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.
