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And among the useful things that they do, besides having a splendid conference every couple of years, is to publish these tables. And volume one is called Symmetry Tables.

And everything that we will derive and all of its properties-- physical and geometrical-- are tabulated in this book. It is, however, a reference book and not a textbook. You don't learn it for the first time from this book. But in terms of generating atomic arrangements from the data that's present in the literature, looking at the arrangement of symmetry elements in space, and how they move atoms around, it is the code book that tells you how to crack the arcane language in which diffraction and structural results are recorded and find out how to unravel it. I call also to your attention, although it will not be germane to this class, there are four other volumes.

Volume two is called Mathematical Tables. And this has all sorts of useful stuff. If you've ever done diffraction, you know that depending on the symmetry of the crystal, there are some planes for which h squared plus k squared plus l squared divided by 2 pi is not a reflection [INAUDIBLE] if the crystal is green, and other arcane rules like that. All of these are summarized in these books. There are quantities that you need to calculate, things like interplanar spacings. Tables are available there. So this is a handy thing primarily for diffraction.

Volume three is called Physical Tables. And this is where you find things like absorption coefficients for x-rays and for neutrons.

Electron crystallography ‐ an introduction

It's where you find the latest values of absorption coefficients, neutron scattering length. And since these things are derived experimentally, the values improve and change from time to time. So this is where you find the most up to date values of physical constants and items that are necessary for diffraction. It never ceases to amaze me how somebody who has the good fortune of having to use the diffraction for a thesis will labor carefully over making the measurements and reducing the data.

And then when it comes to using a wavelength, which is how the final numbers will be determined, goes to an appendix of a book on diffraction that was published 20 years ago. And that's not the most up to date value. Scattering powers of x-rays by the electrons on the atoms are calculated from wave functions, which constantly get better from year to year. And the value of the scattering powers of the function of angle gets better from year to year.

So this is where you want to go if you need any of that physical data. And finally, volume four is-- it's not its title, but it's essentially an update of the Physical Tables, giving later values which came out about 10 years later. OK, this series was getting out of hand. So I have to bend my knees and use two hands when I pick up this one. This is a continuation of the series, essentially. But this one is called International Tables for Crystallography, period, no x-rays because neutrons and electrons are just as important today for doing scattering experiments.

And this is International Tables for Crystallography. No x-ray in there. And there are now something like six volumes out. They're not called one, two, three, and four, but they're called A, B, and C to avoid confusion. And volume A is one called Space Group Symmetry. And then, there are a whole series of other ones. As I say, I think there's six of them that give physical data and all sorts of useful guides. I have mixed feelings about the new series. You will see that it is about three times as large and three times as heavy, which means it's nine times as expensive.

And to me, it's almost the case for most people of a situation where if it wasn't broke, you shouldn't fix it. But nevertheless, if you wanted, you'll find it there, which is something that could not be said before. They've added a few things which are useful, but a lot of additional information which you don't really need. And you pay for that whether you want it or not. Nevertheless, it's been done. You can't buy the old volumes any longer. You have to buy the new volumes. So anyway, this is what you'll find in the library now. Maybe they do still have the old volumes, one through four.

This, we will make reference to in the course of the term. I will give you some copies of certain pages in here as handouts when we need them for purposes of illustration or for use. But I spent the last five minutes just to make you aware of the existence of these books. And these are really the penultimate source of information and numerical quantities that will be used in diffraction, one of the principle applications of crystallography.

I think I have just enough enough-- to start things off, I have a syllabus for the course that is, in very dense form, exactly what we will be covering this term. And I'd like to lead you by the hand through this. All right, what we will be doing in the first half of the term is something that is known as crystallography.

OK, the meaning of the word is almost self-explanatory. The first part is crystal. We're going to be dealing with the crystalline state of matter. To me, amorphous materials, although they may be important, have all the interest of a piece of steak before it's been cooked. The atoms in amorphous materials are fine. But they really get interesting when they organize themselves into an ordered fashion. So the name is self-explanatory.

