G4PenelopeGammaConversionModel.cc

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// $Id$
//
// Author: Luciano Pandola
//
// History:
// --------
// 13 Jan 2010   L Pandola    First implementation (updated to Penelope08)
// 24 May 2011   L Pandola    Renamed (make v2008 as default Penelope)
//
 
#include "G4PenelopeGammaConversionModel.hh"
#include "G4PhysicalConstants.hh"
#include "G4SystemOfUnits.hh"
#include "G4ParticleDefinition.hh"
#include "G4MaterialCutsCouple.hh"
#include "G4ProductionCutsTable.hh"
#include "G4DynamicParticle.hh"
#include "G4Element.hh"
#include "G4Gamma.hh"
#include "G4Electron.hh"
#include "G4Positron.hh"
#include "G4PhysicsFreeVector.hh"
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
 
G4PenelopeGammaConversionModel::G4PenelopeGammaConversionModel(const G4ParticleDefinition*,
                                   const G4String& nam)
  :G4VEmModel(nam),fParticleChange(0),logAtomicCrossSection(0),
   fEffectiveCharge(0),fMaterialInvScreeningRadius(0),
   fScreeningFunction(0),isInitialised(false)
{
  fIntrinsicLowEnergyLimit = 2.0*electron_mass_c2;
  fIntrinsicHighEnergyLimit = 100.0*GeV;
  fSmallEnergy = 1.1*MeV;
  InitializeScreeningRadii();
 
  //  SetLowEnergyLimit(fIntrinsicLowEnergyLimit);
  SetHighEnergyLimit(fIntrinsicHighEnergyLimit);
  //
  verboseLevel= 0;
  // 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
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4PenelopeGammaConversionModel::~G4PenelopeGammaConversionModel()
{
  std::map <G4int,G4PhysicsFreeVector*>::iterator i;
  if (logAtomicCrossSection)
    {
      for (i=logAtomicCrossSection->begin();i != logAtomicCrossSection->end();i++)
        if (i->second) delete i->second;
      delete logAtomicCrossSection;
    }
  if (fEffectiveCharge)
    delete fEffectiveCharge;
  if (fMaterialInvScreeningRadius)
    delete fMaterialInvScreeningRadius;
  if (fScreeningFunction)
    delete fScreeningFunction;
}
 
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4PenelopeGammaConversionModel::Initialise(const G4ParticleDefinition*,
                        const G4DataVector&)
{
  if (verboseLevel > 3)
    G4cout << "Calling  G4PenelopeGammaConversionModel::Initialise()" << G4endl;
 
  // logAtomicCrossSection is created only once, since it is  never cleared
  if (!logAtomicCrossSection)
    logAtomicCrossSection =  new std::map<G4int,G4PhysicsFreeVector*>;
 
  //delete old material data...
  if (fEffectiveCharge)
    {
      delete fEffectiveCharge;
      fEffectiveCharge = 0;
    }
  if (fMaterialInvScreeningRadius)
    {
      delete fMaterialInvScreeningRadius;
      fMaterialInvScreeningRadius = 0;
    }
  if (fScreeningFunction)
    {
      delete fScreeningFunction;
      fScreeningFunction = 0;
    }      
  //and create new ones
  fEffectiveCharge = new std::map<const G4Material*,G4double>;
  fMaterialInvScreeningRadius = new std::map<const G4Material*,G4double>;
  fScreeningFunction = new std::map<const G4Material*,std::pair<G4double,G4double> >;
 
  if (verboseLevel > 0) { 
    G4cout << "Penelope Gamma Conversion model v2008 is initialized " << G4endl
       << "Energy range: "
       << LowEnergyLimit() / MeV << " MeV - "
       << HighEnergyLimit() / GeV << " GeV"
       << G4endl;
  }
 
  if(isInitialised) return;
  fParticleChange = GetParticleChangeForGamma();
  isInitialised = true;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
G4double G4PenelopeGammaConversionModel::ComputeCrossSectionPerAtom(
                                    const G4ParticleDefinition*,
                                    G4double energy,
                                    G4double Z, G4double,
                                    G4double, G4double)
{
  //
  // Penelope model v2008.
  // Cross section (including triplet production) read from database and managed 
  // through the G4CrossSectionHandler utility. Cross section data are from
  // M.J. Berger and J.H. Hubbel (XCOM), Report NBSIR 887-3598
  //
  
  if (energy < fIntrinsicLowEnergyLimit)
    return 0;
 
