Statistical Physics, Second Revised and Enlarged Edition

Aspin-^12 solid 27

frozen outinto thelowest energy(ground) state, andno particleis excitedinto the

higher state.

4. On theotherhandathightemperatures (meaningTθorkkkBTε)wehave
   n 0 =n 1 =N/2. There are equalnumbersinthe two states,i.e. thedifferencein
   energy between them has become an irrelevance. The probability of any particle
   beingineither ofthe two statesisthe same,justlikeinthe penny-tossing problem
   ofAppendixB.

Internal energyU An expression forUcan be written down at once as

U=n 0 ε 0 +n 1 ε 1

=Nε 0 +n 1 ε (3.3)

withn 1 givenby (3.2). ThisfunctionissketchedinFig. 3.2. Thefirst termin (3.3)
isthe ‘zero-point’ energy,U( 0 ),the energyT= 0 .The secondtermisthe‘thermal
energy’,U(th), which depends on the energy level spacingεandkkkBTonly.
One shouldnote that this expression may alsobeobtaineddirectlyfrom the partition
function (3.1) using(2.27). TheZ(0)factorleadstoU(0) andtheZ(th)factor to
theU(th). It is seen from Fig. 3.2 that the transition from low to high temperature
behaviour again occurs aroundthecharacteristic temperatureθ.

Heat capacityC Theheat capacityC(strictlyCV,oringeneraltheheat capacity
at constant energy levels) is obtained by differentiatingUwith respect toT.The
zero-point term goes out, andone obtains

C=NkkkB

(θ/T)^2 exp(−θ/T)

[ 1 +exp(−θ/T)]^2

(3.4)

The result, plottedinFig. 3.3, showsasubstantialmaximum (oforderNkkkB,the nat-
uralunitforC)at a temperature near toθ.Cvanishes rapidly(as exp(−ε/kkkBT)/T^2 )

T

2  3 

U

UUU(0)

UUU(0) +NNNe/2

0 

Fig. 3. 2 The variation ofinternalenergywithtemperatureforaspin-^12 solid.U(0)isthe zero-point energy.