00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006, 2007, 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 \htmlonly 00026 <a href="../latex/refman.pdf">PDF documentation</a> 00027 \endhtmlonly 00028 00029 All equations of state inherit from \ref eos (except for the TOV 00030 solver \ref tov_solve and \ref cold_nstar). The \c hal_eos 00031 library contains all of the classes mentioned in this documentation. 00032 00033 \hline 00034 \section dcsect User's Guide 00035 - \ref hadronic 00036 - \ref quark_matter 00037 - \ref tovtoc 00038 - \ref cnstar 00039 - \ref nrTeos_section 00040 - \ref todo 00041 - \ref bug 00042 - \ref eosref_section 00043 00044 \hline 00045 \section hadronic Hadronic equations of state 00046 00047 The hadronic equations of state are all inherited from 00048 hadronic_eos: schematic_eos, skyrme_eos, rmf_eos, apr_eos, and 00049 gen_potential_eos. 00050 00051 hadronic_eos includes several methods that can be used 00052 to calculate the saturation properties of nuclear matter. 00053 These methods are sometimes overloaded in descendants when 00054 exact formulas are available. 00055 00056 There is also a set of classes to modify the quartic term 00057 of the symmetry energy: rmf4_eos, apr4_eos, skyrme4_eos, and 00058 mdi4_eos all based on sym4_eos_base which can be used 00059 in sym4_eos. 00060 00061 \hline 00062 \section quark_matter Equations of state of quark matter 00063 00064 The equations of state of quark matter are all inherited from 00065 quark_eos: bag_eos is a simple bag model, nambujl_eos is the 00066 Nambu--Jona-Lasinio model. 00067 00068 \hline 00069 \section tovtoc Solution of the Tolman-Oppenheimer-Volkov equations 00070 00071 The class \ref tov_solve provide a solution to the 00072 Tolman-Oppenheimer-Volkov (TOV) equations given an equation of 00073 state. This is particularly useful for static neutron star 00074 structure: given any equation of state one can calculate the mass 00075 vs. radius curve and the properties of any star of a given 00076 mass. An adaptive integration is employed and calculates the 00077 gravitational mass, the baryonic mass (if the baryon density is 00078 supplied), and the gravitational potential. The remaining columns 00079 is the equation of state are also interpolated into the solution, 00080 e.g. if a chemical potential is given, then the radial dependence 00081 of the chemical potential for a 1.4 solar mass star can be 00082 automatically computed. The equation of state may be specified in 00083 arbitrary units so long as an appropriate conversion factor is 00084 supplied. An equation of state for low densities (baryon density < 00085 0.08 \f$ \mathrm{fm}^{-3} \f$ ) is provided and can be 00086 automatically appended to the user-defined equation of state. 00087 00088 This is still experimental. 00089 00090 \hline 00091 \section cnstar Naive Cold Neutron Stars 00092 00093 There is also a class to calculate zero-temperature neutron 00094 stars: \ref cold_nstar. It uses \ref tov_solve to compute 00095 the structure, given a hadronic equation of state (of type 00096 \ref hadronic_eos). It also computes the adiabatic index, the 00097 speed of sound, and determines the possibility of the 00098 direct Urca process as a function of density or radius. 00099 00100 This is still experimental. 00101 00102 \hline 00103 \section nrTeos_section Non-relativistic Finite Temperature Approximations 00104 00105 This is taken from the \ref Prakash87. 00106 00107 The entropy is 00108 \f[ 00109 s = -\sum_k \left[ n_k \ln n_k + \left(1-n_k\right) \ln 00110 \left(1-n_k\right)\right] 00111 \f] 00112 00113 The low-temperature (degenerate) approximation to the entropy is 00114 \f[ 00115 s=\pi^2/3 N(0) T 00116 \f] 00117 where the density of states at the Fermi surface is 00118 \f[ 00119 N(0)=\sum_k \delta(\epsilon_k -\mu) = \frac{3 \rho }{k_F v_F} 00120 \f] 00121 where the Fermi velocity is 00122 \f[ 00123 v_F = \left.