  G4int iZ = (G4int) Z;
 
 //read data files
  if (!logAtomicCrossSection->count(iZ))
    ReadDataFile(iZ);
  //now it should be ok
  if (!logAtomicCrossSection->count(iZ))
     {
       G4ExceptionDescription ed;
       ed << "Unable to retrieve cross section table for Z=" << iZ << G4endl;
       G4Exception("G4PenelopeGammaConversionModel::ComputeCrossSectionPerAtom()",
           "em2018",FatalException,ed);
     }
 
  G4double cs = 0;
  G4double logene = std::log(energy);
  G4PhysicsFreeVector* theVec = logAtomicCrossSection->find(iZ)->second;
 
  G4double logXS = theVec->Value(logene);
  cs = std::exp(logXS);
 
  if (verboseLevel > 2)
    G4cout << "Gamma conversion cross section at " << energy/MeV << " MeV for Z=" << Z << 
      " = " << cs/barn << " barn" << G4endl;
  return cs;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void 
G4PenelopeGammaConversionModel::SampleSecondaries(std::vector<G4DynamicParticle*>* fvect,
                          const G4MaterialCutsCouple* couple,
                          const G4DynamicParticle* aDynamicGamma,
                          G4double,
                          G4double)
{
  //
  // Penelope model v2008.
  // Final state is sampled according to the Bethe-Heitler model with Coulomb
  // corrections, according to the semi-empirical model of
  //  J. Baro' et al., Radiat. Phys. Chem. 44 (1994) 531.
  //
  // The model uses the high energy Coulomb correction from 
  //  H. Davies et al., Phys. Rev. 93 (1954) 788
  // and atomic screening radii tabulated from 
  //  J.H. Hubbel et al., J. Phys. Chem. Ref. Data 9 (1980) 1023
  // for Z= 1 to 92. 
  //
  if (verboseLevel > 3)
    G4cout << "Calling SamplingSecondaries() of G4PenelopeGammaConversionModel" << G4endl;
 
  G4double photonEnergy = aDynamicGamma->GetKineticEnergy();
 
  // Always kill primary
  fParticleChange->ProposeTrackStatus(fStopAndKill);
  fParticleChange->SetProposedKineticEnergy(0.);
 
  if (photonEnergy <= fIntrinsicLowEnergyLimit)
    {
      fParticleChange->ProposeLocalEnergyDeposit(photonEnergy);
      return ;
    }
 
  G4ParticleMomentum photonDirection = aDynamicGamma->GetMomentumDirection();
  const G4Material* mat = couple->GetMaterial();
 
  //check if material data are available
  if (!fEffectiveCharge->count(mat))
    InitializeScreeningFunctions(mat); 
  if (!fEffectiveCharge->count(mat))
    {
      G4ExceptionDescription ed;      
      ed << "Unable to allocate the EffectiveCharge data for " << 
    mat->GetName() << G4endl;
      G4Exception("G4PenelopeGammaConversion::SampleSecondaries()",
          "em2019",FatalException,ed);        
    }
 
  // eps is the fraction of the photon energy assigned to e- (including rest mass)
  G4double eps = 0;
  G4double eki = electron_mass_c2/photonEnergy;
 
  //Do it fast for photon energy < 1.1 MeV (close to threshold)
  if (photonEnergy < fSmallEnergy)
    eps = eki + (1.0-2.0*eki)*G4UniformRand();
  else
    {
      //Complete calculation
      G4double effC = fEffectiveCharge->find(mat)->second;
      G4double alz = effC*fine_structure_const;
      G4double T = std::sqrt(2.0*eki);
      G4double F00=(-1.774-1.210e1*alz+1.118e1*alz*alz)*T
         +(8.523+7.326e1*alz-4.441e1*alz*alz)*T*T
         -(1.352e1+1.211e2*alz-9.641e1*alz*alz)*T*T*T
    +(8.946+6.205e1*alz-6.341e1*alz*alz)*T*T*T*T;
     
