Lecture 5: The Shell Model and Multi-electron Atoms

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Settle down. The weekend doesn't begin until at least after the 3.091 lecture. Work first, then play. A couple of announcements, next Tuesday, quiz two based on the subject matter of homework two, the periodic table tests will be on Thursday, and it's going to be not before too long that we'll have time for the monthly test, the first monthly test.

The mnemonics contest deadline is a week from today at 5:00. Send it to me by e-mail if you want to win that hot tie or that hot scarf. It's come to my attention that people are moving freely about recitations without arranging through my office.

And, some of the recitations sections are uncomfortably large, which defeats the purpose of the recitation in part. I mean, we want you to have access to the instructor and if you have 30 students there instead of 20, that dilutes your access.

So, we insist that you go to the section that you are assigned to. And, if you must change because you've got a UROP, or you've got some other activity. Please do it through my office so that we can make the learning experience good for the majority and not just for the few.

Oh, there was something else I think. Yes, I just learned yesterday that to my somewhat disappointment, you know my favorite transuranic element is 111 because it's unununium. But, it has been isolated by the team in Darmstadt in Germany and they have proposed the name roentgenium after Wilhelm Rontgen, who is the discoverer of x-rays.

And, that's now before the International Union of Pure and Applied Chemistry. So, maybe before the fall is out we'll have to change. I already showed you that this is now darmstadtium. So, we're going to have a whole bunch of, it looks like, Meitnerium.

So, there's going to be a whole bunch of German names along here pretty soon. So, somebody better get busy over in Berkeley. What are they doing there? What's going on? Anybody know? People are sleeping.

OK, so let's get to the lesson. Last day, we studied the Bohr model for one electron atoms. And, the Bohr model showed us that through the quantum condition, all of these quantities were quantized.

They were functions of the quantum number, n. And, later in the lecture, and later in the lecture, we began to look for reconciliation between the Bohr model and Balmer's analysis of Angstrom's spectral line study of the atomic hydrogen.

And, so we looked at the prism spectrograph, and the gas discharge tube, and I showed you the energetics of what goes on in there. And, towards the end of the lecture, we started talking about the actual energetics of collision.

And, I want to return to that. So, let's put up that sketch again where we're looking at the incident electron. And, this is the electron that is accelerating off the cathode. There is electrons all over, and I want to make sure we are not confusing one electron for another.

So, this electron I'm talking about here is this one that is moving off the cathode and towards the anode. And, en route is going to collide with some atomic hydrogen. So, this is the incident electron.

And, it has kinetic energy of, I'm going to denote it E incident, one half mv squared **E=1/2 mv^2**. It's not bound. It's moving along. And let's subscript this as well. Incident velocity, electron rest mass, incident energy.

And, it's going to collide with a hydrogen atom at some point. I'm going to draw the hydrogen schematically like this with a proton in the nucleus, and a lone electron in some orbit here. And, if the energy of the incident electron is greater than the energy required for some transition within the hydrogen, then it's possible that this electron will capture some of the energy from the incident electron, and be excited to some other state.

So, let's put that on. Let's call this some initial state, and let's call this some final state. And, that's only going to occur when this incident energy is greater than the energy of transition. So, we've got two electrons here.

Here's a moving electron, free electron. This one is bound. So, this is in the target, or in the specimen, which in this case is atomic hydrogen. So, let's look at what happens here. The excitation occurs.

And we can get the energy required for this transition by using the Bohr model. The Bohr model gives this value, and gives the value as, that came right out of the postulates, the delta E of transition is going to equal -K.

I'm putting it up generally, z squared to allow for a higher z level, one electron atoms. And, it's one over final quantum number squared minus one over initial quantum number squared. So, that gives you the amount of energy.

And, that energy is going to be taken away from the kinetic energy of the incident electron, which then will make its merry way off after the collision. And, I'm purposely making that vector shorter to indicate that this electron has been slowed.

And, we call this electron the scattered electron scattered by the collision with the target hydrogen. And, it has kinetic energy E sub scattered as one half mv squared scattered. The mass of the electron doesn't change, but its velocity certainly decreases as a result of this collision.

