Statistical Physics, Second Revised and Enlarged Edition

Properties ofan ideal Fermi–Dirac gas 89

AboveT=0, whilstyetinthedegeneratelimitTTTTF,theintegral(8.7) canbe

evaluated using the method outlined in Appendix C. The result is

U=U( 0 )+U(th)

=

3

5

Nμ+(π^2 / 6 )(kkkBT)^2 g(μ)+··· (8.9)

As expected,Uremains dominated by the zero-point termU( 0 ),the second thermal
termU(th)being small.Thisis a reasonable result. Inhand-waving termsitfollows
from thesubtleblurringoff(ε)withrisingT.AtT =0wehave seen that all
states with energies belowμare full, whereas all those above are empty. At a low
temperatureT,from (8.2) or Fig. 8.1,itisevident thatfchangesfrom 1 to 0 over an
energyspan of orderkkkBTaroundthe Fermienergy,the occupation numbers ofstates
outside this span being unchanged. Hence only a number of orderg(μ)×kkkBTofthe
fermionshave their energies changed,andthechangein energyoftheseisoforder
kkkBT.Therefore one would expectU(th)∼(kkkBT)^2 g(μ),asin(8.9).
Equation (8.9) is shown with the explicit factorg(μ),since this displays the correct
physics, asjust explained.Itis preciselythis‘densityofstates at the Fermilevel’which
enters manyof the thermodynamic and the transport properties of a fermiongas. And
(8.9) continues to give the right answer even when we are not talking about an ideal
gasinthreedimensions, andthedensityofstatesis not oftheform (8.3).
Nevertheless, it is also interestingto return to the standard idealgas. Substituting
into (8.9) the appropriate density of states, (8.3), together with the expression (8.5)
for the Fermienergy,weobtainfor thethermalinternalenergy

U(th)=(π^2 / 2 )NkBT×(kkkBT/μ) (8.10)

Thisis a usefulwayofwritingtheidealgas result. We see that thethermalenergy

(omittinga numerical factor) is essentiallythe MB result multiplied bythe ‘Fermi

factor’(kkkBT/μ). And this factor is (a) small and (b) temperature dependent.

Heat capacityCV. The hard work is now done. The heat capacityCVfollows

immediately, since

CV=dU/dT

=dU(th)/dT

=(π^2 / 3 )kkkB^2 Tg(μ) (8.11)

inthedegeneratelimit, obtainedbydifferentiating (8.9). Hence atlow temperatures

onehas alinear, andsmall,heat capacity. For theidealgas (asin equation (8.10))its

magnitude is again of order the classical valueNkkkBmultiplied by the Fermi factor

(kkkBT/μ).The restriction ofthermalexcitation enforcedbythe FD statistics ensures

that theheat capacityis a smalleffect.