Public Member Functions | Public Attributes | Protected Attributes | List of all members
o2scl::eos_had_apr Class Reference

EOS from Akmal, Pandharipande, and Ravenhall. More...

#include <eos_had_apr.h>

Inheritance diagram for o2scl::eos_had_apr:
o2scl::eos_had_temp_eden_base o2scl::eos_had_temp_base o2scl::eos_had_base o2scl::eos_base o2scl::eos_had_sym4_apr

Public Member Functions

 eos_had_apr ()
 Create an EOS object with the default parameter set ( $ A18 + \mathrm{UIX}^{*}+\delta v $).
 
Basic usage
virtual int calc_e (fermion &n, fermion &p, thermo &th)
 Equation of state as a function of density.
 
virtual int calc_temp_e (fermion &n, fermion &pr, double temper, thermo &th)
 Equation of state as a function of densities.
 
double fcomp (double nb)
 Compute the compressibility. More...
 
double fesym_diff (double nb)
 Calculate symmetry energy of matter as energy of neutron matter minus the energy of nuclear matter. More...
 
Other functions
int gradient_qij2 (double nn, double np, double &qnn, double &qnp, double &qpp, double &dqnndnn, double &dqnndnp, double &dqnpdnn, double &dqnpdnp, double &dqppdnn, double &dqppdnp)
 Calculate Q's for semi-infinite nuclear matter. More...
 
virtual const char * type ()
 Return string denoting type ("eos_had_apr")
 
- Public Member Functions inherited from o2scl::eos_had_temp_eden_base
virtual int calc_p (fermion &n, fermion &p, thermo &th)
 Equation of state as a function of the chemical potentials.
 
virtual int calc_temp_p (fermion &n, fermion &p, double T, thermo &th)
 Equation of state as a function of the chemical potentials at finite temperature.
 
- Public Member Functions inherited from o2scl::eos_had_temp_base
virtual int calc_liqgas_dens_temp_e (fermion &n1, fermion &p1, fermion &n2, fermion &p2, double T, thermo &th1, thermo &th2)
 Compute liquid-gas phase transition densities using eos_had_temp_base::calc_temp_e() . More...
 
virtual int calc_liqgas_temp_e (fermion &n1, fermion &p1, fermion &n2, fermion &p2, double nB, double Ye, double T, thermo &th1, thermo &th2, double &chi)
 Compute the liquid-gas phase transition using eos_had_temp_base::calc_temp_e() . More...
 
virtual int calc_liqgas_beta_temp_e (fermion &n1, fermion &p1, fermion &n2, fermion &p2, double nB, double T, thermo &th1, thermo &th2, double &Ye, double &chi)
 Compute the liquid-gas phase transition in beta-equilibrium using eos_had_temp_base::calc_temp_e() . More...
 
virtual double fesym_T (double nb, double T, double delta=0.0)
 Compute the symmetry energy at finite temperature.
 
virtual double fsyment_T (double nb, double T, double delta=0.0)
 Compute the symmetry entropy at finite temperature.
 
virtual double calc_temp_mun_e (double nn, double np, double T)
 Neutron chemical potential as a function of the densities.
 
virtual double calc_temp_mup_e (double nn, double np, double T)
 Proton chemical potential as a function of the densities.
 
virtual double calc_temp_nn_p (double mun, double mup, double T)
 Neutron density as a function of the chemical potentials.
 
virtual double calc_temp_np_p (double mun, double mup, double T)
 Proton density as a function of the chemical potentials.
 
double calc_fr (double nn, double np, double T)
 Compute the free energy as a function of the temperature and the densities.
 
virtual void f_number_suscept_T (double mun, double mup, double T, double &dPdnn, double &dPdnp, double &dPdpp)
 Compute the number susceptibilities as a function of the chemical potentials, $ \partial^2 P / \partial \mu_i \mu_j $ at a fixed temperature.
 
virtual void f_inv_number_suscept_T (double mun, double mup, double T, double &dednn, double &dednp, double &dedpp)
 Compute the 'inverse' number susceptibilities as a function of the densities, $ \partial^2 \varepsilon / \partial n_i n_j $ at a fixed temperature.
 
void check_en (fermion &n, fermion &p, double T, thermo &th, double &en_deriv, double &en_err)
 Check the entropy by computing the derivative numerically.
 
void check_mu_T (fermion &n, fermion &p, double T, thermo &th, double &mun_deriv, double &mup_deriv, double &mun_err, double &mup_err)
 Check the chemical potentials at finite temperature by computing the derivative numerically.
 
virtual void set_fermion_eval_thermo (fermion_eval_thermo &f)
 Computing finite-temperature integrals. More...
 