      G4double F0b = fScreeningFunction->find(mat)->second.second;
      G4double g0 = F0b + F00;
      G4double invRad = fMaterialInvScreeningRadius->find(mat)->second;
      G4double bmin = 4.0*eki/invRad;
      std::pair<G4double,G4double> scree =  GetScreeningFunctions(bmin);
      G4double g1 = scree.first;
      G4double g2 = scree.second;
      G4double g1min = g1+g0;
      G4double g2min = g2+g0;
      G4double xr = 0.5-eki;
      G4double a1 = 2.*g1min*xr*xr/3.;      
      G4double p1 = a1/(a1+g2min);
 
      G4bool loopAgain = false;
      //Random sampling of eps
      do{
    loopAgain = false;
    if (G4UniformRand() <= p1)
      {
        G4double  ru2m1 = 2.0*G4UniformRand()-1.0;
        if (ru2m1 < 0)
          eps = 0.5-xr*std::pow(std::abs(ru2m1),1./3.);
        else
          eps = 0.5+xr*std::pow(ru2m1,1./3.);
        G4double B = eki/(invRad*eps*(1.0-eps));
        scree =  GetScreeningFunctions(B);
        g1 = scree.first;
        g1 = std::max(g1+g0,0.);
        if (G4UniformRand()*g1min > g1) 
          loopAgain = true;
      }
    else
      {
        eps = eki+2.0*xr*G4UniformRand();
        G4double B = eki/(invRad*eps*(1.0-eps));
        scree =  GetScreeningFunctions(B);
        g2 = scree.second;
        g2 = std::max(g2+g0,0.);
        if (G4UniformRand()*g2min > g2)
          loopAgain = true;
      }      
      }while(loopAgain);
      
    }
  if (verboseLevel > 4)
    G4cout << "Sampled eps = " << eps << G4endl;
 
  G4double electronTotEnergy = eps*photonEnergy;
  G4double positronTotEnergy = (1.0-eps)*photonEnergy;
  
  // Scattered electron (positron) angles. ( Z - axis along the parent photon)
 
  //electron kinematics
  G4double electronKineEnergy = std::max(0.,electronTotEnergy - electron_mass_c2) ; 
  G4double costheta_el = G4UniformRand()*2.0-1.0;
  G4double kk = std::sqrt(electronKineEnergy*(electronKineEnergy+2.*electron_mass_c2));
  costheta_el = (costheta_el*electronTotEnergy+kk)/(electronTotEnergy+costheta_el*kk);
  G4double phi_el  = twopi * G4UniformRand() ;
  G4double dirX_el = std::sqrt(1.-costheta_el*costheta_el) * std::cos(phi_el);
  G4double dirY_el = std::sqrt(1.-costheta_el*costheta_el) * std::sin(phi_el);
  G4double dirZ_el = costheta_el;
 
  //positron kinematics
  G4double positronKineEnergy = std::max(0.,positronTotEnergy - electron_mass_c2) ;
  G4double costheta_po = G4UniformRand()*2.0-1.0;
  kk = std::sqrt(positronKineEnergy*(positronKineEnergy+2.*electron_mass_c2));
  costheta_po = (costheta_po*positronTotEnergy+kk)/(positronTotEnergy+costheta_po*kk);
  G4double phi_po  = twopi * G4UniformRand() ;
  G4double dirX_po = std::sqrt(1.-costheta_po*costheta_po) * std::cos(phi_po);
  G4double dirY_po = std::sqrt(1.-costheta_po*costheta_po) * std::sin(phi_po);
  G4double dirZ_po = costheta_po;
 
  // Kinematics of the created pair:
  // the electron and positron are assumed to have a symetric angular
  // distribution with respect to the Z axis along the parent photon
  G4double localEnergyDeposit = 0. ;
 
  if (electronKineEnergy > 0.0)
    {
      G4ThreeVector electronDirection ( dirX_el, dirY_el, dirZ_el);
      electronDirection.rotateUz(photonDirection);
      G4DynamicParticle* electron = new G4DynamicParticle (G4Electron::Electron(),
                               electronDirection,
                               electronKineEnergy);
      fvect->push_back(electron);
    }
  else
    {
      localEnergyDeposit += electronKineEnergy;
      electronKineEnergy = 0;
    }
 
