Class fermion_deriv_nr_tl (o2scl)¶
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template<class
fermion_deriv_t
= fermion_deriv_tl<double>, classfp_t
= double>
classo2scl
::
fermion_deriv_nr_tl
: public o2scl::fermion_deriv_thermo_tl<double>¶ Equation of state for a nonrelativistic fermion.
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
\[ \left\{1+\exp\left[\left(\frac{k^2} {2 m^{*}}-\nu\right)/T\right]\right\}^{-1} \]where \( \nu \) is the effective chemical potential, \( m \) is the rest mass, and \( m^{*} \) is the effective mass. For later use, we define \( E^{*} = k^2/2/m^{*} \) .Uncertainties are given in unc.
Evaluation of the derivatives
The relevant derivatives of the distribution function are
\[ \frac{\partial f}{\partial T}= f(1-f)\frac{E^{*}-\nu}{T^2} \]\[ \frac{\partial f}{\partial \nu}= f(1-f)\frac{1}{T} \]\[ \frac{\partial f}{\partial k}= -f(1-f)\frac{k}{m^{*} T} \]\[ \frac{\partial f}{\partial m^{*}}= f(1-f)\frac{k^2}{2 m^{*2} T} \]We also need the derivative of the entropy integrand w.r.t. the distribution function, which is quite simple
\[ {\cal S}\equiv f \ln f +(1-f) \ln (1-f) \qquad \frac{\partial {\cal S}}{\partial f} = \ln \left(\frac{f}{1-f}\right) = \left(\frac{\nu-E^{*}}{T}\right) \]where the entropy density is\[ s = - \frac{g}{2 \pi^2} \int_0^{\infty} {\cal S} k^2 d k \]The derivatives can be integrated directly or they may be converted to integrals over the distribution function through an integration by parts
\[ \int_a^b f(k) \frac{d g(k)}{dk} dk = \left.f(k) g(k)\right|_{k=a}^{k=b} - \int_a^b g(k) \frac{d f(k)}{dk} dk \]using the distribution function for \( f(k) \) and 0 and \( \infty \) as the limits, we have\[ \frac{g}{2 \pi^2} \int_0^{\infty} \frac{d g(k)}{dk} f dk = \frac{g}{2 \pi^2} \int_0^{\infty} g(k) f (1-f) \frac{k}{E^{*} T} dk \]as long as \( g(k) \) vanishes at \( k=0 \) . Rewriting,\[ \frac{g}{2 \pi^2} \int_0^{\infty} h(k) f (1-f) dk = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{T m^{*}}{k} \left[ h^{\prime} - \frac{h}{k}\right] d k \]as long as \( h(k)/k \) vanishes at \( k=0 \) .Explicit forms
1) The derivative of the density wrt the chemical potential
\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2}{T} f (1-f) dk \]Using \( h(k)=k^2/T \) we get\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} m^{*} f dk \]2) The derivative of the density wrt the temperature
\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2(E^{*}-\nu)}{T^2} f (1-f) dk \]Using \( h(k)=k^2(E^{*}-\nu)/T^2 \) we get\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{f}{T} \left[m^{*} \left(E^{*}-\nu\right) -k^2\right] d k \]3) The derivative of the entropy wrt the chemical potential
\[ \left(\frac{d s}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-\nu)}{T^2} dk \]This verifies the Maxwell relation\[ \left(\frac{d s}{d \mu}\right)_T = \left(\frac{d n}{d T}\right)_{\mu} \]4) The derivative of the entropy wrt the temperature
\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-\nu)^2}{T^3} dk \]Using \( h(k)=k^2 (E^{*}-\nu)^2/T^3 \)\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{m^{*}}{T^2} \left[\left( E^{*}-\nu \right)^2 +\frac{2 k^2}{m^{*}} \left(E^{*}-\nu\right)\right] d k \]5) The derivative of the density wrt the effective mass
\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2}{2 m^{* 2} T} f (1-f) k^2 dk \]Using \( h(k)=k^4/(2 m^{* 2} T) \) we get\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{3 k^2}{2 m^{*}} d k \]Conversion to unitless variables:
After integrating by parts \( u = k^2/2/m^{*}/T \) and \( y=\mu/T \), so
\[ k d k = m^{*} T d u \]or\[ d k = \frac{m^{*} T}{\sqrt{2 m^{*} T u}} d u = \sqrt{\frac{m^{*} T}{2 u}} d u \]1) The derivative of the density wrt the chemical potential
\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g m^{* 3/2} \sqrt{T}}{2^{3/2} \pi^2} \int_0^{\infty} u^{-1/2} f d u \]2) The derivative of the density wrt the temperature
\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g m^{* 3/2} \sqrt{T}} {2^{3/2} \pi^2} \int_0^{\infty} f d u \left[ 3 u^{1/2} - y u^{-1/2}\right] \]4) The derivative of the entropy wrt the temperature
\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g m^{* 3/2} T^{1/2}}{2^{3/2} \pi^2} \int_0^{\infty} f \left[ 5 u^{3/2} - 6 y u^{1/2} + y^2 u^{-1/2}\right] d u \]5) The derivative of the density wrt the effective mass
\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = \frac{3 g m{* 1/2} T^{3/2}}{2^{3/2} \pi^2} \int_0^{\infty} u^{1/2} f d u \]Public Functions
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fermion_deriv_nr_tl
()¶ Create a fermion with mass
m
and degeneracyg
.
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~fermion_deriv_nr_tl
()¶
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void
calc_density_zerot
(fermion_deriv_t &f)¶ Calculate properties as function of density at \( T=0 \).
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void
calc_mu_zerot
(fermion_deriv_t &f)¶ Calculate properties as function of chemical potential at \( T=0 \).
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int
calc_mu
(fermion_deriv_t &f, fp_t temper)¶ Calculate properties as function of chemical potential.
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int
calc_density
(fermion_deriv_t &f, fp_t temper)¶ Calculate properties as function of density.
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int
pair_mu
(fermion_deriv_t &f, fp_t temper)¶ Calculate properties with antiparticles as function of chemical potential.
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int
pair_density
(fermion_deriv_t &f, fp_t temper)¶ Calculate properties with antiparticles as function of density.
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int
nu_from_n
(fermion_deriv_t &f, fp_t temper)¶ Calculate effective chemical potential from density.
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void
set_density_root
(root<> &rp)¶ Set the solver for use in calculating the chemical potential from the density.
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const char *
type
()¶ Return string denoting type (“fermion_deriv_nr”)
Public Members
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fp_t
flimit
¶ The limit for the Fermi functions (default 20.0)
fermion_deriv_nr will ignore corrections smaller than about \( \exp(-\mathrm{f{l}imit}) \) .
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fermion_deriv
unc
¶ Storage for the most recently calculated uncertainties.
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root_cern
def_density_root
¶ The default solver for npen_density() and pair_density()
Protected Functions
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fp_t
pair_fun
(fp_t x, fermion_deriv_t &f, fp_t T)¶ Function to compute chemical potential from density when antiparticles are included.
Protected Attributes
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root *
density_root
¶ Solver to compute chemical potential from density.
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