Lecture 16: Cubic Crystals, Miller Indices, Characterization of Atomic Structure

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Let's see. A couple of announcements. Tomorrow there will be quiz six based on homework six just on the crystallography. There is a little bit at the end of homework six on x-rays. And we will just start to talk about x-rays today, so I think that we will leave that out.

It will just be based on crystallography. And I have asked the recitation instructors to administer the test at the end of the period so you can have some chance to ask questions. Last day we talked about crystallography.

We were introduced to the seven crystal systems and the 14 Bravais lattices which are shown up on the slide taken from the lecture notes. And we saw that if we wanted to describe the arrangement of atoms in a crystal, we could do so by combining the notion of the Bravais lattice of which there are 14 distinct types with a basis.

The basis could be a single atom. Could be a group of atoms. Could be an ion pair. Could be atom pair. And we saw that got us through an elaborate set of crystals. And then towards the end we focused on cubic crystals because cubic crystals are representative of a large number of metals in the Periodic Table, and it is much more tractable mathematically.

We are going to focus on cubic crystals, and I am going to continue along with that today. In particular, I want to start looking at notation. And the reason I want to look at notation is I want to be able to define a point, a line and a plane so that when we start characterizing crystals, I can give you a mathematical notation that represents the face of the cube, the body diagonal of the cube, a line going on the body diagonal, a line going along the edge.

Instead of writing it in words, I want to be able to do so mathematically. And so for that we are going to turn to this handout which has the elements of crystallographic notation. I want to quickly go through this.

I think last day we started talking about position, so I have drawn a little sketch here. This is the unit cell. The unit cell is the cube because we are talking about a cubic system, and this is just the lattice.

I have not put atoms here. So I am just talking about lattice points in space. And, just to remind us, a, b and c are the unit vectors, and in a cubic system a equals b equals c. And each of them is equal to a, the lattice constant.

This is the lattice constant, a. That's the dimension a, the lattice constant. And all angles are right angles, so alpha equals beta equals gamma equals 90 degrees. Just to practice, the origin, as I have shown here, all we do is write zero, zero, zero, as you would in your math class.

But don't put parentheses around it because parentheses are reserved for planes. We are going to use parentheses later on. Crystallographers have their own notation. So here is point a. What do we see? It is in the yz plane.

By the way, I have chosen a set of coordinate axes that are in conformity with the right-hand rule. I don't care which axis you choose to be x. It doesn't matter. I generally like to make z vertical, but I don't care, as long as you use the right-hand rule.

You need to observe the right-hand rule because in physics you are going to find that there are certain vector forces that are the cross-product. And, if you don't use the right-hand rule convention, you will get things identically backwards.

The thumb is your x-axis, y-axis, z-axis, however you want to do it. I have the x-axis coming out of the board, and the y-axis and the z-axis are in the plane of the board. If we come here, this point A is zero units out on the x-axis, it is one unit out on the y-axis and one unit up on the z-axis, so this will be zero, one, one.

And b, as I've drawn it, looks like it is one unit out on the x-axis, it is no units out on a y-axis and a half a unit up on the z-axis. That is one, zero, one-half. That is about the last time I am going to talk about these things.

It is more interesting to talk about lines. For lines, I am going to say let's just follow along on the slide. And the slide imitates what is going on here. Let's see. I've got a couple of them I want to try.

Let's look at, for starters, OB. I want to look at OB. I am going to start at the origin, and I am going to come out to the point B. And I have a vector on this. It is emanating from the origin out to B.

And what does it say to do? It says define a vector from origin to the point on a line, choose the smallest set of integers and no commas. I put the origin, which conveniently here is the origin, out to the point along that direction.

The vector OB, as written, will go from there out to one, zero, one-half. But then it says no commas, enclose in brackets and clear the fractions. They don't like to have fractions when you are talking about lines, so clear the fractions and multiply through, then that will just double it, two, zero, one.

