Formula-based implementation of Oset model. More...

#include <INukeOsetFormula.h>

Inheritance diagram for INukeOsetFormula:

Public Member Functions
void	setupOset (const double &density, const double &pionTk, const int &pionPDG, const double &protonFraction)
	use to set up Oset class (assign pion Tk, nuclear density etc) More...

Public Member Functions inherited from INukeOset
	INukeOset ()
	contructor More...

double	getTotalCrossSection () const
	return total = (qel+cex+abs) cross section More...

double	getCexCrossSection () const
	return cex cross section More...

double	getAbsorptionCrossSection () const
	return absorption cross section More...

double	getCexFraction () const
	return fraction of cex events More...

double	getAbsorptionFraction () const
	return fraction of absorption events More...

Private Member Functions
void	setNucleus (const double &density)
	set nuclear density and Fermi momentum / energy More...

void	setKinematics (const double &pionTk, const bool &isPi0)
	do kinematics More...

void	setDelta ()
	set up Delta More...

void	setSelfEnergy ()
	calculate delta self energy More...

double	deltaReduction () const
	calculalte delta width reduction in nuclear medium More...

void	setCrossSections ()
	calculalte cross sections for each channel More...

Private Attributes
double	fPionMass
	pion mass in MeV More...

double	fPionMass2
	pion mass squared More...

double	fPionMomentum
	pion momentum in MeV More...

double	fPionEnergy
	pion total energy in MeV More...

double	fMomentumCMS
	momentum in CMS in MeV More...

double	fMomentumCMS2
	momentum in CMS squared More...

double	fInvariantMass
	inv mass = sqrt(s mandelstam) in MeV More...

double	fReducedHalfWidth
	reduced delta half width in MeV More...

double	fSelfEnergyTotal
	total delta self energy in MeV More...

double	fSelfEnergyAbsorption
	abs part of delta self energy in MeV More...

double	fDeltaPropagator2
	\|delta propagator\|^2 in MeV-2 More...

double	fFermiMomentum
	Fermi momentum in MeV. More...

double	fFermiEnergy
	Fermi energy in MeV. More...

double	fCouplingFactor
	(coupling constant / pion mass)^2 More...

Static Private Attributes
static const double	fCouplingConstant = 0.36 * 4.0 * kPi
	f*^2 More...

static const double	fNucleonMass = kNucleonMass * 1000.0
	average nucleon mass [MeV] More...

static const double	fNucleonMass2 = fNucleonMass * fNucleonMass
	average nucleon mass squared More...

static const double	fDeltaMass = 1232.0
	delta mass in MeV More...

static const double	fNormalDensity = 0.17
	normal nuclear density [fm-3] More...

static const double	fNormFactor = 197.327 * 197.327 * 10.0
	MeV^-2 -> mb. More...

static const double	fCoefSigma [fNChannels] = {-0.01334, 0.06889, 0.19753}
	s-wave parametrization eq. 3.7 More...

static const double	fCoefB [fNChannels] = {-0.01866, 0.06602, 0.21972}
	s-wave parametrization eq. 3.8 More...

static const double	fCoefD [fNChannels] = {-0.08229, 0.37062,-0.03130}
	s-wave parametrization eq. 3.8 More...

static const double	ImB0 = 0.035
	s-wave parametrization eq. 3.12 More...

static const double	fCoefCQ [fNChannels] = { -5.19, 15.35, 2.06}
	quasi-elastic term (eq. 2.21) More...

static const double	fCoefCA2 [fNChannels] = { 1.06, -6.64, 22.66}
	two-body absorption (eq. 2.21) More...

static const double	fCoefCA3 [fNChannels] = {-13.46, 46.17, -20.34}
	three-body absorption (eq. 2.21) More...

static const double	fCoefAlpha [fNChannels] = {0.382, -1.322, 1.466}
	alpha (eq. 2.21) More...

static const double	fCoefBeta [fNChannels] = {-0.038, 0.204, 0.613}
	beta (eq. 2.21) More...

