G4PenelopeRayleighModel.cc

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// $Id$
//
// Author: Luciano Pandola
//
// History:
// --------
// 03 Dec 2009   L Pandola    First implementation
// 25 May 2011   L.Pandola    Renamed (make v2008 as default Penelope)
//

#include "G4PenelopeRayleighModel.hh"
#include "G4PhysicalConstants.hh"
#include "G4SystemOfUnits.hh"
#include "G4PenelopeSamplingData.hh"
#include "G4ParticleDefinition.hh"
#include "G4MaterialCutsCouple.hh"
#include "G4ProductionCutsTable.hh"
#include "G4DynamicParticle.hh"
#include "G4PhysicsTable.hh"
#include "G4ElementTable.hh"
#include "G4Element.hh"
#include "G4PhysicsFreeVector.hh"


//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....


G4PenelopeRayleighModel::G4PenelopeRayleighModel(const G4ParticleDefinition*,
                                                 const G4String& nam)
  :G4VEmModel(nam),fParticleChange(0),isInitialised(false),logAtomicCrossSection(0),   
   atomicFormFactor(0),logFormFactorTable(0),pMaxTable(0),samplingTable(0)
{
  fIntrinsicLowEnergyLimit = 100.0*eV;
  fIntrinsicHighEnergyLimit = 100.0*GeV;
  //  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 

  //build the energy grid. It is the same for all materials
  G4double logenergy = std::log(fIntrinsicLowEnergyLimit/2.);
  G4double logmaxenergy = std::log(1.5*fIntrinsicHighEnergyLimit);
  //finer grid below 160 keV
  G4double logtransitionenergy = std::log(160*keV); 
  G4double logfactor1 = std::log(10.)/250.;
  G4double logfactor2 = logfactor1*10;
  logEnergyGridPMax.push_back(logenergy);
  do{
    if (logenergy < logtransitionenergy)
      logenergy += logfactor1;
    else
      logenergy += logfactor2;
    logEnergyGridPMax.push_back(logenergy);      
  }while (logenergy < logmaxenergy);
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

G4PenelopeRayleighModel::~G4PenelopeRayleighModel()
{
  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 (atomicFormFactor)
     {
       for (i=atomicFormFactor->begin();i != atomicFormFactor->end();i++)
         if (i->second) delete i->second;
       delete atomicFormFactor;
     }

  ClearTables();
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
void G4PenelopeRayleighModel::ClearTables()
{
   std::map <const G4Material*,G4PhysicsFreeVector*>::iterator i;
 
   if (logFormFactorTable)
     {
       for (i=logFormFactorTable->begin(); i != logFormFactorTable->end(); i++)
         if (i->second) delete i->second;
       delete logFormFactorTable;
       logFormFactorTable = 0; //zero explicitely
     }

   if (pMaxTable)
     {
       for (i=pMaxTable->begin(); i != pMaxTable->end(); i++)
         if (i->second) delete i->second;
       delete pMaxTable;
       pMaxTable = 0; //zero explicitely
     }

   std::map<const G4Material*,G4PenelopeSamplingData*>::iterator ii;
   if (samplingTable)
     {
       for (ii=samplingTable->begin(); ii != samplingTable->end(); ii++)
         if (ii->second) delete ii->second;
       delete samplingTable;
       samplingTable = 0; //zero explicitely
     }     

   return;
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::Initialise(const G4ParticleDefinition* ,
                                         const G4DataVector& )
{
  if (verboseLevel > 3)
    G4cout << "Calling G4PenelopeRayleighModel::Initialise()" << G4endl;

  //clear tables depending on materials, not the atomic ones
  ClearTables();
  
  //create new tables
  //
  // logAtomicCrossSection and atomicFormFactor are created only once,
  // since they are never cleared
  if (!logAtomicCrossSection)
    logAtomicCrossSection = new std::map<G4int,G4PhysicsFreeVector*>;
  if (!atomicFormFactor)
    atomicFormFactor = new std::map<G4int,G4PhysicsFreeVector*>;

  if (!logFormFactorTable)
    logFormFactorTable = new std::map<const G4Material*,G4PhysicsFreeVector*>;
  if (!pMaxTable)
    pMaxTable = new std::map<const G4Material*,G4PhysicsFreeVector*>;
  if (!samplingTable)
    samplingTable = new std::map<const G4Material*,G4PenelopeSamplingData*>;


  if (verboseLevel > 0) {
    G4cout << "Penelope Rayleigh model v2008 is initialized " << G4endl
           << "Energy range: "
           << LowEnergyLimit() / keV << " keV - "
           << HighEnergyLimit() / GeV << " GeV"
           << G4endl;
  }

  if(isInitialised) return;
  fParticleChange = GetParticleChangeForGamma();
  isInitialised = true;
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

G4double G4PenelopeRayleighModel::ComputeCrossSectionPerAtom(const G4ParticleDefinition*,
                                                             G4double energy,
                                                             G4double Z,
                                                             G4double,
                                                             G4double,
                                                             G4double)
{
  // Cross section of Rayleigh scattering in Penelope v2008 is calculated by the EPDL97 
  // tabulation, Cuellen et al. (1997), with non-relativistic form factors from Hubbel 
  // et al. J. Phys. Chem. Ref. Data 4 (1975) 471; Erratum ibid. 6 (1977) 615.

