**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*^{ 3}**r** *d*^{ 3}**p** :

* n_{i}*(

*ε _{j}* =

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

*n*(*p*) *dp* =
2 (4π*p*^{2}*dp* / *h*^{3})
{e^{ [−μ +
mc 2 +
ε(p)] / kT} + 1}^{−1} /cm^{3}.

**a.** *Complete degeneracy*

Consider the 'exponential' part of the distribution,
*F*(*ε*) ≡ {e^{ [ε − (μ −
mc 2)] / kT} + 1}^{−1}.

As *T* → 0, *F*(*ε*) → 1 for
*ε* > *μ* − *mc*^{ 2},
and *F*(*ε*) → 0 for
*ε* < *μ* − *mc*^{ 2}.

We define the *Fermi energy ε*_{F} ≡ *μ* − *mc*^{ 2},
or conversely *μ* = ε_{F} + *mc*^{ 2}.

To find how these quantities depend on number density, let *x ≡ p*/*mc*,
and consider the scaled Fermi momentum *x*_{F} that corresponds to
ε_{F} through the *ε*(*p*) relation above.
Now integrate the total density over momenta as

*n* = (8π/*h*^{3}) ∫_{o}^{pF}
*p*^{2}*dp* =
8π(*mc*/*h*)^{3} ∫_{o}^{xF}
*x*^{2}*dx* = (8π/3) (*mc*/*h*)^{3} *x*_{F}^{3}.

Substituting values for electrons gives
*n*_{e} = 5.865×10^{29} *x*_{F}^{3}.

Carrying out the pressure integral gives

*P* = (8π/3) (*mc*/*h*)^{3} *mc*^{2}
∫_{o}^{xF} *x*^{4}
(1 + *x*^{2})^{−1/2} *dx* ≡ *A f* (*x*_{F})

*A* = (π/3) (*mc*/*h*)^{3} *mc*^{2} =
6.002×10^{22} dyne/cm^{2} for electrons,

*f* (*x*_{F}) = *x*_{F} (2*x*_{F}^{2} − 3)
(1 + *x*_{F}^{2})^{1/2} + 3sinh^{−1}*x*_{F} .

A similar expression may be written for the internal energy *E*(*x*_{F}).

These expressions cover the range from non-relativistic to relativistic.
Converting *x*_{F} to *n*_{e} and then to
*ρ* via *n*_{e}
= *ρN*_{A}/__μ___{e} gives

*P*_{e} = 1.004×10^{13}
(*ρ*/__μ___{e})^{5/3} in the non-relativistic limit, and

*P*_{e} = 1.243×10^{15}
(*ρ*/__μ___{e})^{4/3} in the relativistic limit
(relativistic effects become important at about
*ρ*/__μ___{e} = 10^{6} ).

**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

*M*_{c} = 1.456 (2/__μ___{e})^{2} *M*_{o} ,

where *M*_{o} 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 *kT* ≈ *ε*_{F}.
For electrons, this transition occurs at

*ρ*/__μ___{e} ≈ 6×10^{−9} *T*^{ 3/2}
non-relativistic,

*ρ*/__μ___{e} ≈ 5×10^{−24} *T*^{ 3}
relativistic (the cross-over is at about *T* = 10^{10},
*ρ*/__μ___{e} = 10^{6} ).

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.