Statistical Physics, Second Revised and Enlarged Edition

108 Bose–Einsteingases

back to (4.5) and usinga computer to fold in the complicated dispersion relation.

An approximate fudge (the Debye model) is to maintain the linear dispersion

relation,but to cut offthedensity ofstates aboveaDebye cut-off frequencyνD,

defined so that

3 N=

∫νD

0

∫∫

g(v)dν

Although not in detail correct, the Debye model contains most of the important
physics, anditgives the correctheat capacityoftypicalsolids to, say, 10%. The
resultforUis a modified version of(9.13)

U=V( 12 πh/c^3 S)(kkkBT/h)^4

∫y(T)

0

∫∫

y^3 dy/[exp(y)− 1 ] (9.14)

wherey(T)=hνD/kkkBT. The heat capacityC, shown in Fig. 9.8, is obtained by

differentiating (9.14) withrespect toT.

Theintroduction ofthe cut-off gives a scale temperatureθtotheproblem,defined

bykkkBθ =hνD.For most common solidsθhas a value around room temperature

(lowerforlead,higherfordiamond!). Athightemperatures,Tθory(T)1,

the cut-offensures that theclassicallimitis recovered,yieldingU = 3 NkkkBTand

C= 3 NkkkB(see Exercise 9.6). On the other hand, at low temperatures,Tθor

y(T)→∞, we recover the result analogous toblack-bodyradiation. Theintegralin

(9.14) again becomes a constant, and we haveU∝T^4 and correspondinglyC∝T^3.

This is the so-called DebyeT^3 law, and is in good agreement with the experimental

heat capacityofacrystalline solidatlow temperatures.

Einstein

Debye

3 NkkkB

C

T



Fig. 9.8The variation withTof the lattice heat capacityof a solid in the Debye approximation. Full curve,

the Debye model (from (9.14)). Dotted curve, the Einstein model (Fig. 3.10), for comparison.