Additional Inherited Members
Protected Member Functions inherited from INukeOset
void	setCrossSections (const int &pionPDG, const double &protonFraction)
	calculate avg cross sections according to proton / neutron fraction More...

Protected Attributes inherited from INukeOset
double	fNuclearDensity
	nuclear density in fm-3 More...

double	fPionKineticEnergy
	pion kinetic energy in MeV More...

double	fTotalCrossSection
	el+cex+abs cross section (averaged over proton / neutron fraction) More...

double	fCexCrossSection
	cex cross section (averaged over proton / neutron fraction) More...

double	fAbsorptionCrossSection
	absorption cross section (averaged over proton / neutron fraction) More...

double	fQelCrossSections [fNChannels]
	total qel (el+cex) cross section for each channel More...

double	fCexCrossSections [fNChannels]
	cex cross section for each channel More...

Static Protected Attributes inherited from INukeOset
static const unsigned int	fNChannels = 3
	number of possible channels: pi+n, pi+p, pi0 More...

Detailed Description

Formula-based implementation of Oset model.

Author: Tomasz Golan

Date: 2015

Warning: Applicable for pion with Tk < 350 MeV

Remarks: Based on E. Oset et al., Nucl. Phys. A484 (1988) 557-592

Definition at line 17 of file INukeOsetFormula.h.

Member Function Documentation

double INukeOsetFormula::deltaReduction ( ) const

private

calculalte delta width reduction in nuclear medium

related to Pauli blocking, see sec. 2.3

Definition at line 177 of file INukeOsetFormula.cxx.

 {
   // assuming nucleon at rest
   const double deltaEnergy = fPionEnergy + fNucleonMass;
   // nucleon energy in CMS
   const double energyCMS   = sqrt(fMomentumCMS * fMomentumCMS + fNucleonMass2);
   // fermiEnergy calculated from density
   const double mu0 = (deltaEnergy * energyCMS - fFermiEnergy * fInvariantMass) /
                      fPionMomentum / fMomentumCMS;
 
   // see eq. 2.13
   if (mu0 < -1.0) return 0.0;
   if (mu0 >  1.0) return 1.0;
 
   return (mu0*mu0*mu0 + mu0 + 2) / 4.0;
 }

void INukeOsetFormula::setCrossSections ( )

privatevirtual

calculalte cross sections for each channel

calculalte p- and s-wave absorption cross section
calculalte total qel (el+cex) p- and s-wave cross section
calculalte fraction of cex in total qel (based on isospin relation)

Implements INukeOset.

Definition at line 108 of file INukeOsetFormula.cxx.

 {
 
   // common part for all p-wave cross sections
   const double pXsecCommon = fNormFactor * fCouplingFactor * fDeltaPropagator2 *
                              fMomentumCMS2 / fPionMomentum;
 
   // ----- ABSORPTION ----- //
 
   // constant factor for p-wave absorption cross section
   static const double pAborptionFactor = 4.0 / 9.0;
 
   // absorption p-wave cross section (see eq. 2.24 and eq. 2.21)
   const double pXsecAbsorption = pAborptionFactor * pXsecCommon *
                                  fSelfEnergyAbsorption;
 
   // constant factor for s-wave absorption cross section
   static const double sAbsorptionFactor = 4.0 * kPi * 197.327 * 10.0 * ImB0;
 
   // absorption s-wave cross section (see sec. 3.3)
   const double sXsecAbsorption = sAbsorptionFactor / fPionMomentum * fNuclearDensity *
                                  (1.0 + fPionEnergy / 2.0 / fNucleonMass) / pow (fPionMass / 197.327, 4.0);
 
   // total absorption cross section coming from both s- and p-waves
   fAbsorptionCrossSection = pXsecAbsorption + sXsecAbsorption;
 
   // ----- TOTAL ----- //
 
   // el+cex p-wave cross section (will be multipled by proper factor later)
   const double pXsecTotalQel = pXsecCommon * fReducedHalfWidth;
 
