Detailed Description

This essentially follows the notation in Kubis97, except that our definitions of b and c follow their $\bar{b}$ and $\bar{c}$ , respectively.

Also discussed in Gaitanos04, where they take $m_{\delta}=980$ MeV.

The full Lagragian is:

${\cal L} = {\cal L}_{Dirac} + {\cal L}_{\sigma} + {\cal L}_{\omega} + {\cal L}_{\rho} + {\cal L}_{\delta}$

$\begin{eqnarray*} {\cal L}_{Dirac} &=& \bar{\Psi} \left[ i {{\partial}\!\!\!{\slash}} - g_{\omega} {{\omega}\!\!\!{\slash}} - \frac{g_{\rho}}{2} {{\vec{\rho}}\!\!\!{\slash}}~ \vec{\tau} - M + g_{\sigma} \sigma - \frac{e}{2} \left( 1 + \tau_3 \right) A_{\mu} \right] \Psi \nonumber \\ {\cal L}_{\sigma} &=& {\textstyle \frac{1}{2}} \left( \partial_{\mu} \sigma \right)^2 - {\textstyle \frac{1}{2}} m^2_{\sigma} \sigma^2 - \frac{b M}{3} \left( g_{\sigma} \sigma\right)^3 - \frac{c}{4} \left( g_{\sigma} \sigma\right)^4 \nonumber \\ {\cal L}_{\omega} &=& - {\textstyle \frac{1}{4}} f_{\mu \nu} f^{\mu \nu} + {\textstyle \frac{1}{2}} m^2_{\omega}\omega^{\mu}\omega_{\mu} + \frac{\zeta}{24} g_{\omega}^4 \left(\omega^\mu \omega_\mu\right)^2 \nonumber \\ {\cal L}_{\rho} &=& - {\textstyle \frac{1}{4}} \vec{B}_{\mu \nu} \cdot \vec{B}^{\mu \nu} + {\textstyle \frac{1}{2}} m^2_{\rho} \vec{\rho}^{~\mu} \cdot \vec{\rho}_{~\mu} + \frac{\xi}{24} g_{\rho}^4 \left(\vec{\rho}^{~\mu}\right) \cdot \vec{\rho}_{~\mu} + g_{\rho}^2 f (\sigma, \omega) \vec{\rho}^{~\mu} \cdot \vec{\rho}_{~\mu} \nonumber \\ \end{eqnarray*}$

where the additional terms are

${\cal L}_{\delta} = \bar{\Psi} \left( g_{\delta} \vec{\delta} \cdot \vec{\tau} \right) \Psi + \frac{1}{2} (\partial_{\mu} \vec{\delta})^2 - \frac{1}{2} m_{\delta}^2 \vec{\delta}^{~2}$

The new field equation for the delta meson is

$m_{\delta}^2 \delta = g_{\delta} (n_{s,p} - n_{s,n})$

Idea for Future:: Finish the finite temperature EOS

Definition at line 95 of file eos_had_rmf_delta.h.

Public Member Functions
virtual int	calc_e (fermion &ne, fermion &pr, thermo &lth)
	Equation of state as a function of density.

virtual int	calc_p (fermion &neu, fermion &p, double sig, double ome, double rho, double delta, double &f1, double &f2, double &f3, double &f4, thermo &th)
	Equation of state as a function of chemical potentials.

int	calc_temp_p (fermion &ne, fermion &pr, double temper, double sig, double ome, double lrho, double delta, double &f1, double &f2, double &f3, double &f4, thermo &lth)
	Finite temperature (unfinished)

virtual int	set_fields (double sig, double ome, double lrho, double delta)
	Set a guess for the fields for the next call to calc_e(), calc_p(), or saturation()

virtual int	saturation ()
	Calculate saturation properties for nuclear matter at the saturation density. More...

Public Member Functions inherited from o2scl::eos_had_rmf
virtual int	calc_p (fermion &ne, fermion &pr, thermo &lth)
	Equation of state as a function of chemical potential. More...

virtual int	calc_eq_p (fermion &neu, fermion &p, double sig, double ome, double rho, double &f1, double &f2, double &f3, thermo &th)
	Equation of state and meson field equations as a function of chemical potentials. More...

virtual int	calc_eq_temp_p (fermion &ne, fermion &pr, double temper, double sig, double ome, double rho, double &f1, double &f2, double &f3, thermo &th)
	Equation of state and meson field equations as a function of chemical potentials at finite temperature. More...

virtual int	calc_temp_p (fermion &ne, fermion &pr, double T, thermo &lth)
	Equation of state as a function of chemical potential. More...

int	calc_temp_e (fermion &ne, fermion &pr, double T, thermo &lth)
	Equation of state as a function of densities at finite temperature.

int	fix_saturation (double guess_cs=4.0, double guess_cw=3.0, double guess_b=0.001, double guess_c=-0.001)
	Calculate cs, cw, cr, b, and c from the saturation properties. More...

double	fesym_fields (double sig, double ome, double nb)
	Calculate symmetry energy assuming the field equations have already been solved. More...

double	fcomp_fields (double sig, double ome, double nb)
	Calculate the compressibility assuming the field equations have already been solved. More...

void	fkprime_fields (double sig, double ome, double nb, double &k, double &kprime)
	Calculate compressibilty and `kprime` assuming the field equations have already been solved. More...

int	field_eqs (size_t nv, const ubvector &x, ubvector &y)
	A function for solving the field equations. More...

int	field_eqsT (size_t nv, const ubvector &x, ubvector &y)
	A function for solving the field equations at finite temperature. More...

int	get_fields (double &sig, double &ome, double &lrho)
	Return the most recent values of the meson fields. More...

virtual const char *	type ()
	Return string denoting type ("eos_had_rmf")

	eos_had_rmf ()

virtual int	set_sat_mroot (mroot< mm_funct, ubvector, jac_funct > &mrx)
	Set class mroot object for use calculating saturation density.

void	check_naturalness (eos_had_rmf &re)
	Set the coefficients of a eos_had_rmf object to their limits from naturalness. More...

void	naturalness_limits (double value, eos_had_rmf &re)
	Provide the maximum values of the couplings assuming a limit on naturalness. More...

