Page 315 - Electrical Properties of Materials

Two-state systems 297

So far we have talked about photons generated within the material. Now
if photons of energy hν 31 are shone on to the system from outside, a process
called stimulated emission occurs. Either the photon gets together with an atom
in a lower (E 1 ) state and pushes it up to E 3 ; or, less obviously, it stimulates
the emission by an E 3 -state atom of a photon (hν 31 ). In the latter case one
photon enters the system, and two photons leave it. It was one of Einstein’s
many remarkable contributions to physics to recognize, as early as 1917, that
both these events must be occurring in a thermodynamical equilibrium; he
then went on to prove that the probabilities of a photon stimulating an ‘up’ or
a ‘down’ transition were exactly equal. The proof is simple and elegant.
Consider our system, remembering that we have two states in equilibrium.
The rate of stimulated transitions (R 1→3 ) from the lower to the upper state will
be proportional to both the number of atoms in the lower state and the number
of photons that can cause the transition. So we can write The constant of proportionality
B 13 is the probability of absorbing
R 1→3 = N 1 B 13 ρ(ν 31 )dν. (12.5) a photon, often referred to as the
Einstein B-coefficient.
For the reverse transition, from E 3 to E 1 we have a similar expression for stim-
ulated emission, except that we will write the Einstein B-coefficient as B 31 .
There is also spontaneous emission. The rate for this to occur will be propor-
tional only to the number of atoms in the upper state, since the spontaneous
effect is not dependent on external stimuli. The constant of proportionality or
the probability of each atom in the upper state spontaneously emitting is called
the Einstein A-coefficient, denoted by A 31 . Hence,

R 3→1 = N 3 {A 31 + B 31 ρ(ν 31 )}dν. (12.6)

In equilibrium the rates are equal,

R 1→3 = R 3→1 , (12.7)
that is,

N 1 B 13 ρ(ν 31 )dν = N 3 {A 31 + B 31 ρ(ν 31 )}dν. (12.8)

After a little algebra, using eqn (12.2) to relate N 3 to N 1 , we get
A 31 dv
ρ(ν 31 )dν = . (12.9)
B 13 exp(hν 31 /k B T)– B 31
Comparing this with eqn (12.4), which is a universal truth as far as we can tell,
we find that our (or rather Einstein’s) B-coefficients must be equal,

B 13 = B 31 , (12.10)
that is, stimulated emission and absorption are equally likely. Also,

3
3
8πn hν 31
A 31 = B 31 , (12.11)
c 3
that is, the coefficient of spontaneous emission is related to the coefficient of
stimulated emission.