Equation of State Sub-Library: Version 0.910
Public Member Functions | Data Fields | Protected Member Functions | Protected Attributes
hadronic_eos Class Reference

Hadronic equation of state [abstract base]. More...

#include <hadronic_eos.h>

Inheritance diagram for hadronic_eos:
eos hadronic_eos_eden hadronic_eos_pres hadronic_eos_temp ddc_eos gen_potential_eos schematic_eos sym4_eos tabulated_eos hadronic_eos_temp_eden hadronic_eos_temp_pres mdi4_eos apr_eos skyrme_eos rmf_eos apr4_eos skyrme4_eos rmf4_eos rmf_delta_eos

Detailed Description

Denote the number density of neutrons as $ n_n $, the number density of protons as $ n_p $, the total baryon density $ n_B = n_n + n_p $, the asymmetry $ \alpha \equiv (n_n-n_p)/n_B $, the nuclear saturation density as $ n_0 \approx 0.16~\mathrm{fm}^{-3} $, and the quantity $ \eta \equiv (n-n_0)/3n_0 $. Then the energy per baryon of nucleonic matter matter can be written as an expansion around $ \epsilon =\alpha = 0 $

\[ E(n_B,\alpha) = -B + \frac{\tilde{K}}{2!} {\epsilon}^2 + \frac{Q_0}{3!} {\epsilon}^3 + \alpha^2 \left(S + L \epsilon + \frac{K_{\mathrm{sym}}}{2!} {\epsilon}^2 + \frac{Q_{\mathrm{sym}}}{3!} \epsilon^3 \right) + E_4(n_B,\alpha) + {\cal O}(\alpha^6) \qquad \left(\mathrm{Eq.}~1\right) \]

where $ E_4 $ represents the quartic terms

\[ E_4(n_B,\alpha) = \alpha^4 \left(S_4 + L_4 \epsilon + \frac{K_4}{2!} {\epsilon}^2 + \frac{Q_4}{3!} \epsilon^3 \right) \qquad \left(\mathrm{Eq.}~2\right) \]

(Adapted slightly from Piekarewicz09). From this, one can compute the energy density of nuclear matter $ \varepsilon(n_B,\alpha) = n_B E(n_B,\alpha) $, the chemical potentials $ \mu_i \equiv (\partial \varepsilon) / (\partial n_i ) $ and the pressure $ P = -\varepsilon + \mu_n n_n + \mu_p n_p $. This expansion motivates the definition of several separate terms. The binding energy $ B $ of symmetric nuclear matter ( $ \alpha = 0 $) is around 16 MeV.

The compression modulus is usually defined by $ \chi = -1/V (dV/dP) = 1/n (dP/dn)^{-1} $ . In nuclear physics it has become common to use the incompressibility (or bulk) modulus with an extra factor of 9, $ K=9/(n \chi) $ and refer to $ K $ simply as the incompressibility. Here, we define the function

\[ K(n_B,\alpha) \equiv 9 \left( \frac{\partial P}{\partial n_B} \right) = 9 n_B \left(\frac{\partial^2 \varepsilon} {\partial n_B^2}\right) \]

This quantity is computed by the function fcomp() by computing the first derivative of the pressure, which is more numerically stable than the second derivative of the energy density (and most O2scl EOSs compute the pressure exactly). This function is typically evaluated at the point $ (n_B=n_0,\alpha=0) $ and is stored in comp. This quantity is not always the same as $ \tilde{K} $, defined here as

\[ \tilde{K}(n_B,\alpha) = 9 n_B^2 \left(\frac{\partial^2 E}{\partial n_B^2}\right) = K(n_B,\alpha) - \frac{1}{n_B} 18 P(n_B,\alpha) \]

We denote $ K \equiv K(n_B=n_0,\alpha=0) $ and similarly for $ \tilde{K} $, the quantity in Eq. 1 above. In nuclear matter at saturation, the pressure is zero and $ K = \tilde{K} $. See Chabanat97 for a discussion of this distinction.