And you could say what are they doing? If you can imagine going out one more unit cell, this line would actually go through the point two, zero, one. You are just looking at the projection along here.

You are just multiplying it. And then it says no commas and enclose in brackets. This is called a bracket. These staple-like things are called brackets. That is the direction OB. Let's see. What is the other one we are going to do? AO.

It says you have to move the origin to the base of the vector. If I want to do AO, I have to put the origin up here at the tail of the vector. If this is the origin, I am moving actually what? I am not moving anywhere in the x-axis, but I am moving minus one in the y-axis and minus one in the z-axis in order to get this direction.

I am really going zero in the x, I am going minus one in the y and minus one in the z. And you can see that since we don't use commas, putting minus signs would look cumbersome. In crystallographic notation, the minus sign is represented as a macron.

A macron is a line over. This is zero, one bar, one bar. That is equal to minus one. Crystallographers use the term one bar. This is 0, one bar, one bar for this line A down to origin. That is how that is done.

Then it says we can denote an entire family of directions by carats. Let's do that. Suppose I wanted to say all of the body diagonals. Well, how would I get a body diagonal? If this is the origin and I want to go up the body diagonal, it would be out one unit, out one unit, out one unit.

So one, one, one, would be a body diagonal. But suppose I want to say all body diagonals? That is one, one, one this direction, one, one, one that direction. I want to get this body diagonal. I want to get that body diagonal.

How do I do that? I can do that by putting carats. These are called carats. Those of you who have done some editing and proofreading know these terms. This is all body diagonals. We could do a few others while we are at it.

Suppose I wanted to do all of the face diagonals like this, like this, and then over here like this, like this? Well, what is a face diagonal? Zero, one, one. All the face diagonals. What about all cube edges? What is a cube edge? A cube edge could be this one: zero, zero, one.

This would be all cube edges. What does it mean? It means write this out in full. That is O, O, one; O, O, one bar; O, one, O. I am using O for zero. I am cluing you into crystallographers' talk.

That is how they talk. O and one bar. You are going to be really hip if you end up in an elevator with a bunch of crystallographers. And there are two others here. One, O, O, and one bar, O, O. Well, how many faces are there? Six.

How many face edges? Six. And, look, just going through the permutations and combinations gives you exactly the number that you should have. This is mathematics imitating reality. That is good. And likewise here.

If you go through all the permutations, combinations of zero, one and one bar, you will end up with all of the possibilities for face diagonals, all the possibilities for body diagonals. I think I have a cartoon here that shows all of the different diagonals that are possible when you mix one and one bar.

That is good. So far so good. That is pretty simple. Now let's look at planes. For planes we use something called the Miller indices for describing planes. And they are named after the British mineralogist William Hallowes Miller who, back in the 1830s, gave us this means of describing a plane.

And I think, rather than writing it down, we will just follow along here because you have the notes in front of you. You know from your math that the equation of a plane is x over a plus y over b plus z over c equals one, **x/a + y/b + z/c = 1** and a, b and c are the intercepts with the x, y and z axes respectively.

What Miller did is he said let's let h be one over a, k be one over b and l equals one over c. So this becomes the equation of the plane, hx plus ky plus lz equals one. **hx + ky + lz = 1** And now what you do is take the h, k and l and enclose those in parentheses.

In other words, he is taking the plane index to be the reciprocals of the intercepts with the three axes. And you will see why in a second. Why do it that way? Because every once in a while something like this happens.

Here is a simple one just to get our feet wet. Here a, b and c are the unit vectors, so this is the x-axis in the same convention I have chosen. X is coming out of the plane of the slide, y is moving to the right and z is moving up.

Here is a plane that is shaded in. And what do you see? It cuts the x-axis at x equals plus one-half, it cuts the y-axis at y equals plus one, and it never cuts the z-axis. It is parallel to the z-axis.