But, that's not the end of the story. That's not the end of the story. What happens here is that this electron that's excited is there in an unsustainable basis. It's just a one-shot deal. So, the electron wants to fall back down to the lower energy state, and it does so.

It falls back down, and when it does so, it gives off that energy in the form of a photon. So, we have an emitted photon, and it has energy h nu or hc over lambda, or hc nu bar. And, it is the set of these emitted photons that Angstrom studied.

Angstrom was studying these emitted photons. And, clearly, there is some relationship between the wavelengths that he observed, and energy levels within the atomic hydrogen. And, all of this, this whole process is subject to conservation.

So, let's get that up. Let's make sure that we don't violate any basic law. So, we're going to say that the energy of the incident electron will then be spread across the electron energy of transition within the target.

And, whatever's left over will remain with the electron at some reduced value. And, what I've shown here, I'm applying it to the experiment that Angstrom conducted in which he accelerated electrons.

But, it's the same calculation for any particle. So, this is for all incident particles would be subject to this. All incident particles, so, it could be obviously electrons. We could accelerate protons.

We could accelerate neutrons. We could accelerate alpha particles. We could accelerate argon ions. There's a host of incident particles. Consider them only as bearers of energy. They are energy transfer vehicles, and they transfer energy here, provided that the energy is high enough to cause excitation.

Now, another way of looking at it conceptually is this is a probe beam, a probe beam, or maybe a stimulus. And, the integral of the energy of the particle after interaction and the emitted photons, that collectivity is the response.

This is the response, OK? This is the response. So, what does this mean? This means that we could interrogate some atom of unknown identity, and on the basis of these values, infer, deduce the identity of that species.

And, we talked at the end of last day about how this is applied in analyzing the chemical composition of stars. So, the radiation goes a long way, and we're able to use the information contained therein to conduct analyses.

So, let's see how this will help us reconcile the Bohr model to Balmer's analysis of Angstrom's data. So, let's see. I think we have to go back to the Bohr model. If we go over to Bohr, this thing's decided that it wants to sleep.

Finally. See, it's Friday. The computer has decided it's going to goof off a little bit. So, we're going to use postulate number six in order to help elucidate what's going on here so we can then resume here and say that delta E of the transition is the energy of the final state minus the energy of the initial state.

That's always the case. We always say that the difference goes final minus initial. And, we can represent that as the product, kZ squared, one over n final squared minus one over initial squared. And, conservation of energy says that must be the value of the energy imparted to the emitted photons.

So, the emitted photon, E of the photon, this is emitted when the electron cascades back down. And that will be hc over lambda. So, let's flip this thing around. Let's flip this thing around, and what we'll get that is nu bar, or one over lambda.

Then will be kZ squared over the product of the Planck constant times the speed of light, one over n final squared minus one over n initial squared. Now, for the Balmer series, for Balmer series, we had this set of lines for which the n member was always n equals two.

So, we're dealing with atomic hydrogen. So, let's put Z equals one. And, let's put the final state as n equals two. And, furthermore, if we evaluate k over hc using values that we already know, We get 1.1 times ten to the seven reciprocal meters.

So, if you put all that in, you get 1.1 times ten to the plus seven, one over two squared minus one over ni squared. And, ni has to be greater than nf. So, that necessarily means that it must begin at three.

It could go four, five, etc. Well, that's exactly the Balmer formula. We've got the Balmer lines out of the Bohr model, identical to Balmer. So, Bohr has been able to reconcile with Angstrom's data taken half a century earlier.

But furthermore, Bohr suggests other experiments. For example, doesn't it bother you that the lines end at n equals two? You know that according to the energetics here, the electrons have to keep cascading down to n equals one.

I mean, is there something funny going on in Sweden? Maybe the electrons in Sweden, they are a little bit nervous. Maybe it's winter. Maybe they are afraid to go down n equals one. How come we only get n equals two? Well, let's look for a second here.

Look at this. The sleep function is kicking in. I see the hard drive. There we go. So, let's for a moment take a look at the electromagnetic spectrum. This is taken from your book. And you see, again, I'm thinking about you all the time.

