Implementation of the back-shifted Fermi gas nuclear level density model. More...

#include <BackshiftedFermiGasModel.hh>

Inheritance diagram for marley::BackshiftedFermiGasModel:

Public Member Functions
	BackshiftedFermiGasModel (int Z, int A)

virtual double	level_density (double Ex) override

virtual double	level_density (double Ex, int two_J) override

virtual double	level_density (double Ex, int two_J, marley::Parity Pi) override

Protected Member Functions
double	compute_sigma_F2 (double Ex, double a)
	Helper function used when evaluating the spin cutoff parameter.

Protected Attributes
int	A_
	mass number for this nuclide

double	a_tilde_
	asymptotic level density parameter (MeV^-1)

double	Delta_BFM_
	excitation energy shift (MeV)

double	delta_W_
	shell correction energy (MeV)

double	gamma_
	damping parameter (MeV^-1)

double	sigma_
	spin cut-off parameter

double	sigma_d_global_
	global fit for discrete-region spin cut-off parameter

double	Sn_
	neutron separation energy (MeV)

int	Z_
	atomic number for this nuclide

Detailed Description

Implementation of the back-shifted Fermi gas nuclear level density model.

Level density parameters used in this class are based on the global fits originally performed in A. J. Koning, et al., Nucl. Phys. A810 (2008) pp. 13-76. In addition to being used in MARLEY, this model has been adopted as one of the available options in the RIPL-3 data library and the TALYS nuclear code.

Definition at line 31 of file BackshiftedFermiGasModel.hh.

Constructor & Destructor Documentation

◆ BackshiftedFermiGasModel()

marley::BackshiftedFermiGasModel::BackshiftedFermiGasModel	(	int	Z,
		int	A )

Create a back-shifted Fermi gas model object for a specific nuclide. This constructor will use global fits to initialize all level density parameters.

Parameters

Z	atomic number of the desired nuclide
A	mass number of the desired nuclide

Todo: Add other constructors to the BackshiftedFermiGasModel class to allow the user to set the level density parameters based on local fits

Definition at line 23 of file BackshiftedFermiGasModel.cc.

  : Z_(Z), A_(A)
{
  int N = A_ - Z_;
  double A_third = std::pow(A_, 1.0/3.0);
 
  // Parameters from global level density fit
  static constexpr double alpha = 0.0722396; // MeV^(-1)
  static constexpr double beta = 0.195267; // MeV^(-1)
  static constexpr double gamma_1 = 0.410289; // MeV(-1)
  static constexpr double delta_global = 0.173015; // MeV
 
  // Asymptotic level density parameter
  a_tilde_ = alpha*A_ + beta*std::pow(A_third, 2);
 
  // Damping parameter
  gamma_ = gamma_1 / A_third;
 
  const marley::MassTable& mt = marley::MassTable::Instance();
 
  // Shell correction energy
  delta_W_ = mt.get_mass_excess(Z_, A_)
    - mt.liquid_drop_model_mass_excess(Z_, A_);
 
  // Energy shift
  Delta_BFM_ = delta_global;
  bool z_odd = Z % 2;
  bool n_odd = N % 2;
  // There will be no change to the energy shift if the nucleus is odd-even
  if (z_odd && n_odd) Delta_BFM_ += -12/std::sqrt(A_);
  else if (!z_odd && !n_odd) Delta_BFM_ += 12/std::sqrt(A_);
 
  // Spin cut-off parameter
  sigma_d_global_ = 0.83*std::pow(A_, 0.26);
  Sn_ = mt.get_fragment_separation_energy(Z_, A_, marley_utils::NEUTRON);
}

References A_, a_tilde_, Delta_BFM_, delta_W_, gamma_, marley::MassTable::get_fragment_separation_energy(), marley::MassTable::get_mass_excess(), marley::MassTable::Instance(), marley::MassTable::liquid_drop_model_mass_excess(), sigma_d_global_, Sn_, and Z_.

Member Function Documentation

◆ compute_sigma_F2()

double marley::BackshiftedFermiGasModel::compute_sigma_F2	(	double	Ex,
		double	a )

inlineprotected

Helper function used when evaluating the spin cutoff parameter.

Definition at line 59 of file BackshiftedFermiGasModel.hh.

                                                          {
        double U = Ex - Delta_BFM_;
 
        double sigma_F2 = 0.01389 * std::pow(A_, 5.0/3.0)
          * std::sqrt(a * U) / a_tilde_;
 
        return sigma_F2;
      }

References A_, a_tilde_, and Delta_BFM_.

Referenced by level_density().

◆ level_density() [1/3]

double marley::BackshiftedFermiGasModel::level_density ( double Ex )

overridevirtual

Nuclear level density \( \rho(E_x) \) including all spins and parities.

Parameters

Ex	Excitation energy in MeV

Returns: Level density in MeV^-1

Note: Calls to this function update the spin cut-off parameter sigma_

Todo: Replace Ed here with database of local fits taken from RIPL-3 or TALYS

Todo: Reconsider method used for handling spin cut-off parameter calculation for U <= Ed.

Implements marley::LevelDensityModel.

Definition at line 77 of file BackshiftedFermiGasModel.cc.