   // see eq. 3.7
   const double ksi = (fInvariantMass - fNucleonMass - fPionMass) / fPionMass;
 
   // el+cex s-wave cross section
   const double sXsecTotalQel = quadraticFunction (ksi, fCoefSigma) /
                                fPionMass2 * fNormFactor;
 
   // see eq. 3.8
   const double B = quadraticFunction (ksi, fCoefB);
   const double D = quadraticFunction (ksi, fCoefD);
   const double A = 0.5 + 0.5*D;
   const double C = 1.0 - A;
 
   // see 3.4 (chi = 1 for neutron, chi = -1 for proton, chi = 0 for N=Z)
   const double sTotalQelFactor[fNChannels] = {C - B, A + B, 1.0};
 
   // see eq. 2.18 (chi = 1 for neutron, chi = -1 for proton, chi = 0 for N=Z)
   static const double pTotalQelFactor[fNChannels] = {2.0 / 9.0, 2.0 / 3.0,
                                                     5.0 / 9.0};
 
   // total cross section for each channel = qel_s + qel_p + absorption
   for (unsigned int i = 0; i < fNChannels; i++)
     fQelCrossSections[i] = pTotalQelFactor[i] * pXsecTotalQel + sTotalQelFactor[i] * sXsecTotalQel;
 
   // ----- CEX ----- //
 
   // see 2.18 (chi = 1 for neutron, chi = -1 for proton, chi = 0 for N=Z)
   static const double pCexFactor[fNChannels] = {4.0 / 27.0, 0.0, 5.0 / 27.0};
 
   // see eq. 3.4 (chi = 1 for neutron, chi = -1 for proton, chi = 0 for N=Z)
   const double sCexFactor[fNChannels] = {2.0 * C, 0.0, 2.0 * C};
 
   // total cex cross section coming from both s- and p-waves
   for (unsigned int i = 0; i < fNChannels; i++)
     fCexCrossSections[i] = pCexFactor[i] * pXsecTotalQel + sCexFactor[i] * sXsecTotalQel;
 }

void INukeOsetFormula::setDelta ( )

private

set up Delta

calculale Delta half width (in nuclear matter)
calculalte Delta self energy
calculalte Delta propagator

Definition at line 83 of file INukeOsetFormula.cxx.

 {
   static const double constFactor = 1.0 / 12.0 / kPi;
 
   fCouplingFactor = fCouplingConstant / fPionMass2; // (f*/m_pi)^2, e.g. eq. 2.6
 
   // see eq. 2.7 and sec. 2.3
   fReducedHalfWidth = constFactor * fCouplingFactor * fNucleonMass * fMomentumCMS *
                      fMomentumCMS2 / fInvariantMass * deltaReduction ();
 
   setSelfEnergy (); // calculate qel and absorption part of delta self-energy
 
   // real and imaginary part of delta denominator (see eq. 2.23)
   const double ReDelta = fInvariantMass - fDeltaMass;
   const double ImDelta = fReducedHalfWidth + fSelfEnergyTotal;
 
   fDeltaPropagator2 = 1.0 / (ReDelta * ReDelta + ImDelta * ImDelta);
 }

void INukeOsetFormula::setKinematics	(	const double &	pionTk,
		const bool &	isPi0
	)

private

do kinematics

set pointer to proper mass (charged vs neutral)
calculate energy, momentum (in LAB and CMS) and invariant mass

Definition at line 52 of file INukeOsetFormula.cxx.

 {
   if (isPi0)
   {
     fPionMass  = PDGLibrary::Instance()->Find(kPdgPi0)->Mass() * 1000.0; // [MeV]
     fPionMass2 = fPionMass * fPionMass;
   }
   else
   {
     fPionMass  = PDGLibrary::Instance()->Find(kPdgPiP)->Mass() * 1000.0; // [MeV]
     fPionMass2 = fPionMass * fPionMass;
   }
 
   fPionKineticEnergy = pionTk;
   fPionEnergy        = fPionKineticEnergy + fPionMass;
   fPionMomentum      = sqrt (fPionEnergy * fPionEnergy - fPionMass2);
 
