eos_had_potential.h
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1 /*
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3 
4  Copyright (C) 2006-2017, Andrew W. Steiner
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22 */
23 /** \file eos_had_potential.h
24  \brief File defining \ref o2scl::eos_had_potential
25 */
26 #ifndef O2SCL_GEN_POTENTIAL_EOS_H
27 #define O2SCL_GEN_POTENTIAL_EOS_H
28 
29 #include <iostream>
30 #include <string>
31 #include <cmath>
32 #include <o2scl/constants.h>
33 #include <o2scl/mroot.h>
34 #include <o2scl/eos_had_base.h>
35 #include <o2scl/part.h>
36 #include <o2scl/deriv_gsl.h>
37 #include <o2scl/fermion_nonrel.h>
38 #include <cstdlib>
39 
40 #ifndef DOXYGEN_NO_O2NS
41 namespace o2scl {
42 #endif
43 
44  /** \brief Generalized potential model equation of state
45 
46  The single particle energy is defined by the functional derivative
47  of the energy density with respect to the distribution function
48  \f[
49  e_{\tau} = \frac{\delta \varepsilon}{\delta f_{\tau}}
50  \f]
51 
52  The effective mass is defined by
53  \f[
54  \frac{m^{*}}{m} = \left( \frac{m}{k}
55  \frac{d e_{\tau}}{d k}
56  \right)^{-1}_{k=k_F}
57  \f]
58 
59  In all of the models, the kinetic energy density is
60  \f$\tau_n+\tau_p\f$ where
61  \f[
62  \tau_i = \frac{2}{(2 \pi)^3} \int d^3 k~
63  \left(\frac{k^2}{2 m}\right)f_i(k,T)
64  \f]
65  and the number density is
66  \f[
67  \rho_i = \frac{2}{(2 \pi)^3} \int d^3 k~f_i(k,T)
68  \f]
69 
70  When \ref form is equal to \ref mdi_form or
71  \ref gbd_form, the potential energy
72  density is given by \ref Das03 :
73  \f[
74  V(\rho,\delta) = \frac{Au}{\rho_0} \rho_n \rho_p +
75  \frac{A_l}{2 \rho_0} \left(\rho_n^2+\rho_p^2\right)+
76  \frac{B}{\sigma+1} \frac{\rho^{\sigma+1}}{\rho_0^{\sigma}}
77  \left(1-x \delta^2\right)+V_{mom}(\rho,\delta)
78  \f]
79  where \f$\delta=1-2 \rho_p/(\rho_n+\rho_p)\f$.
80  If \ref form is equal to \ref mdi_form, then
81  \f[
82  V_{mom}(\rho,\delta)=
83  \frac{1}{\rho_0}
84  \sum_{\tau,\tau^{\prime}} C_{\tau,\tau^{\prime}}
85  \int \int
86  d^3 k d^3 k^{\prime}
87  \frac{f_{\tau}(\vec{k}) f_{{\tau}^{\prime}} (\vec{k}^{\prime})}
88  {1-(\vec{k}-\vec{k}^{\prime})^2/\Lambda^2}
89  \f]
90  where \f$C_{1/2,1/2}=C_{-1/2,-1/2}=C_{\ell}\f$ and
91  \f$C_{1/2,-1/2}=C_{-1/2,1/2}=C_{u}\f$. Later
92  parameterizations in this form are given in \ref Chen05.
93 
94  Otherwise if \ref form is equal to \ref gbd_form, then
95  \f[
96  V_{mom}(\rho,\delta)=
97  \frac{1}{\rho_0}\left[
98  C_{\ell} \left( \rho_n g_n + \rho_p g_p \right)+
99  C_u \left( \rho_n g_p + \rho_p g_n \right)
100  \right]
101  \f]
102  where
103  \f[
104  g_i=\frac{\Lambda^2}{\pi^2}\left[ k_{F,i}-\Lambda
105  \mathrm{tan}^{-1} \left(k_{F,i}/\Lambda\right) \right]
106  \f]
107 
108  Otherwise, if \ref form is equal to \ref bgbd_form, \ref bpal_form
109  or \ref sl_form, then the potential energy density is
110  given by \ref Bombaci01 :
111  \f[
112  V(\rho,\delta) = V_A+V_B+V_C
113  \f]
114  \f[
115  V_A = \frac{2 A}{3 \rho_0}
116  \left[ \left(1+\frac{x_0}{2}\right)\rho^2-
117  \left(\frac{1}{2}+x_0\right)\left(\rho_n^2+\rho_p^2\right)\right]
118  \f]
119  \f[
120  V_B=\frac{4 B}{3 \rho_0^{\sigma}} \frac{T}{1+4 B^{\prime} T /
121  \left(3 \rho_0^{\sigma-1} \rho^2\right)}
122  \f]
123  where
124  \f[
125  T = \rho^{\sigma-1} \left[ \left( 1+\frac{x_3}{2} \right) \rho^2 -
126  \left(\frac{1}{2}+x_3\right)\left(\rho_n^2+\rho_p^2\right)\right]
127  \f]
128  The term \f$V_C\f$ is:
129  \f[
130  V_C=\sum_{i=1}^{i_{\mathrm{max}}}
131  \frac{4}{5} \left(C_{i}+2 z_i\right) \rho
132  (g_{n,i}+g_{p,i})+\frac{2}{5}\left(C_i -8 z_i\right)
133  (\rho_n g_{n,i} + \rho_p g_{p,i})
134  \f]
135  where
136  \f[
137  g_{\tau,i} = \frac{2}{(2 \pi)^3} \int d^3 k f_{\tau}(k,T)
138  g_i(k)
139  \f]
140 
141  For \ref form is equal to \ref bgbd_form or \ref form
142  is equal to \ref bpal_form, the form factor is given by
143  \f[
144  g_i(k) = \left(1+\frac{k^2}{\Lambda_i^2}\right)^{-1}
145  \f]
146  while for \ref form is equal to \ref sl_form, the form factor
147  is given by
148  \f[
149  g_i(k) = 1-\frac{k^2}{\Lambda_i^2}
150  \f]
151  where \f$\Lambda_1\f$ is specified in the parameter
152  \c Lambda when necessary.
