main.h

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     \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 H&aring;kansson</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. Guet, and H{\aa}kansson},
00227     \endlatexonly
00228     Nucl. Phys. A \b 386 (1982) 79.
00229 
00230     \anchor Baym71 Baym71:
00231     G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. \b 170 (1971) 299.
00232     
00233     \anchor Beiner75 Beiner75:
00234     \htmlonly
00235     <a href="http://dx.doi.org/10.1016/0375-9474(75)90338-3">
00236     M. Beiner, H. Flocard, Nguyen van Giai, and P. Quentin</a>,
00237     \endhtmlonly
00238     \latexonly
00239     \href{http://dx.doi.org/10.1016/0375-9474(75)90338-3}{
00240     M. Beiner, H. Flocard, Nguyen van Giai, and P. Quentin}, 
00241     \endlatexonly
00242     Nucl. Phys. A \b 238 (1975) 29.
00243 
00244     \anchor Bernard88 Bernard88:
00245     \htmlonly
00246     <a href="http://dx.doi.org/10.1016/0550-3213(88)90127-7">
00247     V. Bernard, R.L. Jaffe, and U.-G. Meissner</a>,
00248     \endhtmlonly
00249     \latexonly
00250     \href{http://dx.doi.org/10.1016/0550-3213(88)90127-7}{
00251     V. Bernard, R.~L. Jaffe, and U.-G. Meissner},
00252     \endlatexonly
00253     Nucl. Phys. B \b 308 (1988) 753.
00254 
00255     \anchor Bombaci01 Bombaci01:
00256     I. Bombaci, "Equation of State for Dense Isospin Asymmetric 
00257     Nuclear Matter for Astrophysical Applications Equation of 
00258     State for Isospin-Asymmetric Nuclear Matter and Neutron Star
00259     Properties", in "Isospin physics in heavy-ion collisions at
00260     intermediate energies" 
00261     \htmlonly
00262     ed. by B-A. Li and W. U. Schr&ouml;der (2001) Nova Science, New York. 
00263     \endhtmlonly
00264     \latexonly
00265     ed. by B-A. Li and W. U. Schr\"{o}der (2001) Nova Science, New York. 
00266     \endlatexonly
00267     
00268     \anchor Brack85 Brack85:
00269     \htmlonly
00270     <a href="http://dx.doi.org/10.1016/0370-1573(86)90078-5">
00271     M. Brack, C. Guet, and H.-B. H&aring;kansson</a>
00272     \endhtmlonly
00273     \latexonly
00274     \href{http://dx.doi.org/10.1016/0370-1573(86)90078-5}{
00275     M. Brack, C. Guet, and H.-B. H{\aa}kansson},
00276     \endlatexonly
00277     Phys. Rep. \b 123 (1985) 275.
00278 
00279     \anchor Buballa99 Buballa99:
00280     \htmlonly
00281     <a href="http://dx.doi.org/10.1016/S0370-2693(99)00533-X">
00282     M. Buballa and M. Oertel</a>,
00283     \endhtmlonly
00284     \latexonly
00285     \href{http://dx.doi.org/10.1016/S0370-2693(99)00533-X}{
00286     M. Buballa and M. Oertel}, 
00287     \endlatexonly
00288     Phys. Lett. B \b 457 (1999) 261.
00289 
00290     \anchor Buballa04 Buballa04:
00291     \htmlonly
00292     <a href="http://dx.doi.org/10.1016/j.physrep.2004.11.004">
00293     M. Buballa</a>,
00294     \endhtmlonly
00295     \latexonly
00296     \href{http://dx.doi.org/10.1016/j.physrep.2004.11.004}{
00297     M. Buballa}, 
00298     \endlatexonly
00299     Phys. Rep. \b 407 (2005) 205-376.
00300     
00301     \anchor Chabanat95 Chabanat95:
00302     E. Chabanat, Ph. D. Thesis 1995
00303 
00304     \anchor Chabanat97 Chabanat97:
00305     \htmlonly
00306     <a href="http://dx.doi.org/10.1016/S0375-9474(97)00596-4">
00307     E. Chabanat, J. Meyer, P. Bonche, R. Schaeffer, and P. Haensel</a>,
00308     \endhtmlonly
00309     \latexonly
00310     \href{http://dx.doi.org/10.1016/S0375-9474(97)00596-4}{
00311     E. Chabanat, J. Meyer, P. Bonche, R. Schaeffer, and P. Haensel}, 
00312     \endlatexonly
00313     Nucl. Phys. A \b 627 (1997) 710.
00314 
00315     \anchor Das03 Das03:
00316     \htmlonly
00317     <a href="http://dx.doi.org/10.1103/PhysRevC.67.034611">
00318     C.B. Das, S. Das Gupta, C. Gale, and B.-A. Li</a>,
00319     \endhtmlonly
00320     \latexonly
00321     \href{http://dx.doi.org/10.1103/PhysRevC.67.034611}{
00322     C.B. Das, S. Das Gupta, C. Gale, and B.-A. Li},
00323     \endlatexonly
00324     Phys. Rev. C \b 67 (2003) 034611.
00325 
00326     \anchor Dobaczewski94 Dobaczewski94:
00327     \htmlonly
00328     <a href="http://dx.doi.org/10.1016/0375-9474(84)90433-0">
00329     J. Dobaczewski, H. Flocard, and J. Treiner</a>,
00330     \endhtmlonly
00331     \latexonly
00332     \href{http://dx.doi.org/10.1016/0375-9474(84)90433-0}{
00333     J. Dobaczewski, H. Flocard, and J. Treiner}, 
00334     \endlatexonly
00335     Nucl. Phys. A \b 422 (1984) 103.
00336 
00337     \anchor Dutta86 Dutta86:
00338     \htmlonly
00339     <a href="http://dx.doi.org/10.1016/0375-9474(86)90283-6">
00340     A.K.Dutta, J.-P.Arcoragi, J.M.Pearson, R.Behrman, F.Tondeur</a>,
00341     \endhtmlonly
00342     \latexonly
00343     \href{http://dx.doi.org/10.1016/0375-9474(86)90283-6}{
00344     A.K.Dutta, J.-P.Arcoragi, J.M.Pearson, R.Behrman, F.Tondeur},
00345     \endlatexonly
00346     Nucl. Phys. A \b 458 (1986) 77.
00347 
00348     \anchor Friedrich86 Friedrich86:
00349     \htmlonly
00350     <a href="http://dx.doi.org/10.1103/PhysRevC.33.335">
00351     J. Friedrich, and P.-G. Reinhard</a>,
00352     \endhtmlonly
00353     \latexonly
00354     \href{http://dx.doi.org/10.1103/PhysRevC.33.335}{
00355     J. Friedrich, and P.-G. Reinhard}, 
00356     \endlatexonly
00357     Phys. Rev. C \b 33 (1986) 335.
00358 
00359     \anchor Gale87 Gale87:
00360     \htmlonly
00361     <a href="http://dx.doi.org/10.1103/PhysRevC.35.1666">
00362     C. Gale, G. Bertsch, and S. Das Gupta</a>,
00363     \endhtmlonly
00364     \latexonly
00365     \href{http://dx.doi.org/10.1103/PhysRevC.35.1666}{
00366     C. Gale, G. Bertsch, and S. Das Gupta},
00367     \endlatexonly
00368     Phys. Rev. C \b 35 (1986) 1666.
00369 
00370     \anchor Hatsuda94 Hatsuda94:
00371     \htmlonly
00372     <a href="http://dx.doi.org/10.1016/0370-1573(94)90022-1">
00373     T. Hatsuda and T. Kunihiro</a>,
00374     \endhtmlonly
00375     \latexonly
00376     \href{http://dx.doi.org/10.1016/0370-1573(94)90022-1}{
00377     T. Hatsuda and T. Kunihiro}, 
00378     \endlatexonly
00379     Phys. Rep. \b 247 (1994) 221.
00380 
00381     \anchor Horowitz01 Horowitz01: 
00382     \htmlonly
00383     <a href="http://dx.doi.org/10.1103/PhysRevLett.86.5647">
00384     Horowitz and Piekrewicz</a>,
00385     \endhtmlonly
00386     \latexonly
00387     \href{http://dx.doi.org/10.1103/PhysRevLett.86.5647}{
00388     Horowitz and Piekarewicz},
00389     \endlatexonly
00390     Phys. Rev. Lett. \b 86 (2001) 5647.
00391 
00392     \anchor Margueron02 Margueron02:
00393     \htmlonly
00394     <a href="http://dx.doi.org/10.1103/PhysRevC.66.014303">
00395     J. Margueron, J. Navarro, and N.V. Giai</a>,
00396     \endhtmlonly
00397     \latexonly
00398     \href{http://dx.doi.org/10.1103/PhysRevC.66.014303}{
00399     J. Margueron, J. Navarro, and N.~V. Giai},
00400     \endlatexonly
00401     Phys. Rev. C \b 66 (2002) 014303.
00402 
00403     \anchor Muller96 Muller96: 
