00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006, 2007, 2008, Andrew W. Steiner 00005 00006 This file is part of O2scl. 00007 00008 O2scl is free software; you can redistribute it and/or modify 00009 it under the terms of the GNU General Public License as published by 00010 the Free Software Foundation; either version 3 of the License, or 00011 (at your option) any later version. 00012 00013 O2scl is distributed in the hope that it will be useful, 00014 but WITHOUT ANY WARRANTY; without even the implied warranty of 00015 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00016 GNU General Public License for more details. 00017 00018 You should have received a copy of the GNU General Public License 00019 along with O2scl. If not, see <http://www.gnu.org/licenses/>. 00020 00021 ------------------------------------------------------------------- 00022 */ 00023 /** \mainpage 00024 00025 \hline 00026 \section dcsect Quick Reference to User's Guide 00027 - \ref hadronic 00028 - \ref quark_matter 00029 - \ref tovtoc 00030 - \ref cnstar 00031 - \ref nrTeos_section 00032 - \ref eossource_section 00033 - \ref eosref_section 00034 00035 \htmlonly 00036 <a href="../latex/refman.pdf">PDF documentation</a> 00037 \endhtmlonly 00038 00039 \hline 00040 \section hadronic Hadronic equations of state 00041 00042 The hadronic equations of state are all inherited from 00043 hadronic_eos: schematic_eos, skyrme_eos, rmf_eos, apr_eos, and 00044 gen_potential_eos. 00045 00046 hadronic_eos includes several methods that can be used 00047 to calculate the saturation properties of nuclear matter. 00048 These methods are sometimes overloaded in descendants when 00049 exact formulas are available. 00050 00051 There is also a set of classes to modify the quartic term 00052 of the symmetry energy: rmf4_eos, apr4_eos, skyrme4_eos, and 00053 mdi4_eos all based on sym4_eos_base which can be used 00054 in sym4_eos. 00055 00056 \hline 00057 \section quark_matter Equations of state of quark matter 00058 00059 The equations of state of \quark matter are all inherited from 00060 quark_eos: bag_eos is a simple bag model, nambujl_eos is the 00061 Nambu--Jona-Lasinio model. 00062 00063 \hline 00064 \section tovtoc Solution of the Tolman-Oppenheimer-Volkov equations 00065 00066 The class \ref tov_solve provide a solution to the 00067 Tolman-Oppenheimer-Volkov (TOV) equations given an equation of 00068 state. This is particularly useful for static neutron star 00069 structure: given any equation of state one can calculate the mass 00070 vs. radius curve and the properties of any star of a given 00071 mass. An adaptive integration is employed and calculates the 00072 gravitational mass, the baryonic mass (if the baryon density is 00073 supplied), and the gravitational potential. The remaining columns 00074 is the equation of state are also interpolated into the solution, 00075 e.g. if a chemical potential is given, then the radial dependence 00076 of the chemical potential for a 1.4 solar mass star can be 00077 automatically computed. The equation of state may be specified in 00078 arbitrary units so long as an appropriate conversion factor is 00079 supplied. An equation of state for low densities (baryon density < 00080 0.08 \f$ \mathrm{fm}^{-3} \f$ ) is provided and can be 00081 automatically appended to the user-defined equation of state. 00082 00083 This is still experimental. 