   if (verboseLevel > 3)
    G4cout << "Calling CrossSectionPerAtom() of G4PenelopeRayleighModel" << G4endl;
 
   G4int iZ = (G4int) Z;

   //read data files
   if (!logAtomicCrossSection->count(iZ))
     ReadDataFile(iZ);
   //now it should be ok
   if (!logAtomicCrossSection->count(iZ))
     {
       G4Exception("G4PenelopeRayleighModel::ComputeCrossSectionPerAtom()",
                   "em2040",FatalException,"Unable to load the cross section table");
     }

   G4double cross = 0;

   G4PhysicsFreeVector* atom = logAtomicCrossSection->find(iZ)->second;
   if (!atom)
     {
       G4ExceptionDescription ed;
       ed << "Unable to find Z=" << iZ << " in the atomic cross section table" << G4endl;
       G4Exception("G4PenelopeRayleighModel::ComputeCrossSectionPerAtom()",
                   "em2041",FatalException,ed);
       return 0;
     }
   G4double logene = std::log(energy);
   G4double logXS = atom->Value(logene);
   cross = std::exp(logXS);

   if (verboseLevel > 2)
    G4cout << "Rayleigh cross section at " << energy/keV << " keV for Z=" << Z << 
      " = " << cross/barn << " barn" << G4endl;
    return cross;
}


//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....
void G4PenelopeRayleighModel::BuildFormFactorTable(const G4Material* material)
{
 
  /*
    1) get composition and equivalent molecular density
  */
  
  G4int nElements = material->GetNumberOfElements();
  const G4ElementVector* elementVector = material->GetElementVector();
  const G4double* fractionVector = material->GetFractionVector();

  std::vector<G4double> *StechiometricFactors = new std::vector<G4double>;
  for (G4int i=0;i<nElements;i++)
    {
      G4double fraction = fractionVector[i];
      G4double atomicWeigth = (*elementVector)[i]->GetA()/(g/mole);
      StechiometricFactors->push_back(fraction/atomicWeigth);
    }
  //Find max
  G4double MaxStechiometricFactor = 0.;
  for (G4int i=0;i<nElements;i++)
    {
      if ((*StechiometricFactors)[i] > MaxStechiometricFactor)
        MaxStechiometricFactor = (*StechiometricFactors)[i];
    }
  if (MaxStechiometricFactor<1e-16)
    {
      G4ExceptionDescription ed;
      ed << "Inconsistent data of atomic composition for " << 
        material->GetName() << G4endl;
      G4Exception("G4PenelopeRayleighModel::BuildFormFactorTable()",
                  "em2042",FatalException,ed);
    }
  //Normalize
  for (G4int i=0;i<nElements;i++)
    (*StechiometricFactors)[i] /=  MaxStechiometricFactor;
 
  // Equivalent atoms per molecule
  G4double atomsPerMolecule = 0;
  for (G4int i=0;i<nElements;i++)
    atomsPerMolecule += (*StechiometricFactors)[i]; 
 
  /*
    CREATE THE FORM FACTOR TABLE
  */
  G4PhysicsFreeVector* theFFVec = new G4PhysicsFreeVector(logQSquareGrid.size());
  theFFVec->SetSpline(true);

  for (size_t k=0;k<logQSquareGrid.size();k++)
    {
      G4double ff2 = 0; //squared form factor
      for (G4int i=0;i<nElements;i++)
        {
          G4int iZ = (G4int) (*elementVector)[i]->GetZ();
          G4PhysicsFreeVector* theAtomVec = atomicFormFactor->find(iZ)->second;
          G4double f = (*theAtomVec)[k]; //the q-grid is always the same            
          ff2 += f*f*(*StechiometricFactors)[i];
        }
      if (ff2)
        theFFVec->PutValue(k,logQSquareGrid[k],std::log(ff2)); //NOTICE: THIS IS log(Q^2) vs. log(F^2)
    }
  logFormFactorTable->insert(std::make_pair(material,theFFVec));

  delete StechiometricFactors;
  
  return;
}


//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::SampleSecondaries(std::vector<G4DynamicParticle*>* ,
                                                const G4MaterialCutsCouple* couple,
                                                const G4DynamicParticle* aDynamicGamma,
                                                G4double,
                                                G4double)
{
  // Sampling of the Rayleigh final state (namely, scattering angle of the photon) 
  // from the Penelope2008 model. The scattering angle is sampled from the atomic 
  // cross section dOmega/d(cosTheta) from Born ("Atomic Phyisics", 1969), disregarding 
  // anomalous scattering effects. The Form Factor F(Q) function which appears in the 
  // analytical cross section is retrieved via the method GetFSquared(); atomic data 
  // are tabulated for F(Q). Form factor for compounds is calculated according to 
  // the additivity rule. The sampling from the F(Q) is made via a Rational Inverse 
  // Transform with Aliasing (RITA) algorithm; RITA parameters are calculated once 
  // for each material and managed by G4PenelopeSamplingData objects.
  // The sampling algorithm (rejection method) has efficiency 67% at low energy, and 
  // increases with energy. For E=100 keV the efficiency is 100% and 86% for 
  // hydrogen and uranium, respectively.

  if (verboseLevel > 3)
    G4cout << "Calling SamplingSecondaries() of G4PenelopeRayleighModel" << G4endl;

