4. Equation of State and Related Descriptions

From Hansen and Kawaler.

4.4D. Fermi-Dirac statisitics

Elementary statistical mechanics leads to the following expression for the number of particles of species i in phase-space volume d 3r d 3p :

ni(p) d 3r d 3p = (d 3r d 3p / h3) ∑ j gj {e [−μi + εj + ε(p)] / kT ± 1}−1 .

εj = mc 2
ε(p) = mc 2 {[1 + (p/mc)2 ]1/2 − 1}
v(p) = ∂ε/∂p = (p/m) [1 + (p/mc)2 ]−1/2
P = (1/3) ∫ n(p) pv dp .

Individual Fermi particles have g = 2 and the ±-sign reverting to +. With isotropy we have

n(p) dp = 2 (4πp2dp / h3) {e [−μ + mc 2 + ε(p)] / kT + 1}−1 /cm3.

a. Complete degeneracy
Consider the 'exponential' part of the distribution, F(ε) ≡ {e [ε − (μmc 2)] / kT + 1}−1.
As T → 0, F(ε) → 1 for ε > μmc 2, and F(ε) → 0 for ε < μmc 2.
We define the Fermi energy εFμmc 2, or conversely μ = εF + mc 2.
To find how these quantities depend on number density, let x ≡ p/mc, and consider the scaled Fermi momentum xF that corresponds to εF through the ε(p) relation above. Now integrate the total density over momenta as

n = (8π/h3) ∫opF p2dp = 8π(mc/h)3oxF x2dx = (8π/3) (mc/h)3 xF3.

Substituting values for electrons gives ne = 5.865×1029 xF3.

Carrying out the pressure integral gives

P = (8π/3) (mc/h)3 mc2oxF x4 (1 + x2)−1/2 dxA f (xF)
A = (π/3) (mc/h)3 mc2 = 6.002×1022 dyne/cm2 for electrons,
f (xF) = xF (2xF2 − 3) (1 + xF2)1/2 + 3sinh−1xF .

A similar expression may be written for the internal energy E(xF).

These expressions cover the range from non-relativistic to relativistic. Converting xF to ne and then to ρ via ne = ρNA/μe gives

Pe = 1.004×1013 (ρ/μe)5/3 in the non-relativistic limit, and
Pe = 1.243×1015 (ρ/μe)4/3 in the relativistic limit (relativistic effects become important at about ρ/μe = 106 ).

b. White dwarfs
At this point in their discussion, H&K briefly discuss the application to white dwarfs, referring to an admittedly approximate constant-density model. We could attempt to apply our 2-layer model with slightly more accurate results. A properly integrated polytropic model results in the classical Chandrasekhar relativistic limit

Mc = 1.456 (2/μe)2 Mo ,

where Mo is the mass of the Sun.

c. Partial degeneracy
As one raises the temperature at any one density, the degeneracy becomes partial in the neighborhood of kTεF. For electrons, this transition occurs at

ρ/μe ≈ 6×10−9 T 3/2 non-relativistic,
ρ/μe ≈ 5×10−24 T 3 relativistic (the cross-over is at about T = 1010, ρ/μe = 106 ).

A good model must treat partial degeneracy for densities within a factor of 10 on either side of this boundary. In the past, modelers have used various power-law expansions or some graded combination of degenerate/non-degenerate limits in the boundary regime, but today they use either pre-prepared tables or direct integration.

Below is H&K's figure showing state regimes on the ρ,T plane.