I got out my pens, and I've colored it in. The doctor says this is good. It's very good. So, here we are with wavelengths on the upper trace showing wavelength increasing from right to left, and energy increases from left to right.

So, high energy is low wavelength. One point here that irritates me: the SI unit of frequency is hertz which means beats per second. It's not the per-second. This is a little sloppiness on the part of the textbook.

And so, here what we've got is at the very, very low end of wavelength, very high energy are gamma rays, cosmic rays, and then over here at the low end of energy are the gentle forms of radiation, radio, TV, microwaves over here, and then in the center as it's drawn is visible light.

And I want to expand that. And, here's the visible spectrum, which in round numbers, I'd like you to know a little bit of this. I think an educated person should know a few things here. And, round numbers, visible spectrum runs from about 400 nm up to about 700 nm.

And so, the blue is low wavelength. It's high energy, and red is high wavelength, low energy. And, here it is staggered from one end to the other. And so, what we have here is actually an indication of what our photodetectors are capable of sensing.

We are tuned to operate here, and it's no accident that we are maximally positioned here in green, we are very sensitive in the green. We don't have uniform sensitivity. It maxes in green. So, the green pointer is a very good pointer to use.

So, now let's go over to the experiment. We can convert this to electron volts, by the way. I think electron volts is a good way to view this. And, you can see that visible spectrum stands roughly two to three electron volts, two to three electron volts.

And, think about it. Think about already this little paradigm here I've shown you, something excites. Something gives off photon. What about the inverse? Photon comes in; something excites. Wouldn't it make sense to have photodetectors that operate at around two to three electron volts? We are going to see later on in 3.091 that semiconductors have band gaps roughly in this regime.

OK, so what's this have to do with n equals two? Well, this is taken from your textbook. And here's the Balmer series ending in n equals two. If you convert these transitions, n equals three to n equals two, that turns out to give you an energy on the order of something between 400 and 700 nanometers.

And, what was the sensor that Angstrom was using? He was using photographic film. And, photographic film is sensitive in the visible. So, the electrons in Sweden were falling down to n equals one.

But, look what happens. When you fall down to n equals one, look at these wavelengths. 100 nm, well, where does the 100 nm put you? Way over here. You are in the ultraviolet. So, the photographic detector doesn't pick them up.

And likewise, some of the electrons were promoted even higher than n equals four, n equals five. And, on their way down, they were landing on n equals three. So you say, well, why didn't he get a set of lines where sometimes you get one over three squared? Well, let's take a look.

Let's take a look. If you go down to n equals three, that's called the Paschen series. And, look at these numbers, these 1,875 nm? 1,875 nm, that puts you way over here in the infrared. So, n equals two, those four lines end up right in the middle of the visible spectrum.

And, that's why the lines that Angstrom measured were in that regime. So, we can generalize all of this in the following equation, nu bar, which is the wave number, goes as R times Z squared one over nf squared minus one over ni squared.

And, this is known as the Rydberg Equation in honor of Professor Rydberg. Another Swede, he was at the University of Lund. And he was also active in spectroscopy. And so, we encapsulate everything that's on this slide here is in this equation, and more so because this allows for higher Z elements besides just elemental hydrogen.

Well, there is still more evidence. There is still more evidence in support of Bohr. Let's take a look at that. Bohr was really riding high. This was good, but it gets better. It gets better. There is the Franck-Hertz experiment.

Franck-Hertz experiment was performed in Berlin around 1912 to 1914. So, it was published around 1914. And, what Franck and Hertz did was operate a gas discharge tube. But, instead of running it with hydrogen, they filled it with mercury vapor.

Now, mercury, as you will learn later in its vapor state exists as the single atom. It's like helium, single atom, whereas hydrogen, there is atomic hydrogen. There is a molecular hydrogen. So, they operated gas discharge tube with mercury vapor.

And, what did they find? What they found was that when they varied - so let's draw this little schematic here. They've got a variable voltage power supply feeding electrodes. We've got mercury in here, cathode, anode.