  G4double photonEnergy0 = aDynamicGamma->GetKineticEnergy();
 
  if (photonEnergy0 <= fIntrinsicLowEnergyLimit)
    {
      fParticleChange->ProposeTrackStatus(fStopAndKill);
      fParticleChange->SetProposedKineticEnergy(0.);
      fParticleChange->ProposeLocalEnergyDeposit(photonEnergy0);
      return ;
    }

  G4ParticleMomentum photonDirection0 = aDynamicGamma->GetMomentumDirection();
  
  const G4Material* theMat = couple->GetMaterial();
  
  //1) Verify if tables are ready
  if (!pMaxTable || !samplingTable)
    {
      G4Exception("G4PenelopeRayleighModel::SampleSecondaries()",
                  "em2043",FatalException,"Invalid model initialization");    
      return;
    }
  
  //2) retrieve or build the sampling table
  if (!(samplingTable->count(theMat)))
    InitializeSamplingAlgorithm(theMat);
  G4PenelopeSamplingData* theDataTable = samplingTable->find(theMat)->second;
  
  //3) retrieve or build the pMax data
  if (!pMaxTable->count(theMat))
    GetPMaxTable(theMat);
  G4PhysicsFreeVector* thePMax = pMaxTable->find(theMat)->second;

  G4double cosTheta = 1.0;
  
  //OK, ready to go!
  G4double qmax = 2.0*photonEnergy0/electron_mass_c2; //this is non-dimensional now

  if (qmax < 1e-10) //very low momentum transfer
    {
      G4bool loopAgain=false;
      do
        {
          loopAgain = false;
          cosTheta = 1.0-2.0*G4UniformRand();
          G4double G = 0.5*(1+cosTheta*cosTheta);
          if (G4UniformRand()>G)
            loopAgain = true;
        }while(loopAgain);
    }
  else //larger momentum transfer
    {
      size_t nData = theDataTable->GetNumberOfStoredPoints();
      G4double LastQ2inTheTable = theDataTable->GetX(nData-1);
      G4double q2max = std::min(qmax*qmax,LastQ2inTheTable);

      G4bool loopAgain = false;
      G4double MaxPValue = thePMax->Value(photonEnergy0);
      G4double xx=0;
      
      //Sampling by rejection method. The rejection function is 
      //G = 0.5*(1+cos^2(theta))
      //
      do{
        loopAgain = false;
        G4double RandomMax = G4UniformRand()*MaxPValue;
        xx = theDataTable->SampleValue(RandomMax);
        //xx is a random value of q^2 in (0,q2max),sampled according to 
        //F(Q^2) via the RITA algorithm
        if (xx > q2max)
          loopAgain = true;
        cosTheta = 1.0-2.0*xx/q2max;
        G4double G = 0.5*(1+cosTheta*cosTheta);
        if (G4UniformRand()>G)
          loopAgain = true;
      }while(loopAgain);
    }
  
  G4double sinTheta = std::sqrt(1-cosTheta*cosTheta);
 
  // Scattered photon angles. ( Z - axis along the parent photon)
  G4double phi = twopi * G4UniformRand() ;
  G4double dirX = sinTheta*std::cos(phi);
  G4double dirY = sinTheta*std::sin(phi);
  G4double dirZ = cosTheta;
  
  // Update G4VParticleChange for the scattered photon 
  G4ThreeVector photonDirection1(dirX, dirY, dirZ);

  photonDirection1.rotateUz(photonDirection0);
  fParticleChange->ProposeMomentumDirection(photonDirection1) ;
  fParticleChange->SetProposedKineticEnergy(photonEnergy0) ;
 
  return;
}


//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::ReadDataFile(const G4int Z)
{
  if (verboseLevel > 2)
    {
      G4cout << "G4PenelopeRayleighModel::ReadDataFile()" << G4endl;
      G4cout << "Going to read Rayleigh data files for Z=" << Z << G4endl;
    }

  char* path = getenv("G4LEDATA");
  if (!path)
    {
      G4String excep = "G4LEDATA environment variable not set!";
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em0006",FatalException,excep);
      return;
    }

  /*
    Read first the cross section file
  */
  std::ostringstream ost;
  if (Z>9)
    ost << path << "/penelope/rayleigh/pdgra" << Z << ".p08";
  else
    ost << path << "/penelope/rayleigh/pdgra0" << Z << ".p08";
  std::ifstream file(ost.str().c_str());
  if (!file.is_open())
    {
      G4String excep = "Data file " + G4String(ost.str()) + " not found!";
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em0003",FatalException,excep);
    }
  G4int readZ =0;
  size_t nPoints= 0;
  file >> readZ >> nPoints;
  //check the right file is opened.
  if (readZ != Z || nPoints <= 0 || nPoints >= 5000)
    {
      G4ExceptionDescription ed;
      ed << "Corrupted data file for Z=" << Z << G4endl;
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em0005",FatalException,ed);
      return;
    }  
  G4PhysicsFreeVector* theVec = new G4PhysicsFreeVector((size_t)nPoints);
  G4double ene=0,f1=0,f2=0,xs=0;
  for (size_t i=0;i<nPoints;i++)
    {
      file >> ene >> f1 >> f2 >> xs;
      //dimensional quantities
      ene *= eV;
      xs *= cm2;
      theVec->PutValue(i,std::log(ene),std::log(xs));
      if (file.eof() && i != (nPoints-1)) //file ended too early
        {
          G4ExceptionDescription ed ;     
          ed << "Corrupted data file for Z=" << Z << G4endl;
          ed << "Found less than " << nPoints << "entries " <<G4endl;
          G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                      "em0005",FatalException,ed);
        }
    }
  if (!logAtomicCrossSection)
    {
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em2044",FatalException,"Unable to allocate the atomic cross section table");
      delete theVec;
      return;
    }
  logAtomicCrossSection->insert(std::make_pair(Z,theVec));
  file.close();

