinTheta > 1.)
    {
      G4cout 
    << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
    << "sinTheta = " 
    << sinTheta
    << "; set to 1"
    << G4endl;
      sinTheta = 1.;
    }
  if (sinTheta < -1.)
    {
      G4cout 
    << " -- Warning -- G4LivermorePolarizedComptonModel::SampleSecondaries "
    << "sinTheta = " 
    << sinTheta
    << "; set to -1" 
    << G4endl;
      sinTheta = -1.;
    }
  // End protection
  
  // Check for entanglement and re-sample phi if necessary
  
  const auto* auxInfo
  = fParticleChange->GetCurrentTrack()->GetAuxiliaryTrackInformation(fEntanglementModelID);
  if (auxInfo) {
    const auto* entanglementAuxInfo = dynamic_cast<const G4EntanglementAuxInfo*>(auxInfo);
    if (entanglementAuxInfo) {
      auto* clipBoard = dynamic_cast<G4eplusAnnihilationEntanglementClipBoard*>
      (entanglementAuxInfo->GetEntanglementClipBoard());
      if (clipBoard) {
        // This is an entangled photon from eplus annihilation at rest.
        // If this is the first scatter of the first photon, place theta and
        // phi on the clipboard.
        // If this is the first scatter of the second photon, use theta and
        // phi of the first scatter of the first photon, together with the
        // theta of the second photon, to sample phi.
        if (clipBoard->IsTrack1Measurement()) {
          // Check we have the relevant track. Not sure this is strictly
          // necessary but I want to be sure tracks from, say, more than one
          // entangled system are properly paired.
          // Note: the tracking manager pops the tracks in the reverse order. We
          // will rely on that.  (If not, the logic here would have to be a bit
          // more complicated to ensure we matched the right tracks.)
          // So our track 1 is clipboard track B.
          if (clipBoard->GetTrackB() == fParticleChange->GetCurrentTrack()) {
            // This is the first scatter of the first photon. Reset flag.
            //            // Debug
            //            auto* track1 = fParticleChange->GetCurrentTrack();
            //            G4cout
            //            << "This is the first scatter of the first photon. Reset flag."
            //            << "\nTrack: " << track1->GetTrackID()
            //            << ", Parent: " << track1->GetParentID()
            //            << ", Name: " << clipBoard->GetParentParticleDefinition()->GetParticleName()
            //            << G4endl;
            //            // End debug
            clipBoard->ResetTrack1Measurement();
            // Store cos(theta),phi of first photon.
            clipBoard->SetComptonCosTheta1(cosTheta);
            clipBoard->SetComptonPhi1(phi);
          }
        } else if (clipBoard->IsTrack2Measurement()) {
          // Check we have the relevant track.
          // Remember our track 2 is clipboard track A.
          if (clipBoard->GetTrackA() == fParticleChange->GetCurrentTrack()) {
            // This is the first scatter of the second photon. Reset flag.
            //            // Debug
            //            auto* track2 = fParticleChange->GetCurrentTrack();
            //            G4cout
            //            << "This is the first scatter of the second photon. Reset flag."
            //            << "\nTrack: " << track2->GetTrackID()
            //            << ", Parent: " << track2->GetParentID()
            //            << ", Name: " << clipBoard->GetParentParticleDefinition()->GetParticleName()
            //            << G4endl;
            //            // End debug
            clipBoard->ResetTrack2Measurement();
            
            // Get cos(theta),phi of first photon.
            const G4double& cosTheta1 = clipBoard->GetComptonCosTheta1();
            const G4double& phi1 = clipBoard->GetComptonPhi1();
            // For clarity make aliases for the current cos(theta),phi.
            const G4double& cosTheta2 = cosTheta;
            G4double& phi2 = phi;
            //            G4cout << "cosTheta1,phi1: " << cosTheta1 << ',' << phi1 << G4endl;
            //            G4cout << "cosTheta2,phi2: " << cosTheta2 << ',' << phi2 << G4endl;
            