  //Generate the positron. Real particle in any case, because it will annihilate. If below
  //threshold, produce it at rest
  if (positronKineEnergy < 0.0)
    {
      localEnergyDeposit += positronKineEnergy;
      positronKineEnergy = 0; //produce it at rest
    }
  G4ThreeVector positronDirection(dirX_po,dirY_po,dirZ_po);
  positronDirection.rotateUz(photonDirection);
  G4DynamicParticle* positron = new G4DynamicParticle(G4Positron::Positron(),
                              positronDirection, positronKineEnergy);
  fvect->push_back(positron);
 
  //Add rest of energy to the local energy deposit
  fParticleChange->ProposeLocalEnergyDeposit(localEnergyDeposit);
  
  if (verboseLevel > 1)
    {
      G4cout << "-----------------------------------------------------------" << G4endl;
      G4cout << "Energy balance from G4PenelopeGammaConversion" << G4endl;
      G4cout << "Incoming photon energy: " << photonEnergy/keV << " keV" << G4endl;
      G4cout << "-----------------------------------------------------------" << G4endl;
      if (electronKineEnergy)
    G4cout << "Electron (explicitely produced) " << electronKineEnergy/keV << " keV" 
           << G4endl;
      if (positronKineEnergy)
    G4cout << "Positron (not at rest) " << positronKineEnergy/keV << " keV" << G4endl;
      G4cout << "Rest masses of e+/- " << 2.0*electron_mass_c2/keV << " keV" << G4endl;
      if (localEnergyDeposit)
    G4cout << "Local energy deposit " << localEnergyDeposit/keV << " keV" << G4endl;
      G4cout << "Total final state: " << (electronKineEnergy+positronKineEnergy+
                      localEnergyDeposit+2.0*electron_mass_c2)/keV <<
        " keV" << G4endl;
      G4cout << "-----------------------------------------------------------" << G4endl;
    }
 if (verboseLevel > 0)
    {
      G4double energyDiff = std::fabs(electronKineEnergy+positronKineEnergy+
                      localEnergyDeposit+2.0*electron_mass_c2-photonEnergy);
      if (energyDiff > 0.05*keV)
    G4cout << "Warning from G4PenelopeGammaConversion: problem with energy conservation: " 
           << (electronKineEnergy+positronKineEnergy+
           localEnergyDeposit+2.0*electron_mass_c2)/keV 
           << " keV (final) vs. " << photonEnergy/keV << " keV (initial)" << G4endl;
    } 
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4PenelopeGammaConversionModel::ReadDataFile(const G4int Z)
{
 if (verboseLevel > 2)
    {
      G4cout << "G4PenelopeGammaConversionModel::ReadDataFile()" << G4endl;
      G4cout << "Going to read Gamma Conversion data files for Z=" << Z << G4endl;
    }
 
  char* path = getenv("G4LEDATA");
  if (!path)
    {
      G4String excep = 
    "G4PenelopeGammaConversionModel - G4LEDATA environment variable not set!";
      G4Exception("G4PenelopeGammaConversionModel::ReadDataFile()",
          "em0006",FatalException,excep);
      return;
    }
 
  /*
    Read the cross section file
  */
  std::ostringstream ost;
  if (Z>9)
    ost << path << "/penelope/pairproduction/pdgpp" << Z << ".p08";
  else
    ost << path << "/penelope/pairproduction/pdgpp0" << Z << ".p08";
  std::ifstream file(ost.str().c_str());
  if (!file.is_open())
    {
      G4String excep = "G4PenelopeGammaConversionModel - data file " + 
    G4String(ost.str()) + " not found!";
      G4Exception("G4PenelopeGammaConversionModel::ReadDataFile()",
          "em0003",FatalException,excep);
    }
 