153 
154  \bug The BGBD and SL EOSs do not work.
155 
156  \future Calculate the chemical potentials analytically.
157  */
159 
160  public:
161 
162  /// \name The parameters for the various interactions
163  //@{
164  double x,Au,Al,rho0,B,sigma,Cl,Cu,Lambda;
165  double A,x0,x3,Bp,C1,z1,Lambda2,C2,z2,bpal_esym;
166  int sym_index;
167  //@}
168 
170 
171  /// Equation of state as a function of density.
172  virtual int calc_e(fermion &ne, fermion &pr, thermo &lt);
173 
174  /// Form of potential
175  int form;
176 
177  /** \brief The "momentum-dependent-interaction" form from
178  \ref Das03
179  */
180  static const int mdi_form=1;
181 
182  /// The modifed GBD form
183  static const int bgbd_form=2;
184 
185  /// The form from \ref Prakash88 as formulated in \ref Bombaci01
186  static const int bpal_form=3;
187 
188  /// The "SL" form. See \ref Bombaci01.
189  static const int sl_form=4;
190 
191  /// The Gale, Bertsch, Das Gupta from \ref Gale87.
192  static const int gbd_form=5;
193 
194  /// The form from \ref Prakash88.
195  static const int pal_form=6;
196 
197  /** \brief Set the derivative object to calculate the
198  chemical potentials
199  */
201  mu_deriv_set=true;
202  mu_deriv_ptr=&de;
203  return 0;
204  }
205 
206  /// The default derivative object for calculating chemical potentials
208 
209  /// Return string denoting type ("eos_had_potential")
210  virtual const char *type() { return "eos_had_potential"; }
211 
212  protected:
213 
214 #ifndef DOXYGEN_INTERNAL
215 
216  /// Non-relativistic fermion thermodyanmics
218 
219  /// True of the derivative object has been set
221 
222  /// The derivative object
224 
225  /// Compute the momentum integral for \ref mdi_form
226  double mom_integral(double pft, double pftp);
227 
228  /** \name The mode for the energy() function [protected] */
229  //@{
230  int mode;
231  static const int nmode=1;
232  static const int pmode=2;
233  static const int normal=0;
234  //@}
235 
236  /// Compute the energy
237  double energy(double x);
238 
239 #endif
240 
241  };
242 
243 #ifndef DOXYGEN_NO_O2NS
244 }
245 #endif
246 
247 #endif
static const int bpal_form
The form from Prakash88 as formulated in Bombaci01.
Generalized potential model equation of state.
double mom_integral(double pft, double pftp)
Compute the momentum integral for mdi_form.
static const int pal_form
The form from Prakash88.
int form
Form of potential.
double energy(double x)
Compute the energy.
static const int sl_form
The "SL" form. See Bombaci01.
static const int gbd_form
The Gale, Bertsch, Das Gupta from Gale87.
virtual int calc_e(fermion &ne, fermion &pr, thermo &lt)
Equation of state as a function of density.
static const int bgbd_form
The modifed GBD form.
A hadronic EOS based on a function of the densities [abstract base].
Definition: eos_had_base.h:932
deriv_gsl def_mu_deriv
The default derivative object for calculating chemical potentials.
int set_mu_deriv(deriv_base<> &de)
Set the derivative object to calculate the chemical potentials.
bool mu_deriv_set
True of the derivative object has been set.
static const int mdi_form
The "momentum-dependent-interaction" form from Das03.
virtual const char * type()
Return string denoting type ("eos_had_potential")
deriv_base * mu_deriv_ptr
The derivative object.
fermion_nonrel nrf
Non-relativistic fermion thermodyanmics.

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