   // assuming nucleon at rest
   fInvariantMass = sqrt (fPionMass2 + 2.0 * fNucleonMass * fPionEnergy +
                         fNucleonMass2);
 
   fMomentumCMS  = fPionMomentum * fNucleonMass / fInvariantMass;
   fMomentumCMS2 = fMomentumCMS * fMomentumCMS;
 }

void INukeOsetFormula::setNucleus ( const double & density )

private

set nuclear density and Fermi momentum / energy

$p_{F} = \left(\frac{3}{2}\pi\rho\right)^{1/3}$

Definition at line 38 of file INukeOsetFormula.cxx.

 {
   static const double constFactor = 3.0 / 2.0 * kPi * kPi;
 
   fNuclearDensity = density;
   fFermiMomentum  = pow (constFactor * fNuclearDensity, 1.0 / 3.0) * 197.327;
   fFermiEnergy    = sqrt (fFermiMomentum*fFermiMomentum + fNucleonMass2);
 }

void INukeOsetFormula::setSelfEnergy ( )

private

calculate delta self energy

based on eq. 2.21

Definition at line 195 of file INukeOsetFormula.cxx.

 {
   const double x = fPionKineticEnergy / fPionMass;
   const double densityFraction = fNuclearDensity / fNormalDensity;
 
   const double alpha = quadraticFunction (x, fCoefAlpha);
   const double  beta = quadraticFunction (x, fCoefBeta);
 
   double  absNN = quadraticFunction (x, fCoefCA2);
   double absNNN = quadraticFunction (x, fCoefCA3);
 
   absNN *= pow (densityFraction, beta);
 
   if (absNNN < 0.0) absNNN = 0.0; // it may happen for Tk < 50 MeV
   else absNNN *= pow (densityFraction, 2.0 * beta);
 
   // absNN and absNNN are multiplied by number of nucleons to get proper
   // cross section normalization
   //fSelfEnergyAbsorption = 2.0 * absNN + 3.0 * absNNN;
   fSelfEnergyAbsorption = absNN + absNNN;
 
   // this one goes to the delta propagator
   fSelfEnergyTotal = absNN + absNNN + quadraticFunction (x, fCoefCQ) * pow (densityFraction, alpha);
 }

void INukeOsetFormula::setupOset	(	const double &	density,
		const double &	pionTk,
		const int &	pionPDG,
		const double &	protonFraction
	)

virtual

use to set up Oset class (assign pion Tk, nuclear density etc)

set up nucleus
set up kinematics
set up Delta (width, propagator)
calculate cross sections

Implements INukeOset.

Definition at line 227 of file INukeOsetFormula.cxx.

 {
   setNucleus (density);
   setKinematics (pionTk, pionPDG == kPdgPi0);
   setDelta();
   setCrossSections ();
   INukeOset::setCrossSections (pionPDG, protonFraction);
 }

Member Data Documentation

const double INukeOsetFormula::fCoefAlpha = {0.382, -1.322, 1.466}

staticprivate

alpha (eq. 2.21)

delta parametrization coefficients (NuclPhys A468 (1987) 631-652)

Definition at line 43 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefB = {-0.01866, 0.06602, 0.21972}

staticprivate

s-wave parametrization eq. 3.8

Definition at line 36 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefBeta = {-0.038, 0.204, 0.613}

staticprivate

beta (eq. 2.21)

delta parametrization coefficients (NuclPhys A468 (1987) 631-652)

Definition at line 44 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefCA2 = { 1.06, -6.64, 22.66}

staticprivate

two-body absorption (eq. 2.21)

delta parametrization coefficients (NuclPhys A468 (1987) 631-652)

Definition at line 41 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefCA3 = {-13.46, 46.17, -20.34}

staticprivate

three-body absorption (eq. 2.21)

delta parametrization coefficients (NuclPhys A468 (1987) 631-652)

Definition at line 42 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefCQ = { -5.19, 15.35, 2.06}

staticprivate

quasi-elastic term (eq. 2.21)

delta parametrization coefficients (NuclPhys A468 (1987) 631-652)

