G4PAIySection.cc

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// $Id: G4PAIySection.cc 96934 2016-05-18 09:10:41Z gcosmo $
//
// 
// G4PAIySection.cc -- class implementation file
//
// GEANT 4 class implementation file
//
// For information related to this code, please, contact
// the Geant4 Collaboration.
//
// R&D: Vladimir.Grichine@cern.ch
//
// History:
//
// 01.10.07, V.Ivanchenko create using V.Grichine G4PAIxSection class
// 26.07.09, V.Ivanchenko added protection for mumerical exceptions for 
//              low-density materials
// 21.11.10 V. Grichine bug fixed in Initialise for reading sandia table from
//            material. Warning: the table is tuned for photo-effect not PAI model.
// 23.06.13 V.Grichine arrays->G4DataVectors
//

#include "G4PAIySection.hh"

#include "globals.hh"
#include "G4PhysicalConstants.hh"
#include "G4SystemOfUnits.hh"
#include "G4ios.hh"
#include "G4Poisson.hh"
#include "G4Material.hh"
#include "G4MaterialCutsCouple.hh"
#include "G4SandiaTable.hh"
#include "G4Exp.hh"
#include "G4Log.hh"

using namespace std;

// Local class constants

const G4double G4PAIySection::fDelta = 0.005; // energy shift from interval border
const G4double G4PAIySection::fError = 0.005; // error in lin-log approximation

const G4int G4PAIySection::fMaxSplineSize = 500;  // Max size of output spline
                                                  // arrays

//
// Constructor
//

G4PAIySection::G4PAIySection()
{
  fSandia = 0;
  fDensity = fElectronDensity = fNormalizationCof = fLowEnergyCof = 0.0;
  fIntervalNumber = fSplineNumber = 0;
  fVerbose = 0;

  betaBohr = fine_structure_const;
  G4double cofBetaBohr = 4.0;
  G4double betaBohr2   = fine_structure_const*fine_structure_const;
  betaBohr4   = betaBohr2*betaBohr2*cofBetaBohr;
  
  fSplineEnergy          = G4DataVector(fMaxSplineSize,0.0);
  fRePartDielectricConst = G4DataVector(fMaxSplineSize,0.0);
  fImPartDielectricConst = G4DataVector(fMaxSplineSize,0.0);
  fIntegralTerm          = G4DataVector(fMaxSplineSize,0.0);
  fDifPAIySection        = G4DataVector(fMaxSplineSize,0.0);
  fdNdxCerenkov          = G4DataVector(fMaxSplineSize,0.0);
  fdNdxPlasmon           = G4DataVector(fMaxSplineSize,0.0);
  fIntegralPAIySection   = G4DataVector(fMaxSplineSize,0.0);
  fIntegralPAIdEdx       = G4DataVector(fMaxSplineSize,0.0);
  fIntegralCerenkov      = G4DataVector(fMaxSplineSize,0.0);
  fIntegralPlasmon       = G4DataVector(fMaxSplineSize,0.0);

  for( G4int i = 0; i < 500; ++i ) 
  {
    for( G4int j = 0; j < 112; ++j ) { fPAItable[i][j] = 0.0; }
  }
}

//
// Destructor

G4PAIySection::~G4PAIySection()
{}

//
// Constructor with beta*gamma square value called from G4PAIModel

void G4PAIySection::Initialize( const G4Material* material,
                                G4double maxEnergyTransfer,
                                G4double betaGammaSq, 
                                G4SandiaTable* sandia)
{
  if(fVerbose > 0)
  {
    G4cout<<G4endl;
    G4cout<<"G4PAIySection::Initialize(...,G4SandiaTable* sandia)"<<G4endl;
    G4cout<<G4endl;
  }
  G4int i, j;

  fSandia          = sandia;
  fIntervalNumber  = sandia->GetMaxInterval();
  fDensity         = material->GetDensity();
  fElectronDensity = material->GetElectronDensity();

  // fIntervalNumber--;

  if( fVerbose > 0 )
  {
    G4cout<<"fDensity = "<<fDensity<<"\t"<<fElectronDensity<<"\t fIntervalNumber = "
          <<fIntervalNumber<< " (beta*gamma)^2= " << betaGammaSq << G4endl;
  }  
  fEnergyInterval = G4DataVector(fIntervalNumber+2,0.0);
  fA1             = G4DataVector(fIntervalNumber+2,0.0);
  fA2             = G4DataVector(fIntervalNumber+2,0.0);
  fA3             = G4DataVector(fIntervalNumber+2,0.0);
  fA4             = G4DataVector(fIntervalNumber+2,0.0);

  for( i = 1; i <= fIntervalNumber; i++ ) 
  {
    if ( sandia->GetSandiaMatTablePAI(i-1,0) < 1.*eV ) 
    { 
      fIntervalNumber--;
      continue;
    }
    if( ( sandia->GetSandiaMatTablePAI(i-1,0) >= maxEnergyTransfer ) 
        || i >= fIntervalNumber ) 
    {
      fEnergyInterval[i] = maxEnergyTransfer;
      fIntervalNumber = i;
      break;
    }
    fEnergyInterval[i] = sandia->GetSandiaMatTablePAI(i-1,0);
    fA1[i]             = sandia->GetSandiaMatTablePAI(i-1,1);
    fA2[i]             = sandia->GetSandiaMatTablePAI(i-1,2);
    fA3[i]             = sandia->GetSandiaMatTablePAI(i-1,3);
    fA4[i]             = sandia->GetSandiaMatTablePAI(i-1,4);

