Stars as Laboratories for Fundamental Physics - MPP Theory Group

594 Appendix D
of antifermions is given by the same expression with µ → −µ. Then µ
is implicitly given by the phase-space integral
∫
n f =
2 d 3 (
)
p 1
(2π) 3 e (E p−µ)/T
+ 1 − 1
, (D.5)
e (E p+µ)/T
+ 1
where the second term represents antifermions and the factor 2 is for the
two spin degrees of freedom. Again, the fermion density is understood
to mean the density of fermions minus that of antifermions.
At vanishing temperature, f p becomes a step function Θ(µ − E p ).
If µ > 0 so that n f > 0, i.e. an excess of fermions over antifermions,
there are no antifermions at all at T = 0. The fermion integral yields
n f = p 3 F/3π 2 ,
(D.6)
where the Fermi momentum is defined by µ 2 0 = p 2 F + m 2 with µ 0 the
zero-temperature chemical potential. The Fermi energy is defined by
E 2 F = p 2 F + m 2 e, i.e. E F = µ 0 .
Equation D.6 is taken as the definition of the Fermi momentum even
at T > 0; it is a useful parameter to characterize the fermion density,
whether or not they are degenerate. Numerically it is
p F = 5.15 keV (Y e ρ) 1/3
(D.7)
for electrons with the mass density ρ in units of g cm −3 .
In general, Eq. (D.5) cannot be made explicit for µ; it has to be
solved numerically or by an approximation method. In the ρ-T -plane,
contours for the electron chemical potential are shown in Fig. D.3.
Above the main plot, the electron density is characterized by p F . On
the right side, the temperature is shown in units of keV. Recall that
10 7 K = 0.8621 keV (Appendix A).
For nonrelativistic electrons the contours in Fig. D.3 are very sensitive
to the exact value of µ. Therefore, in this regime the nonrelativistic
chemical potential
ˆµ ≡ µ − m (D.8)
is a more appropriate parameter. Often ˆµ is referred to as the chemical
potential. This can be very confusing when relativistic effects are
important. In terms of ˆµ, the relativistic Fermi-Dirac distribution is
f p =
1
e (E kin−ˆµ)/T
+ 1 ,
(D.9)
with the kinetic energy E kin = E p −m → p 2 /2m (nonrelativistic limit).