The symmetry energy $ S(n_B,\alpha) $ can be defined as

\[ S(n_B,\alpha) \equiv \frac{1}{2 n_B}\frac{\partial^2 \varepsilon} {\partial \alpha^2} \]

and the parameter $ S $ in Eq. 1 is just $ S(n_0,0) $. Using

\[ \left(\frac{\partial \varepsilon}{\partial \alpha}\right)_{n_B} = \frac{\partial \varepsilon}{\partial n_n} \left(\frac{\partial n_n}{\partial \alpha}\right)_{n_B} + \frac{\partial \varepsilon}{\partial n_p} \left(\frac{\partial n_p}{\partial \alpha}\right)_{n_B} = \frac{n_B}{2} \left(\mu_n - \mu_p \right) \]

this can be rewritten

\[ S(n_B,\alpha) = \frac{1}{4} \frac{\partial}{\partial \alpha} \left(\mu_n - \mu_p\right) \]

where the dependence of the chemical potentials on $ n_B $ and $ \alpha $ is not written explicitly. This quantity is computed by function fesym(). Note that many of the functions in this class are written in terms of the proton fraction $ x_p = (1-\alpha)/2 $ denoted as 'pf' instead of as functions of $ \alpha $. Frequently, $ S(n_B,\alpha) $ is evaluated at $ \alpha=0 $ to give a univariate function of the baryon density. It is sometimes also evaluated at the point $ (n_B=n_0, \alpha=0) $, and this value is denoted by $ S $ above and is typically stored in esym. Alternatively, one can define the symmetry energy by

\[ \tilde{S}(n_B) \approx E(n_B,\alpha=1)-E(n_B,\alpha=0) \]

which is computed by function fesym_diff() . The functions $ S(n_B,\alpha=0) $ and $ \tilde{S}(n_B) $ are equal when $ {\cal O}(\alpha^4) $ terms are zero. In this case, $ \mu_n - \mu_p $ is proportional to $ \alpha $ and so

\[ S(n_B) = \tilde{S}(n_B) = \frac{1}{4} \frac{(\mu_n-\mu_p)}{\alpha} \, . \]

The symmetry energy slope parameter $ L $, can be defined by

\[ L(n_B,\alpha) \equiv 3 n_B \frac{\partial S(n_B,\alpha)} {\partial n_B} = 3 n_B \frac{\partial}{\partial n_B} \left[ \frac{1}{2 n_B} \frac{\partial^2 \varepsilon}{\partial \alpha^2} \right] \]

This can be rewritten as

\[ L(n_B,\alpha) = \frac{3 n_B}{4} \frac{\partial}{\partial n_B} \frac{\partial}{\partial \alpha} \left(\mu_n - \mu_p\right) \]

(where the derivatives can be evaluated in either order) or alternatively using

\[ \left(\frac{\partial \varepsilon}{\partial n_B}\right)_{\alpha} = \frac{\partial \varepsilon}{\partial n_n} \left(\frac{\partial n_n}{\partial n_B}\right)_{\alpha} + \frac{\partial \varepsilon}{\partial n_p} \left(\frac{\partial n_p}{\partial n_B}\right)_{\alpha} = \frac{1}{2} \left(\mu_n + \mu_p \right) \]

$ L $ can be rewritten

\begin{eqnarray*} L(n_B,\alpha) &=& 3 n_B \left[\frac{-1}{2 n_B^2} \frac{\partial^2 \varepsilon}{\partial \alpha^2} + \frac{1}{4 n_B} \frac{\partial^2}{\partial \alpha^2} \left(\mu_n + \mu_p\right)\right] \\ &=& \frac{3}{4}\frac{\partial^2}{\partial \alpha^2} \left(\mu_n + \mu_p\right) - 3 S(n_B,\alpha) \, . \end{eqnarray*}

The third derivative with respect to the baryon density is sometimes called the skewness. Here, we define

\[ Q_0(n_B,\alpha) = 27 n_B^3 \frac{\partial^3}{\partial n_B^3} \left(\frac{\varepsilon}{n_B}\right) = 27 n_B^3 \frac{\partial^2}{\partial n_B^2} \left(\frac{P}{n_B^2}\right) \]

and this function is computed in fkprime() .