    if( fVerbose > 0 ) {
      G4cout<<i<<"\t"<<fEnergyInterval[i]/keV<<"\t"<<fA1[i]<<"\t"<<fA2[i]<<"\t"
            <<fA3[i]<<"\t"<<fA4[i]<<"\t"<<G4endl;
    }
  }
  if( fVerbose > 0 ) { 
    G4cout<<"last i = "<<i<<"; "<<"fIntervalNumber = "
          <<fIntervalNumber<<G4endl;   
  }
  if( fEnergyInterval[fIntervalNumber] != maxEnergyTransfer )
  {
      fIntervalNumber++;
      fEnergyInterval[fIntervalNumber] = maxEnergyTransfer;
  }
  if( fVerbose > 0 )
  {  
    for( i = 1; i <= fIntervalNumber; i++ )
    {
      G4cout<<i<<"\t"<<fEnergyInterval[i]/keV<<"\t"<<fA1[i]<<"\t"<<fA2[i]<<"\t"
        <<fA3[i]<<"\t"<<fA4[i]<<"\t"<<G4endl;
    }
  }  
  if( fVerbose > 0 ) {
    G4cout<<"Now checking, if two borders are too close together"<<G4endl;
  }
  for( i = 1; i < fIntervalNumber; i++ )
  {
    if( fEnergyInterval[i+1]-fEnergyInterval[i] >
         1.5*fDelta*(fEnergyInterval[i+1]+fEnergyInterval[i]) ) continue;
    else
    {
      for( j = i; j < fIntervalNumber; j++ )
      {
              fEnergyInterval[j] = fEnergyInterval[j+1];
              fA1[j]             = fA1[j+1];
              fA2[j]             = fA2[j+1];
              fA3[j]             = fA3[j+1];
              fA4[j]             = fA4[j+1];
      }
      fIntervalNumber--;
    }
  }
  if( fVerbose > 0 )
  {
    for( i = 1; i <= fIntervalNumber; i++ )
    {
      G4cout<<i<<"\t"<<fEnergyInterval[i]/keV<<"\t"<<fA1[i]<<"\t"<<fA2[i]<<"\t"
        <<fA3[i]<<"\t"<<fA4[i]<<"\t"<<G4endl;
    }
  }
  // Preparation of fSplineEnergy array corresponding to min ionisation, G~4

  ComputeLowEnergyCof(material);
            
  G4double   betaGammaSqRef = 
    fLorentzFactor[fRefGammaNumber]*fLorentzFactor[fRefGammaNumber] - 1;

  NormShift(betaGammaSqRef);             
  SplainPAI(betaGammaSqRef);
      
  // Preparation of integral PAI cross section for input betaGammaSq
   
  for( i = 1; i <= fSplineNumber; i++ )
  {
     fDifPAIySection[i] = DifPAIySection(i,betaGammaSq);

     if( fVerbose > 0 ) G4cout<<i<<"; dNdxPAI = "<<fDifPAIySection[i]<<G4endl;
  }
  IntegralPAIySection();   
}

//
// Compute low energy cof. It reduces PAI xsc for Lorentz factors less than 4.
//

void G4PAIySection::ComputeLowEnergyCof(const G4Material* material)
{    
  G4int i, numberOfElements = material->GetNumberOfElements();
  G4double sumZ = 0., sumCof = 0.; 

  static const G4double p0 =  1.20923e+00; 
  static const G4double p1 =  3.53256e-01; 
  static const G4double p2 = -1.45052e-03; 
  
  G4double* thisMaterialZ   = new G4double[numberOfElements];
  G4double* thisMaterialCof = new G4double[numberOfElements];
   
  for( i = 0; i < numberOfElements; i++ )
  {
    thisMaterialZ[i] = material->GetElement(i)->GetZ();
    sumZ += thisMaterialZ[i];
    thisMaterialCof[i] = p0+p1*thisMaterialZ[i]+p2*thisMaterialZ[i]*thisMaterialZ[i];   
  }
  for( i = 0; i < numberOfElements; i++ )
  {
    sumCof += thisMaterialCof[i]*thisMaterialZ[i]/sumZ;
  }
  fLowEnergyCof = sumCof;
  delete [] thisMaterialZ;
  delete [] thisMaterialCof;
  // G4cout<<"fLowEnergyCof = "<<fLowEnergyCof<<G4endl;
}

//
// General control function for class G4PAIySection
//

void G4PAIySection::InitPAI()
{    
   G4int i;
   G4double betaGammaSq = fLorentzFactor[fRefGammaNumber]*
                          fLorentzFactor[fRefGammaNumber] - 1;

   // Preparation of integral PAI cross section for reference gamma
   
   NormShift(betaGammaSq);             
   SplainPAI(betaGammaSq);

   IntegralPAIySection();
   IntegralCerenkov();
   IntegralPlasmon();

   for( i = 0; i<= fSplineNumber; i++)
   {
     fPAItable[i][fRefGammaNumber] = fIntegralPAIySection[i];
      
     if(i != 0)  fPAItable[i][0] = fSplineEnergy[i];     
   }
   fPAItable[0][0] = fSplineNumber;
   
   for( G4int j = 1; j < 112; j++)       // for other gammas
   {
      if( j == fRefGammaNumber ) continue;
      
      betaGammaSq = fLorentzFactor[j]*fLorentzFactor[j] - 1;
      
      for(i = 1; i <= fSplineNumber; i++)
      {
         fDifPAIySection[i] = DifPAIySection(i,betaGammaSq);
         fdNdxCerenkov[i]   = PAIdNdxCerenkov(i,betaGammaSq);
         fdNdxPlasmon[i]    = PAIdNdxPlasmon(i,betaGammaSq);
      }
      IntegralPAIySection();
      IntegralCerenkov();
      IntegralPlasmon();
      
      for(i = 0; i <= fSplineNumber; i++)
      {
        fPAItable[i][j] = fIntegralPAIySection[i];
      }
   } 
}  

//
// Shifting from borders to intervals Creation of first energy points
//

void G4PAIySection::NormShift(G4double betaGammaSq)
{
  G4int i, j;

  for( i = 1; i <= fIntervalNumber-1; i++ )
  {
    for( j = 1; j <= 2; j++ )
    {
      fSplineNumber = (i-1)*2 + j;

      if( j == 1 ) fSplineEnergy[fSplineNumber] = fEnergyInterval[i  ]*(1+fDelta);
      else         fSplineEnergy[fSplineNumber] = fEnergyInterval[i+1]*(1-fDelta); 
      //    G4cout<<"cn = "<<fSplineNumber<<"; "<<"energy = "
      //  <<fSplineEnergy[fSplineNumber]<<G4endl;
    }
  }
  fIntegralTerm[1]=RutherfordIntegral(1,fEnergyInterval[1],fSplineEnergy[1]);

  j = 1;

  for(i=2;i<=fSplineNumber;i++)
  {
    if(fSplineEnergy[i]<fEnergyInterval[j+1])
    {
         fIntegralTerm[i] = fIntegralTerm[i-1] + 
                            RutherfordIntegral(j,fSplineEnergy[i-1],
                                                 fSplineEnergy[i]   );
    }
    else
    {
       G4double x = RutherfordIntegral(j,fSplineEnergy[i-1],
                                           fEnergyInterval[j+1]   );
         j++;
         fIntegralTerm[i] = fIntegralTerm[i-1] + x + 
                            RutherfordIntegral(j,fEnergyInterval[j],
                                                 fSplineEnergy[i]    );
    }
    // G4cout<<i<<"\t"<<fSplineEnergy[i]<<"\t"<<fIntegralTerm[i]<<"\n"<<G4endl;
  } 
  static const G4double nfactor =
    2*pi*pi*hbarc*hbarc*fine_structure_const/electron_mass_c2;
  fNormalizationCof = nfactor*fElectronDensity/fIntegralTerm[fSplineNumber];

