Statistical Physics, Second Revised and Enlarged Edition

88 Fermi–Diracgases

functiong(ε)∝ε^1 /^2 andμfallsslightly.Usingthe results ofAppendixCitcan

be shown that for this case

μ(T)=μ( 0 )[ 1 −(π^2 / 12 )(kkkBT/μ)^2 ...]

Since the variation is small, we shall in future not distinguish betweenμandμ(0),

unless confusion wouldarise. Note that the symbolεFfor the Fermienergy will

be usedinplace ofμin sections 8 .2 and8. 3.

8 .1.3 The thermodynamicfunctions

It remains to calculate thethermodynamicfunctions,U,CV,Pandso on.Weshall

do this explicitlyin the degenerate limit only, and shall sketchgraphs to indicate how

the functions connect to the classical MB limit. Numerical methods are needed to

compute theshape ofsuchgraphs.

Internal energyU. The internal energy can be evaluated from the distribution

functionf(ε),simplyusingthedirect expression

U=

∑

i

niεi

Inthe present context thisbecomes

U=

∫∞

0

∫∫

εg(ε)))f(ε)dε (8.7)

AtT=0,f(ε)isthesimple stepfunction, so that theintegralis readilyevaluatedas

U( 0 )=

∫μ

0

∫∫

εg(ε)dε

Substitutingg(ε)=Cε^1 /^2 for the fermion gas, this gives

U( 0 )=

2

5

Cμ^5 /^2

=

3

5

Nμ (8.8)

The result (8.8)follows without the needto rememberCwhenitis recalledfrom the

previous section thatN=^23 Cμ^3 /^2 from the correspondingintegral(8.4). Thisvalue

ofU(0) represents a very large zero-point energy, an average of 0. 6 μper particle. It

isadirect expression ofthe exclusion principle, that the particles areforcedto occupy

highenergystates even atT=0. (Fermionsdisplaytower-blockmentality!)