gen_potential_eos Class Reference

Generalized potential model equation of state. More...

#include <gen_potential_eos.h>

Inheritance diagram for gen_potential_eos:

hadronic_eos_eden hadronic_eos eos mdi4_eos

Detailed Description

Generalized potential model equation of state.

The single particle energy is defined by the functional derivative of the energy density with respect to the distribution function

\[ e_{\tau} = \frac{\delta \varepsilon}{\delta f_{\tau}} \]

The effective mass is defined by

\[ \frac{m^{*}}{m} = \left( \frac{m}{k} \frac{d e_{\tau}}{d k} \right)^{-1}_{k=k_F} \]

In all of the models, the kinetic energy density is $\tau_n+\tau_p$ where

\[ \tau_i = \frac{2}{(2 \pi)^3} \int d^3 k~ \left(\frac{k^2}{2 m}\right)f_i(k,T) \]

and the number density is

\[ \rho_i = \frac{2}{(2 \pi)^3} \int d^3 k~f_i(k,T) \]

When form == mdi_form or gbd_form, the potential energy density is given by Das03 :

\[ V(\rho,\delta) = \frac{Au}{\rho_0} \rho_n \rho_p + \frac{A_l}{2 \rho_0} \left(\rho_n^2+\rho_p^2\right)+ \frac{B}{\sigma+1} \frac{\rho^{\sigma+1}}{\rho_0^{\sigma}} \left(1-x \delta^2\right)+V_{mom}(\rho,\delta) \]

where $\delta=1-2 \rho_p/(\rho_n+\rho_p)$. If form == mdi_form, then

\[ V_{mom}(\rho,\delta)= \frac{1}{\rho_0} \sum_{\tau,\tau^{\prime}} C_{\tau,\tau^{\prime}} \int \int d^3 k d^3 k^{\prime} \frac{f_{\tau}(\vec{k}) f_{{\tau}^{\prime}} (\vec{k}^{\prime})} {1-(\vec{k}-\vec{k}^{\prime})^2/\Lambda^2} \]

where $C_{1/2,1/2}=C_{-1/2,-1/2}=C_{\ell}$ and $C_{1/2,-1/2}=C_{-1/2,1/2}=C_{u}$. Otherwise if form == gbd_form, then

\[ V_{mom}(\rho,\delta)= \frac{1}{\rho_0}\left[ C_{\ell} \left( \rho_n g_n + \rho_p g_p \right)+ C_u \left( \rho_n g_p + \rho_p g_n \right) \right] \]

where

\[ g_i=\frac{\Lambda^2}{\pi^2}\left[ k_{F,i}-\Lambda \mathrm{tan}^{-1} \left(k_{F,i}/\Lambda\right) \right] \]

Otherwise, if form == bgbd_form, bpalb_form or sl_form, then the potential energy density is given by Bombaci01 :

\[ V(\rho,\delta) = V_A+V_B+V_C \]

\[ V_A = \frac{2 A}{3 \rho_0} \left[ \left(1+\frac{x_0}{2}\right)\rho^2- \left(\frac{1}{2}+x_0\right)\left(\rho_n^2+\rho_p^2\right)\right] \]

\[ V_B=\frac{4 B}{3 \rho_0^{\sigma}} \frac{T}{1+4 B^{\prime} T / \left(3 \rho_0^{\sigma-1} \rho^2\right)} \]

where

\[ T = \rho^{\sigma-1} \left[ \left( 1+\frac{x_3}{2} \right) \rho^2 - \left(\frac{1}{2}+x_3\right)\left(\rho_n^2+\rho_p^2\right)\right] \]

The term $V_C$ is:

\[ V_C=\sum_{i=1}^{i_{\mathrm{max}}} \frac{4}{5} \left(C_{i}+2 z_i\right) \rho (g_{n,i}+g_{p,i})+\frac{2}{5}\left(C_i -8 z_i\right) (\rho_n g_{n,i} + \rho_p g_{p,i}) \]

where

\[ g_{\tau,i} = \frac{2}{(2 \pi)^3} \int d^3 k f_{\tau}(k,T) g_i(k) \]

For form == bgbd_form or form == bpalb_form, the form factor is given by

\[ g_i(k) = \left(1+\frac{k^2}{\Lambda_i^2}\right)^{-1} \]

while for form == sl_form, the form factor is given by

\[ g_i(k) = 1-\frac{k^2}{\Lambda_i^2} \]

where $\Lambda_1$ is specified in the parameter Lambda when necessary.

See Mathematica notebook at gen_potential_eos.nb, and gen_potential_eos.ps.

Bug:
The BGBD EOS doesn't work and the effective mass for the GBD EOS doesn't work
Idea for future:
Calculate the chemical potentials analytically

Definition at line 172 of file gen_potential_eos.h.


The mode for the energy() function [protected]



int mode
static const int nmode = 1
static const int pmode = 2
static const int normal = 0

Public Member Functions

virtual int calc_e (fermion &ne, fermion &pr, thermo &lt)
 Equation of state as a function of density.
int set_mu_deriv (deriv< int, funct< int > > &de)
 Set the derivative object to calculate the chemical potentials.
virtual const char * type ()
 Return string denoting type ("gen_potential_eos").

Data Fields

int form
 Form of potential.
gsl_deriv< int,funct< int > > def_mu_deriv
 The default derivative object for calculating chemical potentials.
nonrel_fermion def_nr_neutron
 Default nonrelativistic neutron.
nonrel_fermion def_nr_proton
 Default nonrelativistic proton.
The parameters for the various interactions
double x
double Au
double Al
double rho0
double B
double sigma
double Cl
double Cu
double Lambda
double A
double x0
double x3
double Bp
double C1
double z1
double Lambda2
double C2
double z2
double bpal_esym
int sym_index

Static Public Attributes

static const int mdi_form = 1
 The "momentum-dependent-interaction" form.
static const int bgbd_form = 2
 The modifed GBD form.
static const int bpalb_form = 3
 The form from Prakash88 as formulated in Bombaci01.
static const int sl_form = 4
 The "SL" form. See Bombaci01.
static const int gbd_form = 5
 The Gale, Bertsch, Das Gupta from Gale87.
static const int bpal_form = 6
 The form from Prakash88.

Protected Member Functions

double mom_integral (double pft, double pftp)
 Compute the momentum integral for mdi_form.
double energy (double x)
 Compute the energy.

Protected Attributes

bool mu_deriv_set
 True of the derivative object has been set.
deriv< int,funct< int > > * mu_deriv_ptr
 The derivative object.

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

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