  // G4cout<<"fNormalizationCof = "<<fNormalizationCof<<G4endl;

          // Calculation of PAI differrential cross-section (1/(keV*cm))
          // in the energy points near borders of energy intervals

   for(G4int k=1;k<=fIntervalNumber-1;k++)
   {
      for(j=1;j<=2;j++)
      {
         i = (k-1)*2 + j;
         fImPartDielectricConst[i] = fNormalizationCof*
                                     ImPartDielectricConst(k,fSplineEnergy[i]);
         fRePartDielectricConst[i] = fNormalizationCof*
                                     RePartDielectricConst(fSplineEnergy[i]);
         fIntegralTerm[i] *= fNormalizationCof;

         fDifPAIySection[i] = DifPAIySection(i,betaGammaSq);
         fdNdxCerenkov[i]   = PAIdNdxCerenkov(i,betaGammaSq);
         fdNdxPlasmon[i]    = PAIdNdxPlasmon(i,betaGammaSq);
      }
   }

}  // end of NormShift 

//
// Creation of new energy points as geometrical mean of existing
// one, calculation PAI_cs for them, while the error of logarithmic
// linear approximation would be smaller than 'fError'

void G4PAIySection::SplainPAI(G4double betaGammaSq)
{
   G4int k = 1;
   G4int i = 1;

   while ( (i < fSplineNumber) && (fSplineNumber < fMaxSplineSize-1) )
   {
      if(fSplineEnergy[i+1] > fEnergyInterval[k+1])
      {
          k++;   // Here next energy point is in next energy interval
          i++;
          continue;
      }
                        // Shifting of arrayes for inserting the geometrical 
                       // average of 'i' and 'i+1' energy points to 'i+1' place
      fSplineNumber++;

      for(G4int j = fSplineNumber; j >= i+2; j-- )
      {
         fSplineEnergy[j]          = fSplineEnergy[j-1];
         fImPartDielectricConst[j] = fImPartDielectricConst[j-1];
         fRePartDielectricConst[j] = fRePartDielectricConst[j-1];
         fIntegralTerm[j]          = fIntegralTerm[j-1];

         fDifPAIySection[j] = fDifPAIySection[j-1];
         fdNdxCerenkov[j]   = fdNdxCerenkov[j-1];
         fdNdxPlasmon[j]    = fdNdxPlasmon[j-1];
      }
      G4double x1  = fSplineEnergy[i];
      G4double x2  = fSplineEnergy[i+1];
      G4double yy1 = fDifPAIySection[i];
      G4double y2  = fDifPAIySection[i+1];

      G4double en1 = sqrt(x1*x2);
      fSplineEnergy[i+1] = en1;

                 // Calculation of logarithmic linear approximation
                 // in this (enr) energy point, which number is 'i+1' now

      G4double a = log10(y2/yy1)/log10(x2/x1);
      G4double b = log10(yy1) - a*log10(x1);
      G4double y = a*log10(en1) + b;
      y = pow(10.,y);

                 // Calculation of the PAI dif. cross-section at this point

      fImPartDielectricConst[i+1] = fNormalizationCof*
                                    ImPartDielectricConst(k,fSplineEnergy[i+1]);
      fRePartDielectricConst[i+1] = fNormalizationCof*
                                    RePartDielectricConst(fSplineEnergy[i+1]);
      fIntegralTerm[i+1] = fIntegralTerm[i] + fNormalizationCof*
                           RutherfordIntegral(k,fSplineEnergy[i],
                                                fSplineEnergy[i+1]);

      fDifPAIySection[i+1] = DifPAIySection(i+1,betaGammaSq);
      fdNdxCerenkov[i+1]   = PAIdNdxCerenkov(i+1,betaGammaSq);
      fdNdxPlasmon[i+1]    = PAIdNdxPlasmon(i+1,betaGammaSq);

                  // Condition for next division of this segment or to pass
                  // to higher energies

      G4double x = 2*(fDifPAIySection[i+1] - y)/(fDifPAIySection[i+1] + y);

      G4double delta = 2.*(fSplineEnergy[i+1]-fSplineEnergy[i])
    /(fSplineEnergy[i+1]+fSplineEnergy[i]);

      if( x < 0 ) 
      {
         x = -x;
      }
      if( x > fError && fSplineNumber < fMaxSplineSize-1 && delta > 2.*fDelta  )
      {
         continue;  // next division
      }
      i += 2;  // pass to next segment

      // Loop checking, 03-Aug-2015, Vladimir Ivanchenko
   }   // close 'while'

}  // end of SplainPAI 


//
// Integration over electrons that could be considered
// quasi-free at energy transfer of interest

G4double G4PAIySection::RutherfordIntegral( G4int k,
                                            G4double x1,
                                              G4double x2   )
{
   G4double  c1, c2, c3;
   // G4cout<<"RI: x1 = "<<x1<<"; "<<"x2 = "<<x2<<G4endl;
   G4double x12 = x1*x2;   
   c1 = (x2 - x1)/x12;
   c2 = (x2 - x1)*(x2 + x1)/(x12*x12);
   c3 = (x2 - x1)*(x1*x1 + x1*x2 + x2*x2)/(x12*x12*x12);
   // G4cout<<" RI: c1 = "<<c1<<"; "<<"c2 = "<<c2<<"; "<<"c3 = "<<c3<<G4endl;   
   
   return  fA1[k]*log(x2/x1) + fA2[k]*c1 + fA3[k]*c2/2 + fA4[k]*c3/3;