Mathematically, we would say the z-intercept is infinity. And who wants to have infinity? Now you see the wisdom of Miller. Miller said let's describe this in terms of the reciprocals. What is the reciprocal of one-half? It is two.

What is the reciprocal of one? It is one. And what is the reciprocal of infinity? Zero. We can describe this plane as the two, one, zero plane. Whenever you see a zero, whatever place the zero lies that means that plane is parallel to that axis, so right away I can say that is a plane that is parallel to the z-axis.

It does not cross the z-axis. So there it is. There is the formula. You take the reciprocal of the intercepts, put them side-by-side, no commas, and this is why we save the parentheses. That is why we don't put our position in parentheses because we are saving parentheses for the plane.

Let's do a few more. This is a lot of fun on a Monday. See, you are excited to be back in class. There is a cool thing that happens when you use Miller indices. You see what I just did? I just drew a direction, and that direction is the two, one, O direction.

And what do you notice about the position of two, one, O versus the two, one, O plane? When you write directions, according to this formula and planes according to Miller indices, the two, one, O direction is perpendicular to the two, one, O plane.

Two, one, O direction is perpendicular to the two, one, O plane. Let's do another one. This is a good one. This is a plane that cuts the y-axis at y equals one. It is parallel to the x-axis and parallel to the z-axis.

It never cuts the x or the z-axes, so that is an O, one, O plane. I mean, you can do that in your head. That is an O, one, O plane, that is an O, one, O plane and so on and so forth. Here is an O, two, O plane.

Why? Well, because it cuts the y-axis at y equals one-half, you take the reciprocal, which is two, so O, two, O. And let's try again. Well, there is O, one, O direction. And what do you know? The O, one, O direction is normal to the O, one, O plane and O, two, O direction is normal to the O, two, O plane.

What do you know about the O, one, O direction and the O, two, O direction? They are the same direction. O anything O is going to be along the y-axis. Now, that is an O, two, O plane. I could shift the origin to here, couldn't I? Then that would be an O, two, O plane and that would be an O, two, O plane.

Oh, here is some more. One, one, one. Well, look at this plane here. It cuts the x-axis at x equals one, the y-axis at y equals one, the z-axis at z equals one. Shade that in. That is the one, one, one plane, but that is also a one, one, one plane.

We can keep shifting the origin. I don't know where the origin of the universe is, so I can move this anywhere along. Here is one. Here is backwards. See, this now is one, one, one bar, because if I put the origin here I am cutting the x-axis at plus one, I am cutting the y-axis at plus one, but I have a negative tilt to this plane so the z-axis is being cut at minus one.

Again, we try to avoid the minus sign. And, instead, we put the macron over the number. In that way we give the minus indication on a number, no commas in parentheses. Now, let's do the test. Here is the one, one, one direction.

And, sure enough, the one, one, one direction cuts the one, one, one plane normally. Let's try over here. There is the one, one, one bar direction. It cuts through the one, one, one bar plane normally.

Fantastic. And we can do families of planes as well. Let's look at families of planes. What kind of families could we come up with if we want to do families of planes with brace brackets? Families of planes use braces.

If I want to say all cubed faces that would be O, O, one. Because O, one, O we saw cuts the y-axis parallel to the x-z plane. That is going to tell you to roll over zero, zero, one and one bar in this manner.

Only instead of brackets you have braces. How many faces on a cube? There are six. How many combinations of zero, one and one bar are there? Six. There you go. By the way, one other thing. If you want to be hip with the crystallographers, this is one way of writing it.

Another way of writing it is one, O, O, which is equivalent to O, O, one. But crystallographers like to put things in ascending order. A crystallographer would never write one, O, O. A crystallographer would write O, O, one.

That is a cool factor there. You never know when you are going to meet a crystallographer on an elevator. It could be the icebreaker because they might be shy. Now, the last thing I want to do is talk about the distance calculation, another virtue of using Miller indices.