  /*
    Then read the form factor file
  */
  std::ostringstream ost2;
  if (Z>9)
    ost2 << path << "/penelope/rayleigh/pdaff" << Z << ".p08";
  else
    ost2 << path << "/penelope/rayleigh/pdaff0" << Z << ".p08";
  file.open(ost2.str().c_str());
  if (!file.is_open())
    {
      G4String excep = "Data file " + G4String(ost2.str()) + " not found!";
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em0003",FatalException,excep);
    }
  file >> readZ >> nPoints;
  //check the right file is opened.
  if (readZ != Z || nPoints <= 0 || nPoints >= 5000)
    {
      G4ExceptionDescription ed;
      ed << "Corrupted data file for Z=" << Z << G4endl;
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em0005",FatalException,ed);
      return;
    }  
  G4PhysicsFreeVector* theFFVec = new G4PhysicsFreeVector((size_t)nPoints);
  G4double q=0,ff=0,incoh=0;
  G4bool fillQGrid = false;
  //fill this vector only the first time.
  if (!logQSquareGrid.size())
    fillQGrid = true;
  for (size_t i=0;i<nPoints;i++)
    {
      file >> q >> ff >> incoh;
      //q and ff are dimensionless (q is in units of (m_e*c)
      theFFVec->PutValue(i,q,ff);
      if (fillQGrid)
        {
          logQSquareGrid.push_back(2.0*std::log(q));
        }
      if (file.eof() && i != (nPoints-1)) //file ended too early
        {
          G4ExceptionDescription ed;      
          ed << "Corrupted data file for Z=" << Z << G4endl;
          ed << "Found less than " << nPoints << "entries " <<G4endl;
          G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                      "em0005",FatalException,ed);
        }
    }
  if (!atomicFormFactor)
    {
      G4Exception("G4PenelopeRayleighModel::ReadDataFile()",
                  "em2045",FatalException,
                  "Unable to allocate the atomicFormFactor data table");
      delete theFFVec;
      return;
    }
  atomicFormFactor->insert(std::make_pair(Z,theFFVec));
  file.close();
  return;
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

G4double G4PenelopeRayleighModel::GetFSquared(const G4Material* mat, const G4double QSquared)
{
  G4double f2 = 0;
  //Input value QSquared could be zero: protect the log() below against 
  //the FPE exception
  //If Q<1e-10, set Q to 1e-10
  G4double logQSquared = (QSquared>1e-10) ? std::log(QSquared) : -23.;
  //last value of the table
  G4double maxlogQ2 = logQSquareGrid[logQSquareGrid.size()-1];
  //If the table has not been built for the material, do it!
  if (!logFormFactorTable->count(mat))
    BuildFormFactorTable(mat);
  
  //now it should  be all right
  G4PhysicsFreeVector* theVec = logFormFactorTable->find(mat)->second;

  if (!theVec)
    {
      G4ExceptionDescription ed;
      ed << "Unable to retrieve F squared table for " << mat->GetName() << G4endl;
      G4Exception("G4PenelopeRayleighModel::GetFSquared()",
                  "em2046",FatalException,ed);
      return 0;
    }
  if (logQSquared < -20) // Q < 1e-9
    {
      G4double logf2 = (*theVec)[0]; //first value of the table
      f2 = std::exp(logf2);
    }
  else if (logQSquared > maxlogQ2)
    f2 =0;
  else
    {
      //log(Q^2) vs. log(F^2)
      G4double logf2 = theVec->Value(logQSquared);
      f2 = std::exp(logf2);

    }
  if (verboseLevel > 3)
    {
      G4cout << "G4PenelopeRayleighModel::GetFSquared() in " << mat->GetName() << G4endl;  
      G4cout << "Q^2 = " <<  QSquared << " (units of 1/(m_e*c); F^2 = " << f2 << G4endl;
    }
  return f2;
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::InitializeSamplingAlgorithm(const G4Material* mat)
{
  G4double q2min = 0;
  G4double q2max = 0;
  const size_t np = 150; //hard-coded in Penelope
  for (size_t i=1;i<logQSquareGrid.size();i++)
    {
      G4double Q2 = std::exp(logQSquareGrid[i]);
      if (GetFSquared(mat,Q2) >  1e-35)
        {
          q2max = std::exp(logQSquareGrid[i-1]);
        }
    }
  
  size_t nReducedPoints = np/4;

  //check for errors
  if (np < 16)
    {
      G4Exception("G4PenelopeRayleighModel::InitializeSamplingAlgorithm()",
                  "em2047",FatalException,
                  "Too few points to initialize the sampling algorithm");
    }
  if (q2min > (q2max-1e-10))
    {
      G4Exception("G4PenelopeRayleighModel::InitializeSamplingAlgorithm()",
                  "em2048",FatalException,
                  "Too narrow grid to initialize the sampling algorithm");
    }