The second derivative of the symmetry energy with respect to the baryon density is

\[ K_{\mathrm{sym}}(n_B,\alpha) = 9 n_B^2 \frac{\partial^2}{\partial n_B^2} S(n_B,\alpha) \]

The third derivative of the symmetry energy with respect to the baryon density is

\[ Q_{\mathrm{sym}}(n_B,\alpha) = 27 n_B^3 \frac{\partial^3}{\partial n_B^3} S(n_B,\alpha) \]

Note that solving for the baryon density for which $ P=0 $ gives, to order $ \alpha^2 $ (Piekarewicz09)

\[ n_B = n_0 \left[ 1 + \frac{6 K}{Q} + \alpha^2 \left( \frac{3 L}{K}-\frac{6 K_{\mathrm{sym}}}{Q} + \frac{6 K Q_{\mathrm{sym}}}{Q^2} \right) \right] \]

The quartic symmetry energy $ S_4(n_B,\alpha) $ can be defined as

\[ S_4(n_B,\alpha) \equiv \frac{1}{24 n_B}\frac{\partial^4 \varepsilon} {\partial \alpha^4} \]

However, fourth derivatives are difficult numerically, and so an alternative quantity is preferable. Instead, one can evaluate the extent to which $ {\cal O}(\alpha^4) $ terms are important from

\[ \eta(n_B) \equiv \frac{E(n_B,1)-E(n_B,1/2)} {3 \left[E(n_B,1/2)-E(n_B,0)\right]} \]

as described in Steiner06 . This function can be expressed either in terms of $ \tilde{S} $ or $ S_4 $

\[ \eta(n_B) = \frac{5 \tilde{S}(n_B) - S(n_B,0)} {\tilde{S}(n_B) + 3 S(n_B,0)} = \frac{5 S_4(n_B,0) + 4 S(n_B,0)} {S_4(n_B,0) + 4 S(n_B,0)} \]

Evaluating this function at the saturation density gives

\[ \eta(n_0) = \frac{4 S + 5 S_4}{4 S + S_4} \]

(Note that $ S_4 $ is referred to as $ Q $ in Steiner06). Sometimes it is useful to separate out the kinetic and potential parts of the energy density when computing $ \eta(n_B) $, and the class sym4_eos_base is useful for this purpose.

Idea for Future:
Could write a function to compute the "symmetry free energy" or the "symmetry entropy"

Definition at line 248 of file hadronic_eos.h.

Public Member Functions

int gradient_qij (fermion &n, fermion &p, thermo &th, double &qnn, double &qnp, double &qpp, double &dqnndnn, double &dqnndnp, double &dqnpdnn, double &dqnpdnp, double &dqppdnn, double &dqppdnp)
 Calculate coefficients for gradient part of Hamiltonian.
virtual const char * type ()
 Return string denoting type ("hadronic_eos")
Equation of state
virtual int calc_p (fermion &n, fermion &p, thermo &th)=0
 Equation of state as a function of the chemical potentials.
virtual int calc_e (fermion &n, fermion &p, thermo &th)=0
 Equation of state as a function of density.
EOS properties
virtual double fcomp (double nb, const double &alpha=0.0)
 Calculate the incompressibility in $ \mathrm{fm}^{-1} $ using calc_e()
virtual double feoa (double nb, const double &alpha=0.0)
 Calculate the energy per baryon in $ \mathrm{fm}^{-1} $ using calc_e()
virtual double fesym (double nb, const double &alpha=0.0)
 Calculate symmetry energy of matter in $ \mathrm{fm}^{-1} $ using calc_dmu_alpha() .
virtual double fesym_err (double nb, double &alpha, double &unc)
 Calculate symmetry energy of matter and its uncertainty.
virtual double fesym_slope (double nb, const double &alpha=0.0)
 The symmetry energy slope parameter.
virtual double fesym_curve (double nb, const double &alpha=0.0)
 The curvature of the symmetry energy.
virtual double fesym_skew (double nb, const double &alpha=0.0)
 The skewness of the symmetry energy.
virtual double fesym_diff (double nb)
 Calculate symmetry energy of matter as energy of neutron matter minus the energy of nuclear matter.
virtual double feta (double nb)
 The strength parameter for quartic terms in the symmetry energy.
virtual double fkprime (double nb, const double &alpha=0.0)
 Calculate skewness of nuclear matter using calc_e()
virtual double fmsom (double nb, const double &alpha=0.0)
 Calculate reduced neutron effective mass using calc_e()
virtual double fn0 (double alpha, double &leoa)
 Calculate saturation density using calc_e()
virtual int saturation ()
 Calculates some of the EOS properties at the saturation density.
Functions for calculating physical properties
double calc_dmu_alpha (double alpha, const double &nb)
 Compute the difference between neutron and proton chemical potentials as a function of the isospin asymmetry.
double calc_musum_alpha (double alpha, const double &nb)
 Compute the sum of the neutron and proton chemical potentials as a function of the isospin asymmetry.
double calc_pressure_nb (double nb, const double &alpha=0.0)
 Compute the pressure as a function of baryon density at fixed isospin asymmetry.
double calc_edensity_nb (double nb, const double &alpha=0.0)
 Compute the energy density as a function of baryon density at fixed isospin asymmetry.
void const_pf_derivs (double nb, double pf, double &dednb_pf, double &dPdnb_pf)
 Compute derivatives at constant proton fraction.
double calc_press_over_den2 (double nb, const double &alpha=0.0)
 Calculate pressure / baryon density squared in nuclear matter as a function of baryon density at fixed isospin asymmetry.
double calc_edensity_alpha (double alpha, const double &nb)
 Calculate energy density as a function of the isospin asymmetry at fixed baryon density.
Other functions
int nuc_matter_p (size_t nv, const ovector_base &x, ovector_base &y, double *&pa)
 Nucleonic matter from calc_p()
int nuc_matter_e (size_t nv, const ovector_base &x, ovector_base &y, double *&pa)
 Nucleonic matter from calc_e()
Set auxilliary objects
virtual int set_mroot (mroot< mm_funct<> > &mr)
 Set class mroot object for use in calculating chemical potentials from densities.
virtual int set_sat_root (root< funct > &mr)
 Set class mroot object for use calculating saturation density.
virtual int set_sat_deriv (deriv< funct > &de)
 Set deriv object to use to find saturation properties.
virtual int set_sat_deriv2 (deriv< funct > &de)
 Set the second deriv object to use to find saturation properties.
virtual int set_n_and_p (fermion &n, fermion &p)
 Set neutron and proton.

