3 Two examples
We now applythe generalresults ofthe previous chapter to two specific examples,
chosenbecause theyare easilysoluble mathematically,whilstyetbeingofdirect
relevance to real physical systems. The first example is an assembly whose local-
izedparticleshavejust two possible states. In the secondthe particles areharmonic
oscillators.
3.1 A Spin-^12 solid
First we derive the statistical physics of an assembly whose particles have just two
states. Then we applythe results to anideal‘spin-^12 solid’. Finallywe can exam-
ine brieflyhow far this model describes the observed properties of real substances,
particularly in the realm of ultra-low temperature physics.
3 .1.1 An assembly of particles with two states
Consider an assembly ofNlocalized weakly interacting particles in which there are
just two one-particle states. Welabelthese statesj= 0 andj= 1 ,withenergies
(undergiven conditions)ε 0 andε 1 .The results oftheprevious chapter canbe used
to write down expressions for the properties of the assembly at given temperatureT.
Thedistribution numbers ThepartitionfunctionZ,(2.24),has onlytwo terms. Itis
Z=exp(−ε 0 /kkkBT)+exp(−ε 1 /kkkBT),which can be conveniently written as
Z=exp(−ε 0 /kkkBT)[ 1 +exp(−ε/kkkBT)] (3.1)where the energyεisdefinedas thedifferencebetween the energylevels(ε 1 −ε 0 ).We
maynote that (3.1) maybewritten asZ=Z( 0 )×Z(th), theproduct oftwofactors.
Thefirstfactor(Z( 0 )=exp(−ε 0 /kkkBT), the so-called zero-point term) depends only
on the ground-state energyε 0 ,whereas the secondfactor (Z(th), thethermalterm)
depends onlyon the relative energyεbetween the two levels.
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