  //This is subroutine RITAI0 of Penelope
  //Create an object of type G4PenelopeRayleighSamplingData --> store in a map::Material*

  //temporary vectors --> Then everything is stored in G4PenelopeSamplingData
  G4DataVector* x = new G4DataVector();
  
  /*******************************************************************************
    Start with a grid of NUNIF points uniformly spaced in the interval q2min,q2max
  ********************************************************************************/
  size_t NUNIF = std::min(std::max(((size_t)8),nReducedPoints),np/2);
  const G4int nip = 51; //hard-coded in Penelope

  G4double dx = (q2max-q2min)/((G4double) NUNIF-1);  
  x->push_back(q2min); 
  for (size_t i=1;i<NUNIF-1;i++)
    {
      G4double app = q2min + i*dx;
      x->push_back(app); //increase 
    }
  x->push_back(q2max);
  
  if (verboseLevel> 3)
    G4cout << "Vector x has " << x->size() << " points, while NUNIF = " << NUNIF << G4endl;

  //Allocate temporary storage vectors
  G4DataVector* area = new G4DataVector();
  G4DataVector* a = new G4DataVector();
  G4DataVector* b = new G4DataVector();
  G4DataVector* c = new G4DataVector();
  G4DataVector* err = new G4DataVector();

  for (size_t i=0;i<NUNIF-1;i++) //build all points but the last
    {      
      //Temporary vectors for this loop
      G4DataVector* pdfi = new G4DataVector();
      G4DataVector* pdfih = new G4DataVector();
      G4DataVector* sumi = new G4DataVector();

      G4double dxi = ((*x)[i+1]-(*x)[i])/(G4double (nip-1));
      G4double pdfmax = 0;
      for (G4int k=0;k<nip;k++)
        {
          G4double xik = (*x)[i]+k*dxi; 
          G4double pdfk = std::max(GetFSquared(mat,xik),0.);
          pdfi->push_back(pdfk);
          pdfmax = std::max(pdfmax,pdfk);       
          if (k < (nip-1))
            {
              G4double xih = xik + 0.5*dxi;
              G4double pdfIK = std::max(GetFSquared(mat,xih),0.);
              pdfih->push_back(pdfIK);
              pdfmax = std::max(pdfmax,pdfIK);
            }
        }
    
      //Simpson's integration
      G4double cons = dxi*0.5*(1./3.);
      sumi->push_back(0.);
      for (G4int k=1;k<nip;k++)
        {
          G4double previous = (*sumi)[k-1];
          G4double next = previous + cons*((*pdfi)[k-1]+4.0*(*pdfih)[k-1]+(*pdfi)[k]);
          sumi->push_back(next);
        }
    
      G4double lastIntegral = (*sumi)[sumi->size()-1];
      area->push_back(lastIntegral);
      //Normalize cumulative function
      G4double factor = 1.0/lastIntegral;
      for (size_t k=0;k<sumi->size();k++)
        (*sumi)[k] *= factor;
      
      //When the PDF vanishes at one of the interval end points, its value is modified
      if ((*pdfi)[0] < 1e-35) 
        (*pdfi)[0] = 1e-5*pdfmax;
      if ((*pdfi)[pdfi->size()-1] < 1e-35)
        (*pdfi)[pdfi->size()-1] = 1e-5*pdfmax;

      G4double pli = (*pdfi)[0]*factor;
      G4double pui = (*pdfi)[pdfi->size()-1]*factor;
      G4double B_temp = 1.0-1.0/(pli*pui*dx*dx);
      G4double A_temp = (1.0/(pli*dx))-1.0-B_temp;
      G4double C_temp = 1.0+A_temp+B_temp;
      if (C_temp < 1e-35)
        {
          a->push_back(0.);
          b->push_back(0.);
          c->push_back(1.);       
        }
      else
        {
          a->push_back(A_temp);
          b->push_back(B_temp);
          c->push_back(C_temp);
        }

      //OK, now get ERR(I), the integral of the absolute difference between the rational interpolation 
      //and the true pdf, extended over the interval (X(I),X(I+1))
      G4int icase = 1; //loop code
      G4bool reLoop = false;
      err->push_back(0.);
      do
        {
          reLoop = false;
          (*err)[i] = 0.; //zero variable
          for (G4int k=0;k<nip;k++)
            {
              G4double rr = (*sumi)[k];
              G4double pap = (*area)[i]*(1.0+((*a)[i]+(*b)[i]*rr)*rr)*(1.0+((*a)[i]+(*b)[i]*rr)*rr)/
                ((1.0-(*b)[i]*rr*rr)*(*c)[i]*((*x)[i+1]-(*x)[i]));
              if (k == 0 || k == nip-1)
                (*err)[i] += 0.5*std::fabs(pap-(*pdfi)[k]);
              else
                (*err)[i] += std::fabs(pap-(*pdfi)[k]);
            }
          (*err)[i] *= dxi;
      
