40 Two examples
3 .2.3 Applications
1.Vibrations ofsolidsThe assembly of identical harmonic oscillators was used by
Einsteininthe earlydays ofquantum theory as a modelfor thethermallattice
vibrations ofasimple atomicsolid.Hederivedthe resultfor theheat capacity
(Fig. 3.10) as a universal function of (T/θ)withθtheso-calledEinsteintem-
perature characteristicofthesolid.Ifwe takeamoleofsolid,withNNAatoms,
then, since eachatom can vibrateinthreedimensions, we shouldrepresentitby
N= 3 NNAone-dimensional oscillators.
TheEinsteinmodelhas goodandbadpoints. The successes are:
1 .Itgives the correct high-temperature (classical) limit, namelyC=NkkkB=
3 NNAkkkBper mole. From the viewpoint of the previous section, this success
arises simply from the correct evaluation ofthe number ofdegrees offreedom
of the solid.
2. It gives the correct extreme quantum limit, namelyC= 0 atT=0, a mystery
inclassicalphysics.
- It is indeed found experimentallythat the heat capacityper mole is a universal
function of(T/θ) for all simple solids, i.e. adjustment of a single parameterθ
makes allresults similar.
However, the bad news is that the form of (3.1 5 )isgrosslyincorrect in the inter-
mediate temperature regime. In particular the experimental lattice heat capacity
is notfrozen out so rapidlyas thetheorypredicts atlow temperatures,but rather
isproportional toT^3 in crystalline solids. The reason for the poor showing of the
Einsteinmodelisthatby no stretchofimaginationdoesasolidconsist oflocal-
izedoscillators whichare weaklycoupled.Ifone atom ofthesolid is moved,a
disturbance is rapidly transmitted to the whole solid; and in fact the key to a more
correct treatmentistomodelthevibrations as a gas ofultra-highfrequencysound
waves (phonons)inthewholesolid,atopictowhichwe returninChapter 9.
2.Vibrations ofgas moleculesThe thermal vibrations of the molecules in a diatomic
gas will bediscussedinChapter 7. In this case, rather unexpectedly, theEinstein
modelapplies verywell. Certainlythevibrations ofone molecule are now weakly
coupled from the vibrations of another molecule in a gas. However, our statistical
treatment sofaris appropriate tolocalizedparticles only,whereasgas molecules
are certainlynon-localized and in consequence indistinguishable. Not surprisingly,
it is to a discussion of gases which we now turn.
3 .3 SUMMARY
Thischapterdiscusses the application oftheBoltzmanndistribution toderive the
thermalproperties oftwo types ofsubstance, a spin-^12 solidandan assemblyof
harmonic oscillators. We consider the somewhat idealized case in which the substance
ismodelledas an assemblyofweakly-interactinglocalizedparticles,for whichsimple
analyticalsolutions canbemade.