00084 00085 \hline 00086 \section cnstar Naive Cold Neutron Stars 00087 00088 There is also a class to calculate zero-temperature neutron 00089 stars: \ref cold_nstar. It uses \ref tov_solve to compute 00090 the structure, given a hadronic equation of state (of type 00091 \ref hadronic_eos). It also computes the adiabatic index, the 00092 speed of sound, and determines the possibility of the 00093 direct Urca process as a function of density or radius. 00094 00095 This is still experimental. 00096 00097 \hline 00098 \section nrTeos_section Non-relativistic Finite Temperature Approximations 00099 00100 This is taken from the \ref Prakash87. 00101 00102 The entropy is 00103 \f[ 00104 s = -\sum_k \left[ n_k \ln n_k + \left(1-n_k\right) \ln 00105 \left(1-n_k\right)\right] 00106 \f] 00107 00108 The low-temperature (degenerate) approximation to the entropy is 00109 \f[ 00110 s=\pi^2/3 N(0) T 00111 \f] 00112 where the density of states at the Fermi surface is 00113 \f[ 00114 N(0)=\sum_k \delta(\epsilon_k -\mu) = \frac{3 \rho }{k_F v_F} 00115 \f] 00116 where the Fermi velocity is 00117 \f[ 00118 v_F = \left.\frac{\partial \epsilon_k}{\partial k}\right|_{k_F} 00119 = \frac{k_F}{m^{*}} 00120 \f] 00121 The latter equation defines the effective mass. 00122 The level density parameter is given by 00123 \f[ 00124 a=\frac{\pi^2 N(0)}{6 \rho} 00125 \f] 00126 Defining the Fermi temperature: 00127 \f[ 00128 T_F=\frac{1}{2} k_F v_F = k_F^2/2/m^{*} 00129 \f] 00130 another expression for the entropy is 00131 \f[ 00132 s=\frac{\pi^2}{2} \rho (T/T_F) 00133 \f] 00134 Expressions for the remaining quantities are 00135 \f[ 00136 P=P(T=0)+\frac{\rho}{3} 00137 a T^2 \left(1+\frac{d \ln v_F}{d \ln k_F }\right) 00138 \f] 00139 \f[ 00140 E/A=E/A(T=0)+a T^2 00141 \f] 00142 \f[ 00143 \mu=\mu(T=0)-\frac{1}{3} a T^2\left(2-\frac{d \ln v_F}{d \ln k_F }\right) 00144 \f] 00145 Typically, the leading correction to 00146 \f[ 00147 s=\frac{\pi^2}{2} \rho (T/T_F) 00148 \f] 00149 is of order \f$ (T/T_F)^2 \f$ unless soft collective modes give 00150 rise to a \f$ (T/T_F)^3 \ln (T/T_F) \f$ correction. 00151 00152 At high temperature (non-degenerate approximation), 00153 a stationary phase approximation gives 00154 \f{eqnarray*} 00155 \rho(T) &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} \cdot k^2 00156 e^{-\epsilon_k/T} \sqrt{2 \pi} \left[\frac{2}{k^2}+ 00157 \frac{1}{T}\frac{\partial v_k}{\partial k}\right]^{-1/2} \\ 00158 &=& e^{\mu/T} \left.f(T)\right|_{k=k_{\rho}} 00159 \f} 00160 where \f$ \gamma \f$ is the spin and isospin degeneracy and the 00161 velocity function is \f$ v_k=\partial \epsilon_k/\partial k \f$ . 00162 The function \f$ f(T) \f$ is evaluated at momentum \f$ k_{\rho} \f$ 00163 which is obtained by solving \f$ T-k v_k/2=0 \f$ . 00164 The chemical potential is obtained by inverting the above 00165 relation for \f$ \rho(T) \f$ : 00166 \f[ 00167 \mu \sim T \ln \rho - T \ln f(T) 00168 \f] 00169 From this value of \f$ \mu \f$ we can derive the entropy density 00170 using 00171 \f[ 00172 T s \sim \sum_k n_k \epsilon_k + \rho T - \mu \rho 00173 \f] 00174 Using the stationary phase method: 00175 \f{eqnarray*} 00176 \sum_k n_k \epsilon_k &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} 00177 \cdot k^2 \epsilon_k e^{-\epsilon_k/T} \sqrt{2 \pi} 00178 \left[\frac{2}{k^2}-\left(\frac{1}{\epsilon_k}-\frac{1}{T}\right) 00179 \frac{\partial v_k}{\partial k}+ 00180 \left(\frac{v_k}{\epsilon_k}\right)^2\right]^{-1/2} \\ 00181 &=& e^{\mu/T} \left.g(T)\right|_{k=k_E} 00182 \f} 00183 where \f$ k_E \f$ is the solution of 00184 \f[ 00185 \frac{2}{k}+v_k \left(\frac{1}{\epsilon_k}-\frac{1}{T}\right)=0 00186 \f] 00187 This provides a first approximation to the energy and together 00188 with the thermodynamic identity gives the pressure. 