}   // end of RutherfordIntegral 


//
// Imaginary part of dielectric constant
// (G4int k - interval number, G4double en1 - energy point)

G4double G4PAIySection::ImPartDielectricConst( G4int    k ,
                                               G4double energy1 )
{
   G4double energy2,energy3,energy4,result;

   energy2 = energy1*energy1;
   energy3 = energy2*energy1;
   energy4 = energy3*energy1;
   
   result = fA1[k]/energy1+fA2[k]/energy2+fA3[k]/energy3+fA4[k]/energy4;  
   result *=hbarc/energy1;
   
   return result;

}  // end of ImPartDielectricConst 


//
// Real part of dielectric constant minus unit: epsilon_1 - 1
// (G4double enb - energy point)
//

G4double G4PAIySection::RePartDielectricConst(G4double enb)
{       
   G4double x0, x02, x03, x04, x05, x1, x2, xx1 ,xx2 , xx12,
            c1, c2, c3, cof1, cof2, xln1, xln2, xln3, result;

   x0 = enb;
   result = 0;
   
   for(G4int i=1;i<=fIntervalNumber-1;i++)
   {
      x1 = fEnergyInterval[i];
      x2 = fEnergyInterval[i+1];
      xx1 = x1 - x0;
      xx2 = x2 - x0;
      xx12 = xx2/xx1;
      
      if(xx12<0)
      {
         xx12 = -xx12;
      }
      xln1 = log(x2/x1);
      xln2 = log(xx12);
      xln3 = log((x2 + x0)/(x1 + x0));
      x02 = x0*x0;
      x03 = x02*x0;
      x04 = x03*x0;
      x05 = x04*x0;
      G4double x12 = x1*x2;
      c1  = (x2 - x1)/x12;
      c2  = (x2 - x1)*(x2 +x1)/(x12*x12);
      c3  = (x2 -x1)*(x1*x1 + x1*x2 + x2*x2)/(x12*x12*x12);

      result -= (fA1[i]/x02 + fA3[i]/x04)*xln1;
      result -= (fA2[i]/x02 + fA4[i]/x04)*c1;
      result -= fA3[i]*c2/2/x02;
      result -= fA4[i]*c3/3/x02;

      cof1 = fA1[i]/x02 + fA3[i]/x04;
      cof2 = fA2[i]/x03 + fA4[i]/x05;

      result += 0.5*(cof1 +cof2)*xln2;
      result += 0.5*(cof1 - cof2)*xln3;
   } 
   result *= 2*hbarc/pi;
   
   return result;

}   // end of RePartDielectricConst 

//
// PAI differential cross-section in terms of
// simplified Allison's equation
//

G4double G4PAIySection::DifPAIySection( G4int              i ,
                                        G4double betaGammaSq  )
{        
  G4double beta, be2,cof,x1,x2,x3,x4,x5,x6,x7,x8,result;
   //G4double beta, be4;
   //G4double be4;
   // G4double betaBohr2 = fine_structure_const*fine_structure_const;
   // G4double betaBohr4 = betaBohr2*betaBohr2*4.0;
   be2 = betaGammaSq/(1 + betaGammaSq);
   //be4 = be2*be2;
   beta = sqrt(be2);
   cof = 1;
   x1 = log(2*electron_mass_c2/fSplineEnergy[i]);

   if( betaGammaSq < 0.01 ) x2 = log(be2);
   else
   {
     x2 = -log( (1/betaGammaSq - fRePartDielectricConst[i])*
                (1/betaGammaSq - fRePartDielectricConst[i]) + 
                fImPartDielectricConst[i]*fImPartDielectricConst[i] )/2;
   }
   if( fImPartDielectricConst[i] == 0.0 ||betaGammaSq < 0.01 )
   {
     x6=0;
   }
   else
   {
     x3 = -fRePartDielectricConst[i] + 1/betaGammaSq;
     x5 = -1 - fRePartDielectricConst[i] +
          be2*((1 +fRePartDielectricConst[i])*(1 + fRePartDielectricConst[i]) +
          fImPartDielectricConst[i]*fImPartDielectricConst[i]);

     x7 = atan2(fImPartDielectricConst[i],x3);
     x6 = x5 * x7;
   }
    // if(fImPartDielectricConst[i] == 0) x6 = 0;
   
   x4 = ((x1 + x2)*fImPartDielectricConst[i] + x6)/hbarc;
   //   if( x4 < 0.0 ) x4 = 0.0;
   x8 = (1 + fRePartDielectricConst[i])*(1 + fRePartDielectricConst[i]) + 
        fImPartDielectricConst[i]*fImPartDielectricConst[i];

   result = (x4 + cof*fIntegralTerm[i]/fSplineEnergy[i]/fSplineEnergy[i]);
   result = std::max(result, 1.0e-8);
   result *= fine_structure_const/be2/pi;
   // low energy correction

   G4double lowCof = fLowEnergyCof; // 6.0 ; // Ar ~ 4.; -> fLowCof as f(Z1,Z2)? 

   result *= (1 - exp(-beta/betaBohr/lowCof));

   //   result *= (1-exp(-beta/betaBohr))*(1-exp(-beta/betaBohr));
   //  result *= (1-exp(-be2/betaBohr2));
   // result *= (1-exp(-be4/betaBohr4));
   //   if(fDensity >= 0.1)
   if(x8 > 0.)
   { 
      result /= x8;
   }
   return result;

} // end of DifPAIySection 

//
// Calculation od dN/dx of collisions with creation of Cerenkov pseudo-photons

G4double G4PAIySection::PAIdNdxCerenkov( G4int    i ,
                                         G4double betaGammaSq  )
{        
   G4double logarithm, x3, x5, argument, modul2, dNdxC; 
   G4double be2, be4;

