154 Dealingwith interactions
1 4.1 Electrons in metals
The elementary treatment of electrons in metals (section 8.2) is simply to describe the
electrons as anidealFD gas. This wasfirstdoneinthe earlydays ofquantum theory
(the Sommerfeldmodel), anditis surprisinglysuccessfulindescribingthe results
of experiments (on both equilibrium properties and transport properties), particularly
ifparameterslikethe numberdensity andthe mass oftheelectrons are taken as
adjustable parameters. In the case ofthesimplest metals, suchas thealkali(group 1,
monovalent) metals sodium and potassium, these adjusted parameters are in fact very
similar to thefree electron parameters. So are they (significantly)for mostliquid
metals. However theybecome quitefancifulfor some other crystalline solids, even
simple elemental ones, such as the group 4 semiconductors germanium or silicon, not
to mention theinsulator carbon (diamond),inwhichtheeffective conduction electron
densityis clearlyzero.
Although the whole basis of the Sommerfeld model was long known to be ques-
tionable,it nevertheless tookscientists atleast 30years tobegintounderstandwhy
the model works so well. The problem is obvious. The Fermi energywhich we calcu-
lated in Chapter 8 for a typical simple metal is of order 5 eV. This is the kinetic energy
scalefor the supposedfree electrons, anditislarge enoughcomparedtoathermal
energyscalekkkBTat any reasonable temperature to make the supposed electron gas
an extremelydegenerate FD gas, as wehave seen. However, this energyis not at
all large comparedwiththe potentialenergyofinteractions whichwe shouldexpect
between an electron and its surroundings. There are two problems here. Firstly, there
istheinteractionbetween a conduction electron andthelatticeions. The conduction
electrons are negativelycharged,havingbeenliberatedfrom atomsleavingbehind
positively charged ions. Therefore there will be a strong attractive Coulomb inter-
actionbetween them, ofexpectedmagnitudeaboute^2 / 4 πε 0 rwhererisabout an
atomic spacing. Puttinginr=0.2 nm, thisgives an energyof7eV,ofthe same order
of magnitude as the kinetic energy. Secondly, there is also the potential energy of
interactionbetween the conduction electrons themselves. They are not truly weakly
interacting, since theyrepel each other with a repulsive interaction which again should
be of the same order of magnitude, since the electrons are typically an interatomic
spacingapart.
Where does this leave us? Clearlyin agreat mess, since this last interaction in
particular blows away the whole ‘weakly interacting’ assumption behind the ideal
FDgas treatment oftheelectrons. However, the reason whyall is notlostis nothard
to see at the hand-wavinglevel. Overall, the metal is electricallyneutral. Therefore
problem 1 (electron–ion attraction) and problem 2 (electron–electron repulsion) must
to agoodapproximation cancelout. Thisis certainlytrue ofthelong-range part of
these interactions, and this is the majorjustification for continuingwith the simple
approach. The idea is that of screening (electrostatic shielding). The ‘other’electrons
cancelout theeffects ofthepositivelatticeions.
Interestingly, this is not yet the end of the story. We are still left with what
would be a very substantialshort-rangelattice potential,anda residualworry about