Hello. Today, we're going to start
talking about spectroscopy. Spectroscopy is the study
of the interaction of light and matter or
radiation and matter. It can be used to study atomic
and molecular structure, and in fact that's how
we know most of what we know about atomic and
molecular structure. But in more interest from an
engineering point of view, it can also be used to measure thermodynamic state properties and species concentrations. There are a number of forms of spectroscopy including
absorption, emission, fluorescence, and scattering, and various
permutations of those. We'll start with discussing the absorption and
emission of radiation. So at the most basic level, spectroscopy is concerned
with how light is absorbed or emitted by atoms and molecules. So here we'll discuss the Einstein picture of absorption and
emission of radiation. So in 1916, Einstein published a seminal paper on the subject of absorption and
emission of radiation. In the paper, he outlined an
extremely simple theory and introduced what
are now called the A and B Einstein coefficients. Interestingly, he used
detailed balance arguments to relate the different
coefficients. I have given the reference to the paper at the
bottom of the slide if you're interested
in going back and finding it and translating
it from German. So the picture is as shown
in this simple diagram. So the two horizontal
lines labeled 1 and 2 are two quantum states of
an atom or a molecule. Now, absorption of radiation involves passing light
through a medium that has these quantum states such that
the energy of the photons h Nu is equal to the energy difference
between State 1 and State 2. That's called a
resonant interaction. In other words, the
frequency of the light is such that it is resonant with the energy difference
between the two states. If that is the case, then that atom or
molecule can absorb light in the form of a transition between
those two states. So the energy of the
molecule would be increased by that
energy difference going from State 1 to State 2. So the rate at which
that occurs which I've called W_12 is equal to the integral of the Einstein B coefficient
from State 1 to State 2 times the
spectral irradiance of the light times the line
shape function of the light, and we will get to that shortly. So the next possibility is that the atom or a molecule is already in the
excited State 2, and then something can happen
cause induced emission. So if a photon with the
appropriate amount of energy, h Nu comes along, it can cause a transition
from State 2 to State 1, and by conservation of energy, two photons must be emitted as a result
of that interaction. In other words, 2h Nu minus h2 is equal to the energy difference
between State 1 and State 2, that's called induced emission. The expression for
induced emission looks essentially the same as the
expression for absorption, it will be W_21 is equal
to the integral of B_21 times the spectral
irradiance times the line shape function. That is the second possibility. So so far we have two Einstein coefficients
of B_12 and a B_21. Then finally, it turns
out that quantum states above the ground level are
unstable in the long-term, long being a
relativistic expression. If you have an atom or a
molecule in an excited state, that is above what it
would be at equilibrium, it will eventually relax down in energy and emit a photon, that's called
spontaneous emission, and Einstein identified the
coefficient for that as A_21, turns out that A_21 is one over the lifetime of
the excited state. Now, I mentioned that Einstein used detailed
balance to relate the coefficients and
the expression on the bottom left is
that expression. So the ratio of the Einstein A
coefficient going from state i to j divided by the Einstein B coefficient
going from i to j is equal to the expression 8 Pi h
Nu cubed over c squared, where c is the speed of light and B
coefficient's going in the opposite direction on our cartoon would be
1 to 2 and 2 to 1, R equal 2 within the ratio of the degeneracies of
those two states. This turned out to be
just a remarkable thing and it made understanding
the absorption, emission of radiation
much simpler. The Einstein coefficients can be calculated from the
first relationship. R squared is the probability for the transition from state i to j and is given by the square of the
electric dipole moment. A dipole moment occurs when you have a charge
distribution in something. A hand radio antenna, the antenna in your iPhone, these are all dipole radiators, meaning that there's
a distribution of charge over some distance, and the dipole
moment is defined in the bottom equation
where it's the sum of all the charges times
their position. So the integral of that over the wave function for the two-states is a spatial wave function for the two states, the integral of Psi i times the electric dipole moment times Psi jdV integrated over the volume of the absolute value of that integral
squared is R squared, the transition, so-called
transition probability. So if you knew that
you could calculate the B coefficients and that is something that people have done. There's another way of expressing transitions and that's the so-called oscillator
strength, f_ij, and it is the rate of the actual transition probability to the transition
probability predicted by the classical harmonic
oscillator model and the equation for that
is given in the bottom. You still need to know the transition
probability R squared. Here are some data for two atoms and four
diatomic molecules. This is from the book
by Radzing and Smirnov. You can get this data
other places as well. As I mentioned a minute ago, the A coefficient is the
inverse of the lifetime and typically tabulations give lifetimes rather
than A coefficient.. The reason for that is that
it lifetimes can be measured, that's what it measure directly. So these range from 16
nanoseconds for sodium to 0.000031 for the C_2 molecule. So they can range over several orders of magnitude
depending on the molecule. Of course, polyatomic
molecules also have Einstein A and B
coefficients and lifetimes. But of course they're much more complicated and as a result,
we won't deal with them. So that's it for this video. In the next video,
we will explore some of the consequences of this. Thanks for listening
and have a great day.