   //G4double cof         = 1.0;

   be2 = betaGammaSq/(1 + betaGammaSq);
   be4 = be2*be2;

   if( betaGammaSq < 0.01 ) logarithm = log(1.0+betaGammaSq); // 0.0;
   else
   {
     logarithm  = -log( (1/betaGammaSq - fRePartDielectricConst[i])*
                        (1/betaGammaSq - fRePartDielectricConst[i]) + 
                        fImPartDielectricConst[i]*fImPartDielectricConst[i] )*0.5;
     logarithm += log(1+1.0/betaGammaSq);
   }

   if( fImPartDielectricConst[i] == 0.0 || betaGammaSq < 0.01 )
   {
     argument = 0.0;
   }
   else
   {
     x3 = -fRePartDielectricConst[i] + 1.0/betaGammaSq;
     x5 = -1.0 - fRePartDielectricConst[i] +
          be2*((1.0 +fRePartDielectricConst[i])*(1.0 + fRePartDielectricConst[i]) +
          fImPartDielectricConst[i]*fImPartDielectricConst[i]);
     if( x3 == 0.0 ) argument = 0.5*pi;
     else            argument = atan2(fImPartDielectricConst[i],x3);
     argument *= x5 ;
   }   
   dNdxC = ( logarithm*fImPartDielectricConst[i] + argument )/hbarc;
  
   if(dNdxC < 1.0e-8) dNdxC = 1.0e-8;

   dNdxC *= fine_structure_const/be2/pi;

   dNdxC *= (1-exp(-be4/betaBohr4));

   //   if(fDensity >= 0.1)
   // { 
   modul2 = (1.0 + fRePartDielectricConst[i])*(1.0 + fRePartDielectricConst[i]) + 
                    fImPartDielectricConst[i]*fImPartDielectricConst[i];
   if(modul2 > 0.)
     {
       dNdxC /= modul2;
     }
   return dNdxC;

} // end of PAIdNdxCerenkov 

//
// Calculation od dN/dx of collisions with creation of longitudinal EM
// excitations (plasmons, delta-electrons)

G4double G4PAIySection::PAIdNdxPlasmon( G4int    i ,
                                        G4double betaGammaSq  )
{        
   G4double cof, resonance, modul2, dNdxP;
   G4double be2, be4;

   cof = 1;

   be2 = betaGammaSq/(1 + betaGammaSq);
   be4 = be2*be2;
 
   resonance = log(2*electron_mass_c2*be2/fSplineEnergy[i]);  
   resonance *= fImPartDielectricConst[i]/hbarc;

   dNdxP = ( resonance + cof*fIntegralTerm[i]/fSplineEnergy[i]/fSplineEnergy[i] );

   dNdxP = std::max(dNdxP, 1.0e-8);

   dNdxP *= fine_structure_const/be2/pi;
   dNdxP *= (1-exp(-be4/betaBohr4));

//   if( fDensity >= 0.1 )
//   { 
   modul2 = (1 + fRePartDielectricConst[i])*(1 + fRePartDielectricConst[i]) + 
     fImPartDielectricConst[i]*fImPartDielectricConst[i];
   if(modul2 > 0.)
     { 
       dNdxP /= modul2;
     }
   return dNdxP;

} // end of PAIdNdxPlasmon 

//
// Calculation of the PAI integral cross-section
// fIntegralPAIySection[1] = specific primary ionisation, 1/cm
// and fIntegralPAIySection[0] = mean energy loss per cm  in keV/cm

void G4PAIySection::IntegralPAIySection()
{
  fIntegralPAIySection[fSplineNumber] = 0;
  fIntegralPAIdEdx[fSplineNumber]     = 0;
  fIntegralPAIySection[0]             = 0;
  G4int k = fIntervalNumber -1;

  for(G4int i = fSplineNumber-1; i >= 1; i--)
  {
    if(fSplineEnergy[i] >= fEnergyInterval[k])
    {
      fIntegralPAIySection[i] = fIntegralPAIySection[i+1] + SumOverInterval(i);
      fIntegralPAIdEdx[i] = fIntegralPAIdEdx[i+1] + SumOverIntervaldEdx(i);
    }
    else
    {
      fIntegralPAIySection[i] = fIntegralPAIySection[i+1] + 
                                   SumOverBorder(i+1,fEnergyInterval[k]);
      fIntegralPAIdEdx[i] = fIntegralPAIdEdx[i+1] + 
                                   SumOverBorderdEdx(i+1,fEnergyInterval[k]);
      k--;
    }
  }
}   // end of IntegralPAIySection 

//
// Calculation of the PAI Cerenkov integral cross-section
// fIntegralCrenkov[1] = specific Crenkov ionisation, 1/cm
// and fIntegralCerenkov[0] = mean Cerenkov loss per cm  in keV/cm

void G4PAIySection::IntegralCerenkov()
{
  G4int i, k;
   fIntegralCerenkov[fSplineNumber] = 0;
   fIntegralCerenkov[0] = 0;
   k = fIntervalNumber -1;

   for( i = fSplineNumber-1; i >= 1; i-- )
   {
      if(fSplineEnergy[i] >= fEnergyInterval[k])
      {
        fIntegralCerenkov[i] = fIntegralCerenkov[i+1] + SumOverInterCerenkov(i);
        // G4cout<<"int: i = "<<i<<"; sumC = "<<fIntegralCerenkov[i]<<G4endl;
      }
      else
      {
        fIntegralCerenkov[i] = fIntegralCerenkov[i+1] + 
                                   SumOverBordCerenkov(i+1,fEnergyInterval[k]);
        k--;
        // G4cout<<"bord: i = "<<i<<"; sumC = "<<fIntegralCerenkov[i]<<G4endl;
      }
   }

}   // end of IntegralCerenkov 

//
// Calculation of the PAI Plasmon integral cross-section
// fIntegralPlasmon[1] = splasmon primary ionisation, 1/cm
// and fIntegralPlasmon[0] = mean plasmon loss per cm  in keV/cm

void G4PAIySection::IntegralPlasmon()
{
   fIntegralPlasmon[fSplineNumber] = 0;
   fIntegralPlasmon[0] = 0;
   G4int k = fIntervalNumber -1;
   for(G4int i=fSplineNumber-1;i>=1;i--)
   {
      if(fSplineEnergy[i] >= fEnergyInterval[k])
      {
        fIntegralPlasmon[i] = fIntegralPlasmon[i+1] + SumOverInterPlasmon(i);
      }
      else
      {
        fIntegralPlasmon[i] = fIntegralPlasmon[i+1] + 
                                   SumOverBordPlasmon(i+1,fEnergyInterval[k]);
        k--;
      }
   }

}   // end of IntegralPlasmon

//
// Calculation the PAI integral cross-section inside
// of interval of continuous values of photo-ionisation
// cross-section. Parameter  'i' is the number of interval.

