#include <sn_nr_fermion.h>
This does not include the rest mass energy in the chemical potential or the rest mass energy density in the energy density to alleviate numerical precision problems at low densities
This implements an equation of state for a nonrelativistic fermion using direct integration. After subtracting the rest mass from the chemical potentials, the distribution function is
where is the effective chemical potential,
is the rest mass, and
is the effective mass. For later use, we define
.
Uncertainties are given in unc.
Evaluation of the derivatives
The relevant derivatives of the distribution function are
We also need the derivative of the entropy integrand w.r.t. the distribution function, which is quite simple
where the entropy density is
The derivatives can be integrated directly or they may be converted to integrals over the distribution function through an integration by parts
using the distribution function for and 0 and
as the limits, we have
as long as vanishes at
. Rewriting,
as long as vanishes at
.
Explicit forms
1) The derivative of the density wrt the chemical potential
Using we get
2) The derivative of the density wrt the temperature
Using we get
3) The derivative of the entropy wrt the chemical potential
This verifies the Maxwell relation
4) The derivative of the entropy wrt the temperature
Using
5) The derivative of the density wrt the effective mass
Using we get
New section
and
, so
or
1) The derivative of the density wrt the chemical potential
2) The derivative of the density wrt the temperature
4) The derivative of the entropy wrt the temperature
5) The derivative of the density wrt the effective mass
Definition at line 221 of file sn_nr_fermion.h.
Public Member Functions | |
sn_nr_fermion (double m=0.0, double g=0.0) | |
Create a fermion with mass m and degeneracy g . | |
virtual int | calc_mu (const double temper) |
Calculate properties as function of chemical potential. | |
virtual int | calc_density (const double temper) |
Calculate properties as function of density. | |
virtual int | pair_mu (const double temper) |
Calculate properties with antiparticles as function of chemical potential. | |
virtual int | pair_density (const double temper) |
Calculate properties with antiparticles as function of density. | |
virtual int | nu_from_n (const double temper) |
Calculate effective chemical potential from density. | |
int | set_density_root (root< const double, funct< const double > > &rp) |
Set the solver for use in calculating the chemical potential from the density. | |
virtual const char * | type () |
Return string denoting type ("sn_nr_fermion"). | |
Data Fields | |
double | flimit |
The limit for the Fermi functions (default 20.0). | |
fermion | unc |
Storage for the most recently calculated uncertainties. | |
deriv_part | dunc |
Storage for the most recently calculated uncertainties. | |
bool | guess_from_nu |
If true, use the present value of the chemical potential as a guess for the new chemical potential. | |
cern_mroot_root< const double, funct< const double > > | def_density_root |
The default solver for npen_density() and pair_density(). | |
Protected Member Functions | |
int | solve_fun (double x, double &yy, const double &temper) |
Function to compute chemical potential from density. | |
int | pair_fun (double x, double &yy, const double &temper) |
Function to compute chemical potential from density when antiparticles are included. | |
Protected Attributes | |
root< const double, funct < const double > > * | density_root |
Solver to compute chemical potential from density. |
double flimit |
The limit for the Fermi functions (default 20.0).
sn_nr_fermion will ignore corrections smaller than about .
Definition at line 235 of file sn_nr_fermion.h.
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