- Public Member Functions inherited from o2scl::eos_had_base
virtual double fcomp (double nb, double delta=0.0)
 Calculate the incompressibility in $ \mathrm{fm}^{-1} $ using calc_e() More...
 
virtual double fcomp_err (double nb, double delta, double &unc)
 Compute the incompressibility and its uncertainty. More...
 
virtual double feoa (double nb, double delta=0.0)
 Calculate the energy per baryon in $ \mathrm{fm}^{-1} $ using calc_e() More...
 
virtual double fesym (double nb, double delta=0.0)
 Calculate symmetry energy of matter in $ \mathrm{fm}^{-1} $ using calc_dmu_delta() . More...
 
virtual double fesym_err (double nb, double delta, double &unc)
 Calculate symmetry energy of matter and its uncertainty in $ \mathrm{fm}^{-1} $. More...
 
virtual double fesym_slope (double nb, double delta=0.0)
 The symmetry energy slope parameter in $ \mathrm{fm}^{-1} $. More...
 
virtual double fesym_curve (double nb, double delta=0.0)
 The curvature of the symmetry energy in $ \mathrm{fm}^{-1} $.
 
virtual double fesym_skew (double nb, double delta=0.0)
 The skewness of the symmetry energy in $ \mathrm{fm}^{-1} $.
 
virtual double feta (double nb)
 The strength parameter for quartic terms in the symmetry energy.
 
virtual double feta_prime (double nb)
 The derivative of the strength parameter for quartic terms in the symmetry energy.
 
virtual double fkprime (double nb, double delta=0.0)
 Calculate skewness of nuclear matter in $ \mathrm{fm}^{-1} $ using calc_e() More...
 
virtual double fmsom (double nb, double delta=0.0)
 Calculate reduced neutron effective mass using calc_e() More...
 
virtual double f_effm_neut (double nb, double delta=0.0)
 Neutron (reduced) effective mass.
 
virtual double f_effm_prot (double nb, double delta=0.0)
 Proton (reduced) effective mass.
 
virtual double f_effm_scalar (double nb, double delta=0.0)
 Scalar effective mass. More...
 
virtual double f_effm_vector (double nb, double delta=1.0)
 Vector effective mass. More...
 
virtual double fn0 (double delta, double &leoa)
 Calculate saturation density using calc_e() More...
 
virtual void f_number_suscept (double mun, double mup, double &dPdnn, double &dPdnp, double &dPdpp)
 Compute the number susceptibilities as a function of the chemical potentials, $ \partial^2 P / \partial \mu_i \mu_j $.
 
virtual void f_inv_number_suscept (double mun, double mup, double &dednn, double &dednp, double &dedpp)
 Compute the 'inverse' number susceptibilities as a function of the densities, $ \partial^2 \varepsilon / \partial n_i n_j $.
 
virtual void saturation ()
 Calculates some of the EOS properties at the saturation density. More...
 
double calc_mun_e (double nn, double np)
 Compute the neutron chemical potential at fixed density. More...
 
double calc_ed (double nn, double np)
 Compute the energy density as a function of the nucleon densities.
 
double calc_pr (double nn, double np)
 Compute the pressure as a function of the nucleon chemical potentials.
 
double calc_mup_e (double nn, double np)
 Compute the proton chemical potential at fixed density. More...
 
double calc_nn_p (double mun, double mup)
 Compute the neutron density at fixed chemical potential. More...
 
double calc_np_p (double mun, double mup)
 Compute the proton density at fixed chemical potential. More...
 
double calc_dmu_delta (double delta, double nb)
 Compute the difference between neutron and proton chemical potentials as a function of the isospin asymmetry. More...
 
double calc_musum_delta (double delta, double nb)
 Compute the sum of the neutron and proton chemical potentials as a function of the isospin asymmetry. More...
 
double calc_pressure_nb (double nb, double delta=0.0)
 Compute the pressure as a function of baryon density at fixed isospin asymmetry. More...
 
double calc_edensity_nb (double nb, double delta=0.0)
 Compute the energy density as a function of baryon density at fixed isospin asymmetry. More...
 
void const_pf_derivs (double nb, double pf, double &dednb_pf, double &dPdnb_pf)
 Compute derivatives at constant proton fraction.
 