          //If err(I) is too large, the pdf is approximated by a uniform distribution
          if ((*err)[i] > 0.1*(*area)[i] && icase == 1) 
            {
              (*b)[i] = 0;
              (*a)[i] = 0;
              (*c)[i] = 1.;
              icase = 2;
              reLoop = true;
            }
        }while(reLoop);

      delete pdfi;
      delete pdfih;
      delete sumi;
    } //end of first loop over i

  //Now assign last point
  (*x)[x->size()-1] = q2max;
  a->push_back(0.);
  b->push_back(0.);
  c->push_back(0.);
  err->push_back(0.);
  area->push_back(0.);

  if (x->size() != NUNIF || a->size() != NUNIF || 
      err->size() != NUNIF || area->size() != NUNIF)
    {
      G4ExceptionDescription ed;
      ed << "Problem in building the Table for Sampling: array dimensions do not match" << G4endl;
      G4Exception("G4PenelopeRayleighModel::InitializeSamplingAlgorithm()",
                  "em2049",FatalException,ed);
    }
  
  /*******************************************************************************
   New grid points are added by halving the sub-intervals with the largest absolute error
  This is done up to np=150 points in the grid
  ********************************************************************************/
  do
    {
      G4double maxError = 0.0;
      size_t iErrMax = 0;
      for (size_t i=0;i<err->size()-2;i++) 
        {
          //maxError is the lagest of the interval errors err[i]
          if ((*err)[i] > maxError)
            {
              maxError = (*err)[i];
              iErrMax = i;
            }
        }
      
      //OK, now I have to insert one new point in the position iErrMax
      G4double newx = 0.5*((*x)[iErrMax]+(*x)[iErrMax+1]);
      
      x->insert(x->begin()+iErrMax+1,newx);
      //Add place-holders in the other vectors
      area->insert(area->begin()+iErrMax+1,0.);
      a->insert(a->begin()+iErrMax+1,0.);
      b->insert(b->begin()+iErrMax+1,0.);
      c->insert(c->begin()+iErrMax+1,0.);
      err->insert(err->begin()+iErrMax+1,0.);
        
      //Now calculate the other parameters
      for (size_t i=iErrMax;i<=iErrMax+1;i++)
        {
          //Temporary vectors for this loop
          G4DataVector* pdfi = new G4DataVector();
          G4DataVector* pdfih = new G4DataVector();
          G4DataVector* sumi = new G4DataVector();
          
          G4double dxLocal = (*x)[i+1]-(*x)[i];
          G4double dxi = ((*x)[i+1]-(*x)[i])/(G4double (nip-1));
          G4double pdfmax = 0;
          for (G4int k=0;k<nip;k++)
            {
              G4double xik = (*x)[i]+k*dxi;
              G4double pdfk = std::max(GetFSquared(mat,xik),0.);
              pdfi->push_back(pdfk);
              pdfmax = std::max(pdfmax,pdfk);   
              if (k < (nip-1))
                {
                  G4double xih = xik + 0.5*dxi;
                  G4double pdfIK = std::max(GetFSquared(mat,xih),0.);
                  pdfih->push_back(pdfIK);
                  pdfmax = std::max(pdfmax,pdfIK);
                }
            }
          
          //Simpson's integration
          G4double cons = dxi*0.5*(1./3.);
          sumi->push_back(0.);
          for (G4int k=1;k<nip;k++)
            {
              G4double previous = (*sumi)[k-1];
              G4double next = previous + cons*((*pdfi)[k-1]+4.0*(*pdfih)[k-1]+(*pdfi)[k]);
              sumi->push_back(next);
            }
          G4double lastIntegral = (*sumi)[sumi->size()-1];
          (*area)[i] = lastIntegral;
          
          //Normalize cumulative function
          G4double factor = 1.0/lastIntegral;
          for (size_t k=0;k<sumi->size();k++)
            (*sumi)[k] *= factor;
          
          //When the PDF vanishes at one of the interval end points, its value is modified
          if ((*pdfi)[0] < 1e-35) 
            (*pdfi)[0] = 1e-5*pdfmax;
          if ((*pdfi)[pdfi->size()-1] < 1e-35)
            (*pdfi)[pdfi->size()-1] = 1e-5*pdfmax;
          
          G4double pli = (*pdfi)[0]*factor;
          G4double pui = (*pdfi)[pdfi->size()-1]*factor;
          G4double B_temp = 1.0-1.0/(pli*pui*dxLocal*dxLocal);
          G4double A_temp = (1.0/(pli*dxLocal))-1.0-B_temp;
          G4double C_temp = 1.0+A_temp+B_temp;
          if (C_temp < 1e-35)
            {
              (*a)[i]= 0.;
              (*b)[i] = 0.;
              (*c)[i] = 1;
            }
          else
            {
              (*a)[i]= A_temp;
              (*b)[i] = B_temp;
              (*c)[i] = C_temp;
            }
          //OK, now get ERR(I), the integral of the absolute difference between the rational interpolation 
          //and the true pdf, extended over the interval (X(I),X(I+1))
          G4int icase = 1; //loop code
          G4bool reLoop = false;
          do
            {
              reLoop = false;
              (*err)[i] = 0.; //zero variable
              for (G4int k=0;k<nip;k++)
                {
                  G4double rr = (*sumi)[k];       
                  G4double pap = (*area)[i]*(1.0+((*a)[i]+(*b)[i]*rr)*rr)*(1.0+((*a)[i]+(*b)[i]*rr)*rr)/
                    ((1.0-(*b)[i]*rr*rr)*(*c)[i]*((*x)[i+1]-(*x)[i]));
                  if (k == 0 || k == nip-1)
                    (*err)[i] += 0.5*std::fabs(pap-(*pdfi)[k]);
                  else
                    (*err)[i] += std::fabs(pap-(*pdfi)[k]);
                }
              (*err)[i] *= dxi;
              