            // Re-sample phi
            // Draw the difference of azimuthal angles, deltaPhi, from
            // A + B * cos(2*deltaPhi), or rather C + D * cos(2*deltaPhi), where
            // C = A / (A + |B|) and D = B / (A + |B|), so that maximum is 1.
            const G4double sin2Theta1 = 1.-cosTheta1*cosTheta1;
            const G4double sin2Theta2 = 1.-cosTheta2*cosTheta2;
            
            // Pryce and Ward, Nature No 4065 (1947) p.435.
            auto* g4Pow = G4Pow::GetInstance();
            const G4double A =
            ((g4Pow->powN(1.-cosTheta1,3))+2.)*(g4Pow->powN(1.-cosTheta2,3)+2.)/
            ((g4Pow->powN(2.-cosTheta1,3)*g4Pow->powN(2.-cosTheta2,3)));
            const G4double B = -(sin2Theta1*sin2Theta2)/
            ((g4Pow->powN(2.-cosTheta1,2)*g4Pow->powN(2.-cosTheta2,2)));
 
            //    // Snyder et al, Physical Review 73 (1948) p.440.
            //    // (This is an alternative formulation but result is identical.)
            //    const G4double& k0 = gammaEnergy0;
            //    const G4double k1 = k0/(2.-cosTheta1);
            //    const G4double k2 = k0/(2.-cosTheta2);
            //    const G4double gamma1 = k1/k0+k0/k1;
            //    const G4double gamma2 = k2/k0+k0/k2;
            //    const G4double A1 = gamma1*gamma2-gamma1*sin2Theta2-gamma2*sin2Theta1;
            //    const G4double B1 = 2.*sin2Theta1*sin2Theta2;
            //    // That's A1 + B1*sin2(deltaPhi) = A1 + B1*(0.5*(1.-cos(2.*deltaPhi).
            //    const G4double A = A1 + 0.5*B1;
            //    const G4double B = -0.5*B1;
            
            const G4double maxValue = A + std::abs(B);
            const G4double C = A / maxValue;
            const G4double D = B / maxValue;
            //    G4cout << "A,B,C,D: " << A << ',' << B  << ',' << C << ',' << D << G4endl;
            
            // Sample delta phi
            G4double deltaPhi;
            const G4int maxCount = 999999;
            G4int iCount = 0;
            for (; iCount < maxCount; ++iCount) {
              deltaPhi = twopi * G4UniformRand();
              if (G4UniformRand() < C + D * cos(2.*deltaPhi)) break;
            }
            if (iCount >= maxCount ) {
              G4cout << "G4LivermorePolarizedComptonModel::SampleSecondaries: "
              << "Re-sampled delta phi not found in " << maxCount
              << " tries - carrying on anyway." << G4endl;
            }
            
            // Thus, the desired second photon azimuth
            phi2 = deltaPhi - phi1 + halfpi;
            // The minus sign is in above statement because, since the two
            // annihilation photons are in opposite directions, their phi's
            // are measured in the opposite direction.
            // halfpi is added for the following reason:
            // In this function phi is relative to the polarisation - see
            // SystemOfRefChange below. We know from G4eplusAnnihilation that
            // the polarisations of the two annihilation photons are perpendicular
            // to each other, i.e., halfpi different.
            // Furthermore, only sin(phi) and cos(phi) are used below so no
            // need to place any range constraints.
            //            if (phi2 > pi) {
            //              phi2 -= twopi;
            //            }
            //            if (phi2 < -pi) {
            //              phi2 += twopi;
            //            }
          }
        }
      }
    }
  }
  
  // End of entanglement
      
  G4double dirx = sinTheta*std::cos(phi);
  G4double diry = sinTheta*std::sin(phi);
  G4double dirz = cosTheta ;
 
  // oneCosT , eom
 
  // Doppler broadening -  Method based on:
  // Y. Namito, S. Ban and H. Hirayama, 
  // "Implementation of the Doppler Broadening of a Compton-Scattered Photon Into the EGS4 Code" 
  // NIM A 349, pp. 489-494, 1994
  
  // Maximum number of sampling iterations
  static G4int maxDopplerIterations = 1000;
  G4double bindingE = 0.;
  G4double photonEoriginal = epsilon * gammaEnergy0;
  G4double photonE = -1.;
  G4int iteration = 0;
  G4double eMax = gammaEnergy0;
 