Data Fields

double eoa
 Binding energy.
double comp
 Compressibility.
double esym
 Symmetry energy.
double n0
 Saturation density.
double msom
 Effective mass (neutron)
double kprime
 Skewness.
fermion def_neutron
 The defaut neutron.
fermion def_proton
 The defaut proton.
Default solvers and derivative classes
gsl_deriv< functdef_deriv
 The default object for derivatives.
gsl_deriv< functdef_deriv2
 The second default object for derivatives.
gsl_mroot_hybrids< mm_funct<> > def_mroot
 The default solver.
cern_mroot_root< functdef_sat_root
 The default solver for calculating the saturation density.

Protected Member Functions

double t1_fun (double barn)
 Compute t1 for gradient_qij().
double t2_fun (double barn)
 Compute t2 for gradient_qij().

Protected Attributes

mroot< mm_funct<> > * eos_mroot
 The EOS solver.
root< funct > * sat_root
 The solver to compute saturation properties.
deriv< funct > * sat_deriv
 The derivative object for saturation properties.
deriv< funct > * sat_deriv2
 The second derivative object for saturation properties.
fermionneutron
 The neutron object.
fermionproton
 The proton object.

Member Function Documentation

virtual double hadronic_eos::fcomp ( double  nb,
const double &  alpha = 0.0 
) [virtual]

This function computes $ K (n_B,\alpha) = 9 n_B \partial^2 \varepsilon /(\partial n_B^2) = 9 \partial P / (\partial n_B) $ . The value of $ K(n_0,0) $, often referred to as the "compressibility", is stored in comp by saturation() and is about 240 MeV at saturation density.

virtual double hadronic_eos::feoa ( double  nb,
const double &  alpha = 0.0 
) [virtual]

This function computes the energy per baryon of matter without the nucleon rest masses at the specified baryon density, nb, and isospin asymmetry alpha.

virtual double hadronic_eos::fesym ( double  nb,
const double &  alpha = 0.0 
) [virtual]

This function computes the symmetry energy,

\[ \left(\frac{1}{2 n_B}\frac{d^2 \varepsilon}{d \alpha^2} \right) = \frac{1}{4} \frac{\partial}{\partial \alpha} \left(\mu_n - \mu_p \right) \]

at the value of $ n_B $ given in nb and $ \alpha $ given in alpha. The symmetry energy at $ \alpha=0 $ at the saturation density and is stored in esym by saturation().

