blockFermiDiracEoS
Fermi-Dirac EoS
Extends from GNU_ScientificLibrary.Icons.Block (Icon for blocks).
Information
Fermi-Dirac Equation of State (EoS), appropriate for a gas of fermions, like the conduction electrons in a metal or plasma.
Parameters for the atomic mass, # of conduction e-'s per atom, and mass density (ρ) set the number density (n) of e-. Together with the temperature, T (in K), the relationship with the chemical potential (x = μ / kT) can be established:
n / T3/2 ~ F1/2(x) = y
Numerical inversion of F1/2(x)=y: x = F1/2-1(y). Then we can determine the energy density:
e ~ T5/2 F3/2(x)
which also gives us the pressure, P = 2e/3. The specific heat capacity is calculated via
c_v = de/dT ~ e [2.5 / T + F1/2(x) dx/dT / F3/2(x)] / ρ
where we have used F'3/2(x)=F1/2(x) and the derivative of the chemical potential (dx/dT) is calculated via
dx/dT ~ -1.5 n / (T5/2 F-1/2(x))
thanks to the first relation above and F'1/2(x)=F-1/2(x). The chemical potential, μ = x k T, is also provided as an output.
The energy density (in J m-3) and chemical potential (in J) outputs are shifted relative to the values at absolute zero:
e - e0 , where e0 = 3 n EF / 5
μ - EF , where EF = h2 (3 π2 n)2/3 / (8 π2 me)
EF and e0 are available as parameters (*.E_Fermi and *.edens0).
The default parameter values are those appropriate for copper.
Parameters
| Type | Name | Default | Description |
|---|---|---|---|
| Real | m_e | 9.10938e-31 | mass of the electron in kg |
| Real | k_dens | 2*(2*const.pi*m_e*const.k/const.h^2)^1.5 | |
| Real | k_pres | k_dens*const.k | |
| Real | eff_am | 63.546 | Da ; Effective atomic mass |
| Integer | N_e | 1 | # conduction e-'s per atom |
| Real | mdens | 8935 | kg/m^3 ; Mass density |
| Real | ndens | N_e*mdens/(eff_am*1.66053906892e-27) | 1/m^3 ; Number density of e- |
| Real | E_Fermi | 0.5*(0.5*const.h/const.pi)^2*(3*const.pi^2)^0.666666666667*ndens^0.666666666667/m_e | J ; Fermi Energy |
| Real | edens0 | 0.6*ndens*E_Fermi | J/m^3 ; Energy density at T=0 |
Connectors
| Type | Name | Default | Description |
|---|---|---|---|
| Modelica.Blocks.Interfaces.RealInput | T | ||
| Modelica.Blocks.Interfaces.RealOutput | e_minus_e0 | ||
| Modelica.Blocks.Interfaces.RealOutput | c_v | ||
| Modelica.Blocks.Interfaces.RealOutput | mu_minus_Ef |
Components
| Type | Name | Default | Description |
|---|---|---|---|
| Real | xmin | ||
| Real | xmax | ||
| Real | x | ||
| Real | y | ||
| Real | F3half | ||
| Real | edens | ||
| Real | mu | ||
| Real | Fmhalf | ||
| Real | dxdT |
Contents
| Name | Description |
|---|---|
 FhalfMinusF |