The first part, crystal, means we're going to deal with the crystalline state. What does the graphy mean? That means mapping or geometry. And let me give you an example of a few other words that have the same sort of structure. Geo-- the Earth-- followed by graph, geography, is the mapping of the Earth. And there are many other terms that involve these two separate parts. Crystallography, though, is very often subdivided into different flavors.

There is something well defined called x-ray crystallography. And this is the experimental determination of the crystallography of a material using diffraction, usually x-rays because they're relatively inexpensive and they're widely available.

Utilization of X-Ray Diffraction

But increasingly, neutron scattering or electron scattering is used for this purpose. And there are a number of very powerful, very exciting sources of neutrons, either from reactor sources of unprecedented intensity or from what's called a spallation source, where an entire synchrotron is built just to direct a beam of particles onto a heavy metal target. And those high energy particles split off neutrons from the nuclei of the target material.

Doesn't really matter what the material is. It helps if it's a heavy metal. The nice thing about these sources of neutron radiation is that they're so expensive they are all national facilities. And the consequence of that is that anybody with a good idea and a project worth doing can apply for beam time. And if it's a good problem, you get it. You have people whose sole function in life is to help you do the experiment and make sure you're doing it properly.

And this is a very, very exciting time to be somebody working with diffraction using these neutron sources. There's another branch of crystallography which is called optical crystallography. And this is the characterization and study of crystalline materials using polarized light. You can identify unknowns using their optical properties if they're transparent about 10 times faster than you can do with x-ray diffraction. It's a technique that today is little used.

But it's a very powerful technique. And all it takes is a microscope, and you're off and running. Some other flavors of crystallography, well, I'll mention the one that we're going to use. What we're going to talk about is something called geometrical crystallography, to distinguish it from these other branches. And this is synonymous with symmetry theory. So that's what we'll do for the first month and a half or so.

All right, let me introduce now some basic concepts. Geometrical crystallography is the study of patterns and their symmetry. So let me give you an example of some very simple patterns that extent in one dimension. And let me put in a figure. The thing that is in the pattern is something that's called the motif. And let me use a plump, little fat comma. And I'll make a chain of these things extending in one dimension. The nice thing about this fat little comma is that it is a figure which, in itself, has no inherent symmetry.

So it is asymmetric, without symmetry. And imagine this is being repeated without limit in both directions, both to the left and to the right. We then draw another pattern with a different sort of motif. And let me use a rectangle with one concave side.

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OK, and I think you get the picture of this one. And imagine that as extending without limit indefinitely to the left and to the right. Then, I'm getting tired of inventing new motifs. So let me use the same motif the second time, but arrange it in a slightly different way. And again, imagine that as extended indefinitely.

OK, having now generated these three patterns in two dimensions but extending periodically in two dimensions. Let me ask the question now. Are any of these patterns the same? Or are they all different? Are any of the patterns the same? Or are they different? Well, that's a-- yeah? They have the same sort of the motif.

They both have the same rectangle with one concave side. And that's a valid answer. Do you have a different answer? Why do you say that? This is the point I was trying to introduce. And that is your choice of answering the question, one is the nature the motif. And you're absolutely correct. This pattern and this pattern are both based on the same motif. But in patterns, we are less concerned with the motif that is in the pattern than we are with the relations between one motif and all of the others. And in that context, the first and the third pattern, although they look entirely different, are really exactly the same sort of pattern.

So let's begin to analyze what sort of operations are in these patterns that take one motif-- and obviously, they're all the same-- and relate it to all of the others. First of all, there is an operation which I'll call translation for obvious reasons. And I'll represent that by a vector, T, since a translation has magnitude and direction but no unique origin.

I could take this pair of objects sitting nose to nose, pick them up, slide them over by T, put them down again. And I have the relation that gives me this neighboring pair. Pick it up again, move it to the right by the same translation in the same direction, put it down again.

And I've got this pair. So that is one operation that can exist in patterns. This is the operation of translation. So let me call that by a vector relation. And it has magnitude. It has direction, but no unique origin, just like a plain old vector. So in other words, I can't say that the translation moves us from here to here or from here to here.