I want to put this up here because it is on the handout, but I want to make sure that it gets on the board. The hkl plane is perpendicular to the hkl direction when we use such a notation. And then I want to go back to the previous slide.

Go back two of them because it is easier to see here. I want to calculate the distance between adjacent planes of the same index. We can calculate the distance between adjacent planes of the same index.

What is the distance between two adjacent planes of some value hkl? That is given by the following formula, which again is a testimony to Miller. The distance, and I am going to subscript it hkl because they are all the same index, is given by a, which is the lattice constant, divided by the square root of h squared plus k squared plus l squared where a is the lattice constant.

Let's test it. If hkl is O, one, O, trivially that separation should be a, the lattice constant. And I think you can see that quite nicely up on the screen here. It is one lattice constant between successive O, one, O planes.

Let's try it with O, two, O. It is going to be the square root of two squared which is a over two. And, sure enough, what is the distance between O, two, O and the adjacent plane comparable to it? It is only half a lattice constant.

In essence, if you look here, every O, two, O plane is only half a lattice constant away. Every O, one, O plane is full lattice constant. What you are really seeing then is that the d goes down, the distance between planes goes down as the Miller index goes up.

What is the lowest Miller index? The lowest Miller index is O, O, one. That is a lattice constant. That is the maximum separation. And I think you can see that as you move hkl to higher values, what you are doing is you are cutting at a higher angular slice.

And, as you cut for a higher angular slice, the distance between successive planes is going to be reduced. Mathematically what is going on, this is getting larger in the denominator, this is remaining the same, so that distance becomes closer and closer together.

And why am I showing you this? Because we are going to use it in the very next unit. Not three years from now. We are going to use it in the next unit when we take this information and characterize crystals.

And we are going to characterize atomic structure by the use of something that is comparable in dimension. If we are going to measure the dimension of the human hair, we are not going to use a yardstick.

What kind of numbers are we looking at for these distances? We are looking at something down around some number of angstroms. We better find some tool that gives me units of some number of angstroms.

And that tool will be x-rays. Now let's look at characterization of atomic structure by x-rays. Today I want to talk about the discovery of x-rays and the underlying physics. And then next day we are going to look at application.

First of all, what are x-rays? X-rays are a form of electromagnetic radiation. What does that mean? That means all of these formulas apply. E equals h nu equals hc over lambda equals hc nu bar. That applies.

And the special characteristic of x-rays is lambda is very short and typically there is no hard rule on this, but just in round numbers we are looking at between one one-hundredth to one hundred angstroms.

It is centered at about one angstrom where one angstrom is equal to ten to the minus ten meters. I know it is not an SI unit, and you know my aversion to the nanometer, but I will put it up here because I am required by the International Union of Pure and Applied Chemistry on the Systeme International to put this contemptible unit here, but I love the angstroms.

This is just my personal preference. Let's plug in a value. Suppose we have one angstrom as the wavelength if I use hc over lambda where lambda equals one angstrom, or ten to the minus ten meters. I end up with 12,400 electron volts per photon.

And this is clearly going to be powerful radiation. Let's take a look here. I pulled this out of the back of the text. This has ionization energies for the first seven elements. If you look here, in green I am showing the first ionization energy.

And this is right off your Periodic Table. 13.6 electron volts. I have rounded it to two significant figures. It is 13.6 for hydrogen, it is 25.0 for helium, it is 5 for lithium and so on. These are tiny, tiny numbers.

These are electron volts. Now what we can do is look down this row. This is the last electron to come off. The second ionization energy of helium is for a one electron atom. Helium has two electrons.

If it loses one, it only has one left, so this is a one electron atom, this is a one electron atom and this is a one electron atom. And look at the energies. This is for the inner most electron. One electron atom, that will conform to the Bohr model.

And, sure enough, look at the ratios of these numbers. These are determined by x-ray photoelectron spectroscopy. And roughly 14 times four is 56, so there it is. One to four to nine to 16. And it is just a manifestation of what you have seen as the ground state energy in the Bohr model goes as the square of the atomic number.