Definition at line 40 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefD = {-0.08229, 0.37062,-0.03130}

staticprivate

s-wave parametrization eq. 3.8

Definition at line 37 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCoefSigma = {-0.01334, 0.06889, 0.19753}

staticprivate

s-wave parametrization eq. 3.7

Definition at line 35 of file INukeOsetFormula.h.

const double INukeOsetFormula::fCouplingConstant = 0.36 * 4.0 * kPi

staticprivate

f*^2

Definition at line 28 of file INukeOsetFormula.h.

double INukeOsetFormula::fCouplingFactor

private

(coupling constant / pion mass)^2

Definition at line 66 of file INukeOsetFormula.h.

const double INukeOsetFormula::fDeltaMass = 1232.0

staticprivate

delta mass in MeV

Definition at line 31 of file INukeOsetFormula.h.

double INukeOsetFormula::fDeltaPropagator2

private

|delta propagator|^2 in MeV-2

Definition at line 59 of file INukeOsetFormula.h.

double INukeOsetFormula::fFermiEnergy

private

Fermi energy in MeV.

Definition at line 63 of file INukeOsetFormula.h.

double INukeOsetFormula::fFermiMomentum

private

Fermi momentum in MeV.

Definition at line 62 of file INukeOsetFormula.h.

double INukeOsetFormula::fInvariantMass

private

inv mass = sqrt(s mandelstam) in MeV

Definition at line 53 of file INukeOsetFormula.h.

double INukeOsetFormula::fMomentumCMS

private

momentum in CMS in MeV

Definition at line 51 of file INukeOsetFormula.h.

double INukeOsetFormula::fMomentumCMS2

private

momentum in CMS squared

Definition at line 52 of file INukeOsetFormula.h.

const double INukeOsetFormula::fNormalDensity = 0.17

staticprivate

normal nuclear density [fm-3]

Definition at line 32 of file INukeOsetFormula.h.

const double INukeOsetFormula::fNormFactor = 197.327 * 197.327 * 10.0

staticprivate

MeV^-2 -> mb.

Definition at line 33 of file INukeOsetFormula.h.

const double INukeOsetFormula::fNucleonMass = kNucleonMass * 1000.0

staticprivate

average nucleon mass [MeV]

Definition at line 29 of file INukeOsetFormula.h.

const double INukeOsetFormula::fNucleonMass2 = fNucleonMass * fNucleonMass

staticprivate

average nucleon mass squared

Definition at line 30 of file INukeOsetFormula.h.

double INukeOsetFormula::fPionEnergy

private

pion total energy in MeV

Definition at line 50 of file INukeOsetFormula.h.

double INukeOsetFormula::fPionMass

private

pion mass in MeV

Definition at line 47 of file INukeOsetFormula.h.

double INukeOsetFormula::fPionMass2

private

pion mass squared

Definition at line 48 of file INukeOsetFormula.h.

double INukeOsetFormula::fPionMomentum

private

pion momentum in MeV

Definition at line 49 of file INukeOsetFormula.h.

double INukeOsetFormula::fReducedHalfWidth

private

reduced delta half width in MeV

Definition at line 56 of file INukeOsetFormula.h.

double INukeOsetFormula::fSelfEnergyAbsorption

private

abs part of delta self energy in MeV

Definition at line 58 of file INukeOsetFormula.h.

double INukeOsetFormula::fSelfEnergyTotal

private

total delta self energy in MeV

Definition at line 57 of file INukeOsetFormula.h.

const double INukeOsetFormula::ImB0 = 0.035

staticprivate

s-wave parametrization eq. 3.12

Definition at line 38 of file INukeOsetFormula.h.

The documentation for this class was generated from the following files:

Generator/src/Physics/HadronTransport/INukeOsetFormula.h
Generator/src/Physics/HadronTransport/INukeOsetFormula.cxx

Public Member Functions

Private Member Functions

Private Attributes

Static Private Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

Member Data Documentation