170 Statistics under extreme conditions
equalto^35 Nμ( 0 )for anidealgas. Not altogether surprisingly, nature seems to seeka
way to lower this high zero-point energy. One such way is for a magnetic transition to
occur, ashappensin some metals. But another wayisfor a transition to a superfluid
state to takeplace, andthisisthetopicofthepresent section.
For a full discussion of superfluidity in FD systems, the reader must consult a
specialistbookonlow-temperature physics (suchas the author’sBasic Superfluids).
Thefollowingwillonlytouchupon afew aspects ofwhatisanexcitingfield forboth
physics and technology.
1 5.1.1 Superconductivity
Manybut not all metals become superconductingat low enough temperatures. The
transition temperatureTTTCis typically a few kelvin (for example 1 K for aluminium,
3 .7 Kfor tin, 7.2 Kforlead; none ofthe monovalent metalsbecome superconducting).
BelowTTTC, a two-fluid model can again be used (as in superfluid^4 He)to describe the
properties. The super-statehas zero entropy,i.e.itisfullyordered.The normalfluid
densityvaries withtemperature as exp(−/kkkBT),whereisacharacteristic energy,
called the energy gap, of order of magnitude 1.7 5 kkkBTTTC. This exponential Boltzmann
factor means that the normalfluid is totallyinsignificant at temperaturesbelow 0. 1
or 0. 2 TTTC. So we can understandthe temperaturedependence ofthespecificheat and
of the entropy, as illustrated in Fig. 1 5 .1.
Howdoes this come about? Well,aglance at the energy scaleshouldwarn thatitis
asubtleeffect. In a typicalmetal,the Fermitemperature ofthe conduction electrons
(Chapter 8) is 5 0 000 K, and the lattice vibrations have a characteristic temperature
oforder 300 K, whereas the transition temperatureis merely 1 K. Theideaisthat
thereisasmallattractiveinteractionbetween conduction electrons, arisingusingthe
intermediary of the lattice phonons, which can win out over the obvious Coulomb
repulsion. Thisis not so unlikelywhen we recallthediscussion ofsection 14.1,
where we saw that at longrange this repulsion is screened out bythe ions. Ina
superconductor, electrons team up in pairs (called Cooper pairs after the originator of
theidea) movingin oppositedirections andwithopposite spins. Injargon, these are
L=0,S= 0 pairs, whereLgives the total angular momentum andSthe total spin.
(If you are an expert on quantum mechanics of identical particles, you can check that
this arrangement will give an antisymmetric wavefunction as requiredtodescribe
fermions!) To be oversimplistic about it, it is clear that two odds must make an even;
two spin-^12 fermions pairedmust give aboson, so thatiftheelectrons conspire to
occupystates only inpairs, theycanfoolthesymmetryofthesystem.
The nature of the interaction between electrons is a second-order one,in that a third
particleisinvolved. Roughly, whathappensisthat one electron (wavevectork)in
travellingthroughthelattice can exciteashort-lived(‘virtual’) phonon; thislattice
disturbance is then sensed in a coherent way by the second electron (wavevector
−k)whichis travellingin exactlythe oppositedirection (to giveL=0). Itisthis
interactionbywhichthe conspiracyto pairedoccupation wasfound(byBardeen,