virtual double hadronic_eos::fesym_err ( double  nb,
double &  alpha,
double &  unc 
) [virtual]

This estimates the uncertainty due to the numerical differentiation, assuming that difference betwen the neutron and proton chemical potentials is computed exactly by calc_dmu_alpha() .

virtual double hadronic_eos::fesym_slope ( double  nb,
const double &  alpha = 0.0 
) [virtual]

This returns the value of the "slope parameter" of the symmetry energy

\[ L=3 n_{B} \left(\frac{\partial E_{sym}}{\partial n_{B}}\right) \]

in inverse Fermis.

where $ n_B $ is the baryon density. This ranges between about zero and 200 MeV for many EOSs.

virtual double hadronic_eos::fesym_diff ( double  nb) [virtual]

This function returns the energy per baryon of neutron matter minus the energy per baryon of nuclear matter. This will deviate significantly from the results from fesym() only if the dependence of the symmetry energy on $ \delta $ is not quadratic.

Reimplemented in apr_eos.

virtual double hadronic_eos::fkprime ( double  nb,
const double &  alpha = 0.0 
) [virtual]

The skewness is defined to be $ 27 n^3 d^3 (\varepsilon/n)/(d n^3) = 27 n^3 d^2 (P/n^2) / (d n^2) $ and is denoted 'kprime'. This definition seems to be ambiguous for densities other than the saturation density and is not quite analogous to the compressibility.

virtual double hadronic_eos::fmsom ( double  nb,
const double &  alpha = 0.0 
) [virtual]

Neutron effective mass (as stored in part::ms) divided by vacuum mass (as stored in part::m) in nuclear matter at saturation density. Note that this simply uses the value of n.ms from calc_e(), so that this effective mass could be either the Landau or Dirac mass depending on the context. Note that this may not be equal to the reduced proton effective mass.

virtual double hadronic_eos::fn0 ( double  alpha,
double &  leoa 
) [virtual]

This function finds the baryon density for which the pressure vanishes.

virtual int hadronic_eos::saturation ( ) [virtual]
Idea for Future:
It would be great to provide numerical uncertainties in the saturation properties.

Reimplemented in rmf_eos, and rmf_delta_eos.

int hadronic_eos::gradient_qij ( fermion n,
fermion p,
thermo th,
double &  qnn,
double &  qnp,
double &  qpp,
double &  dqnndnn,
double &  dqnndnp,
double &  dqnpdnn,
double &  dqnpdnp,
double &  dqppdnn,
double &  dqppdnp 
)
Note:
This is still somewhat experimental.

We want the gradient part of the Hamiltonian in the form

\[ {\cal H}_{\mathrm{grad}} = \frac{1}{2} \sum_{i=n,p} \sum_{j=n,p} Q_{ij} \vec{\nabla} n_i \cdot \vec{\nabla} n_j \]

The expression for the gradient terms from Pethick95 is

\begin{eqnarray*} {\cal H}_{\mathrm{grad}}&=&-\frac{1}{4} \left(2 P_1 + P_{1;f}-P_{2;f}\right) \nonumber \\ && +\frac{1}{2} \left( Q_1+Q_2 \right) \left(n_n \nabla^2 n_n+n_p \nabla^2 n_p\right) \nonumber \\ && + \frac{1}{4}\left( Q_1-Q_2 \right) \left[\left(\nabla n_n\right)^2+\left(\nabla n_p\right)^2 \right] \nonumber \\ && + \frac{1}{2} \frac{d Q_2}{d n} \left( n_n \nabla n_n + n_p \nabla n_p \right) \cdot \nabla n \end{eqnarray*}

This can be rewritten

\begin{eqnarray*} {\cal H}_{\mathrm{grad}}&=&\frac{1}{2}\left(\nabla n\right)^2 \left[ \frac{3}{2} P_1+n \frac{d P_1}{d n}\right] \nonumber \\ && - \frac{3}{4} \left[ \left( \nabla n_n\right)^2 + \left( \nabla n_p \right)^2 \right] \nonumber \\ && -\frac{1}{2} \left[ \right] \cdot \nabla n \frac{d Q_1}{d n} \nonumber \\ && - \frac{1}{4} \left( \nabla n\right)^2 P_2 - \frac{1}{4} \left[ \left( \nabla n_n\right)^2 + \left( \nabla n_p \right)^2 \right] Q_2 \end{eqnarray*}