And, they put up an ammeter in the circuit. And, they measured the current voltage characteristic. So, here's the current. It goes through the cell, electrons moving off the cathode shooting across to the anode.

And they measured the current as a function of voltage. And, what they found was that at low voltage, you get low current and at higher voltage you get higher current as you'd expect, some kind of an ohmic behavior.

Up to some critical voltage, and at that point the current dropped to almost zero. And, the tube started glowing blinding white. This turns out to be about 4.9 volts. They continued raising the voltage, and again, now they've got current going through, the ammeter is registering current.

The tube is glowing, and now, the current is rising until another critical voltage. And then it falls again. And now, the tube is glowing even brighter. And, this is 6.7 volts. And just for your information, you look on the periodic table today and it turns out that the ionization voltage for mercury is about 10.4 eV.

10.4 eV is the ionization energy. So, this is below the ionization energy. So, what's going on here? What they reasoned was, they were exciting bound electrons within mercury, and furthermore that there are critical voltages associated with critical energies.

Conclusion: there are quantized energy levels within mercury, which is to say that the Bohr postulate of quantization applies not only to a one electron atom. But it applies to multi-electron atoms.

Well, if it applies to multi-electron atoms, doesn't that mean it applies to everything? So, they concluded from this experiment, quantization of energies of electrons in multi-electron atoms. So, the Bohr model doesn't apply to multi-electron atoms, but the assumption that energy is quantized in multi-electron atoms, the assumption of energy quantization extends to multi-electron atoms.

So, this is really, really good. Bohr was riding very, very high. But, all good things come to an end. There were problems with the Bohr model too. So, now let's look at limitations. I want to look at limitations of the Bohr model.

Limitations of Bohr. OK, first of all, there's data out there. There's data out there already, first 1887. 1887 Michelson and Morley, this work was distinguished by the fact that it was done in the United States.

You notice, most of everything I've shown you up until now was done in Europe with one exception, the Millikan experiment in Chicago. Everything else is European science. But, Michelson had immigrated to the United States and was really, really clever.

He was brilliant inventor, was doing military research for the Navy back in the early 1880's. He invented the interferometer. And, he was the first to make accurate measurements of the speed of light, the Michelson interferometer.

And, he eventually worked at the Case School of Applied Science in Cleveland, which ultimately became Case Western Reserve University after a number of transitions. So, he was working at Case in Cleveland, and doing very, very high precision spectroscopy.

That was his forte. He was a brilliant experimentalist, knew how to make equipment that was highly accurate. And, what he found was that one of Angstrom's lines, the line associated with the transition three to two, if you look really, really carefully, now remember, Angstrom made the measurements by looking at light hitting a photographic plate.

And, that light doesn't come down nice and sharp. There is a little bit of scatter in the data. So, you get this fat line. But the positions of center to center on the lines follow that set of four numbers that we were looking at yesterday.

Now, what Michelson redid the work, remember, Angstrom did his experiments in 1853. Michelson looks very, very carefully and sees that it's not one line, but that it's two lines, two lines very close together.

So, the three to two line is in fact a doublet. It's a doublet, that is to say, two lines very, very close together. But from a distance, it looks like a line. And then, if you've got really accurate instrumentation, you discover that it's a doublet.

And that's a problem. That's a problem because if you look at this term scheme here, if we've got a doublet, it means that you've got two different transitions. So, the Bohr model is saying you go from n equals two up to n equals three.

And, when you fall from three back down to two, you give off a photon. And that photon has a unique value of wavelength. But, they're measuring two. But they're very close. So, what that suggests is that maybe there are two levels that are really, really close together.

And, sometimes the electron falls from the higher level. And, sometimes it falls from the lower level. But in any event, there must be more than one level, but very close together. So, Bohr is silent on that.

Now, the other possibility is that two is the problem. We don't know. Maybe this has got two lines, and this has one. So, that's the point, first of all. By the way, Michelson was awarded the Nobel Prize in 1907, not for this observation.

He got it for the measurement of the speed of light. Why I'm putting it down is you are going to see everybody today. We are giving out Nobels like they're going out of style. So, everybody gets a Nobel today.

