Hello. Today we're going to talk about Laser-induced
Fluorescence Spectroscopy. I start with this first slide which I've titled
The Simple Picture. The Simple Picture is as follows. Suppose we have a two-level
atom or molecule, no such thing exists, but it's a simple way of
looking at this process. So you have a lower
quantum State 1 and an excited quantum State 2. Drawn on there are the same terms that we
used when we talked about the Einstein picture except we've added in
two additional terms, that's Q_12 and Q_21. So Q_12 is the collisional
rate of excitation, and Q_21 is a collisional rate of de-excitation to
and from State 2. Now, if you think about it, collisions have to be part
of the process at least when you have any
reasonable density at all of a gas because what else
is going to determine the equilibrium distribution of the quantum states
or energy states, the ones that are given
by the Boltzmann ratio. So it's those collisional rates or collisional processes
that determine the starting distribution of quantum states in
particular species, and particular atom
or molecule in a gas. So what happens is
you send a laser in, it could be a non-laser, but nowadays, pretty much all of this is done with lasers. You send a laser in, and what does it do? You have the W_12 term which is the Einstein excitation
term and that's going to excite the molecule from
State 1 to State 2. Once it's in State 2, it can obviously be de-excited
by stimulated emission, it can de-excite by
spontaneous emission, and of course, the
collisional processes are continuing to take place. So if I were to write a rate equation for
the rate at which the population or number of molecules in State
2 changes with time, it would look like the
differential equation, the ordinary
differential equation below the little diagram. So the first term
is W_12 and Q_12, N_1 that's exciting the molecule, and the second term
is minus W_21, A_21, Q_21 times N_2 and that's the
de-excitation process. Now, what are we looking at when we look at laser
induced fluorescence? Well, light out from the
molecule, of course, the laser light that goes on through the excitation
light source, that will go, but also stimulated emission
will be generated. A stimulated emission is a so-called coherent
process and it occurs in the direction
of the light source. So it would be going
in our little image there to the right
with the arrow. What will radiate in all directions is
spontaneous emission. So the signal that we're
going to get is given by the bottom relationship, Epsilon is some kind
of efficiency factor, h Nu is the energy of the photon, A_21 over four Pi is
the contribution within a single steradian of solid angle due to
spontaneous emission, Omega_c is the
collection solid angle, it's a solid angle defined by the lens
system that you have in place that's energy in light from the source to the detector. Then this is going to
get integrated over a some focal volume because the laser beam will have
some finite diameter and the collection objects will overlap the laser
such that there is a finite volume that's
emitting light. Over the frequency, all with an integrand
of N_2 Phi of Nu, so that would be your signal. So what are you measuring? You're making, actually, a direct measurement
of the population of State 2 in N_2 assuming you know
the line shape factors, you know the geometric factors, you know the a coefficients, you know any efficiency factors like loss factors,
that kind of thing. What's the reality? Well, as I just said, one measures directly is the
excited state concentration. You also indirectly measure
the ground state population. Because in the population
balance equation that I showed, there's a term of excitation that has the ground state
population in it. So for all other
things being equal, the larger the ground
state population, the larger the signal would be. To infer thermodynamic
state properties or total concentrations requires understanding the
excitation dynamics. Because LIF is inherently
a transient process, it's described by a
set of rate equations, and those rate equations
are shown on there, they're the same ones as in
the previous slide really, except that we've
taken into account, there may be many more
quantum states or energy states than just the two that we showed in
the previous slide. In addition, there's
a possibility of chemical source
and sink terms. When you change the population distribution
in a molecule, you change its
chemical equilibrium with respect to other molecules. So that can either add to the population of a given
state or reduce it. So as I just mentioned, the laser only couples
one transition, k to l in this diagram, but spontaneous radiation and collisions involve many
other quantum states. Typically, laser light sources involve very short
pulses of light. So in addition to
this complexity, it's a transient problem as I mentioned in the
previous slide. So we might think
of the situation as shown in this diagram, l and k are the two states
that are coupled by the laser but because of
spontaneous emission, because of collision processes, and because of chemistry, all of the other states of
the molecule can be involved. There is something called
the linear or weak limit where one assumes that the excitation laser intensity is small enough that populations
are not greatly perturbed. In that case, a simple
relationship between the total population
of the molecule or atom and the LIF signal
can be written as shown. Here, C is a calibration factor, Alpha and Beta represent the fraction of population
in the ground and excited states respectively that are tied up in the
laser coupled states, and Y is the fluorescent yield. The fluorescent yield defined by the bottom equation is A_21, the Einstein A rate
constant divided by the collisional
de-excitation rate plus the radiative
de-excitation rate. It's called the yields because the more
important collisions are, then the fewer molecules
are available to radiate. So the signal is going to be strongly affected by
collisional rates. Now, what are some of the strategies that
are used with LIF? Well first of all, you can say something about structure by observing and
interpreting the position of the LIF spectra. You can just take a spectra, notice what the populations of the various molecules under
excitation conditions are, and that tells you
a little something about the structure
of the molecule. The other thing you can
do is do what's called a fluorescence scan which is you can take LIF spectra
where the laser is tuned to different
frequencies and you can scan the laser in frequency through a bunch of
different absorption lines. With that kind of data,
you can say quite a bit about the structure
of the molecule. You can also look at the
dynamics of the process, you can track populations of laser excited states during
and after a laser pulse. Concentration, you can measure excited state population and relate to the total population. As we just discussed in the
so-called linear limit, you can make a
direct relationship between the signal and
the total population. You can use Boltzmann's
ratio to obtain a temperature or if the line shape is dominated
by Doppler broadening, by measuring the line shape, you can recover a temperature. You can obtain
pressure several ways, it's a little more
difficult to do directly, you can relate
pressure to measure temperature using
the ideal gas law. If the line shape is
dominated by collisions, then you can recover pressure
from the line shape. Here is an example
from some work that I did on the laser
induced fluorescence of the NO molecule. On the left hand side is an
energy level diagram for NO. When people do NO LIF, they excite from the ground
X-state to the A-state, and on the right are two plots, the plot on the bottom is the ground state
population distribution at a particular time
falling a laser pulse, and the upper plot is the
excited state population. Well, there's four
sets of curves in the ground state
distribution plot and two in the excited
state distribution plot, and that is because
of the structure of the NO molecule it's
much more complex than just the simple diatonic molecule that we talked about, and depending on spin state
and that sort of thing, it has these different states. So what you see in the
top plot is you see, you match along and you
can see a big bulges in the excited
state distribution, and those are states where the laser is directly
exciting to those states. There are more than one
state because the laser overlaps more than
one absorption line. Then the bottom one, it's less obvious, but if you look carefully
say on the bottom, those curve you'll see a
little dip there about state index about 28 or
something like that, I don't remember
exactly what it is, but that's called a hole. When you excite from
the ground state, you're removing population
from that ground state, and so the population of
the particular state you're starting from is reduced
below the equilibrium value. That's what you see in
these distributions. So the idea in this
is that you can use this dynamic information and a particular model to be able to measure the things
that we're interested in; concentration, temperature,
and pressure, and so on. So that's it for this video. Thanks for listening
and have a great day.