Statistical Physics, Second Revised and Enlarged Edition

Phoneybosons 107

inthis case, andtherefore (followingtheargument ofsection 8.1.3)P=U/ 3 V.

The radiation pressure is simply one-third of the energy per unit volume.

6. FinallyletusconsidertherangeofvalidityoftheT^4 law. Nohigh-temperaturelimit

isevident, since photon states exist wellabove thehighest conceivablethermally

excited frequencies. No problem exists here (unlike the phonon case discussed

below).Atlow temperatures,however, we shouldsee thebreakdown ofthedensity

ofstates approximation. TheT^4 lawisbasedon thereplacement ofa sum over

photon states by an integral over a smooth density of states. This will clearly

becomeinvalid inabox oflineardimensionawhenkkkBT<ε,the energy level

spacing, i.e. forkkkBT<ch/a.This corresponds roughlytoT<1K foraboxof

side 1 cm, butT= 1 K is just too low a temperature for radiation to be important

in practice anyway.

9. 3 .2 Phononsandlattice vibrations

The same ideas maybe applied to a discussion of the thermal lattice vibrations of
asimple atomic solid. We have already noted in Chapter 3 the inappropriateness of
thelocalizedoscillator model,since the motion ofone atominthesolid is strongly
coupled to that of another. The weaklycoupled vibrational modes are sound waves,
and hence it is appealing to consider the vibrations of the lump of solid as the motion
ofagas ofsoundwavesinabox. Hence thesimilarityto the previous section, where
abox of electromagnetic waves was discussed.
The weakly interacting gas particles, quantized sound waves, are called phonons.
Andthe totalinternalenergyUofthephonongas canbecalculatedinananalogous
wayto equation (9.13). However, there are three important differences in comparison
with the photon gas:

1. 
   

   Sound waves haveG=3,notG=2. The three polarizations arise from three-
   dimensionalmotion ofthe atomsinthesolid.Inthelong wavelengthlimit, there
   are two transverse polarization modes andonelongitudinalmode. Hence (9.13)
   should be multiplied by a factor 3/2.

   
   


2. 
   

   Thedispersion relationisaltered.Toafirst approximation we can simplyuse
   ε=hν=cSk/ 2 π,wherecSis a suitable average velocityof sound. This will
   simply replacecbycSin (9.13). In practice, as any book on solid-state physics
   willreveal,thisis a passable approximationforlongwavelength(lowk)phonons,
   although even there the velocities of transverse and longitudinal modes are very
   different. However, thelineardispersion mustfundamentallyfailatlarge values
   ofk,apoint relatedto thefollowing.

   
   


3. 
   

   There are only a limited number of phonon modes in the solid. The solid is not an
   elastic continuum,but consists ofNatoms, so that the wavefunction ofaphonon
   only has physicalmeaning(the atomicdisplacement) at theseNatomicsites.
   Hence there are only 3 Npossible modes. Compare the localized oscillator model,
   whichsimilarly considered3Noscillators. Asindicatedabove, the correct phonon
   dispersion relation will fullytake care ofthis problem. However, this meansgoing