14
Dealing with interactions
So far in this book we have dealt almost entirely with assemblies made up of weakly
interactingparticles, eitherlocalizedorgaseous. Thishas reallynotbeen a matter
of choice, but almost of necessity. The basic approach has been founded on the
idea that the energy ofthe assemblyissimplythe sum oftheindividualone-particle
energies,i.e.U=
∑
nnjεεj.Without thissimplification the mathematicsgets out of
hand (although some progress can be made if we start with the canonical or grand
canonicalapproach).
Nevertheless wehave successfullyappliedour statisticalideas to a rather wide
variety of real problems. In the case of real chemical gases, this success is not too
muchofa surprise, sinceithasbeen well known since thedays ofkinetictheory that
gases are almostideal,atanyrate atlow pressures andnot too close to theliquidphase
transition. Later in the chapter, we shall discuss briefly how the small corrections to
theidealgas equation maybecalculated,togive a more realistic equation ofstate
thanPV=NkkkBT.
But chemical gases are not the only situation in which ideal statistics are applied.
There are severalcases where the assumption ofweakly interactingparticles must
cause raised eyebrows, in spite of our earlier protestations. These include (i) treatment
as a Fermi gas of the conduction electrons in metals and semiconductors (section 8.2),
(ii) treatment ofliquid helium-3 as anidealFDgas (section 8.3) and(iii) treatment
of liquid helium-4 as an ideal BEgas (section 9.2). We take the opportunityin this
chapter to explain a little further our understanding about why the simple models turn
out tobeapplicable.
The central idea to developis that ofquasiparticles.We concentrate not on the
‘real’ particle, but on some other entity called a quasiparticle. The quasiparticle is
chosen so thatitis weakly interactingto abetter approximation thanistheoriginal
particle. One way in which we have already seen this type of approach at work is in
our treatment oflatticevibrations ofasolid.The motions ofthe atoms themselves are
verystrongly interactinginasolid,soinsteadwe choose to redefine the totalmotion
in terms of phonons (see section 9.3.2), which can be treated as a weakly interacting
idealgas modelto afair approximation. Let us now seehow this works outfor the
three caseslistedabove.
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