G4double G4PAIySection::SumOverInterval( G4int i )
{         
   G4double x0,x1,y0,yy1,a,b,c,result;

   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];

   if( std::abs( 2.*(x1-x0)/(x1+x0) ) < 1.e-6) return 0.;

   y0 = fDifPAIySection[i];
   yy1 = fDifPAIySection[i+1];
   //G4cout << "## x0= " << x0 << " x1= " << x1 << G4endl;
   c = x1/x0;
   //G4cout << "c= " << c << " y0= " << y0 << " yy1= " << yy1 << G4endl;
   a = log10(yy1/y0)/log10(c);
   //G4cout << "a= " << a << G4endl;
   // b = log10(y0) - a*log10(x0);
   b = y0/pow(x0,a);
   a += 1;
   if(a == 0) 
   {
      result = b*log(x1/x0);
   }
   else
   {
      result = y0*(x1*pow(c,a-1) - x0)/a;
   }
   a++;
   if(a == 0) 
   {
      fIntegralPAIySection[0] += b*log(x1/x0);
   }
   else
   {
      fIntegralPAIySection[0] += y0*(x1*x1*pow(c,a-2) - x0*x0)/a;
   }
   return result;

} //  end of SumOverInterval


G4double G4PAIySection::SumOverIntervaldEdx( G4int i )
{         
   G4double x0,x1,y0,yy1,a,b,c,result;

   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];

   if( std::abs( 2.*(x1-x0)/(x1+x0) ) < 1.e-6) return 0.;

   y0 = fDifPAIySection[i];
   yy1 = fDifPAIySection[i+1];
   c = x1/x0;
   a = log10(yy1/y0)/log10(c);
   // b = log10(y0) - a*log10(x0);
   b = y0/pow(x0,a);
   a += 2;
   if(a == 0) 
   {
     result = b*log(x1/x0);
   }
   else
   {
     result = y0*(x1*x1*pow(c,a-2) - x0*x0)/a;
   }
   return result;

} //  end of SumOverInterval

//
// Calculation the PAI Cerenkov integral cross-section inside
// of interval of continuous values of photo-ionisation Cerenkov
// cross-section. Parameter  'i' is the number of interval.

G4double G4PAIySection::SumOverInterCerenkov( G4int i )
{         
   G4double x0,x1,y0,yy1,a,c,result;

   x0  = fSplineEnergy[i];
   x1  = fSplineEnergy[i+1];

   if( std::abs( 2.*(x1-x0)/(x1+x0) ) < 1.e-6) return 0.;

   y0  = fdNdxCerenkov[i];
   yy1 = fdNdxCerenkov[i+1];
   // G4cout<<"SumC, i = "<<i<<"; x0 ="<<x0<<"; x1 = "<<x1
   //   <<"; y0 = "<<y0<<"; yy1 = "<<yy1<<G4endl;

   c = x1/x0;
   a = log10(yy1/y0)/log10(c);
   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);

   a += 1.0;
   if(a == 0) result = b*log(c);
   else       result = y0*(x1*pow(c,a-1) - x0)/a;   
   a += 1.0;

   if( a == 0 ) fIntegralCerenkov[0] += b*log(x1/x0);
   else         fIntegralCerenkov[0] += y0*(x1*x1*pow(c,a-2) - x0*x0)/a;
   //  G4cout<<"a = "<<a<<"; b = "<<b<<"; result = "<<result<<G4endl;   
   return result;

} //  end of SumOverInterCerenkov

//
// Calculation the PAI Plasmon integral cross-section inside
// of interval of continuous values of photo-ionisation Plasmon
// cross-section. Parameter  'i' is the number of interval.

G4double G4PAIySection::SumOverInterPlasmon( G4int i )
{         
  G4double x0,x1,y0,yy1,a,c,result;

   x0  = fSplineEnergy[i];
   x1  = fSplineEnergy[i+1];

   if( std::abs( 2.*(x1-x0)/(x1+x0) ) < 1.e-6) return 0.;

   y0  = fdNdxPlasmon[i];
   yy1 = fdNdxPlasmon[i+1];
   c = x1/x0;
   a = log10(yy1/y0)/log10(c);

   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);

   a += 1.0;
   if(a == 0) result = b*log(x1/x0);
   else       result = y0*(x1*pow(c,a-1) - x0)/a;   
   a += 1.0;

   if( a == 0 ) fIntegralPlasmon[0] += b*log(x1/x0);
   else         fIntegralPlasmon[0] += y0*(x1*x1*pow(c,a-2) - x0*x0)/a;
   
   return result;

} //  end of SumOverInterPlasmon

//
// Integration of PAI cross-section for the case of
// passing across border between intervals

G4double G4PAIySection::SumOverBorder( G4int      i , 
                                       G4double en0    )
{               
  G4double x0,x1,y0,yy1,a,/*c,*/d,e0,result;

   e0 = en0;
   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];
   y0 = fDifPAIySection[i];
   yy1 = fDifPAIySection[i+1];

   //c = x1/x0;
   d = e0/x0;   
   a = log10(yy1/y0)/log10(x1/x0);

   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);
   
   a += 1;
   if(a == 0)
   {
      result = b*log(x0/e0);
   }
   else
   {
      result = y0*(x0 - e0*pow(d,a-1))/a;
   }
   a++;
   if(a == 0)
   {
      fIntegralPAIySection[0] += b*log(x0/e0);
   }
   else 
   {
      fIntegralPAIySection[0] += y0*(x0*x0 - e0*e0*pow(d,a-2))/a;
   }
   x0 = fSplineEnergy[i - 1];
   x1 = fSplineEnergy[i - 2];
   y0 = fDifPAIySection[i - 1];
   yy1 = fDifPAIySection[i - 2];

   //c = x1/x0;
   d = e0/x0;   
   a = log10(yy1/y0)/log10(x1/x0);
   //  b0 = log10(y0) - a*log10(x0);
   b = y0/pow(x0,a);
   a += 1;
   if(a == 0)
   {
      result += b*log(e0/x0);
   }
   else
   {
      result += y0*(e0*pow(d,a-1) - x0)/a;
   }
   a++;
   if(a == 0) 
   {
      fIntegralPAIySection[0] += b*log(e0/x0);
   }
   else
   {
      fIntegralPAIySection[0] += y0*(e0*e0*pow(d,a-2) - x0*x0)/a;
   }
   return result;