  G4int shellIdx = 0;
 
  if (verboseLevel > 3) {
    G4cout << "Started loop to sample broading" << G4endl;
  }
 
  do
    {
      iteration++;
      // Select shell based on shell occupancy
      shellIdx = shellData->SelectRandomShell(Z);
      bindingE = shellData->BindingEnergy(Z,shellIdx);
      
      if (verboseLevel > 3) {
    G4cout << "Shell ID= " << shellIdx 
           << " Ebind(keV)= " << bindingE/keV << G4endl;
      }
      eMax = gammaEnergy0 - bindingE;
      
      // Randomly sample bound electron momentum (memento: the data set is in Atomic Units)
      G4double pSample = profileData->RandomSelectMomentum(Z,shellIdx);
 
      if (verboseLevel > 3) {
       G4cout << "pSample= " << pSample << G4endl;
     }
      // Rescale from atomic units
      G4double pDoppler = pSample * fine_structure_const;
      G4double pDoppler2 = pDoppler * pDoppler;
      G4double var2 = 1. + onecost * E0_m;
      G4double var3 = var2*var2 - pDoppler2;
      G4double var4 = var2 - pDoppler2 * cosTheta;
      G4double var = var4*var4 - var3 + pDoppler2 * var3;
      if (var > 0.)
    {
      G4double varSqrt = std::sqrt(var);        
      G4double scale = gammaEnergy0 / var3;  
          // Random select either root
      if (G4UniformRand() < 0.5) photonE = (var4 - varSqrt) * scale;               
      else photonE = (var4 + varSqrt) * scale;
    } 
      else
    {
      photonE = -1.;
    }
   } while ( iteration <= maxDopplerIterations && 
         (photonE < 0. || photonE > eMax || photonE < eMax*G4UniformRand()) );
 
  // End of recalculation of photon energy with Doppler broadening
  // Revert to original if maximum number of iterations threshold has been reached
  if (iteration >= maxDopplerIterations)
    {
      photonE = photonEoriginal;
      bindingE = 0.;
    }
 
  gammaEnergy1 = photonE;
 
  //
  // update G4VParticleChange for the scattered photon 
  //
  // New polarization
  G4ThreeVector gammaPolarization1 = SetNewPolarization(epsilon,
                            sinThetaSqr,
                            phi,
                            cosTheta);
 
  // Set new direction
  G4ThreeVector tmpDirection1( dirx,diry,dirz );
  gammaDirection1 = tmpDirection1;
 
  // Change reference frame.
  SystemOfRefChange(gammaDirection0,gammaDirection1,
            gammaPolarization0,gammaPolarization1);
 
  if (gammaEnergy1 > 0.)
    {
      fParticleChange->SetProposedKineticEnergy( gammaEnergy1 ) ;
      fParticleChange->ProposeMomentumDirection( gammaDirection1 );
      fParticleChange->ProposePolarization( gammaPolarization1 );
    }
  else
    {
      gammaEnergy1 = 0.;
      fParticleChange->SetProposedKineticEnergy(0.) ;
      fParticleChange->ProposeTrackStatus(fStopAndKill);
    }
 
  //
  // kinematic of the scattered electron
  //
  G4double ElecKineEnergy = gammaEnergy0 - gammaEnergy1 -bindingE;
 
  // SI -protection against negative final energy: no e- is created
  // like in G4LivermoreComptonModel.cc
  if(ElecKineEnergy < 0.0) {
    fParticleChange->ProposeLocalEnergyDeposit(gammaEnergy0 - gammaEnergy1);
    return;
  }
 
  G4double ElecMomentum = std::sqrt(ElecKineEnergy*(ElecKineEnergy+2.*electron_mass_c2));
 
  G4ThreeVector ElecDirection((gammaEnergy0 * gammaDirection0 -
                   gammaEnergy1 * gammaDirection1) * (1./ElecMomentum));
 
  G4DynamicParticle* dp = 
    new G4DynamicParticle (G4Electron::Electron(),ElecDirection.unit(),ElecKineEnergy) ;
  fvect->push_back(dp);
 