It's all the same thing-- magnitude and direction, no unique origin. In fact, all of these patterns have translational periodicity. There's a translation in this bottom pattern and another translation from here to here in the middle pattern. The thing that makes a crystal a crystal is that it is an arrangement of atoms or molecules which is related one part to another by the operation of translation.

If you don't have translational periodicity, you do not have a crystal. So that comes to the essence of what crystallography is about. You can imagine, in one sense, the generation of this pattern by a rubber stamp sort of operation. Suppose I have a rubber stamp. And I put on the rubber stamp the pair of motifs like this. Pick it up, move it over, chunk. And I can stamp out the pattern in that fashion. Notice that my statement about no unique origin in these terms can be stated that it doesn't matter where the two motifs are on the stamp.

As long as I move the stamp through the same distance and the same direction, I get the same pattern. Now, that's not bad for an introduction. But I want to be more general than this because when I deal in terms of a rubber stamp operation, that is a transformation that involves taking one little chunk of a two dimensional space, picking it up, and putting it down in another location to another unique location in space. So I'm going to now make another generalization that operations, which we've begun to define, act on all of space.

So I don't want you to think of this repetition in terms of a rubber stamp, although we could get the pattern that way and it's conceptually appealing. But I'm going to say now that this string of motifs has translational periodicity if, when I pick it up, move it by T in a particular direction, and drop the whole infinite chain back down again, it is mapped into congruence with itself. Which leads me to another definition-- an object or a space possesses symmetry when there is an operation or a set of operations that maps it into congruence with itself.

In other words, in plain words, you can't tell that it's been moved. OK, is there anything else that is a transformation which leaves the set invariant? OK, if we look at the first pattern, there are [? And it will be mapped into coincidence with itself. And that is an operation, and another sort of distinct operation of transformation. And this is one that I could call rotation for obvious reasons. And there are two things I have to tell you about a rotation operation. The first one is the point about which the rotation takes place, and that's going to be some point.

And let me call this point here A. So this will be some labelled point that is the location of the rotation axis. But then, the other thing that I have to tell you is the angle through which I'm going to rotate. And I'll append to the A as a subscript the angle of rotation. So this particular operation, called a twofold rotation because it rotates through half of a circle, would be the operation A pi. This point is A.

We rotate through an angle pi. This pattern here has also rotational symmetry. In addition to the translation, there is a rotation operation, A pi, in the lower pattern. So the follow who is unfortunate enough not to have a seat-- and I should have given you this one a long time ago. I'll give that to you as your reward for giving the best answer. And you get a seat wherever you would like to place it.

The first and the final pattern are the same in the sense that they contain two operations, translation and rotation. This pattern is a much more interesting one. This also has a rotational symmetry, A pi. It also is based on a translation. But now, there's another operation that we can do to leave the pattern invariant.

There exists [? It's a reflection sort of operation. So this is a new type of transformation. So we'll add that to our list. And the symbol that's usually used to indicate the locus of this operation is m, standing from mirror. And that does it for these particular patterns. Three sorts of operations-- translation, rotation, and reflection. And in fact, that is all you can have in a two dimensional space-- not necessarily a rotation that's restricted to degrees.

If these patterns are translational periodic in more than one direction, you can have higher symmetries. One of the things I would like to suggest to you is that you look around you in everyday life at the sort of patterns that enrich your environment. I see a one dimensionally periodic pattern there, the black and white stripes. It's translationally periodic, going up and down.

It also has mirror planes running through the black stripes and the white stripes. I see another two dimensional pattern back there. That has translation. But you could rotate-- no, you can't do anything. That just has translation, nothing else. Get a new shirt. That's not terribly interesting. There's another one there that's so complex I don't think I can look at it without climbing all over him and drawing some translational vectors and things like that.

But that's a nice periodic pattern. That's a good one. But there's lots of stuff like that. Look at the grills in the ventilators. They have mirror planes. They are translationally periodic in one direction.

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