Statistical Physics, Second Revised and Enlarged Edition

Statics or dynamics? 131

absorption rate andto thedensityofphoton statesinto whichtheemission occurs.

Thef()term is responsible forstimulated emission. The emission rate is greatly

increasedintoaphoton state oftheright energy,ifthere are alreadyphotons presentin

the state (thosefriendly bosons again!). Thatisthe essence oflaser action,inwhich

atoms deliberately prepared to be in the upper atomic state and then de-excite by

emitting photonsinto a cavity whichsupports a particular photon state. Hence the

intense andcoherent radiation.

Looked at from the viewpoint of statistical physics, the existence of stimulated

emissionis a necessity. In a strongradiationfield,thestimulatedterm ensures that the

dynamic equilibrium reached is one in which transitions up and down are frequent

andNNN 2 =N 1 ,corresponding to infinite temperature. Disorder is rife, and the entropy

is maximized.Alaser needsNNN 2 >N 1 ,implyinganegative ‘temperature’,i.e. a non-

equilibrium prepared situation, one of lower entropy. Without thef()term in(12.3)

one could cheat the second law; simply applying radiation would pump all the atoms

upinto the top state, achievingthisdesirablesituationbyequilibrium means.

1 2.1.2 Transitions with electrons

Theprincipleofdetailedbalance maybe usedto establishthe same sort ofconnection

between Boltzmann statistics andFermi–Dirac statistics. Consider as an examplea

metal which contains some magnetic impurities. The conduction electrons in the

metalcanbemodelledas anidealFD gas, whichweknowinthermalequilibrium

will have thedistribution of(8.2). Themagneticimpurities can againbe consideredas

anumber of atoms with two energy states separated by.As before, their equilibrium

distributionistobegivenby (12.1).

The transitionsin question areinelastic scatteringprocesses ofan electronfrom

the impurity atom. In an ‘up’ transition (of the atom, as previously) an atom in the

lower state collides withan electron ofenergy(ε+)leaving the atominthe upper

state andtheelectron withenergyε.The reverse ‘down’ transition sees the atomgo

from state 2 to state 1, whilst the electron energy increases fromεto(ε+).

Detailedbalance shouldapply to every transition, whatever electron energyε

isinvolved.Since theelectrons arefermions obeyingthe exclusion principle, the

requirements for the up transition are, (i) an atom in state 1, (ii) a filled electron state

ofenergy(ε+),and(iii) an emptyelectron state ofenergyε.Hence,inthespirit

of theprevious section, we would expect the transition rates to be

R(up)=N 111 f(ε+)[ 1 −f(ε)]g(ε+)g(ε)X

and

R(down)=NNN 222 f(ε)[ 1 −f(ε+)]g(ε+)g(ε)X

(12.4)

where f(ε)is thedistributionand g(ε)the density of states for electrons of

energyε.ThefactorXis againthe appropriate coupling strengthfor the transition

under examination. Now usingdetailedbalanceR(up)=R(down),andassuming