Statistical Physics, Second Revised and Enlarged Edition

Localized harmonic oscillators 37

enablesZtobe summedexplicitlyas ageometric progression. Wehave

Z=

∑

j

exp

[

−

(

j

((

+

1

2

)

hv/kkkBT

]

from(2.24)and(3.11)

=

∑

j

exp

[

−

(

j

((

+

1

2

)

θ/T

]

definingθfromhν=kkkBθ

=exp(−θ/ 2 T)·

∑

j

exp(−−jθ/T) separating the common factor

=exp(−θ/ 2 T)·[ 1 −exp(−θ/T)]−^1 doing the sum (see below!)

=Z( 0 )·Z(th) say, compare(3.1) (3.12)

Theevaluation of the suminthe Z(th)factor aboveis straightforward.Ifwe
writey = exp(−hν/kkkBT),then therequiredsumis(1+y+y^2 +y^3 +···),
which is readily summed to infinity to give ( 1 −y)−^1. (Check it by multiplying
bothsidesby 1 −yifyou are unsure). The separationin (3.12)into a zero-point term
Z(0) andathermaltermZ(th)hasasimilar significance to our earlierfactorization
of(3.1).
NowthatZisevaluatedin terms ofTandthe oscillators’scale temperatureθ,there
is no problem in workingout expressions for the thermal quantities. For example,U
is obtained by taking logarithms of (3.12) and differentiating as in (2.27). The result

U=

1

2

Nhv+

Nhv

exp(θ/T)− 1

(3.13)

isplottedinFig. 3.9. Thefirst termisthe zero-point energy,derivedfromZ(0), and
the second term is the thermal energyarisingfromZ(th). At high temperatures, as the
figure demonstrates, the internal energy becomes

U=NkkkBT (3.14)

to a ratherhighdegree ofaccuracy. (Expandthe exponentialin (3.13) to test this!)
We returnbelow to thissimplebutimportant expression.
The heat capacity is obtained by differentiating (3.13)

C=

dU

dT

=NkkkB

(θ/T)^2 exp(θ/T)

[exp(θ/T)− 1 ]^2

(3.15)

This is plotted in Fig. 3.10, and as expected from (3.14) one mayobserve thatC=NkkkB
at high temperatures(T>θ).