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