} 


G4double G4PAIySection::SumOverBorderdEdx( G4int      i , 
                                       G4double en0    )
{               
  G4double x0,x1,y0,yy1,a,/*c,*/d,e0,result;

   e0 = en0;
   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];
   y0 = fDifPAIySection[i];
   yy1 = fDifPAIySection[i+1];

   //c = x1/x0;
   d = e0/x0;   
   a = log10(yy1/y0)/log10(x1/x0);
   
   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);
   
   a += 2;
   if(a == 0)
   {
      result = b*log(x0/e0);
   }
   else 
   {
      result = y0*(x0*x0 - e0*e0*pow(d,a-2))/a;
   }
   x0 = fSplineEnergy[i - 1];
   x1 = fSplineEnergy[i - 2];
   y0 = fDifPAIySection[i - 1];
   yy1 = fDifPAIySection[i - 2];

   //c = x1/x0;
   d = e0/x0;   
   a = log10(yy1/y0)/log10(x1/x0);

   if(a < 20.) b = y0/pow(x0,a);

   a += 2;
   if(a == 0) 
   {
      result += b*log(e0/x0);
   }
   else
   {
      result += y0*(e0*e0*pow(d,a-2) - x0*x0)/a;
   }
   return result;

} 

//
// Integration of Cerenkov cross-section for the case of
// passing across border between intervals

G4double G4PAIySection::SumOverBordCerenkov( G4int      i , 
                                             G4double en0    )
{               
   G4double x0,x1,y0,yy1,a,e0,c,d,result;

   e0 = en0;
   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];
   y0 = fdNdxCerenkov[i];
   yy1 = fdNdxCerenkov[i+1];

   //  G4cout<<G4endl;
   //G4cout<<"SumBordC, i = "<<i<<"; en0 = "<<en0<<"; x0 ="<<x0<<"; x1 = "<<x1
   //     <<"; y0 = "<<y0<<"; yy1 = "<<yy1<<G4endl;
   c = x1/x0;
   d = e0/x0;
   a = log10(yy1/y0)/log10(c);
 
   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);
   
   a += 1.0;
   if( a == 0 ) result = b*log(x0/e0);
   else         result = y0*(x0 - e0*pow(d,a-1))/a;   
   a += 1.0;

   if( a == 0 ) fIntegralCerenkov[0] += b*log(x0/e0);
   else         fIntegralCerenkov[0] += y0*(x0*x0 - e0*e0*pow(d,a-2))/a;

   //G4cout<<"a = "<<a<<"; b = "<<b<<"; result = "<<result<<G4endl;
   
   x0  = fSplineEnergy[i - 1];
   x1  = fSplineEnergy[i - 2];
   y0  = fdNdxCerenkov[i - 1];
   yy1 = fdNdxCerenkov[i - 2];

   //G4cout<<"x0 ="<<x0<<"; x1 = "<<x1
   //    <<"; y0 = "<<y0<<"; yy1 = "<<yy1<<G4endl;

   c = x1/x0;
   d = e0/x0;
   a  = log10(yy1/y0)/log10(x1/x0);
  
   //   G4cout << "a= " << a << G4endl;
   if(a > 20.0) b = 0.0;
   else         b = y0/pow(x0,a); 

   //G4cout << "b= " << b << G4endl;

   a += 1.0;
   if( a == 0 ) result += b*log(e0/x0);
   else         result += y0*(e0*pow(d,a-1) - x0 )/a;
   a += 1.0;
   //G4cout << "result= " << result << G4endl;

   if( a == 0 )   fIntegralCerenkov[0] += b*log(e0/x0);
   else           fIntegralCerenkov[0] += y0*(e0*e0*pow(d,a-2) - x0*x0)/a;

   //G4cout<<"a = "<<a<<"; b = "<<b<<"; result = "<<result<<G4endl;    

   return result;

} 

//
// Integration of Plasmon cross-section for the case of
// passing across border between intervals

G4double G4PAIySection::SumOverBordPlasmon( G4int      i , 
                                             G4double en0    )
{               
   G4double x0,x1,y0,yy1,a,c,d,e0,result;

   e0 = en0;
   x0 = fSplineEnergy[i];
   x1 = fSplineEnergy[i+1];
   y0 = fdNdxPlasmon[i];
   yy1 = fdNdxPlasmon[i+1];

   c = x1/x0;
   d = e0/x0;   
   a = log10(yy1/y0)/log10(c);

   G4double b = 0.0;
   if(a < 20.) b = y0/pow(x0,a);
   
   a += 1.0;
   if( a == 0 ) result = b*log(x0/e0);
   else         result = y0*(x0 - e0*pow(d,a-1))/a;   
   a += 1.0;

   if( a == 0 ) fIntegralPlasmon[0] += b*log(x0/e0);
   else         fIntegralPlasmon[0] += y0*(x0*x0 - e0*e0*pow(d,a-2))/a;
   
   x0 = fSplineEnergy[i - 1];
   x1 = fSplineEnergy[i - 2];
   y0 = fdNdxPlasmon[i - 1];
   yy1 = fdNdxPlasmon[i - 2];

   c = x1/x0;
   d = e0/x0;
   a = log10(yy1/y0)/log10(c);
 
   if(a < 20.) b = y0/pow(x0,a);

   a += 1.0;
   if( a == 0 ) result += b*log(e0/x0);
   else         result += y0*(e0*pow(d,a-1) - x0)/a;
   a += 1.0;

   if( a == 0 )   fIntegralPlasmon[0] += b*log(e0/x0);
   else           fIntegralPlasmon[0] += y0*(e0*e0*pow(d,a-2) - x0*x0)/a;
   
   return result;

} 

//
//

G4double G4PAIySection::GetStepEnergyLoss( G4double step )
{  
  G4int iTransfer ;
  G4long numOfCollisions;
  G4double loss = 0.0;
  G4double meanNumber, position;

  // G4cout<<" G4PAIySection::GetStepEnergyLoss "<<G4endl;



  meanNumber = fIntegralPAIySection[1]*step;
  numOfCollisions = G4Poisson(meanNumber);

