Statistical Physics, Second Revised and Enlarged Edition

130 Twonewideas

NNN 2

NNN 1

hv = Δ hv = Δ

Energy states

of

an atom

Absorption

of a photon

Emission

of a photon

Δ

Fig. 12.1Matter and radiation.

Bose–Einsteindistribution

f(ε)= 1 /[exp(ε/kkkBT)− 1 ] (9.9) and(12.2)

Let us consider the mutual equilibrium between the atoms and the photons. Clearly
there canbe a stronginteractionbetween the atoms andthose photons whichhave an
energyε(=hν)=. There can be an absorptionprocess (atom in state1+photon
of energy→atom in state 2) which conserves energy; and an emission process
whichisthe reverse (atomin state 2 →atomin state1+photon) (see Fig. 12.1).
Einstein’s argument was to invoke theprinciple ofdetailed balance.This states that
in equilibrium the average transition rates of absorption and emission processes must
be equal(andthis mustbe truefor anytransition anditsinverse –hence ‘detailed’
balance). Now the upward transition rate, the number of absorptionprocessesper unit
time, is plausibly given by:R(up)=N 111 f()g()X.The process needs an atom in
state 1, andit needsaphoton oftheright energy (g(ε)isthephotondensityofstates).
ThefactorXdescribes the basic coupling strength, given the correct input of atom
andphoton.
What can we sayaboutR(down), theemission rate? It requires an atominthe
upper state, so it must beproportional toNNN 2. And that is enough to determine it fully!
Detailedbalance tellsusthatR(down)=R(up), andthe equilibriumdistributions
((12.1) and(12.2))implythat

NNN 2 /N 1 [=exp(−/kkkBT)]=f()/[ 1 +f()]

Hence

R(down)=NNN 2 [ 1 +f()]g()X (12.3)

Equation (12.3)has alot to revealabout theemissionprocesses. Thefirstisthat the
same coupling parameterXshould be used for absorption and emission, an idea that
shouldcome as no surprise to anyone whohas studiedmuchquantum mechanics
(the same matrixelementisinvolved),but Einstein wasinventingthesubject! But
of continuing interest is the( 1 +f())factor in(12.3). The ‘1’ term relates to what
iscalledspontaneous emission,emission whichwilloccur evenifthere are no other
photonspresent. The treatment shows thatithas a rate whichisrelatedbothto the