Statistical Physics, Second Revised and Enlarged Edition

84 Fermi–Diracgases

8.1 Properties ofanidealFermi–Dirac gas

As we have seen, FD statistics is needed, (i) when we are dealing with a gas of weakly
interacting particleshaving spin^12 (or^32 ,^52 ...),and(ii)when the gashas adegeneracy
parameterA> 1 .Since

A=(N/V)(h^2 / 2 πMkBT)^3 /^2 (8.1) and ( 6. 6 a)

onefindsin practice that FD statisticsis neededonly inafew cases of high density

N/V, of low temperatureTor of low massM.Important applications are:

1. Conduction electrons in metals at all reasonable temperatures, and also in

semiconductors withahighenoughcarrierdensityN/V.

2. Liquid^3 Heatlow temperatures.


3. Dense-matter problems in astrophysics, such as in neutron and white dwarf stars.

Since each of these involves spin-^12 particles, we shall explicitly consider spin-^12

fermions only inthischapter. Thegeneralizationis straightforward.

8 .1.1 The Fermi–Dirac distribution

The distribution function was derived in Chapter 5. The result (( 5 .10) and ( 5 .13)) is

usuallywritten as

fffFD(ε)= 1 /{exp[(ε−μ)/kkkBT]+ 1 } (8.2)

For the rest ofthischapter, thesubscript FD will beomittedintheinterests ofclarity.In

(8.2), we have made the identification of the ‘Fermi energy’,μ.This is simplyanother

way of characterizing the parametersα(equation ( 5 .10)) orB(equation ( 5 .13)) as

B=exp(−α)=exp(−μ/kkkBT).Thesymbolμisappropriate sinceit turns out that

this quantityis preciselythe same as the chemical potential of thegas.

The form of (8.2) is shown in Fig. 8.1, where it is plotted at three different tem-

peratures. Itis nothardto understand.Thedistributionfunctionf(ε)isdefinedas

the number of particles per state of energyεin thermal equilibrium. Equation (8.2)

bears the markofthe exclusion principle, sinceit guarantees thatf ≤1from the

+ 1 inthedenominator. So curve 1inthefigure, correspondingto alow temperature,

shows as expected thatf = 1 for states withε<μbutf =0forε>μ.There

isachange-over region ofenergy widthaboutkkkBTaroundμinwhichf changes

from1to 0 .AsTisraisedsomewhat (curve 2), thischange-over regiongets wider,

although the extremes of the distribution are virtually unaltered. WhenTisraised

further (curve 3) thewholeform ofthedistributionisaffected, tending towardsthe

simple exponentialMBdistributioninthehighTlimit.