  //   G4cout<<"numOfCollisions = "<<numOfCollisions<<G4endl;

  while(numOfCollisions)
  {
    position = fIntegralPAIySection[1]*G4UniformRand();

    for( iTransfer=1; iTransfer<=fSplineNumber; iTransfer++ )
    {
        if( position >= fIntegralPAIySection[iTransfer] ) break;
    }
    loss += fSplineEnergy[iTransfer] ;
    numOfCollisions--;
    // Loop checking, 03-Aug-2015, Vladimir Ivanchenko
  }
  // G4cout<<"PAI energy loss = "<<loss/keV<<" keV"<<G4endl; 

  return loss;
}

//
//

G4double G4PAIySection::GetStepCerenkovLoss( G4double step )
{  
  G4int iTransfer ;
  G4long numOfCollisions;
  G4double loss = 0.0;
  G4double meanNumber, position;

  // G4cout<<" G4PAIySection::GetStepCreLosnkovs "<<G4endl;



  meanNumber = fIntegralCerenkov[1]*step;
  numOfCollisions = G4Poisson(meanNumber);

  //   G4cout<<"numOfCollisions = "<<numOfCollisions<<G4endl;

  while(numOfCollisions)
  {
    position = fIntegralCerenkov[1]*G4UniformRand();

    for( iTransfer=1; iTransfer<=fSplineNumber; iTransfer++ )
    {
        if( position >= fIntegralCerenkov[iTransfer] ) break;
    }
    loss += fSplineEnergy[iTransfer] ;
    numOfCollisions--;
    // Loop checking, 03-Aug-2015, Vladimir Ivanchenko
  }
  // G4cout<<"PAI Cerenkov loss = "<<loss/keV<<" keV"<<G4endl; 

  return loss;
}

//
//

G4double G4PAIySection::GetStepPlasmonLoss( G4double step )
{  
  G4int iTransfer ;
  G4long numOfCollisions;
  G4double loss = 0.0;
  G4double meanNumber, position;

  // G4cout<<" G4PAIySection::GetStepCreLosnkovs "<<G4endl;



  meanNumber = fIntegralPlasmon[1]*step;
  numOfCollisions = G4Poisson(meanNumber);

  //   G4cout<<"numOfCollisions = "<<numOfCollisions<<G4endl;

  while(numOfCollisions)
  {
    position = fIntegralPlasmon[1]*G4UniformRand();

    for( iTransfer=1; iTransfer<=fSplineNumber; iTransfer++ )
    {
        if( position >= fIntegralPlasmon[iTransfer] ) break;
    }
    loss += fSplineEnergy[iTransfer] ;
    numOfCollisions--;
    // Loop checking, 03-Aug-2015, Vladimir Ivanchenko
  }
  // G4cout<<"PAI Plasmon loss = "<<loss/keV<<" keV"<<G4endl; 

  return loss;
}

//

void G4PAIySection::CallError(G4int i, const G4String& methodName) const
{
  G4String head = "G4PAIySection::" + methodName + "()";
  G4ExceptionDescription ed;
  ed << "Wrong index " << i << " fSplineNumber= " << fSplineNumber;
  G4Exception(head,"pai001",FatalException,ed);
}

//
// Init  array of Lorentz factors
//

G4int G4PAIySection::fNumberOfGammas = 111;

const G4double G4PAIySection::fLorentzFactor[112] =     // fNumberOfGammas+1
{
0.0,
1.094989e+00, 1.107813e+00, 1.122369e+00, 1.138890e+00, 1.157642e+00,
1.178925e+00, 1.203082e+00, 1.230500e+00, 1.261620e+00, 1.296942e+00, // 10
1.337032e+00, 1.382535e+00, 1.434181e+00, 1.492800e+00, 1.559334e+00,
1.634850e+00, 1.720562e+00, 1.817845e+00, 1.928263e+00, 2.053589e+00, // 20
2.195835e+00, 2.357285e+00, 2.540533e+00, 2.748522e+00, 2.984591e+00,
3.252533e+00, 3.556649e+00, 3.901824e+00, 4.293602e+00, 4.738274e+00, // 30
5.242981e+00, 5.815829e+00, 6.466019e+00, 7.203990e+00, 8.041596e+00,
8.992288e+00, 1.007133e+01, 1.129606e+01, 1.268614e+01, 1.426390e+01, // 40
1.605467e+01, 1.808721e+01, 2.039417e+01, 2.301259e+01, 2.598453e+01,
2.935771e+01, 3.318630e+01, 3.753180e+01, 4.246399e+01, 4.806208e+01, // 50
5.441597e+01, 6.162770e+01, 6.981310e+01, 7.910361e+01, 8.964844e+01,
1.016169e+02, 1.152013e+02, 1.306197e+02, 1.481198e+02, 1.679826e+02, // 60
1.905270e+02, 2.161152e+02, 2.451581e+02, 2.781221e+02, 3.155365e+02,
3.580024e+02, 4.062016e+02, 4.609081e+02, 5.230007e+02, 5.934765e+02, // 70
6.734672e+02, 7.642575e+02, 8.673056e+02, 9.842662e+02, 1.117018e+03,
1.267692e+03, 1.438709e+03, 1.632816e+03, 1.853128e+03, 2.103186e+03, // 80
2.387004e+03, 2.709140e+03, 3.074768e+03, 3.489760e+03, 3.960780e+03,
4.495394e+03, 5.102185e+03, 5.790900e+03, 6.572600e+03, 7.459837e+03, // 90
8.466860e+03, 9.609843e+03, 1.090714e+04, 1.237959e+04, 1.405083e+04,
1.594771e+04, 1.810069e+04, 2.054434e+04, 2.331792e+04, 2.646595e+04, // 100
3.003901e+04, 3.409446e+04, 3.869745e+04, 4.392189e+04, 4.985168e+04,
5.658206e+04, 6.422112e+04, 7.289153e+04, 8.273254e+04, 9.390219e+04, // 110
1.065799e+05
};

//
// The number of gamma for creation of  spline (near ion-min , G ~ 4 )
//

const G4int G4PAIySection::fRefGammaNumber = 29; 

//   
// end of G4PAIySection implementation file 
//