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:
Numerical inversion of F
1/2(x)=y: x = F
1/2-1(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.
.GNU_ScientificLibrary.Blocks.Physics.FermiDiracEoS
FhalfMinusF