Statistical Physics, Second Revised and Enlarged Edition

158 Dealingwith interactions

the one-electron value at theexpense ofother states withinafewkkkBθDof theFermi

level, as shown in Fig. 14.1b. Incidentally, this figure also demonstrates why it is that

athighenoughtemperatures (T>θD)theenhancementisshed,i.e.η=1. This

occursbecause the experimental‘samplingwindow’ oftheg(ε)curveisthethermal

energy scalekkkBT. When this window exceedskkkBθD,the whole of the deformed curve

is sampledrather than simplyg(μ).Andsince theinteractionsdo not generate more

states,but simplymove the energies oftheoldone-electron states, the averagedensity

of states is unchanged. This shedding effect is observed experimentally. It is a fine

challengetorelate thedispersion curve (Fig. 14.1a) to Fig. 14.1b.Itshows the same

physics! The states are evenlyspaced ink,as ever. The flatteningof theε−kcurve

at the Fermi level implies that there aremore k-states per unit energy range than

before, as statedabove. Thejoiningbackto theoriginalcurve ensures ahigher slope

at energies a little removed fromμand hence a diminished densityof states as shown

in Fig. 14.1b.

To summarize this section. Wehave seen that there are two types ofcorrection

which need to be made to the ideal freegas model in describingconduction electrons.

Fortunately, both effects are capable of being treated as ‘small’because (i) the overall

electricalneutralityofthe metalguarantees an accurate cancellation ofthelongrange

parts of the electron–electron repulsion and the electron–lattice ion attraction and (ii)

thehighkinetic energy ofconduction electrons near theion core allows us todealwith

amuchsmallereffectivelatticepotentialthan the truepotential.Thefirst consideration

is the effect on the free electron model of the small effective electron–latticepotential,

namelytointroduce thelattice symmetryinto thebandstructure orε−krelationfor

theelectrons. As a result, althoughFD statistics are stillvalid,theymustbe applied

with the new and intricateε−krelation, not with the simple free gas one. This

isthe ‘one-electron approximation’. The secondconsiderationis to recognize that

there maywell beother effectsgoingon whichare ofa ‘many-body’character. The

residual electron–electron interactions are an obvious candidate for such effects,but

in practice thelargest many-bodyeffects arisefrom electron–phononinteractions.

1 4.2 Liquid helium-3: A Fermi liquid

In section 8.3, we introduced the topic of liquid helium-3. The suggestion made there

was that the liquid could be described as a free fermion gas, that the gas is not of bare

(^3) He atomsbut ofquasiparticles whichhave an effective mass ofseveralbare (^3) He
masses. This statement has the correct flavour, but needless to say the full truth isa
little more complicated. Let us explore thisin some moredetail.
14.2.1 Landau Fermi liquid theory
As statedin section 8.3, our understanding ofso-calledFermiliquids owes muchto
the Russian theorist, Lev Landau. Afuller treatment ofthis section maybefoundin