double calc_press_over_den2 (double nb, double delta=0.0)
 Calculate pressure / baryon density squared in nuclear matter as a function of baryon density at fixed isospin asymmetry. More...
 
double calc_edensity_delta (double delta, double nb)
 Calculate energy density as a function of the isospin asymmetry at fixed baryon density. More...
 
int nuc_matter_p (size_t nv, const ubvector &x, ubvector &y, double nn0, double np0)
 Solve for the chemical potentials given the densities. More...
 
int nuc_matter_e (size_t nv, const ubvector &x, ubvector &y, double mun0, double mup0)
 Solve for the densities given the chemical potentials. More...
 
virtual void set_mroot (mroot<> &mr)
 Set class mroot object for use in calculating chemical potentials from densities. More...
 
virtual void set_sat_root (root<> &mr)
 Set class mroot object for use calculating saturation density. More...
 
virtual void set_sat_deriv (deriv_base<> &de)
 Set deriv_base object to use to find saturation properties.
 
virtual void set_sat_deriv2 (deriv_base<> &de)
 Set the second deriv_base object to use to find saturation properties. More...
 
virtual void set_n_and_p (fermion &n, fermion &p)
 Set neutron and proton.
 
void gradient_qij (fermion &n, fermion &p, thermo &th, double &qnn, double &qnp, double &qpp, double &dqnndnn, double &dqnndnp, double &dqnpdnn, double &dqnpdnp, double &dqppdnn, double &dqppdnp)
 Calculate coefficients for gradient part of Hamiltonian. More...
 
void check_mu (fermion &n, fermion &p, thermo &th, double &mun_deriv, double &mup_deriv, double &mun_err, double &mup_err)
 Check the chemical potentials by computing the derivatives numerically.
 
void check_den (fermion &n, fermion &p, thermo &th, double &nn_deriv, double &np_deriv, double &nn_err, double &np_err)
 Check the densities by computing the derivatives numerically.
 
- Public Member Functions inherited from o2scl::eos_base
virtual void set_thermo (thermo &th)
 Set class thermo object.
 
virtual const thermoget_thermo ()
 Get class thermo object.
 

Public Attributes

bool parent_method
 If true, use the methods from eos_had_base for fcomp() and fesym_diff() (default true) More...
 
- Public Attributes inherited from o2scl::eos_had_temp_base
fermion_eff def_fet
 Default fermion thermodynamics object.
 
- Public Attributes inherited from o2scl::eos_had_base
double eoa
 Binding energy (without the rest mass) in $ \mathrm{fm}^{-1} $.
 
double comp
 Compression modulus in $ \mathrm{fm}^{-1} $.
 
double esym
 Symmetry energy in $ \mathrm{fm}^{-1} $.
 
double n0
 Saturation density in $ \mathrm{fm}^{-3} $.
 
double msom
 Effective mass (neutron)
 
double kprime
 Skewness in $ \mathrm{fm}^{-1} $.
 
fermion def_neutron
 The defaut neutron. More...
 
fermion def_proton
 The defaut proton. More...
 
deriv_gsl def_deriv
 The default object for derivatives. More...
 
deriv_gsl def_deriv2
 The second default object for derivatives. More...
 
mroot_hybrids def_mroot
 The default solver. More...
 
root_cern def_sat_root
 The default solver for calculating the saturation density. More...
 
- Public Attributes inherited from o2scl::eos_base
thermo def_thermo
 The default thermo object.
 

Protected Attributes

fermion_nonrel nrf
 Non-relativistic fermion thermodyanmics.
 
int choice
 The variable indicating which parameter set is to be used.
 
double par [22]
 Storage for the parameters.
 
int lp
 An integer to indicate which phase was used in calc_e()
 
- Protected Attributes inherited from o2scl::eos_had_temp_base
fermion_eval_thermofet
 Fermion thermodynamics (default is def_fet)
 
- Protected Attributes inherited from o2scl::eos_had_base
mrooteos_mroot
 The EOS solver.
 
rootsat_root
 The solver to compute saturation properties.
 
deriv_basesat_deriv
 The derivative object for saturation properties.
 
deriv_basesat_deriv2
 The second derivative object for saturation properties.
 
fermionneutron
 The neutron object.
 
fermionproton
 The proton object.
 
- Protected Attributes inherited from o2scl::eos_base
thermoeos_thermo
 A pointer to the thermo object.
 