                                                              {
 
  // Effective excitation energy
  double U = Ex - Delta_BFM_;
 
  // Level density parameter
  double a;
  if (U <= 0) { // Equivalently, Ex <= Delta_BFM_
    // Use first-order Taylor expansion for small energies
    a = a_tilde_ * (1 + gamma_ * delta_W_);
  }
  else {
    a = a_tilde_ * (1 + (delta_W_ / U) * (1 - std::exp(-gamma_ * U)));
  }
  const double Ed = 0.;
 
  // Compute the spin cut-off parameter sigma_
 
  // To avoid numerical problems, we will always use the discrete spin cutoff
  // parameter for U <= Ed.
  // The TALYS manual suggests using this for Ex <= Ed, but this isn't a huge
  // change. The actual TALYS code may make this same choice.
  if (U <= Ed) sigma_ = sigma_d_global_;
  else {
 
    if (Ex >= Sn_) {
      double sigma_F2 = compute_sigma_F2(Ex, a);
      sigma_ = std::sqrt(sigma_F2);
    }
    else {
 
      // sigma_F2 evaluated for Ex == Sn for the linear interpolation
      double sigma_F2_Sn = compute_sigma_F2(Sn_, a);
 
      // Ed < Ex < Sn_
      double sigma_d2 = std::pow(sigma_d_global_, 2);
      sigma_ = std::sqrt(sigma_d2 + (Ex - Ed)
        * (sigma_F2_Sn - sigma_d2) / (Sn_ - Ed));
    }
  }
 
  // For very small excitation energies, take the limit of the total
  // level density as U -> 0 to prevent numerical issues.
  if (U <= 0) {
    static const double exp1 = std::exp(1);
    return exp1 * a / (12 * sigma_);
  }
 
  double aU = a * U;
  double sqrt_aU = std::sqrt(aU);
  return std::pow(12 * sigma_ * (std::sqrt(2 * sqrt_aU)*U*std::exp(-2 * sqrt_aU)
    + std::exp(-aU - 1)/a), -1);
}

References a_tilde_, compute_sigma_F2(), Delta_BFM_, delta_W_, gamma_, sigma_, sigma_d_global_, and Sn_.

Referenced by level_density(), and level_density().

◆ level_density() [2/3]

double marley::BackshiftedFermiGasModel::level_density	(	double	Ex,
		int	two_J )

overridevirtual

Level density \( \rho(E_x, J) \) for a specific nuclear spin.

Parameters

Ex	Excitation energy in MeV
two_J	Two times the nuclear spin

Returns: Level density in MeV^-1

Implements marley::LevelDensityModel.

Definition at line 67 of file BackshiftedFermiGasModel.cc.

                                                                         {
  double rho = level_density(Ex);
  // Spin-cutoff parameter sigma_ is updated by previous call to
  // this->level_density(Ex)
  double two_sigma2 = 2 * std::pow(sigma_, 2);
  return ((two_J + 1) / two_sigma2) * std::exp(-0.25 * std::pow(two_J + 1, 2)
    / two_sigma2) * rho;
}

References level_density(), and sigma_.

◆ level_density() [3/3]

double marley::BackshiftedFermiGasModel::level_density	(	double	Ex,
		int	two_J,
		marley::Parity	Pi )

overridevirtual

Level density \( \rho(E_x, J, \Pi) \) for a specific nuclear spin and parity.

Parameters

Ex	Excitation energy in MeV
two_J	Two times the nuclear spin
Pi	The nuclear parity

Returns: Level density in MeV^-1

The current implementation assumes parity equipartition.

Implements marley::LevelDensityModel.

Definition at line 61 of file BackshiftedFermiGasModel.cc.

{
  return 0.5 * level_density(Ex, two_J);
}

References level_density().

Member Data Documentation

◆ A_

int marley::BackshiftedFermiGasModel::A_

protected

mass number for this nuclide

Definition at line 69 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), and compute_sigma_F2().

◆ a_tilde_

double marley::BackshiftedFermiGasModel::a_tilde_

protected

asymptotic level density parameter (MeV^-1)

Definition at line 73 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), compute_sigma_F2(), and level_density().

◆ Delta_BFM_

double marley::BackshiftedFermiGasModel::Delta_BFM_

protected

excitation energy shift (MeV)

Definition at line 76 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), compute_sigma_F2(), and level_density().

◆ delta_W_

double marley::BackshiftedFermiGasModel::delta_W_

protected

shell correction energy (MeV)

Definition at line 75 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), and level_density().

◆ gamma_

double marley::BackshiftedFermiGasModel::gamma_

protected

damping parameter (MeV^-1)

Definition at line 74 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), and level_density().

◆ sigma_

double marley::BackshiftedFermiGasModel::sigma_

protected

spin cut-off parameter

Definition at line 71 of file BackshiftedFermiGasModel.hh.

Referenced by level_density(), and level_density().

◆ sigma_d_global_

double marley::BackshiftedFermiGasModel::sigma_d_global_

protected

global fit for discrete-region spin cut-off parameter

Definition at line 79 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), and level_density().

◆ Sn_

double marley::BackshiftedFermiGasModel::Sn_

protected

neutron separation energy (MeV)

Definition at line 80 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel(), and level_density().

◆ Z_

int marley::BackshiftedFermiGasModel::Z_

protected

atomic number for this nuclide

Definition at line 68 of file BackshiftedFermiGasModel.hh.

Referenced by BackshiftedFermiGasModel().

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

/home/sjg/Desktop/marley/include/marley/BackshiftedFermiGasModel.hh
/home/sjg/Desktop/marley/src/BackshiftedFermiGasModel.cc

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ BackshiftedFermiGasModel()

Member Function Documentation

◆ compute_sigma_F2()

◆ level_density() [1/3]

◆ level_density() [2/3]

◆ level_density() [3/3]

Member Data Documentation

◆ A_

◆ a_tilde_

◆ Delta_BFM_

◆ delta_W_

◆ gamma_

◆ sigma_

◆ sigma_d_global_

◆ Sn_

◆ Z_