              //If err(I) is too large, the pdf is approximated by a uniform distribution
              if ((*err)[i] > 0.1*(*area)[i] && icase == 1) 
                {
                  (*b)[i] = 0;
                  (*a)[i] = 0;
                  (*c)[i] = 1.;
                  icase = 2;
                  reLoop = true;
                }
            }while(reLoop);
          delete pdfi;
          delete pdfih;
          delete sumi;
        }
    }while(x->size() < np);

  if (x->size() != np || a->size() != np || 
      err->size() != np || area->size() != np)
    {
      G4Exception("G4PenelopeRayleighModel::InitializeSamplingAlgorithm()",
                  "em2050",FatalException,
                  "Problem in building the extended Table for Sampling: array dimensions do not match ");
    }

  /*******************************************************************************
   Renormalization
  ********************************************************************************/
  G4double ws = 0;
  for (size_t i=0;i<np-1;i++)
    ws += (*area)[i];
  ws = 1.0/ws;
  G4double errMax = 0;
  for (size_t i=0;i<np-1;i++)
    {
      (*area)[i] *= ws;
      (*err)[i] *= ws;
      errMax = std::max(errMax,(*err)[i]);
    }

  //Vector with the normalized cumulative distribution
  G4DataVector* PAC = new G4DataVector();
  PAC->push_back(0.);
  for (size_t i=0;i<np-1;i++)
    {
      G4double previous = (*PAC)[i];
      PAC->push_back(previous+(*area)[i]);
    }
  (*PAC)[PAC->size()-1] = 1.;
                     
  /*******************************************************************************
  Pre-calculated limits for the initial binary search for subsequent sampling
  ********************************************************************************/

  //G4DataVector* ITTL = new G4DataVector();
  std::vector<size_t> *ITTL = new std::vector<size_t>;
  std::vector<size_t> *ITTU = new std::vector<size_t>;

  //Just create place-holders
  for (size_t i=0;i<np;i++)
    {
      ITTL->push_back(0);
      ITTU->push_back(0);
    }

  G4double bin = 1.0/(np-1);
  (*ITTL)[0]=0;
  for (size_t i=1;i<(np-1);i++)
    {
      G4double ptst = i*bin; 
      G4bool found = false;
      for (size_t j=(*ITTL)[i-1];j<np && !found;j++)
        {
          if ((*PAC)[j] > ptst)
            {
              (*ITTL)[i] = j-1;
              (*ITTU)[i-1] = j;
              found = true;
            }
        }
    }
  (*ITTU)[ITTU->size()-2] = ITTU->size()-1;
  (*ITTU)[ITTU->size()-1] = ITTU->size()-1;
  (*ITTL)[ITTL->size()-1] = ITTU->size()-2;

  if (ITTU->size() != np || ITTU->size() != np)
    {
      G4Exception("G4PenelopeRayleighModel::InitializeSamplingAlgorithm()",
                  "em2051",FatalException,
                  "Problem in building the Limit Tables for Sampling: array dimensions do not match");
    }


  /********************************************************************************
    Copy tables
  ********************************************************************************/
  G4PenelopeSamplingData* theTable = new G4PenelopeSamplingData(np);
  for (size_t i=0;i<np;i++)
    {
      theTable->AddPoint((*x)[i],(*PAC)[i],(*a)[i],(*b)[i],(*ITTL)[i],(*ITTU)[i]);
    }

  if (verboseLevel > 2)
    {
      G4cout << "*************************************************************************" << 
        G4endl;
      G4cout << "Sampling table for Penelope Rayleigh scattering in " << mat->GetName() << G4endl;
      theTable->DumpTable();
    } 
  samplingTable->insert(std::make_pair(mat,theTable));

 
  //Clean up temporary vectors
  delete x;
  delete a;
  delete b;
  delete c;
  delete err;
  delete area;
  delete PAC;
  delete ITTL;
  delete ITTU;

  //DONE!
  return;
  
}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::GetPMaxTable(const G4Material* mat)
{
  if (!pMaxTable)
    {
      G4cout << "G4PenelopeRayleighModel::BuildPMaxTable" << G4endl;
      G4cout << "Going to instanziate the pMaxTable !" << G4endl;
      G4cout << "That should _not_ be here! " << G4endl;
      pMaxTable = new std::map<const G4Material*,G4PhysicsFreeVector*>;
    }
  //check if the table is already there
  if (pMaxTable->count(mat))
    return;

  //otherwise build it
  if (!samplingTable)
    {
      G4Exception("G4PenelopeRayleighModel::GetPMaxTable()",
                  "em2052",FatalException,
                  "SamplingTable is not properly instantiated");
      return;
    }

  if (!samplingTable->count(mat))
    InitializeSamplingAlgorithm(mat);      
  