or

\begin{eqnarray*} {\cal H}_{\mathrm{grad}}&=&\frac{1}{4} \left( \nabla n\right)^2 \left[3 P_1 + 2 n \frac{d P_1}{d n}-P_2\right] \nonumber \\ && - \frac{1}{4} \left( 3 Q_1+Q_2 \right) \left[ \left( \nabla n_n\right)^2 + \left( \nabla n_p \right)^2 \right] \nonumber \\ && - \frac{1}{2} \frac{d Q_1}{d n} \left[ n_n \nabla n_n + n_p \nabla n_p \right] \cdot \nabla n \end{eqnarray*}

or

\begin{eqnarray*} {\cal H}_{\mathrm{grad}}&=&\frac{1}{4} \left( \nabla n\right)^2 \left[3 P_1 + 2 n \frac{d P_1}{d n}-P_2\right] \nonumber \\ && - \frac{1}{4} \left( 3 Q_1+Q_2 \right) \left[ \left( \nabla n_n\right)^2 + \left( \nabla n_p \right)^2 \right] \nonumber \\ && - \frac{1}{2} \frac{d Q_1}{d n} \left[ n_n \left( \nabla n_n \right)^2 + n_p \left( \nabla n_p \right)^2 + n \nabla n_n \cdot \nabla n_p \right] \end{eqnarray*}

Generally, for Skyrme-like interactions

\begin{eqnarray*} P_i &=& \frac{1}{4} t_i \left(1+\frac{1}{2} x_i \right) \nonumber \\ Q_i &=& \frac{1}{4} t_i \left(\frac{1}{2} + x_i \right) \, . \end{eqnarray*}

for $ i=1,2 $ .

This function uses the assumption $ x_1=x_2=0 $ to calculate $ t_1 $ and $ t_2 $ from the neutron and proton effective masses assuming the Skyrme form. The values of $ Q_{ij} $ and their derivatives are then computed.

The functions set_n_and_p() and set_thermo() will be called by gradient_qij(), to facilitate the use of the n, p, and th parameters.

double hadronic_eos::calc_pressure_nb ( double  nb,
const double &  alpha = 0.0 
)

Used by fcomp().

double hadronic_eos::calc_edensity_nb ( double  nb,
const double &  alpha = 0.0 
)

This function calls hadronic_eos::calc_e() with the internally stored neutron and proton objects.

double hadronic_eos::calc_press_over_den2 ( double  nb,
const double &  alpha = 0.0 
)

Used by fkprime().

double hadronic_eos::calc_edensity_alpha ( double  alpha,
const double &  nb 
)

Used by fesym().

This function calls hadronic_eos::calc_e() with the internally stored neutron and proton objects.

virtual int hadronic_eos::set_mroot ( mroot< mm_funct<> > &  mr) [virtual]
Note:
While in principle this allows one to use any mroot object, in practice some of the current EOSs require gsl_mroot_hybrids because it automatically avoids regions where the equations are undefined.
virtual int hadronic_eos::set_sat_root ( root< funct > &  mr) [virtual]
Note:
While in principle this allows one to use any mroot object, in practice some of the current EOSs require gsl_mroot_hybrids because it automatically avoids regions where the equations are undefined.
virtual int hadronic_eos::set_sat_deriv2 ( deriv< funct > &  de) [virtual]

Computing the slope of the symmetry energy at the saturation density requires two derivative objects, because it has to take an isospin derivative and a density derivative. Thus this second deriv object is used in the function fesym_slope().


Field Documentation

The value of gsl_deriv::h is set to $ 10^{-3} $ in the hadronic_eos constructor.

Definition at line 608 of file hadronic_eos.h.

The value of gsl_deriv::h is set to $ 10^{-3} $ in the hadronic_eos constructor.

Definition at line 615 of file hadronic_eos.h.

Used by calc_e() to solve nuc_matter_p() (2 variables) and by calc_p() to solve nuc_matter_e() (2 variables).

Definition at line 622 of file hadronic_eos.h.

Used by fn0() (which is called by saturation()) to solve saturation_matter_e() (1 variable).

Definition at line 630 of file hadronic_eos.h.


The documentation for this class was generated from the following file:
 All Data Structures Namespaces Files Functions Variables Typedefs Enumerations Friends

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).

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