It's not like Oprah; we're not giving everybody a car. But we're going to give everybody a Nobel, OK? So, that's the first piece of bad news for Bohr. These are headaches for Bohr. OK, number two: number two is 1896.

1896: Pete Zeeman, who is a postdoc at Leyden in the Netherlands, he was a postdoc at Leyden in the Netherlands. And, what he did, these people were so creative. I mean, they were so excited about science.

So, he works with a gas discharge tube, and he takes the gas discharge tube and he puts it into a magnetic field. He makes the magnetic field by wrapping a conductor around the tube and passing current through the conductor.

And, by playing with the value of the current through the coil, he changes the intensity of magnetic field, and then goes through the experiments with the prism spectrograph. OK, so he measures gas discharge tube, same thing all over again, gas discharge tube in a magnetic field.

And, what he observes is the following. Just schematically, if this is the photographic plate, and you've got multiple lines on the photographic plate, this is no magnetic field, what we've seen before.

What he finds is that in the magnetic field, some of these lines actually, and this is not to scale, some of these lines actually show up as doublets, triplets, but only in the applied magnetic field.

So, we call this line splitting, which is even different from what Morley said. Michelson and Morley observed that no magnetic field, if you look carefully, there's multiple lines. What Zeeman sees is that single line, no magnetic field.

Apply a magnetic field, you see a line splitting. And, furthermore, he found that the intensity of the magnetic field affected the degree of splitting. The higher the magnetic field, the greater the degree of splitting, line splitting proportional to intensity of magnetic field.

So, this is called the Zeeman effect. And, that's not bad for a postdoc because he got a Nobel in 1902, and they were only offered for the first time in 1901, so talk about a meteoric rise. People obviously on the Nobel committee thought this was pretty good work.

So, how's the Bohr model going to describe this? It can't. It's incapable of describing this effect. There is more. People are looking; they're working hard. So, 1913, I'm even going to give you the month.

November 1913, why am I giving you the month? Because the paper was published in July of 1913, Bohr's paper, July 1913. November 1913 there is a fellow by the name of Starck, and he's doing some work at Aachen in Germany.

And, analogous to what Zeeman was doing, Starck takes gas discharge tube and he puts it between plates that are connected to a power supply. So then, he subjects the gas in the tube to an electric field.

And, what does he observe? Line splitting. And, how is the line splitting? The more intense the applied electric field, the greater the line splitting. So, same thing. Line splitting in electric field.

It's known as the Starck effect. And, you guessed it: he gets a Nobel. Everybody's getting Nobels. 1919 he gets a Nobel for the Starck effect. So, all of this is adding up to, there's good news here.

Bohr was able to give us a sense of what was going on in the Angstrom experiment, but there's a lot of stuff here indicating that it's an incomplete model. So, 1916, a theoretician comes to the rescue, Arnold Sommerfeld.

He was a professor of physics in Munich. And he proposed some modifications to the Bohr model. He liked the Bohr model. He didn't want to throw it out. He liked the concept of a planetary model.

But then maybe he looked to the heavens. You know Kepler's laws of planetary motion. What's the orbit that the planets describe? It's an ellipse, ellipsoidal. So, Sommerfeld said, how about this? Just suppose, and this is not to scale.

It's going to be exaggerated so you get a sense of it. Suppose in addition to a spherical orbit, suppose the electron might describe an elliptical orbit. Now, overall, this is roughly the same. It's not super elliptical.

It's just mildly elliptical, but just enough that it's not exactly spherical. So, this could give you sort of have your cake and eat it too. You could say, this is the principal quantum number of Bohr.

It's n. But, within n there's a little bit of fine structure. Fine structure, that is to say, there are details there that the first pass didn't quite capture. So, suppose we have more than one. Maybe we have a plurality of elliptical orbits.

So, we have elliptical orbits. But, they are roughly of the same dimension as the original circular orbit. And, he says we have sort of a band. And he calls this a shell containing multiple orbits, which he termed orbitals to distinguish them from the circular orbits of Bohr.

