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eos_had_potential.h
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1 /*
2  -------------------------------------------------------------------
3 
4  Copyright (C) 2006-2014, Andrew W. Steiner
5 
6  This file is part of O2scl.
7 
8  O2scl is free software; you can redistribute it and/or modify
9  it under the terms of the GNU General Public License as published by
10  the Free Software Foundation; either version 3 of the License, or
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13  O2scl is distributed in the hope that it will be useful,
14  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16  GNU General Public License for more details.
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18  You should have received a copy of the GNU General Public License
19  along with O2scl. If not, see <http://www.gnu.org/licenses/>.
20 
21  -------------------------------------------------------------------
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$C_{1/2,-1/2}=C_{-1/2,1/2}=C_{u}\f$.
92  Otherwise if \ref form is equal to \ref gbd_form, then
93  \f[
94  V_{mom}(\rho,\delta)=
95  \frac{1}{\rho_0}\left[
96  C_{\ell} \left( \rho_n g_n + \rho_p g_p \right)+
97  C_u \left( \rho_n g_p + \rho_p g_n \right)
98  \right]
99  \f]
100  where
101  \f[
102  g_i=\frac{\Lambda^2}{\pi^2}\left[ k_{F,i}-\Lambda
103  \mathrm{tan}^{-1} \left(k_{F,i}/\Lambda\right) \right]
104  \f]
105 
106  Otherwise, if \ref form is equal to \ref bgbd_form, \ref bpalb_form
107  or \ref sl_form, then the potential energy density is
108  given by \ref Bombaci01 :
109  \f[
110  V(\rho,\delta) = V_A+V_B+V_C
111  \f]
112  \f[
113  V_A = \frac{2 A}{3 \rho_0}
114  \left[ \left(1+\frac{x_0}{2}\right)\rho^2-
115  \left(\frac{1}{2}+x_0\right)\left(\rho_n^2+\rho_p^2\right)\right]
116  \f]
117  \f[
118  V_B=\frac{4 B}{3 \rho_0^{\sigma}} \frac{T}{1+4 B^{\prime} T /
119  \left(3 \rho_0^{\sigma-1} \rho^2\right)}
120  \f]
121  where
122  \f[
123  T = \rho^{\sigma-1} \left[ \left( 1+\frac{x_3}{2} \right) \rho^2 -
124  \left(\frac{1}{2}+x_3\right)\left(\rho_n^2+\rho_p^2\right)\right]
125  \f]
126  The term \f$V_C\f$ is:
127  \f[
128  V_C=\sum_{i=1}^{i_{\mathrm{max}}}
129  \frac{4}{5} \left(C_{i}+2 z_i\right) \rho
130  (g_{n,i}+g_{p,i})+\frac{2}{5}\left(C_i -8 z_i\right)
131  (\rho_n g_{n,i} + \rho_p g_{p,i})
132  \f]
133  where
134  \f[
135  g_{\tau,i} = \frac{2}{(2 \pi)^3} \int d^3 k f_{\tau}(k,T)
136  g_i(k)
137  \f]
138 
139  For \ref form is equal to \ref bgbd_form or \ref form
140  is equal to \ref bpalb_form, the form factor is given by
141  \f[
142  g_i(k) = \left(1+\frac{k^2}{\Lambda_i^2}\right)^{-1}
143  \f]
144  while for \ref form is equal to \ref sl_form, the form factor
145  is given by
146  \f[
147  g_i(k) = 1-\frac{k^2}{\Lambda_i^2}
148  \f]
149  where \f$\Lambda_1\f$ is specified in the parameter
150  \c Lambda when necessary.
151 
152  \bug The BGBD EOS doesn't work and the
153  effective mass for the GBD EOS doesn't work
154 
155  \future Calculate the chemical potentials analytically
156  */
158 
159  public:
160 
161  /// \name The parameters for the various interactions
162  //@{
163  double x,Au,Al,rho0,B,sigma,Cl,Cu,Lambda;
164  double A,x0,x3,Bp,C1,z1,Lambda2,C2,z2,bpal_esym;
165  int sym_index;
166  //@}
167 
169 
170  /// Equation of state as a function of density.
171  virtual int calc_e(fermion &ne, fermion &pr, thermo &lt);
172 
173  /// Form of potential
174  int form;
175 
176  /// The "momentum-dependent-interaction" form
177  static const int mdi_form=1;
178 
179  /// The modifed GBD form
180  static const int bgbd_form=2;
181 
182  /// The form from \ref Prakash88 as formulated in \ref Bombaci01
183  static const int bpalb_form=3;
184 
185  /// The "SL" form. See \ref Bombaci01.
186  static const int sl_form=4;
187 
188  /// The Gale, Bertsch, Das Gupta from \ref Gale87.
189  static const int gbd_form=5;
190 
191  /// The form from \ref Prakash88.
192  static const int bpal_form=6;
193 
194  /** \brief Set the derivative object to calculate the
195  chemical potentials
196  */
198  mu_deriv_set=true;
199  mu_deriv_ptr=&de;
200  return 0;
201  }
202 
203  /// The default derivative object for calculating chemical potentials
205 
206  /// Return string denoting type ("eos_had_potential")
207  virtual const char *type() { return "eos_had_potential"; }
208 
209  protected:
210 
211 #ifndef DOXYGEN_INTERNAL
212 
213  /// Non-relativistic fermion thermodyanmics
215 
216  /// True of the derivative object has been set
218 
219  /// The derivative object
221 
222  /// Compute the momentum integral for \ref mdi_form
223  double mom_integral(double pft, double pftp);
224 
225  /** \name The mode for the energy() function [protected] */
226  //@{
227  int mode;
228  static const int nmode=1;
229  static const int pmode=2;
230  static const int normal=0;
231  //@}
232 
233  /// Compute the energy
234  double energy(double x);
235 
236 #endif
237 
238  };
239 
240 #ifndef DOXYGEN_NO_O2NS
241 }
242 #endif
243 
244 #endif
static const int bpal_form
The form from Prakash88.
Generalized potential model equation of state.
double mom_integral(double pft, double pftp)
Compute the momentum integral for mdi_form.
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 bpalb_form
The form from Prakash88 as formulated in 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:825
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.
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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