Electrons in metals 155electron–electron interactions. In the 19 6 0s it was realized that theproblem of the
large lattice potential could be attacked in a rather clever imaginative way, all asa
consequence ofthe Pauliexclusion principle. Thisiswhere the quasiparticleideas
comein, althoughthelanguage was not appliedin context at thetime. Thepointis
the ion itself is not a simple mass with a positive charge. In sodium, for instance,
it consists ofa nucleus with11 protons, surroundedby10closelybound(‘core’)
electrons,fillingthe 1s, 2s and2p orbitals. The one conduction electron per atom was
given to the gas-like pool (the so-called Fermi sea) from the outer (partially filled)
3 sshelloftheisolatedatom. Now the Pauliprincipletellsusthat these conduction
electronsinthemetalmust ofcourse be‘unlike’the boundcoreelectronsineach
atom. In technical language, they must be in states which are orthogonal to the core
electron states. In practice, this means that theyhave wavefunctions verymuch like
the 3s functions when theyare near an ion core. Now here is the clever part. The 3s
wave function is a very wiggly affair near the nucleus – and our conduction electron
isforcedinto these wigglesbythe Paulirequirementforittobedissimilarfrom the
filled core states. And wiggles in wave functions mean high curvature(∂^2 ψ/∂x^2 etc.)
andhencehighkinetic energy. So the wavefunctionismade to contribute ahigh
kinetic energy injust those regions ofspace close to the nucleus where weknow
the potential energy must be large and attractive. There is a pretty cancellation here.
This approachiscalleda pseudo-potentialapproach.Thedetailedmathematics and
justification ofitiswell beyondthe scope ofa small bookon statisticalphysics. But
the idea is important. What we do is to convince ourselves that the realproblem (ofa
true conduction electron, moving throughalarge albeitshort-range attractivelattice
potential)has considerablesimilarityto, andin particular the same one-particle ener-
gies as, a pseudo-problem (of an essentially free electron moving through a much
smallerlattice potential). The pseudo problemissolvable, asinthe 1930 treatments,
byperturbationfrom thefree electron modelwiththe perturbingpotentialnow a small
quantity which can be of either sign. It becomes an adjustable parameter (with the
adjustments madeby experimentalists orbytheorists, whoonagoodday andfora
simple metalcome upwiththe same answers).
Although this to an extent justifies the FD gas approach, we are still left with the
residualeffects ofour two potentialenergyproblems.
First, let us continue to ignore anyeffects of electron–electron interactions. In other
words, we adopt a one-electron approach which is tractable according to an ideal gas
model.Thebasic assumption that there are one-particle statesinwhichto set the
problem remains; we maywriteU=
∑
nnjεεj,and we may enumerate the states by
fitting wavesintoboxes asinChapter 4. Whatdiffersfrom thefree electron gas,
however, turns out tobe veryprofound.Itisinthedispersion relationship, neededto
convertk(from the waves in the box) into energyε(to use the FD distribution). Even
asmallperiodic perturbation, suchas thatfrom a crystalline (pseudo-)lattice,has a
dramaticeffect on the states near a Brillouin zoneboundary,wherekforthe electron
has a relationship with the lattice periodicity. The result is that the Fermi surface
(see Fig. 8.2) nolonger remainsasimplesphere (except almostfor the monovalent
alkalimetals)butbecomes carvedup andsomewhatdistortedor ‘sandpapered’bythe