Statistical Physics, Second Revised and Enlarged Edition

106 Bose–Einsteingases

TT = 1.25TTT 0

TT=TTT 0

TT = 0.8TTT 0

0 1 23456 7

(units of kkkBTTT 0 /h)

u()

Fig. 9. 7 The spectraldistribution ofenergyinblack-bodyradiation. ThePlanckformula (9.12)for the
energy densityas afunction offrequencyisshown at threedifferent temperatures.

all possible frequencies. We have for the (internal) energyper unit volume

U/V=( 8 πh/c^3 )

∫∞

0

∫∫

ν^3 dν/[exp(hν/kkkBT)− 1 ]

=( 8 πh/c^3 )(kkkBT/h)^4

∫∞

0

∫∫

y^3 dy/[exp(y)− 1 ] (9.13)

Thedefiniteintegralin (9.13)has the (‘well-known’) value ofπ^4 /15. The energy
Uis represented bythe areas under the curves in Fig. 9.7, which displaythe rapid
variation ofUandT.
4 .OnceU/Visknown, two other propertiesfollow. Oneisthe energy flux radiated
from a black body, a very accessible experimental quantity. It is defined as the
energy per secondleaving a small holeofunit areainthewallofthebox, assuming
noinwardflux. (Ifthe temperatures ofthebox andofits surroundings are the same,
then of course there will be nonetflux – usually a difference between flux in and
flux outisthedirectly measurable quantity.) Since allthephotons are moving with
the same speedc,the number crossingunit areainunittimeis^14 (N/V)c,where
N/Vis the (average) number density of photons. The factor^14 comesfromthe
appropriate angularintegration, asinthe corresponding standardproblemingas
kinetictheory. Hencetheenergycrossingunitareainunittimeis^14 (U/V)c≡σT^4.
This result is the Stefan–Boltzmann law,and the value ofσdeduced from(9.13)
isin excellent agreement withexperiment.

5. The second propertyis the pressure, the ‘radiation pressure’. We have alreadyseen
   (section 8.1.3) that for a gas of massive particles,PV= 2 U/3, this relationship
   followingfrom thedispersion relationε∝k^2 ∝V−^2 /^3. For thephoton gas, the
   dispersion relation is different, namelyε=hν=ck/ 2 π.Henceε∝k∝V−^1 /^3