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_thermo (fermion_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	fesym_diff (double nb)
	Calculate symmetry energy of matter as energy of neutron matter minus the energy of nuclear matter in $\mathrm{fm}^{-1}$ . More...

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, $\partial^2 \varepsilon / \partial n_i n_j$ .

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...

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...

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	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 thermo &	get_thermo ()
	Get class thermo object.

Public Attributes
double	md
	The mass of the scalar-isovector field.

double	cd
	The coupling of the scalar-isovector field to the nucleons.

double	del
	The value of the scalar-isovector field.

Public Attributes inherited from o2scl::eos_had_rmf
size_t	calc_e_steps
	The number of separate calls to the solver that the `calc_e` functions take (default 20) More...

bool	calc_e_relative
	If true, solve for relative densities rather than absolute densities (default false) More...

bool	zm_mode
	Modifies method of calculating effective masses (default false)

int	verbose
	Verbosity parameter. More...

bool	err_nonconv
	If true, throw exceptions when the function calc_e() does not converge (default true)

double	mnuc
	The scale . More...

double	ms
	$\sigma$ mass (in $\mathrm{fm}^{-1}$ )

double	mw
	$\omega$ mass (in $\mathrm{fm}^{-1}$ )

double	mr
	$\rho$ mass (in $\mathrm{fm}^{-1}$ )

double	cs

double	cw

double	cr

double	b

double	c

double	zeta

double	xi

double	a1

double	a2

double	a3

double	a4

double	a5

double	a6

double	b1

double	b2

double	b3

mroot_hybrids	def_sat_mroot
	The default solver for calculating the saturation density. 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 $\mathrm{fm}^{-1}$ . More...

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 $\mathrm{fm}^{-1}$ .

bool	err_nonconv
	If true, call the error handler if msolve() or msolve_de() does not converge (default true)

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...

fermion	def_neutron
	The defaut neutron. More...

fermion	def_proton
	The defaut proton. More...

Public Attributes inherited from o2scl::eos_base
thermo	def_thermo
	The default thermo object.

Protected Member Functions
virtual int	calc_e_solve_fun (size_t nv, const ubvector &ex, ubvector &ey)
	The function for calc_e()

virtual int	zero_pressure (size_t nv, const ubvector &ex, ubvector &ey)
	Compute matter at zero pressure (for saturation())

Protected Member Functions inherited from o2scl::eos_had_rmf
int	fix_saturation_fun (size_t nv, const ubvector &x, ubvector &y)
	The function for fix_saturation()

virtual int	calc_temp_e_solve_fun (size_t nv, const ubvector &ex, ubvector &ey)
	The function for calc_temp_e()

int	calc_cr (double sig, double ome, double nb)
	Calculate the `cr` coupling given `sig` and `ome` at the density 'nb'. More...

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. More...

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. More...

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().

Private Member Functions
virtual int	set_fields (double sig, double ome, double lrho)
	Forbid setting the guesses to the fields unless all four fields are specified.

Additional Inherited Members
Public Types inherited from o2scl::eos_had_base
typedef boost::numeric::ublas::vector< double >	ubvector

Protected Attributes inherited from o2scl::eos_had_rmf
double	n_baryon
	Temporary baryon density.

double	n_charge
	Temporary charge density. More...

double	sigma

double	omega

double	rho

double	fe_temp
	Temperature for solving field equations at finite temperature.

bool	ce_neut_matter
	For calc_e(), if true, then solve for neutron matter.

bool	ce_prot_matter
	For calc_e(), if true, then solve for proton matter.

bool	guess_set
	True if a guess for the fields has been given.

mroot< mm_funct, ubvector, jac_funct > *	sat_mroot
	The solver to compute saturation properties.

double	ce_temp
	Temperature storage for calc_temp_e()

Protected Attributes inherited from o2scl::eos_had_temp_base
fermion_thermo *	fet
	Fermion thermodynamics (default is def_fet)

Protected Attributes inherited from o2scl::eos_had_base
mroot *	eos_mroot
	The EOS solver.

root *	sat_root
	The solver to compute saturation properties.

deriv_base *	sat_deriv
	The derivative object for saturation properties.

deriv_base *	sat_deriv2
	The second derivative object for saturation properties.

fermion *	neutron
	The neutron object.

fermion *	proton
	The proton object.

Protected Attributes inherited from o2scl::eos_base
thermo *	eos_thermo
	A pointer to the thermo object.

Member Function Documentation

◆ saturation()

virtual int o2scl::eos_had_rmf_delta::saturation ( )

virtual

This requires initial guesses to the chemical potentials, etc.

Reimplemented from o2scl::eos_had_rmf.

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

/home/awsteiner/wcs/o2scl/src/eos/eos_had_rmf_delta.h

Detailed Description

Public Member Functions

Public Attributes

Protected Member Functions

Private Member Functions

Additional Inherited Members

Member Function Documentation

◆ saturation()