00404     \htmlonly
00405     <a href="http://dx.doi.org/10.1016/0375-9474(96)00187-X">
00406     H. Muller and B. D. Serot</a>,
00407     \endhtmlonly
00408     \latexonly
00409     \href{http://dx.doi.org/10.1016/0375-9474(96)00187-X}{
00410     H. Muller and B.~D. Serot}, 
00411     \endlatexonly
00412     Nucl. Phys. A \b 606 (1996), 508.
00413 
00414     \anchor Osni94 Osni94:
00415     \htmlonly
00416     <a href="http://dx.doi.org/10.1103/PhysRevC.50.460">
00417     M. Onsi, H. Przysiezniak, J.M. Pearson</a>,
00418     \endhtmlonly
00419     \latexonly
00420     \href{http://dx.doi.org/10.1103/PhysRevC.50.460}{
00421     M. Onsi, H. Przysiezniak, J.M. Pearson},
00422     \endlatexonly
00423     Phys. Rev. C \b 50 (1994) 460.
00424 
00425     \anchor Pethick95 Pethick95:
00426     \htmlonly
00427     <a href="http://dx.doi.org/10.1016/0375-9474(94)00506-I">
00428     C.J. Pethick, D.G. Ravenhall, and C.P. Lorenz</a>,
00429     \endhtmlonly
00430     \latexonly
00431     \href{http://dx.doi.org/10.1016/0375-9474(94)00506-I}{
00432     C.~J. Pethick, D.~G. Ravenhall, and C.~P. Lorenz}, 
00433     \endlatexonly
00434     Nucl. Phys. A \b 584 (1995) 675.
00435 
00436     \anchor Prakash87 Prakash87:
00437     \htmlonly
00438     M. Prakash, T.L. Ainsworth, J.P. Blaizot, and H. Wolter,
00439     in Windsurfing the Fermi Sea, Volume II
00440     edited by T.T.S. Kuo and J. Speth, Elsevier 1987, pg. 357.
00441     \endhtmlonly
00442     \latexonly
00443     M. Prakash, T.~L. Ainsworth, J.~P. Blaizot, and H. Wolter,
00444     in Windsurfing the Fermi Sea, Volume II
00445     edited by T.~T.~S. Kuo and J. Speth, Elsevier 1987, pg. 357.
00446     \endlatexonly
00447 
00448     \anchor Prakash88 Prakash88:
00449     \htmlonly
00450     <a href="http://dx.doi.org/10.1103/PhysRevLett.61.2518">
00451     M. Prakash, T. L. Ainsworth, and J. M. Lattimer</a>,
00452     \endhtmlonly
00453     \latexonly
00454     \href{http://dx.doi.org/10.1103/PhysRevLett.61.2518}{
00455     M. Prakash, T.~L. Ainsworth, and J.~M. Lattimer},
00456     \endlatexonly
00457     Phys. Rev. Lett. \b 61 (1988) 2518.
00458 
00459     \anchor Prakash97 Prakash97:
00460     \htmlonly
00461     <a href="http://dx.doi.org/10.1016/S0370-1573(96)00023-3">
00462     Madappa Prakash, I. Bombaci, Manju Prakash, P.J. Ellis, J.M. Lattimer,
00463     and R. Knorren</a>,
00464     \endhtmlonly
00465     \latexonly
00466     \href{http://dx.doi.org/10.1016/S0370-1573(96)00023-3}{
00467     Madappa Prakash, I. Bombaci, Manju Prakash, P.~J. Ellis, J.~M. Lattimer,
00468     and R. Knorren}, 
00469     \endlatexonly
00470     Phys. Rep. \b 280 (1997) 1.
00471 
00472     \anchor Reinhard95 Reinhard95:
00473     \htmlonly
00474     <a href="http://dx.doi.org/10.1016/0375-9474(94)00770-N">
00475     P.-G. Reinhard and H. Flocard</a>,
00476     \endhtmlonly
00477     \latexonly
00478     \href{http://dx.doi.org/10.1016/0375-9474(94)00770-N}{
00479     P.-G. Reinhard and H. Flocard}, 
00480     \endlatexonly
00481     Nucl. Phys. A \b 584 (1995) 467.
00482 
00483     \anchor Reinhard99 Reinhard99: 
00484     \htmlonly
00485     <a href="http://dx.doi.org/10.1103/PhysRevC.60.014316">
00486     P.-G. Reinhard, D.J. Dean, W. Nazarewicz, J. Dobaczewski, 
00487     J.A. Maruhn, M.R. Strayer</a>,
00488     \endhtmlonly
00489     \latexonly
00490     \href{http://dx.doi.org/10.1103/PhysRevC.60.014316}{
00491     P.-G. Reinhard, D.J. Dean, W. Nazarewicz, J. Dobaczewski, 
00492     J.A. Maruhn, M.R. Strayer},
00493     \endlatexonly
00494     Phys. Rev. C \b 60, (1999) 014316.
00495     
00496     \anchor Shapiro83 Shapiro83:
00497     S. L. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs,
00498     and Neutron Stars: The Physics of Compact Objects", John
00499     Wiley and Sons, New York, 1983.
00500 
00501     \anchor Tondeur84 Tondeur84:
00502     \htmlonly
00503     <a href="http://dx.doi.org/10.1016/0375-9474(84)90444-5">
00504     F. Tondeur, M. Brack, M. Farine, and J.M. Pearson</a>,
00505     \endhtmlonly
00506     \latexonly
00507     \href{http://dx.doi.org/10.1016/0375-9474(84)90444-5}{
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00543 */

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