  // sample deexcitation
  //  
  if (verboseLevel > 3) {
    G4cout << "Started atomic de-excitation " << fAtomDeexcitation << G4endl;
  }
  
  if(fAtomDeexcitation && iteration < maxDopplerIterations) {
    G4int index = couple->GetIndex();
    if(fAtomDeexcitation->CheckDeexcitationActiveRegion(index)) {
      std::size_t nbefore = fvect->size();
      G4AtomicShellEnumerator as = G4AtomicShellEnumerator(shellIdx);
      const G4AtomicShell* shell = fAtomDeexcitation->GetAtomicShell(Z, as);
      fAtomDeexcitation->GenerateParticles(fvect, shell, Z, index);
      std::size_t nafter = fvect->size();
      if(nafter > nbefore) {
    for (std::size_t i=nbefore; i<nafter; ++i) {
      //Check if there is enough residual energy 
      if (bindingE >= ((*fvect)[i])->GetKineticEnergy())
            {
              //Ok, this is a valid secondary: keep it
          bindingE -= ((*fvect)[i])->GetKineticEnergy();
            }
      else
            {
          //Invalid secondary: not enough energy to create it!
          //Keep its energy in the local deposit
          delete (*fvect)[i]; 
              (*fvect)[i]=0;
            }
    } 
      }
    }
  }
  //This should never happen
  if(bindingE < 0.0) 
    G4Exception("G4LivermoreComptonModel::SampleSecondaries()",
        "em2050",FatalException,"Negative local energy deposit");   
  
  fParticleChange->ProposeLocalEnergyDeposit(bindingE);
 
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4double G4LivermorePolarizedComptonModel::SetPhi(G4double energyRate,
                         G4double sinSqrTh)
{
  G4double rand1;
  G4double rand2;
  G4double phiProbability;
  G4double phi;
  G4double a, b;
 
  do
    {
      rand1 = G4UniformRand();
      rand2 = G4UniformRand();
      phiProbability=0.;
      phi = twopi*rand1;
      
      a = 2*sinSqrTh;
      b = energyRate + 1/energyRate;
      
      phiProbability = 1 - (a/b)*(std::cos(phi)*std::cos(phi));
    }
  while ( rand2 > phiProbability );
  return phi;
}
 
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4ThreeVector G4LivermorePolarizedComptonModel::SetPerpendicularVector(G4ThreeVector& a)
{
  G4double dx = a.x();
  G4double dy = a.y();
  G4double dz = a.z();
  G4double x = dx < 0.0 ? -dx : dx;
  G4double y = dy < 0.0 ? -dy : dy;
  G4double z = dz < 0.0 ? -dz : dz;
  if (x < y) {
    return x < z ? G4ThreeVector(-dy,dx,0) : G4ThreeVector(0,-dz,dy);
  }else{
    return y < z ? G4ThreeVector(dz,0,-dx) : G4ThreeVector(-dy,dx,0);
  }
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4ThreeVector G4LivermorePolarizedComptonModel::GetRandomPolarization(G4ThreeVector& direction0)
{
  G4ThreeVector d0 = direction0.unit();
  G4ThreeVector a1 = SetPerpendicularVector(d0); //different orthogonal
  G4ThreeVector a0 = a1.unit(); // unit vector
 
  G4double rand1 = G4UniformRand();
  
  G4double angle = twopi*rand1; // random polar angle
  G4ThreeVector b0 = d0.cross(a0); // cross product
  
  G4ThreeVector c;
  
  c.setX(std::cos(angle)*(a0.x())+std::sin(angle)*b0.x());
  c.setY(std::cos(angle)*(a0.y())+std::sin(angle)*b0.y());
  c.setZ(std::cos(angle)*(a0.z())+std::sin(angle)*b0.z());
  
