156 Dealingwith interactions
lattice perturbation. Themodelisnolongerisotropic,but theε−krelationcanbe
markedly different in different crystallographic directions. So it is that the parameters
ofthemodelcanbechangeddramatically, particularlyinpolyvalent metals suchas
aluminium. TheoriginalFermisphereischoppedupinto a number ofpiecesbythe
pathologicalε−krelation, notably the existence of energy gaps at the Brillouin zone
boundaries. In many metalsorin semimetalslikebismuth,the Fermisurface then con-
tains small pieces, whichcanhave electron-likeorhole-likecharacter;forinstance,
adispersion relation over the relevant range of the formε=A+^2 k^2 / 2 m 1 would
correspondto electrons ofeffective massm 1 ,whereasε=B−^2 k^2 / 2 m 2 would
correspond to holes (because of the minus sign) of effective massm 2 .But it is impor-
tant to stress that the whole statistical treatment remains valid. There is still a Fermi
surface, andthethermalproperties aregovernedprecisely bythedensityofstates
g(μ)at the Fermi level. The complication comes merelyin the calculation ofg(ε).
This one-electron approach has been the basis of much successful modelling of the
thermalandtransport properties ofrealmetals. Butitisstillonlyan approximation
because of the continuingneglect of what are called many-bodyinteractions. These
are still in fact quite significant in many metals, particularly in the transition metals.
Theycanbeobservedexperimentally in severalways, ofwhichthefollowingis one:
- Measure the electronic contribution to the heat capacity, from low-temperature
experiments. Thisisidentifiedas thelinear termγTinthe experimentalC=
γT+βT^3 ;theT^3 term is thephonon contribution (Chapter 9), whilst the linear
term is the required electronic contribution.
2.Calculatefromtheε−kcurvestheelectrondensityofstatesg(μ)attheFermilevel;
in practice this can often be done rather accurately, since there are manychecks in
the calculation, such as information from specialist Fermi surface measurements.
Hence calculate a so-called‘bandstructure’heat capacityγusingthisdensityof
states(see(8.11)). - Note that the measuredvalueis(often)bigger than thecalculatedonebyan
enhancementfactor(η,say)whichisgreater than 1, andin practice canbeas
large as 2 or 3. For instanceηis about 1.2 even in the ideal sodium,it is 1. 5 in
aluminium, about 2.1inleadandevenlargerinthe transition metalpalladium.
There are (at least) three reasons known for such many-bodyeffects, namely
electron–electron interactions, electron–phonon interactions and interactions involv-
ingmagnetic excitations. Wehave mentionedelectron–electroninteractions above.
But actuallytheyare quite small (except in some transition metals, also the only
candidates for the magnetic enhancement). The reason for this is found in the Pauli
principle, whichtendstokeepelectrons apart,quite apartfrom the Coulombrepul-
sion. Hence onlythe long-range part is relevant and this has been used up in screening
the lattice ions(above).
Themajor effectinsimple metals comesfrom theelectron–phononinteraction.
One electron passing through the lattice can excite a short-lived phonon which is then
sensed(absorbed!)by a secondelectron. This rather subtleinteractionis responsible
for superconductivity, to which we shall return brieflyin section 15.1.1. It is also