Page 317 - Electrical Properties of Materials

Lineshape function 299

population’. The other point we should clear up, before describing some real
system with inverted populations, is the one concerned with temperature. From
eqn (12.1) the locus of the line representing the populations of the various
energy levels,
dE k B T
=– (12.16)
dN N
(shown as a dotted curve in Fig. 12.1), has a negative slope proportional to T/N.
∗
∗
Now look at Fig. 12.2. First consider the populations N and N .Theyare in a
3 1
steady state in the sense that as long as the pump continues steadily, they do not
change with time. But for these two levels with a finite energy difference there
is virtually no difference in population. Therefore, if we regard eqn (12.16) as
a way of defining temperature, for a well-pumped two-level system the tem-
perature is infinite. If we now consider the energy-level populations at E 3 and
E 2 in Fig. 12.2, we see that

∗
N > N 2 , (12.17)
3
and the dE/dN locus has a positive slope, which by eqn (12.16) corresponds
to a negative temperature.
Again, this is a fairly reasonable shorthand description of there being more
atoms in an upper state than in a lower one. Now if you imagine a natural-
state system pumped increasingly until it attains an infinite temperature and
then eventually an inverted population, you will see there is some sense in the
statement that a negative temperature is hotter than a positive one.

12.3 Lineshape function

So far we have assumed that energy levels are infinitely narrow. In practice
they are not, and they cannot be as we have already discussed in Section 3.10.
All states have a finite lifetime, and one can use the uncertainty relationship in
the form
We may now identify t with
E t = . (12.18) t spont .
Since E = hν, we shall find for the uncertainty in frequency (which we identify
with the frequency range between half-power points, also called the linewidth),
1
ν = . (12.19)
2πt spont
Unfortunately, the uncertainty relationship will not yield the shape of the
line function. To find that we need to use other kinds of physical arguments.
But before trying to do that, let us define the lineshape, g(ν). We define it so
that g(ν)dν is the probability that spontaneous emission from an upper to a
lower level will yield a photon between ν and ν +dν. The total probability
must then be unity, which imposes the normalization condition
∞

g(ν)dν = 1. (12.20)
–∞
Let us stick for the moment to spontaneous decay (or natural decay) and
derive the linewidth by a circuit analogy.