  G4ThreeVector c0 = c.unit();
 
  return c0;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4ThreeVector G4LivermorePolarizedComptonModel::GetPerpendicularPolarization
(const G4ThreeVector& gammaDirection, const G4ThreeVector& gammaPolarization) const
{
  // 
  // The polarization of a photon is always perpendicular to its momentum direction.
  // Therefore this function removes those vector component of gammaPolarization, which
  // points in direction of gammaDirection
  //
  // Mathematically we search the projection of the vector a on the plane E, where n is the
  // plains normal vector.
  // The basic equation can be found in each geometry book (e.g. Bronstein):
  // p = a - (a o n)/(n o n)*n
  
  return gammaPolarization - gammaPolarization.dot(gammaDirection)/gammaDirection.dot(gammaDirection) * gammaDirection;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4ThreeVector G4LivermorePolarizedComptonModel::SetNewPolarization(G4double epsilon,
                                  G4double sinSqrTh, 
                                  G4double phi,
                                  G4double costheta) 
{
  G4double rand1;
  G4double rand2;
  G4double cosPhi = std::cos(phi);
  G4double sinPhi = std::sin(phi);
  G4double sinTheta = std::sqrt(sinSqrTh);
  G4double cosSqrPhi = cosPhi*cosPhi;
  //  G4double cossqrth = 1.-sinSqrTh;
  //  G4double sinsqrphi = sinPhi*sinPhi;
  G4double normalisation = std::sqrt(1. - cosSqrPhi*sinSqrTh);
 
  // Determination of Theta 
  G4double theta;
 
  // Dan Xu method (IEEE TNS, 52, 1160 (2005))
  rand1 = G4UniformRand();
  rand2 = G4UniformRand();
 
  if (rand1<(epsilon+1.0/epsilon-2)/(2.0*(epsilon+1.0/epsilon)-4.0*sinSqrTh*cosSqrPhi))
    {
      if (rand2<0.5)
    theta = pi/2.0;
      else
    theta = 3.0*pi/2.0;
    }
  else
    {
      if (rand2<0.5)
    theta = 0;
      else
    theta = pi;
    }
  G4double cosBeta = std::cos(theta);
  G4double sinBeta = std::sqrt(1-cosBeta*cosBeta);
  
  G4ThreeVector gammaPolarization1;
 
  G4double xParallel = normalisation*cosBeta;
  G4double yParallel = -(sinSqrTh*cosPhi*sinPhi)*cosBeta/normalisation;
  G4double zParallel = -(costheta*sinTheta*cosPhi)*cosBeta/normalisation;
  G4double xPerpendicular = 0.;
  G4double yPerpendicular = (costheta)*sinBeta/normalisation;
  G4double zPerpendicular = -(sinTheta*sinPhi)*sinBeta/normalisation;
 
  G4double xTotal = (xParallel + xPerpendicular);
  G4double yTotal = (yParallel + yPerpendicular);
  G4double zTotal = (zParallel + zPerpendicular);
  
  gammaPolarization1.setX(xTotal);
  gammaPolarization1.setY(yTotal);
  gammaPolarization1.setZ(zTotal);
  
  return gammaPolarization1;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4LivermorePolarizedComptonModel::SystemOfRefChange(G4ThreeVector& direction0,
                            G4ThreeVector& direction1,
                            G4ThreeVector& polarization0,
                            G4ThreeVector& polarization1)
{
  // direction0 is the original photon direction ---> z
  // polarization0 is the original photon polarization ---> x
  // need to specify y axis in the real reference frame ---> y 
  G4ThreeVector Axis_Z0 = direction0.unit();
  G4ThreeVector Axis_X0 = polarization0.unit();
  G4ThreeVector Axis_Y0 = (Axis_Z0.cross(Axis_X0)).unit(); // to be confirmed;
 
  G4double direction_x = direction1.getX();
  G4double direction_y = direction1.getY();
  G4double direction_z = direction1.getZ();
  
  direction1 = (direction_x*Axis_X0 + direction_y*Axis_Y0 + direction_z*Axis_Z0).unit();
  G4double polarization_x = polarization1.getX();
  G4double polarization_y = polarization1.getY();
  G4double polarization_z = polarization1.getZ();
 
  polarization1 = (polarization_x*Axis_X0 + polarization_y*Axis_Y0 + polarization_z*Axis_Z0).unit();
 
}
 
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo......
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo......
 
void 
G4LivermorePolarizedComptonModel::InitialiseForElement(const G4ParticleDefinition*, 
                          G4int Z)
{
  G4AutoLock l(&LivermorePolarizedComptonModelMutex);
  if(!data[Z]) { ReadData(Z); }
  l.unlock();
}