\frac{\partial \epsilon_k}{\partial k}\right|_{k_F} 00124 = \frac{k_F}{m^{*}} 00125 \f] 00126 The latter equation defines the effective mass. 00127 The level density parameter is given by 00128 \f[ 00129 a=\frac{\pi^2 N(0)}{6 \rho} 00130 \f] 00131 Defining the Fermi temperature: 00132 \f[ 00133 T_F=\frac{1}{2} k_F v_F = k_F^2/2/m^{*} 00134 \f] 00135 another expression for the entropy is 00136 \f[ 00137 s=\frac{\pi^2}{2} \rho (T/T_F) 00138 \f] 00139 Expressions for the remaining quantities are 00140 \f[ 00141 P=P(T=0)+\frac{\rho}{3} 00142 a T^2 \left(1+\frac{d \ln v_F}{d \ln k_F }\right) 00143 \f] 00144 \f[ 00145 E/A=E/A(T=0)+a T^2 00146 \f] 00147 \f[ 00148 \mu=\mu(T=0)-\frac{1}{3} a T^2\left(2-\frac{d \ln v_F}{d \ln k_F }\right) 00149 \f] 00150 Typically, the leading correction to 00151 \f[ 00152 s=\frac{\pi^2}{2} \rho (T/T_F) 00153 \f] 00154 is of order \f$ (T/T_F)^2 \f$ unless soft collective modes give 00155 rise to a \f$ (T/T_F)^3 \ln (T/T_F) \f$ correction. 00156 00157 At high temperature (non-degenerate approximation), 00158 a stationary phase approximation gives 00159 \f{eqnarray*} 00160 \rho(T) &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} \cdot k^2 00161 e^{-\epsilon_k/T} \sqrt{2 \pi} \left[\frac{2}{k^2}+ 00162 \frac{1}{T}\frac{\partial v_k}{\partial k}\right]^{-1/2} \\ 00163 &=& e^{\mu/T} \left.f(T)\right|_{k=k_{\rho}} 00164 \f} 00165 where \f$ \gamma \f$ is the spin and isospin degeneracy and the 00166 velocity function is \f$ v_k=\partial \epsilon_k/\partial k \f$ . 00167 The function \f$ f(T) \f$ is evaluated at momentum \f$ k_{\rho} \f$ 00168 which is obtained by solving \f$ T-k v_k/2=0 \f$ . 00169 The chemical potential is obtained by inverting the above 00170 relation for \f$ \rho(T) \f$ : 00171 \f[ 00172 \mu \sim T \ln \rho - T \ln f(T) 00173 \f] 00174 From this value of \f$ \mu \f$ we can derive the entropy density 00175 using 00176 \f[ 00177 T s \sim \sum_k n_k \epsilon_k + \rho T - \mu \rho 00178 \f] 00179 Using the stationary phase method: 00180 \f{eqnarray*} 00181 \sum_k n_k \epsilon_k &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} 00182 \cdot k^2 \epsilon_k e^{-\epsilon_k/T} \sqrt{2 \pi} 00183 \left[\frac{2}{k^2}-\left(\frac{1}{\epsilon_k}-\frac{1}{T}\right) 00184 \frac{\partial v_k}{\partial k}+ 00185 \left(\frac{v_k}{\epsilon_k}\right)^2\right]^{-1/2} \\ 00186 &=& e^{\mu/T} \left.g(T)\right|_{k=k_E} 00187 \f} 00188 where \f$ k_E \f$ is the solution of 00189 \f[ 00190 \frac{2}{k}+v_k \left(\frac{1}{\epsilon_k}-\frac{1}{T}\right)=0 00191 \f] 00192 This provides a first approximation to the energy and together 00193 with the thermodynamic identity gives the pressure. 00194 00195 \section todo_section Other Todos 00196 00197 \todo Right now, the equation of state classes depend on the 00198 user to input the correct value of \c non_interacting 00199 for the particle inputs. This is not very graceful... 00200 \todo Document the "n15" models. What where they for? 00201 00202 \hline 00203 \section eosref_section Bibliography 00204 00205 Some of the references which contain links should direct you to 00206 the work referred to directly through dx.doi.org. 00207 00208 \anchor Akmal98 Akmal98: 00209 \htmlonly 00210 <a href="http://dx.doi.org/10.1103/PhysRevC.58.1804"> 00211 Akmal, Pandharipande, and Ravenhall</a>, 00212 \endhtmlonly 00213 \latexonly 00214 \href{http://dx.doi.org/10.1103/PhysRevC.58.1804}{ 00215 Akmal, Pandharipande, and Ravenhall}, 00216 \endlatexonly 00217 Phys. Rev. C \b 58, 1804 (1998). 00218 00219 \anchor Bartel79 Bartel79: 00220 \htmlonly 00221 <a href="http://dx.doi.org/10.1016/0375-9474(82)90403-1"> 00222 J. Bartel, P. Quentin, M. Brack, C. Guet, and Håkansson</a>, 00223 \endhtmlonly 00224 \latexonly 00225 \href{http://dx.doi.org/10.1016/0375-9474(82)90403-1}{ 00226 J. Bartel, P. Quentin, M. Brack, C. 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