Statistical Physics, Second Revised and Enlarged Edition

86 Fermi–Diracgases

Attheabsolute zero,the Fermifunctionf(ε)takes thesimpleform notedearlier, that
f=1forε<μ( 0 )butf=0 forε>μ( 0 ). Hence(8.4)becomes

N=

∫μ( 0 )

0

∫∫

g(ε)dε

whichusing (8.3)forg(ε)maybeimmediately evaluatedto give

μ( 0 )=(^2 / 2 M)( 3 π^2 N/V)^2 /^3 (8.5)

Method 2. Someone who really understands the fitting waves into boxes ideas can

use a prettyshortcuthere. The statesink-space whicharefilledatT= 0 canbe

represented as in Fig. 8.2. The low energystates with energyless thanμ( 0 )are all

filled, and sinceε=h^2 k^2 / 2 Mthese filled states correspond to those withkless

than some value (calledthe FermiwavevectorkkkF) correspondingtoμ(0). States with

k>kkkFare unfilled. The sphere of radiuskkkF,representingthe sharp boundarybetween

filled and unfilled states, is called the Fermi surface.

The second approach to the determination ofμ(0) is to recognize that this Fermi

surface must containjust the correct numberNofstates. Hencewe musthave

N=V/( 2 π)^3 × 4 πkkkF^3 / 3 × 2 (8.6)

where thefirstfactoristhebasicdensity ofk-statesink-space, the secondisthe

appropriate volumeink-space (that containedbythe Fermisurface), andthefinal 2

is the spin factor for spin^12. From(8.6)it follows that

kkkF=( 3 π^2 N/V)^1 /^3

andthereforeμ( 0 )=^2 kkkF^2 / 2 Mis preciselyasgiven above in (8.5).

kkky

kkkx

Fig.8. 2 The Fermi surface. Illustratingthe occupation of states ink-space atT= 0 ; the sphere has radius

kkkF(•=occupied; =unoccupied).