Why did he call it a shell model? He said, think about the eggshell. The eggshell: we can describe it but the eggshell itself has some thickness. So, this is a sense of what Sommerfeld was giving us.

So, he says, how am I going to capture these ideas quantitatively? Well, the shell could be captured by the quantity n, but these orbitals, some of them circular, some of them elliptical; I'm going to need new quantum numbers in order to capture these new ideas.

And so, let's take a look at the quantum numbers that Sommerfeld gave us. So, the first one is N, and it is as it was in the case of the Bohr model. So, this is called the principal quantum number or the shell quantum number, principal or shell quantum number.

And, it talks about size. It talks about size, the size, that is to say the distance from the nucleus to the shell. And, it takes values one, two, three, and so on up to infinity. And at infinity, we're talking about a free electron.

And, in parallel, there is another notation that's used that spectroscopists came up with, their own notation. They don't like numbers. The spectroscopists used letters. But the spectroscopists, they're a funny bunch.

They're very suspicious. So, here's what happened. If you look on this chart here, for a while we thought Balmer was the bottom of the barrel. But that's n equals two. And then, when the detectors got better, we saw that n equals one.

So, the spectroscopists said what if somebody comes along with an even better detector and finds transitions to energy levels below that which we are calling one? So, we are going to get ready. We are going to give letters to these, and they started with the letter K because it's somewhere in the middle of the alphabet.

So, if we ever discover an energy level lower than n equals one, we can use a letter. It will be the J. So, there's K, L, M, and this notation is used to this day. When we talk about x-rays, x-rays generated by electrons falling down to ground state are called K x-rays.

So, this is the other notation for the values that n can take. Now, let's get into the contributions of Sommerfeld. He called the next quantum number lowercase l. And, this is the orbital quantum number, the orbital quantum number, and it describes the shape of the orbital.

Remember, I said we needed some way to distinguish purely circular from elliptical or other shape orbitals. And, the orbital quantum number takes values zero, one, two, up to the value of n minus one.

And, the spectroscopists use letters. They refuse to use numbers. They're very numerophobic (sic). So, the use lowercase letters to distinguish the orbital quantum number from the principal quantum number.

And so, when l equals zero, it's lowercase s for sharp because these would be very sharp lines. Then, p for l equals one, which is principal, d for l equals two, which stands for diffuse, and f which stands for fundamental because by the time you get up to f level, you've got such compression that all of the spectral lines are starting to look very similar to those of hydrogen, which is the fundamental spectrum of all time.

And then, after that, they go g, h because they ran out of ideas. So, that was it, OK? So, now let's take a look. When l equals zero, this means you have a spherical or circular, a spherical orbit or a circular orbit.

And that's easy to remember because O is, think of it as a perfect sphere. All right, when L equals one, this is elliptical. You have an elliptical orbit. You can think of that, zero is zero, and one, think of that as there's some elongation, OK? And, when L equals two, when L equals two, I'll show you these later in about a week's time.

They're just much more complex. You get dumbbell shaped, doughnut shaped, and all sorts of things. So, just say they're more complex. So, those are the numbers associated with certain shapes of the orbital.

Then, the next quantum number he gave us was m, which was the magnetic quantum number. It's the magnetic quantum number, and it speaks to the question of orientation. And, the admissible values of m are l, l minus one through zero down to minus l.

So, spanning from plus l to minus l, and of course l is subject to values of n. So, in the case of n equals one, when n equals one, l can equal only zero. So, if l is zero, then this value, m must always be zero.

And, indeed this confirms the observations. There was no line splitting in a magnetic field for the ground state. It wasn't seen. What happens when n equals two? When n equals two, well, l can take values of one and zero.

So, what do we have? We have now n equals two, l equals one, so, m can now go one, zero, and minus one. So, this is the one case where there is some value in thinking about the Cartesian equivalent of this.

It's very dangerous when you're talking about quantum ideas to try to ascribe physical attributes to them because in many cases, it just doesn't work. But, in this case, I think it does. So, let's look at three values of the magnetic quantum number when we have an elliptical orbit, right? So, one way of thinking about it is here are the three principal coordinate directions, X, Y, and Z.