  G4PenelopeSamplingData *theTable = samplingTable->find(mat)->second;
  size_t tablePoints = theTable->GetNumberOfStoredPoints();

  size_t nOfEnergyPoints = logEnergyGridPMax.size();
  G4PhysicsFreeVector* theVec = new G4PhysicsFreeVector(nOfEnergyPoints);

  const size_t nip = 51; //hard-coded in Penelope

  for (size_t ie=0;ie<logEnergyGridPMax.size();ie++)
    {
      G4double energy = std::exp(logEnergyGridPMax[ie]);
      G4double Qm = 2.0*energy/electron_mass_c2; //this is non-dimensional now
      G4double Qm2 = Qm*Qm;
      G4double firstQ2 = theTable->GetX(0);
      G4double lastQ2 = theTable->GetX(tablePoints-1);
      G4double thePMax = 0;
      
      if (Qm2 > firstQ2)
        {
          if (Qm2 < lastQ2)
            {
              //bisection to look for the index of Qm
              size_t lowerBound = 0;
              size_t upperBound = tablePoints-1;
              while (lowerBound <= upperBound)
                {
                  size_t midBin = (lowerBound + upperBound)/2;
                  if( Qm2 < theTable->GetX(midBin))
                    { upperBound = midBin-1; }
                  else
                    { lowerBound = midBin+1; }
                }
              //upperBound is the output (but also lowerBounf --> should be the same!)
              G4double Q1 = theTable->GetX(upperBound);
              G4double Q2 = Qm2;
              G4double DQ = (Q2-Q1)/((G4double)(nip-1));
              G4double theA = theTable->GetA(upperBound);
              G4double theB = theTable->GetB(upperBound);
              G4double thePAC = theTable->GetPAC(upperBound);
              G4DataVector* fun = new G4DataVector();
              for (size_t k=0;k<nip;k++)
                {
                  G4double qi = Q1 + k*DQ;
                  G4double tau = (qi-Q1)/
                    (theTable->GetX(upperBound+1)-Q1);
                  G4double con1 = 2.0*theB*tau; 
                  G4double ci = 1.0+theA+theB;
                  G4double con2 = ci-theA*tau;
                  G4double etap = 0;
                  if (std::fabs(con1) > 1.0e-16*std::fabs(con2))
                    etap = con2*(1.0-std::sqrt(1.0-2.0*tau*con1/(con2*con2)))/con1;
                  else
                    etap = tau/con2;
                  G4double theFun = (theTable->GetPAC(upperBound+1)-thePAC)*
                    (1.0+(theA+theB*etap)*etap)*(1.0+(theA+theB*etap)*etap)/
                    ((1.0-theB*etap*etap)*ci*(theTable->GetX(upperBound+1)-Q1));
                  fun->push_back(theFun);
                }
              //Now intergrate numerically the fun Cavalieri-Simpson's method
              G4DataVector* sum = new G4DataVector;
              G4double CONS = DQ*(1./12.);
              G4double HCONS = 0.5*CONS;
              sum->push_back(0.);
              G4double secondPoint = (*sum)[0] + 
                (5.0*(*fun)[0]+8.0*(*fun)[1]-(*fun)[2])*CONS;
              sum->push_back(secondPoint);
              for (size_t hh=2;hh<nip-1;hh++)
                {
                  G4double previous = (*sum)[hh-1];
                  G4double next = previous+(13.0*((*fun)[hh-1]+(*fun)[hh])-
                                            (*fun)[hh+1]-(*fun)[hh-2])*HCONS;
                  sum->push_back(next);
                }
              G4double last = (*sum)[nip-2]+(5.0*(*fun)[nip-1]+8.0*(*fun)[nip-2]-
                                             (*fun)[nip-3])*CONS;
              sum->push_back(last);      
              thePMax = thePAC + (*sum)[sum->size()-1]; //last point
              delete fun;
              delete sum;
            }
          else
            {
              thePMax = 1.0;
            }    
        }
      else
        {
          thePMax = theTable->GetPAC(0);
        }

      //Write number in the table
      theVec->PutValue(ie,energy,thePMax);
  }
  
  pMaxTable->insert(std::make_pair(mat,theVec));
  return;

}

//....oooOO0OOooo........oooOO0OOooo........oooOO0OOooo........oooOO0OOooo....

void G4PenelopeRayleighModel::DumpFormFactorTable(const G4Material* mat)
{
  G4cout << "*****************************************************************" << G4endl;
  G4cout << "G4PenelopeRayleighModel: Form Factor Table for " << mat->GetName() << G4endl;
  //try to use the same format as Penelope-Fortran, namely Q (/m_e*c) and F
  G4cout <<  "Q/(m_e*c)                 F(Q)     " << G4endl;
  G4cout << "*****************************************************************" << G4endl;
  if (!logFormFactorTable->count(mat))
    BuildFormFactorTable(mat);
  
  G4PhysicsFreeVector* theVec = logFormFactorTable->find(mat)->second;
  for (size_t i=0;i<theVec->GetVectorLength();i++)
    {
      G4double logQ2 = theVec->GetLowEdgeEnergy(i);
      G4double Q = std::exp(0.5*logQ2);
      G4double logF2 = (*theVec)[i];
      G4double F = std::exp(0.5*logF2);
      G4cout << Q << "              " << F << G4endl;
    }
  //DONE
  return;
}

---