  //I have to know in advance how many points are in the data list
  //to initialize the G4PhysicsFreeVector()
  size_t ndata=0;
  G4String line;
  while( getline(file, line) )
    ndata++;
  ndata -= 1; //remove one header line
  //G4cout << "Found: " << ndata << " lines" << G4endl;
 
  file.clear();
  file.close();
  file.open(ost.str().c_str());
  G4int readZ =0;
  file >> readZ; 
 
  if (verboseLevel > 3)
    G4cout << "Element Z=" << Z << G4endl;
 
  //check the right file is opened.
  if (readZ != Z)
    {
      G4ExceptionDescription ed;
      ed << "Corrupted data file for Z=" << Z << G4endl;      
      G4Exception("G4PenelopeGammaConversionModel::ReadDataFile()",
          "em0005",FatalException,ed);
    }
 
  G4PhysicsFreeVector* theVec = new G4PhysicsFreeVector(ndata);
  G4double ene=0,xs=0;
  for (size_t i=0;i<ndata;i++)
    {
      file >> ene >> xs;
      //dimensional quantities
      ene *= eV;
      xs *= barn;
      if (xs < 1e-40*cm2) //protection against log(0)
    xs = 1e-40*cm2;
      theVec->PutValue(i,std::log(ene),std::log(xs));      
    }
  file.close();
 
  if (!logAtomicCrossSection)
    {
      G4ExceptionDescription ed;
      ed << "Problem with allocation of logAtomicCrossSection data table " << G4endl;
      G4Exception("G4PenelopeGammaConversionModel::ReadDataFile()",
          "em2020",FatalException,ed);
      delete theVec;
      return;
    }
  logAtomicCrossSection->insert(std::make_pair(Z,theVec));
 
  return;
 
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4PenelopeGammaConversionModel::InitializeScreeningRadii()
{
  G4double temp[99] = {1.2281e+02,7.3167e+01,6.9228e+01,6.7301e+01,6.4696e+01,
               6.1228e+01,5.7524e+01,5.4033e+01,5.0787e+01,4.7851e+01,4.6373e+01,
               4.5401e+01,4.4503e+01,4.3815e+01,4.3074e+01,4.2321e+01,4.1586e+01,
               4.0953e+01,4.0524e+01,4.0256e+01,3.9756e+01,3.9144e+01,3.8462e+01,
               3.7778e+01,3.7174e+01,3.6663e+01,3.5986e+01,3.5317e+01,3.4688e+01,
               3.4197e+01,3.3786e+01,3.3422e+01,3.3068e+01,3.2740e+01,3.2438e+01,
               3.2143e+01,3.1884e+01,3.1622e+01,3.1438e+01,3.1142e+01,3.0950e+01,
               3.0758e+01,3.0561e+01,3.0285e+01,3.0097e+01,2.9832e+01,2.9581e+01,
               2.9411e+01,2.9247e+01,2.9085e+01,2.8930e+01,2.8721e+01,2.8580e+01,
               2.8442e+01,2.8312e+01,2.8139e+01,2.7973e+01,2.7819e+01,2.7675e+01,
               2.7496e+01,2.7285e+01,2.7093e+01,2.6911e+01,2.6705e+01,2.6516e+01,
               2.6304e+01,2.6108e+01,2.5929e+01,2.5730e+01,2.5577e+01,2.5403e+01,
               2.5245e+01,2.5100e+01,2.4941e+01,2.4790e+01,2.4655e+01,2.4506e+01,
               2.4391e+01,2.4262e+01,2.4145e+01,2.4039e+01,2.3922e+01,2.3813e+01,
               2.3712e+01,2.3621e+01,2.3523e+01,2.3430e+01,2.3331e+01,2.3238e+01,
               2.3139e+01,2.3048e+01,2.2967e+01,2.2833e+01,2.2694e+01,2.2624e+01,
               2.2545e+01,2.2446e+01,2.2358e+01,2.2264e+01};
 