Choice of phase

static const int best =0
 use LDP for densities less than 0.16 and for higher densities, use the phase which minimizes energy (default)
 
static const int ldp =1
 LDP (no pion condensation)
 
static const int hdp =2
 HDP (pion condensation)
 
int pion
 Choice of phase (default best)
 
int last_phase ()
 Return the phase of the most recent call to calc_e() More...
 

Model selection

static const int a18_uix_deltav =1
 With three body forces and relativistic corrections.
 
static const int a18_uix =2
 With three body forces.
 
static const int a18_deltav =3
 With relativistic corrections.
 
static const int a18 =4
 No three body forces or relativistic corrections.
 
void select (int model_index=1)
 Select model. More...
 
double get_par (size_t n)
 Get the value of one of the model parameters.
 
void set_par (size_t n, double x)
 Set the value of one of the model parameters.
 

Additional Inherited Members

- Public Types inherited from o2scl::eos_had_base
typedef boost::numeric::ublas::vector< double > ubvector
 
- Protected Member Functions inherited from o2scl::eos_had_temp_base
int nuc_matter_temp_e (size_t nv, const ubvector &x, ubvector &y, double nn0, double np0, double T)
 Solve for nuclear matter at finite temperature given density.
 
int nuc_matter_temp_p (size_t nv, const ubvector &x, ubvector &y, double mun0, double mup0, double T)
 Solve for nuclear matter at finite temperature given mu.
 
int liqgas_dens_solve (size_t nv, const ubvector &x, ubvector &y, fermion &n1, fermion &p1, fermion &n2, fermion &p2, double T, thermo &th1, thermo &th2)
 Solve for the liquid gas phase transition as a function of the densities.
 
int liqgas_solve (size_t nv, const ubvector &x, ubvector &y, fermion &n1, fermion &p1, fermion &n2, fermion &p2, double nB0, double Ye0, double T, thermo &th1, thermo &th2)
 Solve for the liquid-gas phase transition at fixed baryon density and electron fraction.
 
int liqgas_beta_solve (size_t nv, const ubvector &x, ubvector &y, fermion &n1, fermion &p1, fermion &n2, fermion &p2, double nB0, double T, thermo &th1, thermo &th2, fermion &e)
 Solve for the liquid-gas phase transition in beta-equilibrium.
 
double calc_entropy_delta (double delta, double nb, double T)
 Compute the entropy.
 
double calc_dmu_delta_T (double delta, double nb, double T)
 Compute the difference between the neutron and proton chemical potentials.
 
- Protected Member Functions inherited from o2scl::eos_had_base
double t1_fun (double barn)
 Compute t1 for gradient_qij().
 
double t2_fun (double barn)
 Compute t2 for gradient_qij().
 

Detailed Description

The EOS of Akmal, Pandharipande, and Ravenhall, from Akmal98eo (APR).

The Hamiltonian is:

\[ {\cal H}_{APR} = {\cal H}_{kin} + {\cal H}_{pot} \]

\[ {\cal H}_{kin} = \left\{ \frac{\hbar^2}{2 m} + \left[ p_3 + \left( 1 - x \right) p_5 \right] n e^{-p_4 n} \right\} \tau_n + \left\{ \frac{\hbar^2}{2 m} + \left[ p_3 + x p_5 \right] n e^{-p_4 n} \right\} \tau_p \]

\[ {\cal H}_{pot} = g_1 \left[ 1 - \left( 1 - 2 x \right)^2 \right] + g_2 \left( 1 - 2 x \right)^2 \]

The following are definitions for $ g_i $ in the low-density phase (LDP) or the high-density phase (HDP):

\[ g_{1,LDP} = -n^2 \left[ p_1 + p_2 n + p_6 n^2 + \left( p_{10} + p_{11} n \right) e^{-p_9^2 n^2} \right] \]

\[ g_{2,LDP} = -n^2 \left[ \frac{p_{12}}{n} + p_7 + p_8 n + p_{13} e^{-p_9^2 n^2} \right] \]

\[ g_{1,HDP} = g_{1,LDP} -n^2 \left[ p_{17} \left( n - p_{19} \right) + p_{21} \left( n - p_{19} \right)^2 e^{p_{18} \left( n - p_{19} \right) } \right] \]

\[ g_{2,HDP} = g_{2,LDP} -n^2 \left[ p_{15} \left( n - p_{20} \right) + p_{14} \left( n - p_{20} \right)^2 e^{p_{16} \left( n - p_{20} \right)} \right] \]

Note
APR seems to have been designed to be used with non-relativistic neutrons and protons with equal masses of 939 MeV. This gives a saturation density very close to 0.16.