Now, I don't know where real X lies, but once I choose X, I can use the right-hand rule, and I know where Y and Z have to be. They're orthogonal. So, if this is X, this is Y, this is Z. I have a non-spherical, non-axisymmetric orbital.

So, for simplicity, I'm going to just make it elongated, OK? Well, what does this say? It says m can go one, zero, minus one, which to my mind represents the fact that there are three principal coordinate directions in which you can orient that.

So, I think this is a nice example of mathematics imitating reality. OK, after that, it gets very messy. And then, the last quantum number, this is where Sommerfeld left it. But I'll finish by introducing the last quantum number.

The last quantum number came a little bit later. It's a lowercase s, which is the spin quantum number. And, it represents the fact that the electron is considered to have spin. And it takes values plus or minus one half.

And here's the basis for this value here. I want to put this all together so that I'm doing a little bit of back to the future because this is 1916. That's where Sommerfeld leaves it. This comes about 1925, but I want you to have all quantum numbers.

And then, we can move forward. And what introduced this was an experiment conducted by Stern and Gerlach in Frankfurt, and that was done around 1921. We can wake this up; we'll have a look at it. This is cute.

I think it's cute. We'll see. You don't laugh at any of my jokes, but this is Sommerfeld on the occasion of his 80th birthday. And as is the custom for her birthday so momentous as this, people from all over the world get together for a party.

And they accompany this with a two or three day affair in which people give scientific papers in honor of the celebrant. So, this is taken from the book that was published on the occasion of Sommerfeld's 80th birthday.

And, since Sommerfeld was the one that proposed that orbits could be spherical or elliptical, they as a gag said, well, here's Sommerfeld, and then here's Sommerfeld somewhat distended. So, this is l equals zero, and this is l equals one, and this was accompanied by the inscription that said Sommerfeld was the one who taught us that the circle is the degenerate form of the ellipse.

And, that was really funny. That's 1920s geek humor. That was roaring. But clearly it doesn't age well. You're not sensing this. OK, here's the Stern-Gerlach experiment, very, very cool experiment.

They were doing studies in the early 20s to try to validate Maxwell's equations. So, what they've got over here is a furnace with a crucible of liquid metal. And, the metal has obviously a vapor pressure above it.

And, there's a slit here. And, they've got a beam. It's like a physical vapor deposition. And so, they're shooting metal in a beam over to a cooled substrate here. And, they're analyzing the deposit.

OK, so if you shoot a beam and it's got a slit, you should just get a shadow of the slit. So now, let's get interesting. Let's test Maxwell's equations. So, they send the beam through a divergent magnetic field.

You can see the South Pole comes to a point. And the North Pole is a curve. So, the field lines had to diverge. So, they were asking questions like, what happens if you take a beam of metal and send it through? And here's what they found.

What they found was really, really interesting. If you take a metal and send it with no magnetic field, and you look at what you get on the substrate, this is the shadow of the slit. In other words, if you send it through a narrow slit, this will be the deposit of the metal.

When B does not equal zero, they observe a deflection. So, now, the question is, if here's the rest position of the beam, in which direction should the deflection go? And, so some would argue, well, it should go up, and some argued that it should go down.

And, what they found was they found two, a single beam split into two. And, this really vexed people. In fact, here are the data. This is a very grainy reproduction from their paper. The dark line is a trace of the deposit, in this case, it's silver.

They are sending silver through with no magnetic field and now, here it is with the magnetic field on. And this is a grating. And they are measuring separation. So, the silver beam splits into two.

It goes up and down. So, Goudsmit and Uhlenbeck come along. Let's get these guys on here first. This is Stern and Gerlach, 1921, and, yeah, so they're working on that one. And then comes Goudsmit and Uhlenbeck also at Leiden, again.

They were two graduate students. And they said, we can explain that. Watch this. Crazy as it sounds, here's the electron moving around. They said the electron just doesn't move, it spins. They said what is the electron is spinning as it moves around the nucleus, all right? And, if you use the right-hand rule, if it's spinning clockwise, it's spinning up.