Everything is making sense, but what I draw your attention to is, look, this is the innermost held electron of nitrogen. It is 668 electron volts and I have 12,400. This has enormous power. This has the power to ionize inner shell electrons.

This is like a bowling alley. All the pins are going flying here. This is incredible energy. What are the underlying principles here? Well, I think we have to start with a history lesson. We will go back to November 8, 1895.

It was a dark and stormy night, really. It was a dark and stormy night on November 8, 1895 in Würzburg, Bavaria, which is now part of Germany. There was a physics professor there by the name of Wilhelm Roentgen.

And he was doing research, as many people were at that time, on gas discharge tubes. We have seen gas discharge tubes a number of times in 3.091, and Roentgen was part of that army of researchers. And what Roentgen's specialty was, he was pursuing gas discharge spectral analysis under reduced pressure.

He was pumping down with a vacuum pump to get the gas pressure very, very low, and very, very high voltage. That is what his take was. He wanted to look at high voltage and very, very low pressure.

And let's take a look at what his apparatus looked like. Well, we will start with the tube. As you have seen many times, we will put the cathode to the left and we will put the anode to the right.

And he had the laboratory on the second floor. His apartment was on the first floor. And down in the basement he had a bunch of batteries. Chromic acid in open beakers down in the basement. Let's indicate that with the classical symbol.

And this altogether put out about 20 volts. And he was a professor of physics who understood some electrical engineering. He had a choke coil here. The choke coil would take the current, store up, and eight times a second it would discharge.

And it would discharge at 35,000 volts. He could take 20 volts and up it to 35,000 by the use of this choke coil. Eight times a second he had 35,000 volts coming off the cathode here. You have bam, bam, bam.

But speed that up about eight times. That is what is going on. And so with 35,000 volts he launches electrons off the cathode, and they go zooming across what is very, very low pressure gas and crash into the anode.

In the meantime, to keep this thing dynamically at low pressure, this goes to a vacuum pump to keep this thing pumped down as low as he can. And he was measuring what was going on inside this gas tube.

And so he had a detector. His detector was over here off to the side. The detector consisted of either a piece of cloth or a paper towel which had been painted with a chemical that would glow when it was struck by something.

What would the something be? The something would be electrons or photons that have an energy high enough to cause excitation of something in the paint and then reemission in the visible. It doesn't do any good to excite electrons in the detector to have them reemit outside the visible.

That is what is going on. This was paper or cloth, and it was painted with an aqueous solution of barium plantinocyanide which glows green when it is excited. And this is called a scintillation screen.

It comes from the Latin word for spark. Scintilla means spark in Latin. When the screen is struck by some form of energy it will glow. And so he had dinner. He was up on the second floor cranking away after dinner.

It was dark and raining. He had the windows open. And a streetlight was coming in. Oh, by the way, Roentgen suffered from Daltonism. He was red-green colored blind. And this was a green screen. And with 35,000 volts, even at reduced pressure, given the capacity of those days for vacuum pumps, he still had a lot of glow from the tube.

The tube is glowing, he has a streetlight in the background, so he decides to close the drapes and take out the streetlight. And then, furthermore, he decides to darken the room so he puts a cardboard box over the tube.

Now the tube is enclosed in a cardboard box so it is not glowing visibly to him. It is glowing inside the box. And he continues to see scintillation, continues to see things glowing. He says that is something interesting.

The other thing that happened which was kind of interesting, his graduate student who had prepared some of the apparatus -- I guess he was practicing painting the scintillation screen. What the student had done was to paint the letter A on a paper towel, and he left the paper towel on the lab bench.

Now, Rˆntgen has the tube enclosed in cardboard, this thing is glowing and he looks down and sees the letter A and the letter A is glowing. He is going, wait a minute, this is crazy. He takes a piece of black paper and puts it here in between the tube and the screen, and the screen continues to glow.