The variables $ \nu_n$ and $ \nu_p$ contain the expressions $ (-\mu_n+V_n)/T $ and $ (-\mu_p+V_p)/T $ respectively, where $ V $ is the potential part of the single particle energy for particle i (i.e. the derivative of the Hamiltonian w.r.t. density while energy density held constant). Equivalently, $ \nu_n$ is just $ -k_{F_n}^2/ 2 m^{*} $.

The selection between the LDP and HDP is controlled by the value of pion. The default is to use the LDP at densities below 0.16 $ \mathrm{fm}^{-3} $, and for larger densities to just use whichever minimizes the energy.

The finite temperature approximations from Prakash97 are used in testing.

Note
Since this EOS uses the effective masses and chemical potentials in the fermion class, the values of part::non_interacting for neutrons and protons are set to false in many of the functions.
The parameter array is unit indexed, so that par[0] is unused. This choice makes the connection between the code and the paper a bit more transparent.
Idea for Future:

There might be room to improve the testing of the finite temperature part a bit.

There is some repetition between calc_e() and calc_temp_e() that possibly could be removed.

Include the analytic relations from Constantinou et al.

Definition at line 124 of file eos_had_apr.h.

Member Function Documentation

◆ fcomp()

double o2scl::eos_had_apr::fcomp ( double  nb)

See general notes at eos_had_base::fcomp(). This computes the compressibility (at fixed proton fraction = 0.5) exactly, unless parent_method is true in which case the derivative is taken numerically in eos_had_base::fcomp().

◆ fesym_diff()

double o2scl::eos_had_apr::fesym_diff ( double  nb)
virtual

This function returns the energy per baryon of neutron matter minus the energy per baryon of nuclear matter. This will deviate significantly from the results from fesym() only if the dependence of the symmetry energy on $ \delta $ is not quadratic.

Reimplemented from o2scl::eos_had_base.

◆ gradient_qij2()

int o2scl::eos_had_apr::gradient_qij2 ( double  nn,
double  np,
double &  qnn,
double &  qnp,
double &  qpp,
double &  dqnndnn,
double &  dqnndnp,
double &  dqnpdnn,
double &  dqnpdnp,
double &  dqppdnn,
double &  dqppdnp 
)

For general discussion, see the documentation to eos_had_base::qs().

For APR, we set $ x_1=x_2=0 $ so that $ Q_i=P_i/2 $ and then

\begin{eqnarray*} P_1 &=& \left(\frac{1}{2} p_3-p_5 \right) e^{-p_4 n} \nonumber \\ P_2 &=& \left(\frac{1}{2} p_3+p_5 \right) e^{-p_4 n} \end{eqnarray*}

This gives

\begin{eqnarray*} Q_{nn}&=&\frac{1}{4} e^{-p_4 \rho} \left[ -6 p_5 - p_4 (p_3 - 2 p_5) (n_n + 2 n_p) \right] \nonumber \\ Q_{np}&=&\frac{1}{8} e^{-p_4 \rho} \left[ 4 (p_3 - 4 p_5) - 3 p_4 (p_3 - 2 p_5) (n_n + n_p)\right] \nonumber \\ Q_{pp}&=&\frac{1}{4} e^{-p_4 \rho} \left[ -6 p_5 - p_4 (p_3 - 2 p_5) (n_p + 2 n_n) \right] \end{eqnarray*}

◆ last_phase()

int o2scl::eos_had_apr::last_phase ( )
inline

This function always returns either eos_had_apr::ldp or eos_had_apr::hdp .

Definition at line 170 of file eos_had_apr.h.

◆ select()

void o2scl::eos_had_apr::select ( int  model_index = 1)

Valid values for model_index are:
1 - A18+UIX*+deltav (preferred by Akmal, et. al. - this is the default)
2 - A18+UIX*
3 - A18+deltav
4 - A18
If any other integer is given, the error handler is called.

Note
This cannot be virtual because it is called by the constructor

Member Data Documentation

◆ parent_method

bool o2scl::eos_had_apr::parent_method

This can be set to true to check the difference in the compressibility and symmety energy between the exact expressions and the numerical values from class o2scl::eos_had_base.

Idea for Future:
This variable is probably unnecessary, as the syntax
ccout << apr.eos_had_base::fcomp(0.16) << endl;

works just as well.

Definition at line 302 of file eos_had_apr.h.


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

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).