But maybe an electron, we don't know where the absolute up or the absolute down is. Some electrons might be spinning anticlockwise, in which case they are spinning down. And, silver is atomic number 47.

So, it has an odd number of electrons. So, it can't get a spin up electron canceling a spin-down electron. It has a net non-paired spin, and they said, well, since we don't know what the absolute is, half of the silvers are spin up, and half of the silvers are spin down.

And that's why you get this. So, let's give a fourth quantum number. Let's call it plus or minus one half. And, you could say this is crazy. But, it explains those results. And, they actually were here during World War II.

Both of them, they worked down just around the corner here in the Rad Lab, radiation laboratory where radar was discovered. It's around building four, first floor. There's a plaque there indicating it.

OK, well let's jump ahead. I want to talk a little bit about hydrogen and energy and I want to talk about the Hindenburg. The Hindenburg was a giant, rigid airship. To give you a sense of scale, it was almost as long as the Titanic: 135 feet in diameter, 804 feet long.

This is a 747 dwarfed. 7 million cubic feet of gas, 112 tons of useful lift. But, they inflated it with hydrogen because the Nazis had come to power. Most of the helium was to be found in the United States.

And, Congress passed the Helium Control Act, which forbade the sale of helium to Germany. So, the engineers said, OK, we'll use hydrogen. So, they filled this thing with molecular hydrogen. And, I mean, it was a fantastic boon.

I mean, these are posters from that era: two and a half days to Europe, only two and a half days, much faster than going by ship. Here's a sketch. This is in German, and now over the North Atlantic, there's the tip of Manhattan.

There's a nice shot of the airship over Manhattan. So, it made ten transatlantic flights in '36 carrying almost 1,000 passengers, cruising speed 78 miles an hour. They only flew in summer. So, it's the arrival of the first flight to the US on May 6, 1937.

And, it's docking at Lakehurst, New Jersey. The docking was moved. If you go to the top of the Empire State building and you look out, you'll see these protrusions. Those protrusions were put there in the expectation that zeppelin airships, dirigible airships would dock at the Empire State building.

People would get off, take the elevator, and next thing you know, you're on Fifth Avenue. When they tried to dock these things, the air currents were so turbulent that it was unsafe to do so. So, they made this docking station across the river in Lakehurst.

So, this is what happens: 36 dead. That's not what's amazing. It's amazing that the majority of people disembarked. What I want to tell you is it was not an explosion. The Hindenburg did not explode.

There was an aluminum grand piano on there. They had ballrooms. They had restaurants. You were there for two and a half days. You're dressed. It's elegant. The aluminum piano melted. They figured what happened was there was electric discharge in the vicinity of a tiny hydrogen leak.

Hydrogen is a tiny molecule, and some hydrogen must have been leaking. They think it may have been something like a St. Elmo's fire or something like that, just a high electrical discharge and the misfortune that this thing caught.

So, here's the analysis. This skin was made of resin, and it was finished with a lacquer dope to make it water repellent. And, to make it glow, they used aluminum powder in the lacquer dope. And then, they put iron oxide on the inside.

Lord knows why. So, you basically have the elements of a thermite reaction. This is what the shuttle uses, in its solid rocket boosters. So, once this thing caught, it was the skin that went. And of course this meant the end of rigid airships and commercial air transportation, and to point out that it wasn't the hydrogen.

If that had exploded, it would have been a very different scene. This is 1956, a U.S. Navy airship dirigible filled with helium; same thing, but it had a two ply cotton envelope. Gasoline fire: away it goes.

So, it was the skin. In fact, 100% hydrogen will quench flame. It's like running a carburetor too rich. You choke. You need fuel plus oxygen. You cannot get 7 million cubic feet of oxygen to the site to sustain the explosion.

Besides, you're working with only 20% oxygen in air. This, I blew it up because I thought this would look sort of like modern art. So, there it is. This is a Roman candle. It's not an explosion. If this had been an explosion it would have been totally different.

So, this is stoichiometry. This is a limited reaction. There was not enough oxygen to keep it going. So, there it is. Have a nice weekend.

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