He had a deck of playing cards in his lab. Don't laugh. That is how Mendeleev discovered periodicity. He takes a playing card and puts it in front and the tube continues to glow. And then he takes a book and puts it in between and the tube continues to glow.

He is going wait a minute. Then he grabs a piece of lead foil. He puts the lead foil in here. And what happens is that where the lead foil is the screen is not glowing, but then he sees the outline of the bones of his hand on the lead holding the lead foil.

All of this stuff is going on, and he just goes to town. He says, you know what, this cannot be electrons because electrons cannot live in air. They will be stopped by air. And he knew about radio waves from Hertz and said could this be some form of mysterious radiation that is capable, and he was really afraid to say this, of penetrating matter? It is 1895.

Are you going to come out publicly and say I have a form of radiation that can penetrate matter? He gets a magnet out. He moves a magnet and it doesn't change, so this stuff is not sensitive to magnetic field.

But, if he puts a magnet here, of course he gets changes. He cannot see this stuff. He can only see evidence of it. It is mysterious. He calls it x, the unknown quantity. It is x radiation. This is November 8, 1895.

And, to show you what a great scientist he was, rather than rushing to publication, he spends all of November and December repeating the experiment and trying it in different ways until he is convinced that the effect is real.

And then around Christmas time of 1895 he sends off a manuscript. And it is published in the first week of January of 1896 and takes the world by storm. He is announcing that there is a form of radiation that can penetrate matter.

And, furthermore, he has an x-ray of a human hand. This is the 1890s and people are very prude, very private, and he has a form of radiation that can look inside the human body. This is really shocking stuff.

Already on January 16, 1896 there is an article in The New York Times about this stuff, a mysterious form of radiation. By the way, when he was doing these experiments how was he recording his results? He was recording his results on photographic film, and his photographic film was fogging.

He had his photographic film stored in a cabinet on the other side of the lab, and half of the time he would take the photographic film out and it was fogged. He said whatever this stuff is, it is penetrating everything.

It is penetrating the doors of his file cabinet. They weren't steel cased. They were probably wooden file cabinets. So, widespread adoption, some of it silly. What are some of the silly ones? In London there was a manufacturer who announced, remember this is prudish Victorian England, that he could sell you x-ray proof ladies underwear.

In France, there was no such thing. The French don't care about such matters. Remember, the French Daguerre had invented halide photography. So someone in France offered to x-ray the human soul. This is very good.

I am ashamed to say what happened in the United States. The New York City College of Physicians and Surgeons announced that they would use x-rays to project anatomical diagrams from textbooks onto the brains of medical students, thereby creating, and I quote, an enduring impression.

In Iowa, somebody offered to x-ray copper pennies and thereby turn them into gold. That is what happened. But, seriously, what really did happen very quickly was the use of x-rays as a diagnostic tool.

Already in February of 1896 there was a Scottish physician who used x-rays. It is simple. You could do this in a high school. I mean what is it? It is just a gas discharge tube, a pair of electrodes with a feed through, and he has 35,000 volts, I mean that is the kind of stuff that comes off the chassis of your television set.

This is trivial stuff. People were putting these things everywhere. They were sitting in offices in open view. Nobody knew. This physician had a seamstress come to him. She was involved in an industrial accident in a mill.

A needle broke off in her hand. Now, how would you find the needle in a hand? You cannot. X-ray, they identified the location of the needle, performed the surgery and boom. For dental applications it was immediately adopted.

They were already using it in 1899 as a form of treatment for cancer. Very interesting. It took over. What is the relative physics? What is going on here? Where do we have to go? I say we have to look at the anode because that is where the electrons are crashing into.

That is the site of the collision. Something is happening at that anode. Let's look inside the anode. Suppose it is copper, what could we say about it? This is where the electron collision is. That is the maximum impact.