  //copy temporary vector in class data member
  for (G4int i=0;i<99;i++)
    fAtomicScreeningRadius[i] = temp[i];
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
void G4PenelopeGammaConversionModel::InitializeScreeningFunctions(const G4Material* material)
{
  // This is subroutine GPPa0 of Penelope
  //
  // 1) calculate the effective Z for the purpose
  //
  G4double zeff = 0;
  G4int intZ = 0;
  G4int nElements = material->GetNumberOfElements();
  const G4ElementVector* elementVector = material->GetElementVector();
 
  //avoid calculations if only one building element!
  if (nElements == 1)
    {
      zeff = (*elementVector)[0]->GetZ();
      intZ = (G4int) zeff;
    }
  else // many elements...let's do the calculation
    {
      const G4double* fractionVector = material->GetVecNbOfAtomsPerVolume();
      
      G4double atot = 0;
      for (G4int i=0;i<nElements;i++)
    {
      G4double Zelement = (*elementVector)[i]->GetZ();
      G4double Aelement = (*elementVector)[i]->GetAtomicMassAmu();
      atot += Aelement*fractionVector[i];
      zeff += Zelement*Aelement*fractionVector[i]; //average with the number of nuclei
    }
      atot /= material->GetTotNbOfAtomsPerVolume();
      zeff /= (material->GetTotNbOfAtomsPerVolume()*atot);
      
      intZ = (G4int) (zeff+0.25);
      if (intZ <= 0)
    intZ = 1;
      if (intZ > 99)
    intZ = 99;
    }
 
  if (fEffectiveCharge)
    fEffectiveCharge->insert(std::make_pair(material,zeff));
 
  //
  // 2) Calculate Coulomb Correction
  //
  G4double alz = fine_structure_const*zeff;
  G4double alzSquared = alz*alz;
  G4double fc =  alzSquared*(0.202059-alzSquared*
                 (0.03693-alzSquared*
                  (0.00835-alzSquared*(0.00201-alzSquared*
                           (0.00049-alzSquared*
                            (0.00012-alzSquared*0.00003)))))
                 +1.0/(alzSquared+1.0));
  //
  // 3) Screening functions and low-energy corrections
  //
  G4double matRadius = 2.0/ fAtomicScreeningRadius[intZ-1];
  if (fMaterialInvScreeningRadius)
    fMaterialInvScreeningRadius->insert(std::make_pair(material,matRadius));
 
  std::pair<G4double,G4double> myPair(0,0);
  G4double f0a = 4.0*std::log(fAtomicScreeningRadius[intZ-1]);
  G4double f0b = f0a - 4.0*fc;
  myPair.first = f0a;
  myPair.second = f0b;
 
  if (fScreeningFunction)
    fScreeningFunction->insert(std::make_pair(material,myPair));
 
  if (verboseLevel > 2)
    {
      G4cout << "Average Z for material " << material->GetName() << " = " << 
    zeff << G4endl;
      G4cout << "Effective radius for material " << material->GetName() << " = " << 
    fAtomicScreeningRadius[intZ-1] << " m_e*c/hbar --> BCB = " << 
    matRadius << G4endl;
      G4cout << "Screening parameters F0 for material " << material->GetName() << " = " << 
    f0a << "," << f0b << G4endl;
    }
  return;
}
 
//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
 
std::pair<G4double,G4double> 
G4PenelopeGammaConversionModel::GetScreeningFunctions(G4double B)
{
  // This is subroutine SCHIFF of Penelope
  //
  // Screening Functions F1(B) and F2(B) in the Bethe-Heitler differential cross 
  // section for pair production
  //
  std::pair<G4double,G4double> result(0.,0.);
  G4double BSquared = B*B;
  G4double f1 = 2.0-2.0*std::log(1.0+BSquared);
  G4double f2 = f1 - 6.66666666e-1; // (-2/3)
  if (B < 1.0e-10)
    f1 = f1-twopi*B;
  else
    {
      G4double a0 = 4.0*B*std::atan(1./B);
      f1 = f1 - a0;
      f2 += 2.0*BSquared*(4.0-a0-3.0*std::log((1.0+BSquared)/BSquared));
    }
  G4double g1 = 0.5*(3.0*f1-f2);
  G4double g2 = 0.25*(3.0*f1+f2);
 
  result.first = g1;
  result.second = g2;
 
  return result;
}