00189 00190 \hline 00191 \section eossource_section Example source code 00192 00193 \subsection exlist_eos_subsect Example list 00194 00195 - \ref ex_cold_nstar_sect 00196 00197 \subsection ex_cold_nstar_sect Cold neutron star example 00198 00199 \dontinclude ex_cold_nstar.cpp 00200 \skip Example: 00201 \until End of example 00202 00203 \hline 00204 \section todo_section Other Todos 00205 00206 \future Right now, the equation of state classes depend on the 00207 user to input the correct value of \c non_interacting 00208 for the particle inputs. This is not very graceful... 00209 00210 \hline 00211 \section eosref_section Bibliography 00212 00213 Some of the references which contain links should direct you to 00214 the work referred to directly through dx.doi.org. 00215 00216 \anchor Akmal98 Akmal98: 00217 \htmlonly 00218 <a href="http://dx.doi.org/10.1103/PhysRevC.58.1805"> 00219 Akmal, Pandharipande, and Ravenhall</a>, 00220 \endhtmlonly 00221 \latexonly 00222 \href{http://dx.doi.org/10.1103/PhysRevC.58.1805}{ 00223 Akmal, Pandharipande, and Ravenhall}, 00224 \endlatexonly 00225 Phys. Rev. C \b 58, 1805 (1998). 00226 00227 \anchor Bartel79 Bartel79: 00228 \htmlonly 00229 <a href="http://dx.doi.org/10.1016/0375-9474(82)90403-1"> 00230 J. Bartel, P. Quentin, M. Brack, C. Guet, and Håkansson</a>, 00231 \endhtmlonly 00232 \latexonly 00233 \href{http://dx.doi.org/10.1016/0375-9474(82)90403-1}{ 00234 J. Bartel, P. Quentin, M. Brack, C. Guet, and H{\aa}kansson}, 00235 \endlatexonly 00236 Nucl. Phys. 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Bombaci, "Equation of State for Dense Isospin Asymmetric 00265 Nuclear Matter for Astrophysical Applications Equation of 00266 State for Isospin-Asymmetric Nuclear Matter and Neutron Star 00267 Properties", in "Isospin physics in heavy-ion collisions at 00268 intermediate energies" 00269 \htmlonly 00270 ed. by B-A. Li and W. U. Schröder (2001) Nova Science, New York. 00271 \endhtmlonly 00272 \latexonly 00273 ed. by B-A. Li and W. U. Schr\"{o}der (2001) Nova Science, New York. 00274 \endlatexonly 00275 00276 \anchor Brack85 Brack85: 00277 \htmlonly 00278 <a href="http://dx.doi.org/10.1016/0370-1573(86)90078-5"> 00279 M. Brack, C. Guet, and H.-B. Håkansson</a> 00280 \endhtmlonly 00281 \latexonly 00282 \href{http://dx.doi.org/10.1016/0370-1573(86)90078-5}{ 00283 M. Brack, C. Guet, and H.-B. H{\aa}kansson}, 00284 \endlatexonly 00285 Phys. 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Reinhard</a>, 00371 \endhtmlonly 00372 \latexonly 00373 \href{http://dx.doi.org/10.1103/PhysRevC.33.335}{ 00374 J. Friedrich, and P.-G. Reinhard}, 00375 \endlatexonly 00376 Phys. Rev. C \b 33 (1986) 335. 00377 00378 \anchor Gaitanos04 Gaitanos04: 00379 \htmlonly 00380 <a href="http://dx.doi.org/10.1016/j.nuclphysa.2003.12.001"> 00381 T. Gaitanos, M. Di Toro, S. Typel, V. Barana, C. Fuchs, 00382 V. Greco, and d H. H. Wolter</a>, 00383 \endhtmlonly 00384 \latexonly 00385 \href{http://dx.doi.org/10.1016/j.nuclphysa.2003.12.001}{ 00386 T. Gaitanos, M. Di Toro, S. Typel, V. Barana, C. Fuchs, 00387 V. Greco, and d H. H. Wolter}, 00388 \endlatexonly 00389 Nucl. Phys. A \b 732 (2004) 24. 00390 00391 \anchor Gale87 Gale87: 00392 \htmlonly 00393 <a href="http://dx.doi.org/10.1103/PhysRevC.35.1666"> 00394 C. Gale, G. Bertsch, and S. Das Gupta</a>, 00395 \endhtmlonly 00396 \latexonly 00397 \href{http://dx.doi.org/10.1103/PhysRevC.35.1666}{ 00398 C. Gale, G. Bertsch, and S. 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