Let's just see what might be going on. I want to show you the power of estimation. I have shown you that with nitrogen it is 668 electron volts to get to the inner shell electrons. What about in the case of the anode? What is going on there? Well, I can say what is the energy of the 1s electron in copper 28 plus? That is a one electron system.

And you know that in real copper with all of its electrons this value is going to be an over-estimate because the presence of all the other electrons is going to sap some of the protonic Coulombic force.

This will be an upper bound of what that energy could be. And we know that is simply equal to the energy of the 1s in hydrogen times z squared. If I take 13.6 electron volts times 29 squared, I get 11.4 thousand electron volts, which is still less than the 35 thousand electron volts that Roentgen had.

Roentgen is running a huge bowling alley here. He is able to knock out 1s electrons from copper. And what happens when you knock out inner shell electrons? What is the next thing that happens? Let's look at an energy level diagram.

Let's suppose that this is the anode. This is the anode in the gas tube. This is an energy level diagram. This is going to be n equals infinity. And we will just do a few of them. This is n equals four, n equals three, n equals two and n equals one.

Not to scale. I am trying to show that they are a little bit farther apart as you go lower and lower. And so this is ground state energy E1, this is n equals two shell, n equals three shell, n equals four shell.

And out here it is zero. And so what do I have? I have some incident electron that comes zooming in here. This is incident. This is the one that has been accelerated. The energy of this electron is all kinetic.

And that is equal to product of the charge on the electron which is elemental e times a plate voltage which is 35 thousand volts. That is how I get 35 thousand electron volts as the energy. And we have just shown that it has enough energy that it can even knock out an inner shell electron.

It can even dislodge an inner shell electron. Let's see what would happen if that were the case. It dislodges an inner shell electron. What is the next thing? I have a vacancy down here. What will happen? One electron will fall from n equals two to n equals one.

And when it does so it gives off radiation. Well, if it falls from n equals two to n equals one, there is a slim chance it might even fall from n equals three to n equals one or maybe even n equals four to n equals one.

Well, heck, if it can go from two to one, there is a vacancy in two. I might get three to two, I might get four to two, and you get the picture. You get a whole cascade here. Each one of these gives off photons, photon emission.

You have expulsion of k shell all of the electrons, but you can go all the way down to the k shell electrons and then cascade and accompanied by photoemission. The difference is that instead of being in the visible or the infrared, these energy levels are so, so tightly bound that this is in the x region of the spectrum.

And we have labels for these different photons. And, first of all, the spectroscopists don't like the numbers. This is the chemist notation, n equals one, n equals two, n equals three, n equals four.

The spectroscopists call the ground state k shell, this is m shell, this is n shell. I left out a letter in the alphabet. K, l shell, m shell, n shell. We can label this as a photon that came from a cascade down to the k shell.

And to distinguish this k shell electron from the k shell electron involving a transition from n equals three down to n equals one, we have a second indicator. And that is the subscript alpha. This means the cascade down to k shell and the alpha represents how far the electron traveled from one shell away.

That means from n equals two to n equals one, this is a k beta photon, this is a k gamma photon. This is like a Lyman series. This is like a Balmer series because it always ends at n equals two. N equals three to n equals two gives us a photon, and that photon is going to be called L alpha.

N equals four to n equals two will give us a photon, and that is going to be called L beta. I am going to get a whole spectrum. Let's take a look at what that spectrum might look like. I could calculate any one of these.

I could say lambda of copper k alpha would equal what? It is the same as you have seen before, hc over what? The energy of n equals two shell to energy of n equals one shell. **lambda Cu K alpha = hc/(E2-E1) What can we do? We can measure this.

We know h and c. We can calculate what those energy levels are. Let's plot this out. I could make a plot of wavelength intensity and I will get an entire spectrum of lines. This is high wavelength so that means energy is increasing from right to left.

The highest energy is over here. What's the highest energy up there? The highest energy looks like n equals four down to n equals one. And that is going to be over here. This could be k gamma, then k beta, k alpha.

