Properties ofan ideal Fermi–Dirac gas 851
2310ff() Fig. 8. 1 The Fermi–Dirac distribution function at three different temperatures. Curve 1,kkkBT= 0. 02 μ.
Curve2,kkkBT= 0. 1 μ,Curve3,kkkBT= 0. 5 μ.
8 .1.2 The Fermienergy
To use theFDdistribution (8.2), we needtoknow the Fermienergyμ.This parameter
(beingrelated toα) is fixed bythe number conditionN=
∑
ni,asexplained in
section 5 .4.4. Thevalue ofμwill depend on the macrostate conditions (N,V,T),
andinparticularitwill beafunction ofT. We startbycalculatingits value atT=
0. There are two obvious ways to proceed, and it is worth being aware of both
ofthem.
Method 1. Use the density of statesg(ε).The definition and the form ofg(ε)should
bewell-knownbythis stage. The states aredescribedbyfitting wavesintoboxes, as
inChapter 4, togive the statesink.A transformation usingthedispersion relationis
then used to give the density of states inε.Theprocedure is almost identical to that
givenin section 4.3forhelium gas,leading to the result (4.9). Theonlymodifications
to (4.9) are, (i)that the massMshouldrefer not to Hebut to therelevantparticle
mass, and (ii) that a spin factorG=2 for the spin-^12 fermions should multiply the
result. Hence
g(ε)δε=V 4 π( 2 M/h^2 )^3 /^2 ε^1 /^2 δε (8.3)The determinationofμ(0), the Fermi energy atT = 0 ,follows from the fixed
numberNofparticlesinthe macrostate. In thedensity ofstates approximation we
have (directly from thedefinitions ofg(ε)andofthefillingfactorf(ε))
N=
∫∞
0∫∫
g(ε)))f(ε)dε (8.4)