And then the L lines are three to two, four to two, so they are over here. This is L alpha, L beta and so on. And these different wavelengths are associated with different energies. And these different energies are associated with what? They are associated with the binding inside the atom, inside the target.

Every target has its own number of protons, its own number of electrons, so this set of lines is characteristic of the target. And I had to do a whole year of x-ray crystallography as a junior back at the University of Toronto.

We used copper as the target. And I will know on my deathbed that the wavelength of copper k alpha is 1.5418 angstroms. If I walk into the room and see 1.5418 angstroms, that is copper. It comes from copper.

And any other element is going to give me a different set of lines. What I am learning here is I have a way of identifying elements. This is now turning into something that could be a technique of chemical analysis.

And we will return next day to see what else we can do with it. For the last five minutes, I am going to show you a little bit more about Roentgen and his discovery. But I wanted to just draw attention to our winner from Friday who was unable to attend the lecture.

And so I want to ask Nivair Gabriel to stand and accept her acknowledgment for the fine job she did on the Actinides thing. And she is wearing her scarf. Why don't you come down and people can see you wearing the really super hot American Chemical Society scarf.

[APPLAUSE] Now, here is the dark and stormy night the following morning. It is sunny. This is Roentgen's laboratory on the second floor of his flat in Würzburg. It is a museum today. And you can see the tubes and the lead wires going down to the basement and so on.

Hold it. I don't want a lot of noise. This is the first radiograph ever. This is Roentgen's wife Bertha. He noticed that there was a shadow with the lead. Roentgen was smart. He had to share the glory so he said, honey, would you mind coming upstairs, I would like to take your picture.

And so Bertha put her hand in the path of the x-ray, and for 15 full minutes waited while those x-rays went through. You can see the ring on her finger and so on. She endured in that 15 minutes greater than a lifetime dose of what we would consider safe today.

People just did not know. Anyway. This gave birth to medical radiography. And I would, just out of curiosity, like to know if there is anybody in this room who has never had a medical or dental x-ray.

I see very few. I see a few hands. To you I say congratulations and I hope you keep up the good work. Here is the second thing. We have radiography that came out of Roentgen's discovery. Roentgen had a torsion balance.

Like the Justice, she has the two scales. On one scale you put the unknown and the other scale you put the calibrated mass. He had a set of brass weights here in a wooden box, and he closed the wooden box and irradiated the box with the x-rays and took this photograph.

And what do you see? You see that different materials have different electron densities and, therefore, the wood is transparent, more or less, to x-rays. Whereas, the brass is less so. You have a differentiation.

The way we see things is by differentiation. Here you have a differentiation. He could identify what was in the box when the box was closed. That gives rise to security technology. Every time you go to the airport you are subjected to this, and it started with Roentgen.

Roentgen also had a gun. It was a musket. He wasn't holding up convenience stores. What are you laughing at? He had a gun. He went hunting. He took an x-radiograph of his gun. And this is used today.

It gives birth to the technology of failure analysis. In fact, there are certain types of weld, typically involving very dangerous circumstances such as pressure vessels that are going to contain steam at very, very high pressures.

The code is that every inch of the welded seam must be inspected by x-radiation in order to find what? Internal cracks. Because, if there is a crack, there is a difference in electron density. And so you will see contrast in the x-ray photograph.

From that experiment come three technologies. And for his work, Roentgen was the recipient of the very first Nobel prize in physics. It was awarded in 1901. He was the unanimous choice of the world physics community for what he had done.

And you are going to see next day more that the discovery of x-rays enabled. But one last thing, what a fine human being he was. You know what he did with his winnings? He donated them to the University of Würzburg to be made an endowment for student scholarships.

And, to this day, physics students at the University of Würzburg are recipients of scholarships that were endowed by Roentgen with his Nobel prize winnings in 1901. I will see you on Wednesday.

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